Influence of Porosity, Fiber Radius and Fiber Orientation on the Transport and Acoustic Properties of Random Fiber Structures

Summary The ability of air-saturated ﬁbrous media to mitigate sound waves is controlled by their transport properties that are themselves controlled by the geometrical characteristics of their microstructure such as the open porosity, ﬁber radius, and ﬁber orientation. Here, micro-macro relationships are proposed to link these microstructural features to the macroscopic transport properties of random ﬁber structures. These transport properties are the tortuosity, the viscous and thermal static permeabilities, and the viscous and thermal characteristic lengths. First, representative elementary volumes (REVs) of random ﬁber structures are generated for di ﬀ erent triplets of porosity, ﬁber radius and ﬁber orientation. The ﬁbers are allowed to overlap and are motionless (rigid-frame assumption). The ﬁber orientation is derived from a second order orientation tensor. Second, the transport equations are numerically solved on the REVs which are seen as periodic unit cells. These solutions yield the transport properties governing the sound propagation and dissipation in the respective ﬁbrous media. From these solutions, micro-macro relationships are derived to estimate the transport properties when the geometry of the ﬁber structure is known. Finally, these relationships are used to study the inﬂuence of the microstructural features on the acoustic properties of random ﬁber structures.


Introduction
The ability to tune the porosity,t he fiber radius, and the fiber orientation of fibers tructures maket hem attractive materials for sound absorption-based applications. These three geometrical characteristics affect the transport properties that govern the visco-inertial and thermal losses of sound wavesinfiber structures. These transport properties are the tortuosity,the viscous and thermal static permeabilities, and the viscous and thermal characteristic lengths. Consequently,i ti si mportant to understand howt he geometrical characteristics influence the anisotropic transport properties of random fiber structures.
In sound absorbing fibrous media, visco-thermal dissipation phenomena are mainly determined by permeability/resistivity of the porous sample, which givesrise to empirical models with versatile applications [1,2]. As ageneral rule, the lowfrequencylimit (ω → 0) of the dynamic viscous permeability k(ω)o ft he fibrous sample always Received18October 2016, accepted 4September 2017 decreases with the increase of average angle θ between the fibera xis and the sound wave propagation direction [3,4]. This microstructural effect results in alower mean velocity − → v in aviscous fluid flowsolution of the Stokes equations, where φ is the open porosity, η is the dynamic viscosity of the saturating fluid and − → ∇p is am acroscopic pressure gradient acting as as ource term in the generalized Darcy law. The lowfrequencylimit of the dynamic viscous permeability is called the static viscous permeability k 0 .Itis linked to the static airflowresistivity as σ = η/k 0 ,which is usually used in the acoustic literature [1]- [4].
At high frequencies (ω →∞ ), the imaginary part of k(ω)dominates and the real distance traveled by the wave between twop oints is the rectilinear distance between them multiplied by √ α ∞ because of the tortuosity α ∞ of the path [5]. The tortuosity is directly measurable from conductivity experiments or simulations [6]. This fact was first pointed out by Rayleigh [7] and Brown [6]. The tortuosity also decreases with the average angle reduction (α ∞ ≥ 1 → 1, when θ → 0) and with the porosity increase (α ∞ ≥ 1 → 1, when φ → 1).T he variation of Re[k(ω)] arises from ac ombination of the inertial effect plus adependence of the viscous boundary layer thickness δ v = 2η/ρ 0 ω, ρ 0 being the density of the fluid at rest. At high frequencies, the real part of k(ω)isgiven by where Λ is the viscous characteristic length. The Λ parameter and its significance as aweighted measure of the pore volume-to-surface ratio wasfi rst emphasized by Johnson et al. [8,9]. Diffusion controlled reactions can be simulated in microstructural models to provide estimates of the trapping constant Γ,o rt he so-called static thermal permeability k 0 = 1/Γ (k 0 k 0 ) [ 10], which represents the lowf requencylimit of the thermal response function, where τ is the macroscopic excess temperature in fluid phase, k (ω)i st he dynamic thermal permeability, p is the macroscopic pressure and κ is the thermal conduction coefficient.
Johnson et al. [9] and Lafarge et al. [10] showed that k(ω)a nd k (ω)c an be adequately described by approximate butrobust semi-phenomenological models based on more readily measurable physical properties (φ, α ∞ , Λ, Λ , k 0 , k 0 ). Here, Λ denotes the generalized hydraulic radius also known as the thermal characteristic length in the context of sound absorbing materials [11]. The results of Zhou and Sheng [5], on the universality properties of the dynamic permeability k(ω), suggest that the acoustic properties of fibrous media can be deduced from al imited amount of geometrical characteristics (fiber information orientation Ω zz ,fiber radius r f ,porosity φ,hydraulic radius Λ )a nd resulting transport information (static viscous permeability k 0 ,tortuosity α ∞ ,viscous characteristic leng th Λ,static thermal permeability k 0 ).
