hydraulic permeability of fibrous porous media

International Journal of Heat and Mass Transfer 54 (2011) 4009–4018

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International Journal of Heat and Mass Transfer

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Hydraulic permeability of fibrous porous media

Dahua Shou, Jintu Fan ⇑, Feng DingInstitute of Textiles and Clothing, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

a r t i c l e i n f o

Article history:Received 7 October 2010Received in revised form 17 March 2011Available online 6 May 2011

Keywords:Fibrous mediaHydraulic permeabilityAnalytical model

0017-9310/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2011.04.022

⇑ Corresponding author. Tel.: +852 2766 6472; fax:E-mail address: [email protected] (J. Fan).

a b s t r a c t

In this study, the hydraulic permeability for viscous flow through fibrous media of high porosity is inves-tigated theoretically. Fibrous media in one-dimensional (1D), two-dimensional (2D) or three-dimensional(3D) structure are approximated as consisting of repetitive unit cells based on Voronoi Tessellationapproximation and volume averaging method. In the new model, the hydraulic permeability of fibrousmedia is described as a function of porosity and fiber radius as well as geometrical formation factorsincluding the degree of randomness (viz. randomness of fiber distribution) and fiber orientation. In par-ticular, the slip effect for flow through superfine fibrous media is analytically studied. The prediction ofthe new model for fibrous media with porosity greater than 0.7 is highly consistent with the analytical,experimental and numerical results found in the literature. It is further demonstrated that randomlypacked fibrous media have larger permeability than regular ones, the hydraulic permeability of fibrousmedia increases with increasing of through-plane orientation, but is less dependent on in-plane fiber ori-entation, and the slip effect on the longitudinal hydraulic permeability is greater than the perpendicularone.

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1. Introduction

Fibrous porous media have received great attention in a widevariety of applications including fuel cells, functional clothing, fil-tration, thermal insulation, paper products, medical science, andbiological transport phenomena [1–4]. One of the most importantmeasures of characterizing the transport phenomena in porousmedia is hydraulic permeability. Although numerous studies havebeen conducted on this well-known topic, the accurate determina-tion of hydraulic permeability of fibrous media taking into accountof their complicated geometrical structures and slip effect remainsto be solved. In this work, geometrical structure of fibrous porousmedia is simplified as consisting of arrays of cylinders in three cat-egories of formations, viz. 1D structure in which all fibers are par-allel with one another; 2D structure in which fibers lie in parallelplanes with directional or random orientations; and 3D structurein which fibers are directionally or randomly oriented in space.

It is generally assumed that the flow through fibrous media,which is dominated by viscosity at low Reynolds number, can bewell described by Stoke’s law [5]. In the creeping regime, the ef-fects of gravity and inertia become negligible, and the viscoushydraulic permeability of fibrous porous media is defined byDarcy’s law:

ll rights reserved.

+852 2773 1432.

hui ¼ Klrp; ð1Þ

where K is Darcy hydraulic permeability, l is the fluid viscosity,rpis the pressure gradient and hui is the average fluid velocity. Darcy’slaw gives a general description of fluid velocity as a function ofpressure gradient, fluid viscosity and hydraulic permeability. Jack-son and James [6] proposed a simple dimensionless scaling estimatefor hydraulic permeability, viz.

K=r2 ¼ huil=r2rp ¼ f ðeÞ ð2Þ

taking into account the influence of the fiber radius r at a givenporosity, which gives a good qualitative agreement to experimentaland numerical results.

Prediction of the hydraulic permeability of fibrous porous med-ia has been widely studied in the past. In 1950s, Kuwabara [7] pre-dicted the hydraulic permeability of flow perpendicular to 1Dcircular fibers based on the unit cell approach, while Happel [8] im-proved the work of Kuwabara by solving this problem with zeroshear at the cell boundary instead of zero vorticity as in Ref. [7].However, their analytical models were dependent on simplifiedand regular structures with over idealization. Sangani and Yao[9] obtained numerical results for the hydraulic permeability of1D randomly distributed array of fibers towards perpendicularand parallel flows, but the accuracy of their results was limitedby the computer power at that time. In 1996, Higdon and Ford[10] used a rigorous numerical method, the spectral boundary ele-ment formulation, to calculate the hydraulic permeability of 3D

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Nomenclature

a square’s edge lengthAi constants to be determinedci correction factorD space dimensionDc volume averaged dragDc,eff effective volume averaged dragh normal to the square borderi ordinal numberK permeabilityKi fractional permeabilityKeff,i effective fractional permeabilityKn Knudsen numberl radial distanceL length of the fiberm mass of the unit cellMa Mach numbern normal to the fiber surfaceN population of fibersp pressurer fiber radiusR radius of circular unit cell

Re Reynolds numberS area of a unit cellu fluid velocityup parallel velocityus slip tangential velocity

Greek symbolsa degree of randomnessb scale parameter/i fiber fraction/eff ;i effective fiber fractionl fluid viscositye porosityq densityk mean free pathw stream functionh azimuthC Gamma distribution function

4010 D. Shou et al. / International Journal of Heat and Mass Transfer 54 (2011) 4009–4018

networks of fibers on regular cubic lattices. Later, Clague and Phil-lips [11] performed hydraulic permeability calculations for orderedperiodic array of fibers using a slender body theory. Sobera andKleijn [1] analytically studied the hydraulic permeability of bothorderly and disorderly 1D and 2D fibrous media. They proposed ageometrical scale analysis method which effectively modified thescaling rule previously suggested by Clague and Phillips [11] andtheir model was validated by simulation results. Koponen et al.[12] applied Lattice Boltzmann method to simulate flow in 3D ran-dom fibrous media across a wide range of porosity and developed asemi-empirical model for the hydraulic permeability of fibrousmedia. Tomadakis and Robertson [3] numerically determined thetortuosities of flows through 1D, 2D and 3D randomly distributedfibrous materials based on random walk diffusive trajectories ofmolecules, and calculated the viscous permeability of fibrous med-ia based on electrical conduction principles. Tamayol and Bahrami[13] applied the integral technique to determine permeability of1D square array of fibers, and later studied 2D and 3D cases bythe scale analysis method [14]. Very recently, Tahir and Tafreshi[15] solved the Stokes flow equation in virtual fibrous media usingfinite volume method and found the transverse hydraulic perme-ability was little affected by in-plane fiber orientation, but in-creases with increasing through-plane fiber orientation.

Although many researchers have performed computationalsimulation to predict the transport properties for fibrous mediawithin the limit of computer power, little analytical study has beencarried out so far to elucidate the influence of geometrical forma-tion factors on hydraulic permeability. This is becoming moreimportant, with a view of the wide application of superfine fibrousmaterials, within which the additional effect of slip flow should beconsidered.

Here, we therefore aim at developing a comprehensive analyti-cal model of hydraulic permeability taking into account the differ-ent geometrical formation factors and slip effect. The structure ofthe rest paper is organized as follows. In Section 2, we presentthe unit cell method, the Voronoi Tessellation method, and theaverage volume method, and show how to apply these methodsto calculate the hydraulic permeability of fibrous media with dif-ferent geometrical formation factors and slip effect. In Section 3,analytical solutions for hydraulic permeability of fibrous media

are validated against the theoretical, experimental, and numericalresults from the literature, and are discussed with an emphasis onthe influence of the varying geometrical formation factors and slipeffect. This paper ends with the summary of the main conclusionsin Section 4.

2. Theoretical background

In this section, we develop the model of hydraulic permeabilityof fibrous media from simple and regular structures to more com-plicated ones step by step. First, unit cell method is applied to de-scribe the transport properties of 1D regular fibrous media,considering the slip flow boundary conditions. And then we usethe Voronoi Tessellation method to characterize the hydraulic per-meability of fully randomly distributed 1D fibers, and also take intoaccount different degrees of randomness. Finally, we employ theaverage volume method to extend the 1D model to 2D and 3Dcases, and also to model the different orientation distributionsfor more realistic fibrous media.

2.1. Unit cell method

Predicting the exact hydraulic permeability requires detaileddescription of the geometry of the fibrous media, which is notpractically feasible for the complex real fibrous media. In this work,fibrous media are therefore assumed to be composed of periodicalunit cells, which represent the geometrical nature of the micro-structure of the material. We investigate the 1D system first asevery fibrous medium is a mixture of long and thin 1D fibers, sothe 1D model has caught the most important intrinsic nature ofthe system and it can also be extended to model the 2D and 3Dmedia.