The use of even-order tensors wasintroduced by Advani and Tucker to describe the probability distribution function of fiber orientation in fiber materials [12]. The versatile modelling capability of tensors makes them appropriate to elucidate the effect of angular orientation on the sound propagation and dissipation mechanisms; this will be shown in this paper.
At horough reviewo ft he literature wasc onducted by Tomadakis and Robertson [13] who compared manye xperimental and theoretical studies on the viscous permeability of various types of fiber structures. The data from these studies were presented in terms of dimensionless viscous permeability versus porosity to facilitate the comparison with theoretical predictions. Theyw ere categorized by the type of the fiber structure and flowc onfiguration. The structures formed by cylindrical overlapping fibers distributed randomly in 1, 2, or 3d irections were considered; the fibers being allowed to overlap freely in all the three cases. All one-and two-directional structures examined are statistically anisotropic, therefore the permeability wasderivedboth parallel and perpendicular to their characteristic directions. It wasf ound that the conjecture of Johnson et al. [8,9] (k 0 = φΛ 2 /8α ∞ )p rovides very good permeability estimates in most cases, resulting in an overall ratio of the theoretical prediction to measurement close to 1.25 for the over500 experimental points utilized. However, the predictions of all examined fiber structures and flowc onfigurations also revealed as ignificant effect of fiber directionality on permeability (Figure 8o f [ 13]), an effect that can also be regarded as as pecificm ean to functionalize the fibrous material.
Starting from the comment that some non-wovenfibrous materials yield av ery lowa bsorption contrast with X-rays, and therefore that usual computer tomography does not yield satisfying 3D images of the materials, Schladitz et al. [14] used 2D images of sections parallel and orthogonal to the flowd irection obtained by classical light microscopy. The material wasfi rst infiltrated with ar esin. Then, sections were cut, ground and polished. Repeated simulations of astochastic model combin ed with image processing techniques gave evidence that the anisotropyparameter β of spatially stationary random system of lines (Poisson line process)can be successfully estimated from the number of fibers observed in sections parallel and orthogonal to the flowd irection. This procedure corresponds to an experimental method available to estimate the orientation of fibers when 3D images are missing or failing to yield satisfying information. Additional mapping between the tensorial formalism and the Poisson line process leads to acommon framework for the generation of disordered fiber structures, illustrating the crucial role of the parameter governing the orientation of fibers in transversely isotropic fibrous materials (Ω zz as defined in Section II.B). The Poisson line process model wass ubsequently used by Jensen and Raspet [15] to investigate thermoacoustic properties of overlapping fibrous materials in order to test the prediction of analytical models [16,17]. The parameters of the models (shape factors and relaxation times)a re selected to best fit the numerical simulations that were made using alattice Boltzmann approach.
In this paper,n ew numerical data are presented and used to systematically describe the anisotropic transport properties of sound absorbing fibrous media from geometrical information only.Rigid frame models will be examined with special attention to fiber orientation effect, and al arge range of porosities will be studied. Section 2 presents the generation of Representative Elementary Vo lumes (REVs)o fr andomly overlapping fiber structures based on aparametrized fiber orientation. While Section 3 deals with the identification of the geometrical properties from the REVs, Section 4d eals with the identification of the transport properties. Finally,Section 5summarizes the main micro-macro relationships developed in Sections 3 and 4, and these relations are applied to study the influence of the microstructural features on the acoustic properties of random fiber structures.

Random fiber structures
Likei np revious studies [14,15], as imple model for the fiber structure with the smallest possible number of parameters is considered. Following Schladitz et al. [14], the following assumptions were made. Compared to the sample size, the fibers are long and their curvature is negligible. There is no interaction between the fibers. Due to the production process, the fiber structure is macroscopically homogeneous and isotropic in the horizontal (xy-)plane. That is, the distribution properties of the random model are invariant by translations as well as rotations around the z-axis. Therefore, adetailed study of the effect of correlations between fibers is out of the scope of this work.