The unit cell for regular 1D fibrous media (viz. regular arrays ofparallel fibers) is shown in Fig. 1. By assuming that the flow pathsurrounds the fibers and based on the conservation of mass andmomentum, we can obtain an approximate solution as the analyt-ical model of the hydraulic permeability of fibrous media.

Recently, superfine fibers are used in many applications. Forsuperfine fibers ranging from 50 nm to 5 lm in this work, the con-

Fig. 1. A unit cell in regular array of parallel fibers. The square is the unit cell withthe same area as the circle in dotted line.

D. Shou et al. / International Journal of Heat and Mass Transfer 54 (2011) 4009–4018 4011

tinuum non-slip assumption, that the flow in contact with the solidboundary of fibers is stationary, is not strictly correct, for it has notconsidered the molecular characteristic of the flow when the fiberradius is comparable to the mean free path of fluid molecules.Therefore, we assume that slip flow, viz. the normal componentof velocity vanishes while the tangential velocity is not zero, is per-mitted to occur on the surface of the fibers. Note that rarefactionmay be a possibility in micro porous media. According to relation-ship between Reynolds number Re, Mach number Ma and Knudsennumber Kn in Ref. [16], we can infer that Ma

Re�Kn � 0:05. In our work,we have Re� 1 and Kn < 1, and thus Ma� 0.05, which is muchlower than the threshold (i.e. 0.3) of gas compression and rarefac-tion. Therefore, the effect of the compressibility and rarefaction canbe neglected.

For the flow around a fiber, the tangential velocity is propor-tional to the tangential stress when partial slip occurs and thefirst-order slip boundary condition applies [5]:

us ¼ k@u@n

; ð3Þ

where us is the slip tangential velocity on the surface of fibers, and ncorresponds to the normal direction of fiber surface. Wang [17]studied slip flow through square and triangular arrays of fibers byeigenfunction expansions and collocation. As an easier way, ourwork directly considers the boundary condition of slip flow on the

A2 ¼ð1� eÞð1þ 2KnÞhui

�0:5 lnð1� eÞ þ 0:25� e� 0:25ð1� eÞ2 þ 2Knð�0:5 lnð1� eÞ � 0:25þ 0:25ð1� eÞ2Þ: ð15Þ

fiber surface based on unit cell method. Since the Knudsen number(Kn) is defined as Kn ¼ k

r, where k is the mean free path of air and isalso defined as the slip coefficient, the slip tangential velocity can beexpressed as

us ¼ Kn@u@n

r: ð4Þ

For convenience and without losing generality, the unit cell isassumed to be a square with an edge length a. The porosity e forthis arrangement of, both the unit cell and the whole fibrous mediacan be determined by

e ¼ 1� pr2

a2 : ð5Þ

Under the condition of low Reynolds number, the steady flowthrough the unit cell can be described by the Stokes equation:

�rpþ lr2u ¼ 0 ð6Þ

and the continuity equation:

r � u ¼ 0: ð7Þ

To Stokes’ approximation, the stream function for the flow per-pendicular to the fibers satisfies the biharmonic equation:

r4w ¼ 0; ð8Þ

where w is the stream function using cylindrical coordinates (l,h)with velocity components (ul, uh) as

ul ¼1l@w@h

; uh ¼ �@w@l: ð9Þ

On the fiber surface the normal component of velocity vanishes:

1l@w@h¼ 0 for l ¼ r: ð10Þ

According to Eq. (4), the slip tangential velocity around the fibersurface can be expressed as

@w@l¼ r Kn l

@

@l1l@w@l

� �for l ¼ r: ð11Þ

Consider an imaginary circular cell, which is coaxial with the gi-ven fiber and has the same area as that of the square cell as shownin Fig. 1. The radius of the imaginary circular cell is R ¼ rffiffiffiffiffiffi

1�ep . For the

continuous flow, on the surface of the circular cell, we have

1R@w@h¼ hui cos h ð12Þ

and the vorticity is assumed to be zero for the flow perpendicular tothe fibers, [7] viz.

r2w ¼ 0: ð13Þ

An approximate solution of Eq. (8) as in Ref. [8] is

w ¼ ðA0lþ A1l�1 þ A2l lnðl=RÞ þ A3l3Þ sin h; ð14Þ

where A0, A1, A2, and A3 are constants to be determined by fulfillingthe boundary conditions as described by Eqs. (10)–(13). For thesolution of A2, which is relevant to the present work, we have

Since the drag acting on a unit length of fiber can be expressed as[5]:

F ¼ pR2rp ð16Þ

or obtained by integrating stress components over the fiber surface[7], viz.

F ¼ 4plA2 ð17Þ

by applying Darcy’s law as described by Eq. (2) as well as Eqs. (15)–(17), we can obtain dimensionless hydraulic permeability in theunit cell for flow perpendicular to the fibers:

K=r2 ¼ �0:5 lnð1� eÞ þ 0:25� e� 0:25ð1� eÞ2 þ 2Knð�0:5 lnð1� eÞ � 0:25þ 0:25ð1� eÞ2Þ4ð1� eÞð1þ 2KnÞ : ð18Þ

Fig. 2. A unit cell in randomly distributed array of parallel fibers. The Voronoi is theunit cell with the same area as the circle in dotted line.


From Stokes Eq. (6), we then use cylindrical coordinates (l, h)and compute the velocity up for flow parallel with the 1D fibers.

According to Eq. (4), the slip tangential velocity on the fiber sur-face can be expressed as

up ¼ r Kn@up

@lfor l ¼ r: ð19Þ

We have @up

@m ¼ 0 on the bounder of the square due to the sym-metry of the cell, where m is the normal to the square border.The velocity of the square cell is approximated by the circular cellwith radius R, and it can be found that

@up

@l¼ 0 for l ¼ R: ð20Þ

The velocity satisfies Stokes equation (6) and the boundary con-ditions Eqs. (19) and (20) can be expressed as

up ¼1

4lrp½2R2 lnðl=rÞ � ðl2 � r2Þ þ 2KnðR2 � r2Þ�: ð21Þ

The mean velocity is therefore

hui ¼ 1pR2

Z R

r2plup dl: ð22Þ

Substituting Eq. (22) into Eq. (2), we readily obtain the dimension-less hydraulic permeability in the unit cell for flow parallel withfibers:

K=r2 ¼ �0:25 lnð1� eÞ þ 0:25� e� 0:25ð1� eÞ2 þ e2Kn2ð1� eÞ : ð23Þ

2.2. Voronoi Tessellation method

Now let’s consider the slightly more complicated case, in whichfibers of 1D media are placed randomly. Flow through the unit cellbecomes more complex because of the disorder of the voids. To de-scribe the randomness of fibers location, the Voronoi Tessellationmethod [18] is applied.

Consider the cross-sections of parallel 1D fibers in a plane asshown in Fig. 2, each fiber is assumed to be surrounded by a polyg-onal cell whose boundaries are defined by the perpendicular bisec-tors of the lines joining each fiber with its nearest neighbor asdescribed by Voronoi Tessellation method.

The mean area of the polygonal unit cell hSi is determined from:

hSi ¼Z

Sf ðSÞdS ¼ pr2

1� e; ð24Þ

where S is area of a unit cell, and f(S) is the probability density dis-tribution function of the polygonal with area S. For fibrous mediawith random distribution, f(S) can be described by the Gamma dis-

K=r2 ¼R

f ðSÞ½�0:5S2 lnðpr2

S Þ þ pr2S� 0:75S2 � 0:25ðpr2Þ2 þ 2Knð�0:5S2

4pr2ð1þ 2KnÞhSi

¼ �0:64 lnð1� eÞ þ 0:263� e� 0:25ð1� eÞ2 þ 2Knð�0:64 lnð1� e4ð1� eÞð1þ 2KnÞ

tribution [19], and the mean square of the area of the cells for ran-domly distributed fibrous media is

hS2i ¼Z

f ðSÞS2dS ¼ ahSi2; ð25Þ

where f ðSÞ ¼ Sb�1

CðbÞbhSi

� �bexpð� b

hSi SÞ is a Gamma distribution function,C(b) is a Gamma distribution, b is the scale parameter determinedby a, and a is a simple parameter, for fully random fiber distribu-tion, a = 1.28 [18]; for regular fiber distribution a = 1. Hence, theparameter a can provide a measure of the degree of randomnessof fiber distribution.