Orientation distribution function
Fort he purpose of the present research, the random fiber structures result from the successive generation of rigid uniform cylinders of the same diameter.The fibers are introduced at random locations with au niform number of fibers per unit volume. Aw ay to describe the orientation of afi ber is to associate au nit vector − → p to the fiber,a s shown in Figure 1. Ar andom fiber structure is therefore an arrangement of fibers for which the orientation distribution function is adefined function Ψ(ϕ, θ)o ftwo variables describing the orientation of asingle fiber.Examples are giveni nF igures 1a nd 2. Note that in these figures, the x, y, z axes respectively correspond to unit vectors − → e 1 , − → e 2 , − → e 3 .
It is worth recalling that Tomadakis and Robertson [13] simulated one-, two-, and three-dimensional randomly overlapping fiber structures. However, these structures only corresponds to the three specificconfigurations of the fiber orientations shown in Figure 1b,c,d.

Orientation tensor
The use of tensors to describe fiber orientation of composite fibers waspresented in aseries of papers [18]- [21] and reviewed by Advani and Tucker [12]. The second-order orientation tensor is obtained by forming dyadic products of the vector − → p and then integrating the products with the distribution function overall possible directions, where the set of all possible directions of − → p corresponds to the unit sphere, and the integral overthe surface of the unit sphere is noted by Afi ber oriented at anya ngle (ϕ, θ)i su ndistinguishable from afi ber oriented at angle (ϕ + π, π − θ), so Ψ must satisfy Ψ must be normalized, since every fiberhas some orientation, Because the distribution function is even [Equation (6)], only the even-order tensors are of interest (the odd-order integrals are zero). Using Equation (7),t he integral over all − → p weighted by Ψ( − → p ), which appears in Equation (4) becomes, for adiscrete set of fibers, where N f is the total number of fibers, θ (i) is the vertical orientation angle, and ϕ (i) is horizontal orientation angle.
[Ω]c onstitutes the most concise nontrivial description of the orientation. Assuming at ransversely isotropic material, [Ω]i sc ompletely determined by Ω zz .V arying the value of Ω zz from planar (Ω zz = 0) to aligned (Ω zz = 1) random fibers, one can study the influence of fiber orientation on the transport properties of fibrous media. This was simply done by adjusting the mean µ θ and standard deviation σ θ of anormal distribution of angle θ with auniform random orientation of angle ϕ.All the corresponding coefficients are reported in Table I. The choice of the distributions on θ and ϕ is based on the experimental knowledge acquired by the authors on several random fibrous materials using SEM images. From the authors' knowledge, the following modelling assumptions can be formulated: (i) the horizontal angle follows au niform distribution between 0a nd 180 • ;( ii) as the number of analysed fibers increases, the probability density function of the vertical angle is approximately normal (with am ean value generally centered at 90 • in most cases). An example of such ad istribution obtained by SEM image analysis is givenelsewhere [22].

Generation of random fiber structures
Here is ashort description of the algorithm which is fully detailed elsewhere [22], and displayed in Figure 3. The algorithm is used to generate ar epresentative elementary volume (REV)for agiven fiber orientation coefficient Ω zz . The algorithm allows fibers to overlap as in [13], [14] and [15]. This can be questionable since in reality the fibers ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 103 (2017) p 1 =sin cos p 2 =sin sin  Table I. Coefficient Ω zz of the second-order transversely isotropic fiber orientation tensor and the corresponding mean µ θ and standard deviation σ θ adjusted assumming that θ is described by anormal distribution function and ϕ by auniform distribution function.  At the initial iteration, the box size is set to 100 µm, and will be iteratively increased by av alue ΔL = 10 µm.
This domain typically contains as olid volume of fibers equal to To identify the number of fibers of radius r f that are required to meet this volume V (i) f ,the number of fibers randomly generated in the box, following the predetermined Ω zz coefficient, is iteratively increased. The iteration is stopped when for j fibers the solid volume of fibers V f is the length of the k th fiber generated during the proc ess at iteration j.F rom the found volume V  j fibers is compared to the expected value φ.I ft he relative error of the actual porosity is smaller than tolerance ε, then the convergence criterion on iteration i is finally met, and the current elementary volume box of size L (i) is the REV -its actual porosity is φ REV and it contains j fibers. From this REV,n umerical calculations are performed to retrieve the geometrical and transport properties. In practice, since there are variations between randomly generated fibrous networks for the same number of fibers and Ω zz coefficient, the actual porosity that is used in the convergence criterion is an averaged value over1000 realizations of the conditional loop over j -see counted loop over m in Figure 3. As ag eneral trend, the size of the representative elementary volume L REV is linearly decreasing with increasing porosity (atagiven ε), and the smaller is ε the higher is L REV .F or the range of studied porosities, with 0.75 ≤ φ ≤ 0.99, the value L REV ≤ 500 µm is large enough to ensure that ε = 0.003.
Note that the results presented in this study were conducted on aREV with L REV = 500 µm, anumber of fibers N f = 68 and afi ber radius varying between 4.3 µma nd 19.23 µmt oc overt he porosity range 0.75 ≤ φ ≤ 0.99 (within ε = 0.003). Fori nstance, at φ = 0.9, r f = 11.1 µmand L REV ≈ 45r f .Inpractice, anumerical approach wasu sed to determine the final volume and wet surface area of the resulting fiber webs, because there generally exists some overlaps between fibers.