The effective transport property is regarded as the equivalentproperty of a simple homogeneity (1D regular fibrous media) to re-sponse transport as that of the original 1D randomly distributed fi-brous system, which means they share the same total Darcypermeability (or momentum of flow). Therefore, ‘‘hui’’, both themean velocity of flow through random fibrous media and the equiv-alent regular one, is calculated based on conservation of momen-tum, and it is the mass-weighted mean velocity of each cell, viz.

hui ¼PN

i¼1miui

Nhmi ¼PN

i¼1ðqSiLÞui

NhqSLi ¼R

f ðSÞSudShSi ; ð26Þ

where N is the number of fibers in the cross sectional area, i de-scribes the ith fiber in the random fibrous media, mi is the massof the unit cell containing the ith fiber (mi = qSiL, q is the densityof the unit cell, Si is the cross sectional area of the unit cell and Lis the length of the fiber), and hmi is the mean mass of unit cells.

Substitute Eqs. (18), (23), (25), and (26) to Eq. (2), we can get

the dimensionless hydraulic permeability of flow K=r2 ¼ lR

f ðSÞSu dS

r2rphSi

through 1D fibrous media within a degree of randomness a. Partic-ularly, for flow perpendicular to the fibers in the 1D fully random(viz. a = 1.28) fibrous media, the dimensionless hydraulic perme-ability is

lnðpr2

S Þ � 0:25S2 þ 0:25ðpr2Þ2Þ�dS

Þ � 0:097þ 0:25ð1� eÞ2Þ ð27Þ


and for flow parallel with the fibers, the dimensionless hydraulicpermeability is

K=r2 ¼R

f ðSÞ½�0:5S2 lnðpr2

S Þ þ pr2S� 0:75S2 � 0:25ðpr2Þ2 þ KnðS2 � 2pr2Sþ ðpr2Þ2Þ�dS2pr2hSi

¼ �0:64 lnð1� eÞ þ 0:263� e� 0:25ð1� eÞ2 þ ðe2 þ 0:28ÞKn2ð1� eÞ : ð28Þ

2.3. Volume averaging method

Now, we consider the general cases of 2D and 3D fibrousmedia with varying fiber orientation and distribution. The fiberarrangement is highly anisotropic at the macroscopic scale,while local transport properties are dependent on the localcharacteristics. The volume averaging method [20], determiningthe permeability by adding the contribution of fibers in eachdirection into a total drag, is well suited for predicting themacroscopic transport properties of fibrous media. One has tosolve a closure problem to map the local permeability intovolume averaging permeability and thus determine the macro-scopic hydraulic permeability within a representative elemen-tary volume.

In this work, we assume a fibrous medium consists of repetitivecubic cells, each of which gives a representation of the micro struc-ture of the fibrous medium. In each cubic cell, fibers may be lo-cated and oriented in different directions. The overall fiberorientation in each cubic cell can be characterized by the fractionallength of the fibers oriented in x, y, or z direction, respectively (seeFig. 3). And the fiber fractional length is determined by the fiberfraction in the same direction of the cubic cell. This assumptionis used to model 2D and 3D cases by combing the contribution of1D fibers in x, y, or z directions, and it is successfully validated inSection 3.

The volume averaged drag coefficient Dc of the cubic cell can becalculated as

Dc ¼P

Fili

Vði ¼ x; y; or zÞ; ð29Þ

where V is the volume of the cubic cell, i denotes the direction offibers, and Fi is the dimensionless force per cubic cell of fibers in idirection. For 1D fibrous media of fiber volume fraction /i, Clague

Fig. 3. 3D view of fibrous media based on cubic lattice.

and Phillips [11] calculated the hydrodynamic force acting the fi-bers, given by:

Fi ¼pr2

/iKið/iÞ; ð30Þ

where /i is the fiber fractions of the cubic cell and Kið/iÞÞ is the frac-tional permeability coefficients in i direction. Substituting Eq. (30)into Eq. (29) gives

Dc ¼X 1

Kið/iÞ: ð31Þ

However, the hydraulic drag forces of fibers in different direc-tions are interdependent, due to the co-existence of fibers in threedirections. Approximate corrections are necessary since, in 2D or3D fibrous media, the flow dragged by the fibers oriented in onedirection is also partially shielded by the fibers in other principledirection(s), which is not considered by the previous work usingvolume averaging method [21–23]. In our dilute system, whendetermining permeability of fibers in one direction, we assumethe fibers in other direction(s) are negligibly influential excepttheir space existence in the cubic cell. Therefore, the effective fiberfraction ueff,i corresponding to one direction, which should bemodified by removing the fiber fractions in other direction(s),can be expressed as