Identification of the geometrical properties
The simplest geometrical properties of ar andom fiber structure are the characteristic fiber radius r f ,open porosity φ,and thermal characteristic length Λ .Since r f and φ are twoinput parameters of the REV generation (see Section 2.3), theyare either imposed or obtained by measurements. Theya re imposed when one wants to study their effects on the acoustic properties, and theya re measured when one wants to predict the acoustic or transport properties of an existing material. In the latter case, the characteristic fiber radius can be obtained from image analysis or the manufacturing process, and the open porosity measured from existing methods [24]- [26]. Additionaly,it is worth mentionning that for polydisperse and bidisperse fiber structures, it wasshown that the weighted fiber radius can be used as the characteristic fiber radius [27]. The only geometrical property that remains to be identified is Λ .This property is defined as twice the ratio between the fluid phase volume V p and the wet solid surface area of the fibers S w , In this study,each constructed REV is discretized by volume finite elements. Consequently, V p and S w are directly computed on each discretized REV,and Λ deduced from the previous equation. The numerically computed ratios between the thermal characteristic length and the characteristic fiber radius Λ /r f are plotted in Figure 4asafunc- The continuous curvecorresponds to the theoretical model, Equation (11).T he thick curvec orresponds to the corrected model for overlapping fibers, Equation (12). tion of the porosity for the various fibero rientations. As expected, all of the generated fiber structures followt he same behavior,w ith an onlinear increase of Λ /r f as the fibrous media become more porous. Λ /r f is roughly independent on the angular orientation Ω zz .A ssuming that the fibers do not overlap, one can derive from Equation (10),atheoretical expression for this normalized length in function of the open porosity.Itisgiven by If the fibers are allowed to overlap, one can showf rom Equation (10) that an additional term would appear in the denominator of Equation (11).T his term would depend on the r f /L ratio, the number of intersections, and their corresponding shape (ex.: shape formed by intersection of twoo blique fibers). The latter properties are difficult to evaluate on the studied REVs; however, the influence of the intersections can be taken into account by acorrection term c in the denominator of Equation (11).Consequently, ab etter model to fit the numerical data obtained for our randomly overlapping fiberstructures is givenby This expression is plotted on Figure 4. One can note that Equation (12) fits with the numerically calculated normalized characteristic thermal lengths, in which the value of c represents the fiberintersection intensity.Byusing asimple fit from the simulation results, we obtain c = 0.0036. Thus, it can be concluded that agood approximation of Λ can be deduced from r f and φ by Equation (12) when the characteristic fiber radius and porosity are imposed, or known from measurements.

Identification of the transport properties
In this section, the macroscopic effective coefficients for the basic transport processes by conduction, convection and diffusion-controlled reactions in random networks of fibers are studied. The governing equations and their solution methods are briefly recalled in this section. Theyare detailed in an earlier paper [22]. In all cases, the macroscopic coefficients are deduced by integrating the local fields, obtained by solving the transport equations at the pore scale. Since the webs are macroscopically homogeneous, theyare considered as infinite periodic media, made of identical unit cells. The unit cells are the REVs generated as described in section II. Periodicity conditions along the x, y,a nd z axes were applied when computing the transport properties.

Tortuosity and viscous characteristic length
4.1.1. Theoretical framework Electric conduction is governed by the following set of equations, where − → E and π are respectively the local values of the electric field and microscopic potential in the fluid, and − → e is a specificmacroscopic electric field. − → E satisfies the no-flux boundary condition at the wall ∂Ω when the solid phase is assumed to be insulating, where − → n is the unit normal vector to ∂Ω. π is assumed to be spatially periodic with aperiod Ω in the three directions of space. The quantities − → E and − → e are related by the symmetric positive definite tortuosity tensor α ∞ij , which depends only upon the geometry of the medium. Typically,for an isotropic random medium, α ∞ij is aspherical tensor equal to α ∞ I.F or transversely isotropic fiber webs, the x and y directions play equivalent roles, buta different behavior along the z axis is expected. In the following, α ∞xy denotes the average of the tortuosities along the x and y axes, which were found equal within the statistical fluctuations, and α ∞z denotes the tortuosity in the vertical direction. The mean value overt he three axes is denoted α ∞ . The viscous characteristic length Λ,introduced by Johnson et al. [8] and applicable to anykind of porous media, is defined as It is an effective pore-volume-to-surface ratio wherein each volume or area element is weighted according to the local value of the electric field − → E,w hich would exist in the absence of as urface mechanism. Λ is ac haracteristic parameter of the geometry of the porous medium. This length can be derivedf rom the numerical solution of the Laplace'sequation in the pore space and used for the analysis of transport properties. The value of Λ obtained when setting − → e along the x and y directions is denoted by Λ xy , while Λ z corresponds to the z direction.

Numerical results and discussions
The calculations of the tortuosity tensor α ∞ij were performed for the fiber structures described above.T he purpose is to obtain ad escription for all types of random fibrous media using their single-geometry characteristics such as the porosity φ and the fiberorientation Ω zz assuming transverse isotropy. Astudy of the possible anisotropyofthe results is first conducted. Figure 5shows that, while the averaged tortuosity α ∞ is fewly sensitive to Ω zz ,the transverse tortuosity α ∞xy and longitudinal tortuosity α ∞z are very sensitive to Ω zz .A sag eneral trend, it can be noted that fibers that are orthogonal to the direction of wave propagation yield higher tortuosity values than fibers that are parallel to the direction of wave propagation. One popular empirical model for the determination of the tortuosity of porous media is the Archie'sl aw given by α ∞ = (1/φ) γ ,where γ is aconstant which depends on the microstructure of the porous medium. In our randomly overlapping fiber structures of transverse isotropy, the microstructure depends on the direction of the flow( along z or xy)a nd the fiber orientation coefficient Ω zz .T oextend Archie'sl aw to our numerical results, the following expression is proposed: (valid for 0.75 φ 1), (17) where P (Ω zz )i sapolynomial of the second order whose coefficients are obtained by an onlinear curve-fitting on our numerical results in aleast-square sense. For α ∞xy and α ∞z ,the polynomial coefficients are giveninT able II. Except for strongly aligned fiber networks with Ω zz = 0.9 and Ω zz = 1, the effect of fibero rientation is barely visible on α ∞ as shown in Figure 5c. Consequently,aconstant value P (Ω zz ) = 0.7659 can be used in Equation (17) Figure 5. (Colour online)T he tortuosities (a) α ∞xy ,( b) α ∞z ,a nd (c) α ∞ as af unction of porosity φ.S ame convention of colors and symbols as in Figure 4. The symbols refer to numerically computed values. Dotted lines are estimates obtained by the micro-macro relationship, Equation (17).Thick gray lines refer to Tomadakis and Robertson'smodel (Equation (11) and Table 1of [13]). α ∞ .T his value is also obtained by the same curve-fitting on all the numerical data.
Tortuosity estimates obtained with Equation (17) are plotted in Figure 5(dotted lines). These estimates are also compared with results obtained by Tomadakis and Robertson [13]. Based on random-walk simulations on three specifict ypes of randomly overlapping fibers tructures (1-d, 2-d and 3-d), theyderivedacurve-fitrelationship based on ageneralization of the Archie'slaw.These fiber structures are similar to our configurations with Ω zz = 1, Ω zz = 0, and Ω zz = 1/3, respectively.Asshown in Figure 5, good correlations are obtained for these three specificcases. The worst comparison is for α ∞z with Ω zz = 0( 2-d, red circles). In this case, the Tomadakis and Robertson'sm odel predicts tortuosity values smaller than our results. The comparison of the results of this work to the data of the literature reveals that the random-walk simulation results provide accurate predictions of the tortuosity of random fibrous structures in most cases -e xcept for the flowp erpendicular to two-directional randomly overlapping fiber structures -with the deviation increasing at lowporosities (Figure 5b with Ω zz = 0, red circles). The latter corresponds to the situation where the cross-sectional area of the pore space varies relatively fast as one movesa way from the throat. This indicates that the diffusional estimates of the tortuosity might not be accurate enough if the throat region differs significantly from the straight-tubelikemodel As in the case of the tortuosity α ∞ ,t he purpose of the next analysis is to obtain adescription for the viscous characteristic length Λ for all random fibrous media as afunction of the microstructural features. The numerical calculations were performed on the same REVs that were used for the tortuosity tensor analysis. The numerically computed thermal characteristic length Λ normalized by the values of Λ z and Λ xy are shown in Figure 6asafunction of the porosity φ.One can note that the upper (Λ /Λ ≈ 2) and lower (Λ /Λ=1) bounds found here are consistent with those published in [11] (Equation 49)f or fibers perpendicular and parallel to the plane wave propagation direction ( − → e 3 ).
Along direction − → e 3 ,one can observethat there is no dependence of Λ /Λ z with φ.T his is almost the case along the transverse direction for Λ /Λ xy ;however,asthe angular orientation increases, the ratio tends to increase with decreasing porosity.I ndeed, Λ is an effective pore size of dynamically connected pore regions that contribute the most to fluid transport (ane ff ective surface-to-pore volume wherein each area or volume element is weighted according to the local value of − → E, Equations 13 and 14). This weighting eliminates contributions from the isolated regions of the pore space that do not contribute to transport. This effect is strong when the fibers are perpendicular to flowdirection, and increases as the porosity of the fibrous material decreases.
Forac ertain types of porous media, Johnson et al. [8] have shown that the viscous characteristic length can be expressed by where F = α ∞ /φ is the formation factor.S ubstituting the expression of the formation factor in Equation (18), together with relationship of Equation (17),t he previous equation readily givesthe following micro-macro relationship: Here P (Ω zz )i st he same polynomial as in Equation (17) with coefficients defined in Table II for Λ z and Λ xy . Estimates of Λ /Λ z and Λ /Λ xy obtained with Equation (19),a nd the respective coefficients of Table II, are plotted in Figure 6(dotted lines). These estimates are also compared with results obtained by Tomadakis and Robertson [13] for the three specificr andomly overlapping 1-d, 2-d and 3-d fiber structures. Again, good correlations are obtained, except for Λ /Λ z for the 2-d structure (red circles in Figure 6a). In this case, the ratio is lower than the expected ratio of 2. The reasons previously givenf or the tortuosity still apply here to explain this discrepancy.  Figure 4. The symbols refer to numerically computed values. Dotted lines are estimates obtained by the micro-macro relationship, Equation (19).T hick gray lines refer to Tomadakis and Robertson'sm odel (Equation (10),E quation (11) and Table 1 of [13]).
It is worth mentioning that the results by Tomadakis and Robertson also showthat the Λ /Λ xy ratio is no more constant as the fibero rientation coefficient approaches 1. Forh igher orientation coefficients, it tends to increase as porosity decreases.
The results above confirm the accuracyo ft he numerical model and indicate that it captures the essential physics of the fluid-structure interaction as frequencytends to infinite. We therefore conclude that the proposed micro-macro relationships may be regarded as generally valid.