/eff ;i ¼/i

1� ð/� /iÞ¼ /i

eþ /i: ð32Þ

A correction factor is also needed as effective fiber fraction ueff,i isfor fibrous media without fibers in other direction(s). For the wholecubic cell with fibers in all directions, the volume-weighted effec-tive permeability Keff,i for the considered direction is calculated as:

Keff ;i ¼ ciKið/eff ;iÞ; ð33Þ

where ci = e + /i is the correction factor, indicating the volume ratioof the cubic cell without and with fibers in other direction(s).

From the effective volume averaged drag coefficientDc;eff ¼

P 1Keff ;ið/eff ;iÞ

, we can then obtain the dimensionless hydraulicpermeability using the following equation, viz.

K=r2 ¼ 1r2Dc;eff

¼X r2

ðeþ /iÞKi/i

eþ/i

� �24

35�1

: ð34Þ

3. Results and discussion

In this section, the prediction of the hydraulic permeabilityusing the above-described model is compared with analyticalexpressions, numerical results and experimental data available inliterature. For easy comparison, the results in drag coefficients,Kozeny constants or other relevant parameters reported in litera-ture are all converted into dimensionless hydraulic permeability.

This section is divided into four parts: 1D array of fibers, 2D fi-brous media, 3D fibrous media, and the effect of Knudsen number.

(a)

0.65 0.7 0.75 0.8 0.85 0.9 0.95 110-2

10-1

100

101

102

103

ε

K/r2

Regular Moderately random Highly random Fully random Ref. [1] Ref. [25] (square) Ref. [25] (hexagonal) Ref. [24] Ref. [14] (model) Ref. [14] (experiment) Ref. [3]

Flow perpendicular to 1D fibrous media

(b)

0.65 0.7 0.75 0.8 0.85 0.9 0.95 110-1

100

101

102

103

104

ε

K/r2

Regular Moderately random Highly random Fully random Ref. [9] (regular) Ref. [9] (fully random)Ref. [26] Ref. [27] (numerical) Ref. [27] (model) Ref. [13] Ref. [3]

Flow parallel to 1D fibrous media

Fig. 4. Dimensionless (a) perpendicular and (b) parallel hydraulic permeability as afunction of porosity for 1D fibrous media with varying degrees of randomness. Theresults of Refs. [1,3,9,13,14,24–27] are also added for comparison.

100

101

102

103

104

K/r

2

Regular Moderately randomHighly random Fully random Ref. [28] Ref. [29] Ref. [30] Ref. [14] Ref. [3]



Hydraulic permeability of fibrous media is studied in continuumregime (Kn < 10�2) in Sections 3.1–3.3, and the slip flow(10�2 < Kn < 1) is discussed in Section 3.4. In this work, the degreeof randomness of fiber arrangement is represented by the value ofa: for simulating moderately random fibrous medium, a = 1.07;and for simulating highly random fibrous medium, a = 1.14. The fi-ber orientation and medium dimension can be effectively con-trolled by /x, /y, and /z. When /y = vz = 0, it is a 1D fibrousmedium; when /z = 0, /x – 0and /y – 0, it becomes a 2D fibrousmedium; when /x – 0, /y – 0 and /z – 0, it becomes a 3D fibrousmedium. For 2D fibrous media, from low to high level of in-planeorientation, /x changes from 0 to 1/2. For 3D fibrous media, fromlow to high level of through-plane orientation, /z changes from 0to 1 while maintaining /x = /y.

0.65 0.7 0.75 0.8 0.85 0.9 0.95 110-2

10-1

ε

Fig. 5. Dimensionless perpendicular hydraulic permeability as a function ofporosity for 2D fibrous media with varying degrees of randomness. The results ofRefs. [3,14,28–30] are also added for comparison.