Theoretical framework
Permeability can be derivedf rom the solution of Stokes equations, where − → v , p and η are the velocity,pressure, dynamic viscosity of the fluid, respectively,and − → G = − → ∇p m is amacroscopic pressure gradient acting as as ource term. The velocity − → v verifies the non-slip boundary condition on the wet surface of the fibers Because of the spatially periodic character on the large scale of the porous media, − → v and p are assumed to be spatially periodic functions with aperiod equal to the cell size Ω.
The system of Equations (20) -( 21)w ith the periodic boundary condition is numerically solved for as pecified macroscopic pressure gradient − → G on the REVs, which is set to be equal to aprescribed constant vector.Since Equations (20) and (21) form al inear system of equations, it can be demonstrated that − → v is al inear function of − → G. These quantities are related by the permeability tensor k 0ij , where k 0ij is asymmetric positive definite tensor.Here the components k * 0ij are derivedfrom where v i are the components of the local velocity field. Similarly to the tortuosity tensor α ∞ij , k 0ij takes the same values along the x and y axes and can be different along z.

Numerical results and discussions
The permeability of each generated REV described in Section 2having been calculated, the next objective is to obtain ad escription for all types of random fibrous media using their single geometrical characteristics r f , φ and Ω zz .Asfor the electric conduction problem (tortuosity α ∞ and viscous length Λ), as tudy of the possible effects of anisotropyo nt he results is conducted. We start however by presenting ac lassical model, which is used in the following for the analysis of our data. Several classical models aim at representing the dependence of permeability on the geometrical fiberweb characteristics. The most classical model is the Kozeny-Carman equation (see Equation (6) of [28]) where ζ is the Kozeny" constant" which depends on the particle shape and size forming the solid skeleton. As noted in this model, the normalized permeability k 0 /r 2 f is proportional to φ 3 /(1−φ) 2 .Consequently,the normalized permeabilities computed on the REVs are plotted in Figure 7infunction The numerical results showthat the behavior of the permeability tensor k 0ij can be different along the (x, y)a nd z axes. Figure 7s hows that the through-plane normalized permeability k 0z /r 2 f is more sensitive to fiber orientation than the in-plane normalized permeability k 0xy /r 2 f .  (25) and (26).Thick blue lines refer to Tarnow'swork (Equations (37) and (59) of [3]). Thick pink lines refer to Sangani and Ya o'swork (Tables II and III of [29]). In viewo fF igure 7a, in-plane normalized permeability k 0xy /r 2 f varies linearly in function of φ 3 /(1 − φ) 2 -t his is consistent with Kozeny-Carman equation. Figure 7b suggests that ratio k 0z /r 2 f also depends on the fiber orientation Ω zz .Indeed, the ratio k 0z /r 2 f increases significantly for larger fiber alignment in the direction of the wave propagation − → e 3 .Numerical fits can be obtained for k 0xy /r 2 f and k 0 /r 2 f in the form of log 10 k 0
The effect of fiber orientation appears to be negligeable on the in-plane normalized permeability k 0xy /r 2 f when compared to the through-plane normalized permeability k 0z /r 2 f .Some global numerical fitswit Equation (25) can be found for k 0xy and k 0 as linear forms, see Table III for the corresponding coefficients A and B.
Asimple expression for estimating the normalized permeability k 0z /r 2 f in function of φ 3 /(1 − φ) 2 and fiber orientation Ω zz can takethe form of log 10 k 0 Estimates of k 0 /r 2 f obtained with Equations (25) and (26) are plotted in Figure 7( dotted lines). As for the tortuosity and the viscous characteristic length, comparison with Equation (12) of Tomadakis and Robertson did not lead to good correlation on all the studied porosity range. Their permeability estimate may be regarded as valid provided that the throat region of the pore space varies relatively slowasone movesa wayfrom the throat. However, good comparisons were obtained with results obtained by Tarnow [ 3] and Sangani and Ya o [ 29] for the 1-d case of ar andom fiber array; this corresponds to the case of Ω zz = 1(black squares). This tends to validate our numerical results and micro-macro relationships givenbyEquations (25) and (26).
In summary,t he permeability of ar andom fibrous medium can be related to the fiberradius r f ,the porosity φ,and the angular orientation Ω zz .Itcan therefore be predicted directly from the knowledge of the microstructural features.

Static thermal permeability
The thermal terminology is used here butt he following developments are also valid for diffusion of Brownian particles whose size is small with respect to atypical size of the medium. Isothermal heat diffusion and Brownian motion in porous media are governed by aPoisson equation, where τ is the local field. When the frame has asufficiently large thermal capacity,t he excess temperature τ can be considered to vanish at the fiber walls, and the boundary condition is The excess temperature field τ is spatially periodic. The mean value of the excess temperature field in the fluid space between fibers is directly related to the definition of the (scalar)static thermal permeability, Alternatively,the diffusion controlled trapping constant of the porous frame is givenbyΓ=1/k 0 . Based on the numerically calculated values of k 0 on the REVs, the normalized thermal permeability k 0 /r 2 f is shown in Figure 8asafunction of the porosity for the various fiber orientations. Beacause the diffusion of heat does not provide anypreferred direction, the static thermal permeability k 0 normalized by the square of the fiber radius r 2 f can generally be written as af unction independant of fiber orientation. In their work, Olnya nd Panneton [30] provide the following relation between the low-frequency Champoux-Allard description of the thermal characteristic length Λ lf and the static thermal permeability (Section II.B of [30]), Substituing Λ lf by Equation (12),o ne can derive ab asic expression for the normalized thermal permeability in function of the open porosity.Itisgiven by (valid for 0.75 φ 1).
Va lues of m 1 = 0.0691 and m 2 = 0.0216 are obtained by curve-fitting in al east square sense on the simulation results.
Estimates of k 0 /r 2 f obtained with Equations (31) are plotted in Figure 8(thick line). Forthe static thermal permeability,f ew results are available in the litterature for the types of random fiber structures under study.However, Umnova et al. [31] developed atheoretical expression for a1-d square array of fibers. We present in Figure 8acomparison between this analytical result and our numerical finite element solution. We see that both estimates agree well.
In summary,the static thermal permeability of arandom fibrous medium can be related to the fiber radius r f ,t he porosity φ,a nd the angular orientation Ω zz .I tc an therefore be predicted directly from the knowledge of the microstructural features.