3.1. 1D array of fibers

The dimensionless hydraulic permeability as a function ofporosity for viscous flow perpendicular to 1D array of fibers is pre-sented in Fig. 4a. The solid line is calculated using Eq. (18) for a reg-ularly (a = 1) distributed fibrous medium; the dotted line, thedash-dot line and dashed line are calculated using Eq. (23) for fully(a = 1.28), highly (a = 1.14) and moderately (a = 1.07) random fi-

brous media containing infinite amount of 1D array of fibers,respectively. From Fig. 4a, our results reveal that the hydraulic per-meability for flow perpendicular to the axis of 1D array of fibers in-creases with increasing degree of randomness. We also compareour calculations with the theoretical results of Sobera and Kleijn[1], Tomadakis and Robertson [3], and Tamayol and Bahrami[14], experimental results of Kirsch and Fuchs [24], and computa-tional and analytical results of Acrivos and Sangani [25]. Soberaand Kleijn [1] obtained hydraulic permeability of randomly distrib-uted system with a large amount of parallel cylinders by modifiedgeometrical scaling estimate. Tomadakis and Robertson [3] ob-tained analytical models of 1D, 2D and 3D randomly distributed fi-brous media. Tamayol and Bahrami [14] invested transversepermeability of 1D case analytically and experimentally. Kirschand Fuchs [24] measured the hydraulic permeability of square ar-rays of Kapron fibers, while Acrivos and Sangani [25] performedanalytical and numerical studies of viscous permeability of squareand hexagonal array of cylinders. It can be seen from Fig. 4a that,the calculation using our model is in excellent agreement withthe theoretical prediction of Sobera and Kleijn [1] for 1D fully ran-dom distributed fibrous media, and analytical and experimentaldata of Tamayol and Bahrami [14], as well as experimental mea-surements of Kirsch and Fuchs [24]. Predictions of permeabilityof both square and hexagonal array of fibers in Ref. [25] are iden-tical with our model, which reveals the fiber arrangement of highporous regular structures is negligible.

The experimental and theoretical results of hydraulic perme-ability for the flow parallel to the unidirectional fibers are shownin Fig. 4b. Like in Fig. 4a, the hydraulic permeability of unidirec-tional fibers increases with increasing degree of the randomly of fi-bers. Sullivan [26] reported data for the longitudinal viscous flowthrough randomly distributed wool, and Sangani and Yao [9]solved the Stokes equation numerically for longitudinal viscousflow through regular and random arrays of 1D cylinders. Tamayoland Bahrami [27] calculated parallel flow through regular array offibers numerically and theoretically, and they also obtained an-other analytical model based on integral technique [13]. Thenumerical results of Sangani and Yao [9] and all data of Tamayoland Bahrami [13,27] are in good agreement with our model of reg-ular packing, while the experimental results of Sullivan [26] areslightly greater than our perdition, which may be due to the factthat the bundles of loose fibers become looser during the course

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 110-1

100

101

102

103

104

ε

K/r2

OrthogonalModerately unidirectionalUnidirectional

Flow perpendicular to 2D fibrous media with different in-plane fiber orientations

Fig. 6. Comparison of dimensionless hydraulic permeability of 2D fibrous mediawith different in-plane fiber orientations.


of his experiment. However, the prediction of Tomadakis and Rob-ertson [3] is larger than both experiments and our model for longi-tudinal flow through 1D random structure.

From Fig. 4, we can find that randomly distributed fibrous med-ia are more permeable than less random ones for flow both per-pendicular and parallel to the direction of fiber array. This can beunderstood by the fact that average cell size increases with

0.7 0.75 0.8 0.85 0.9 0.95 110

-1

100

101

102

103

104

ε

K/r2

0.95 0.96 0.97

5

10

15

20

ε

K/r2

Regular Moderately random Highly random Fully random Ref. [10] Ref. [31] Ref. [32] Ref. [14] Ref. [3]

3D regular2D regular1D regularRef. [10]


Fig. 7. Dimensionless hydraulic permeability as a function of porosity for a 3Dfibrous media with different degrees of randomness. 1D 2D, and 3D models forregular structures, and the results of Refs. [3,10,14,31,32] are also added forcomparison.

K=r2 ¼�0:64 ln 1�e

1þe

� �þ 1�e

1þe

� �� 0:737� 0:25 1�e

1þe

� �2þ 2Kn �0:64 ln

��

4ð1� eÞð1þ 2KnÞ

increasing degree of randomness, which leads to larger averagepermeability of all cells and hence the total permeability of the fi-brous media.