Application
In this section, the micro-macro relationships developed earlier are used to investigate the influence of the randomness of the fiber orientation on the acoustical properties. Here, three different nonwoventransversely isotropic fiber assemblies are studied. The first assembly has an open porosity of 0.90 and is composed of fibers having an average diameter of 25 µm( this corresponds to al ayer of natural milkweed hollowfi bers compacted up to approximately 33.5 kg/m 3 ). Note that the open porosity is the inter fiberporosity since the hollowpart of the fibers it is too small to significantly influence the acoustic behaviour. The fibers are then considered to be solid. Similarly,inthe second assembly,the average diameter is 25 µm; however, this time the open porosity is 0.99 (this correspond to a layer of natural milkweed hollowfi bers compacted up to approximately 3.35 kg/m 3 ). The porosity of the third assembly is also 0.99; however, the average fiber diameter is reduced to 10 µm(this corresponds to alight glass fiberof 25 kg/m 3 ).
Based on the microstructural features r f and φ of the assemblies, the previous micro-macro relationships are used to evaluate the geometrical and transport properties at different angular orientations Ω zz along the z-axis. These relationships are givenb yE quation (12) for Λ ,E quation (17) for α ∞ ,E quation (19) for Λ,E quation (26) for k 0 ,and Equation (31) for k 0 .Knowing the fiber radius r f (inconnection with the type of fiber),the fiber orientation Ω zz (inconnection with the type of nonwovenmanufacturing process), and the open porosity φ (inconnection with the rate of compaction), these relations completely definethe input macroscopic parameters to use in an equivalent fluid model. Here, the six-parameter Johnson-Lafarge equivalent fluid model is used. In this model, the equivalent dynamic density and bulk modulus are respectively givenby and where ν is the thermal diffusity of the fluid, P 0 is the static pressure, and ω is the angular frequency. From the previous dynamic properties, the normalized characteristic impedance and the complexwavenumber are respectively defined as and where c 0 is the speed of sound in the fluid. Finally,f rom these twoacoustic properties, the normal incidence sound absorption coefficient and transmission loss of afi brous slab of thickness h are respectively givenby and TL = 20 log 10 cos (qh) + j 2 Z + 1 Z sin (qh) . (37) Figure 9presents the normalized characteristic impedance Z and complexw avenumber q for the three materials at different angular orientations Ω zz .O ne can note that Z and q seem sensitive to the angular orientation for lower porosities, or for smaller fiber diameters. However, for a diameter of 25 µm at ahigh porosity of 0.99, both Z and q seem not sensitive to the angular orientation, see Figures 9c and 9d. Only based on these complexa coustic properties, this conclusion may be misleading since for the second case (where Z and q seem not sensitive to orientation), the angular orientation strongly affects the sound absorption coefficient of a100-mm thick (4 inches)l ayer as shown in Figure 10c. In fact, comparing Figure 9with

Conclusions
Randomly overlapping fiber structures have been generated from the knowledge of three parameters describing their microstructure: the characteristic fiber radius r f ,the open porosity φ,and the fiber orientation coefficient Ω zz . Their macroscopic geometrical and transport properties were numerically calculated on Representative Elementary Vo lumes (REVs)ofthe fiber structures. Their dependence on the microstructural parameters were expressed in terms of micro-macro relationships. These relationships are givenb yE quation (12) for the thermal characteristic length Λ ,Equation (17) for the tortuosity tensor α ∞ij , Equation (19) for the viscous characteristic length tensor Λ ij ,E quations (25) and 26 for the static viscous permeability tensor k 0ij ,and Equation (31) for the static thermal permeability k 0 . The numerical results and the proposed micro-macro relationships were validated by comparison with existing results found in the literature. Contrary to existing results, the newresults coverthe whole range of fiber orientations. However, the results are limited to open porosities greater or equal to 0.75. Moreover, some questions remain open concerning the overlapping of the fibers. In fact, in areal fibrous material, the fibers do not really overlap. Nevertheless, due to the possible crushing of the fibers one on the other,i tw as argued that the overlapping may present more realistic predictions than the non-overlapping case. This argument is also supported by the good comparisons between overlapping model predictions and some experimental data in the reviewbyT omadakis and Robertson [13].
Finally,t he micro-macro relationships were used to study the influence of the fiber orientation Ω zz on the acoustic properties of three specificnonwovenfibrous materials of different porosities and fiber radii. It wasf ound that small variations on the characteristic impedance Z and wave number q,with respect to Ω zz ,donot systematically imply small variations on global acoustic indicators such as the sound absorption coefficient α and the sound transmission loss TL -the reverse being also true. Consequently,inorder to avoid misinterpretation of the complex influence of the microstructural features on the acoustic behavior,t he desired acoustic characteristics, or the microstructural parameters, need to be clearly defined first. In this case, the micro-macro relationships may be very useful to investigate this complexi nfluence between the microstructure and the macroscopic behaviors.