3.2. 2D fibrous media

The hydraulic permeability for the flow perpendicular to lay-ered isotropic fibrous structures (/x = /y = 1/2) is calculated andis compared with those experimentally measured by pastresearchers in Fig. 5. Predictions of hydraulic permeability of 2Dregular structure by Tamayol and Bahrami [14] and randomarrangement by Tomadakis and Robertson [3] are also comparedin Fig. 5. Ingmanson [28] investigated the viscous flow through2D glass, nylon and paper fibers, with the fibers compressed intomats as the pressure drop across them increased. Wheat [29] alsomeasured the permeability of fibrous mats, but his results weremostly influenced by slip flow, except for the two data points inFig. 5. Labrecque [30] was interested in the effect of fiber cross sec-tion on hydraulic permeability, also supplying some results for amat of circular fibers. As can be seen from Fig. 5, experimental datatend to fall in between the line for the fully randomly (a = 1.28)distributed 2D array of fibers and the line for the moderately ran-domly (a = 1.07) distributed 2D array of fibers. Note that, at iden-tical porosity and fiber radius, the hydraulic permeability ofmore randomly distributed 2D fiber arrays is higher than that ofless randomly distributed 2D fiber arrays, but the difference isnot as significant as that in 1D fiber arrays. The following equationis the compact model of hydraulic permeability of common layeredfibrous media of random distribution:

1�e1þe

�� 0:097þ 0:25 1�e

1þe

� �2��

: ð35Þ

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 110-1

100

101

102

103

104

ε

K/r2

2D random3D randomUnidirectional


Fig. 8. Comparison of dimensionless hydraulic permeability of 3D fibrous mediawith different through-plane fiber orientations.

(a)

0.65 0.7 0.75 0.8 0.85 0.9 0.95 110-15

10-10

10-5

ε

K(m

2 )

Kn=0.001Kn=0.01Kn=0.1Kn=1

Flow perpendicular to 1D fibrous mediawith different Kn number

(b) 10-5

Kn=0.001Kn=0.01

Flow parallel to 1D fibrous mediawith different Kn number


Fig. 6 presents the calculated hydraulic permeability of 2D fullyrandom fibrous media (deduced from 1D fully random fibrousmedia) with different in-plane fiber orientations. In this case, thefraction of fibers in z-direction /z is 0 in order to maintain 2D lay-ered structure, while the fractions of fibers in x-direction /x and y-direction /y are generally not 0. As the fraction of fibers in x-direc-tion /x increases from 0 to 1/3, and then to 1/2, the fibrous mediumchanges from a unidirectional in-plane fiber orientation (which canbe considered as a 1D case) to a moderately unidirectional case,and then to a random in-plane fiber orientation (orthogonal),respectively. Our calculation plotted in Fig. 6 indicates that thehydraulic permeability is slightly influenced by the variation of/x, i.e. in-plane fiber orientation. Although this was predicted bypast researchers based on a numerical simulation [15], it was notverified with theoretical rigor until our present work.

3.3. 3D Fibrous media

Fig. 7 compares the calculated results using our model for 3Disotropic fibrous media (/x = /y = /z = 1/3) and the experimentaland numerical data available in the literature. Model of 3D ran-dom structure by Tomadakis and Robertson [3] is also comparedin Fig. 7. Higdon and Ford [10] used a boundary element methodto estimate the viscous hydraulic permeability of 3D networks of

(a)

0.65 0.7 0.75 0.8 0.85 0.9 0.95 110-1

100

101

102

103

ε

K/r

2

Kn=0.001Kn=0.01Kn=0.1Kn=1

Flow perpendicular to 1D fibrous mediawith different Kn number

(b)

0.65 0.7 0.75 0.8 0.85 0.9 0.95 110-1

100

101

102

103

104

ε

K/r

2

Kn=0.001Kn=0.01Kn=0.1Kn=1

Flow parallel to 1D fibrous mediawith different Kn number

Fig. 9. Comparison of dimensionless (a) perpendicular and (b) parallel hydraulicpermeability of 1D fibrous media with different Kn.

0.65 0.7 0.75 0.8 0.85 0.9 0.95 110-15

10-10

ε

K(m

2 )Kn=0.1Kn=1

Fig. 10. Comparison of (a) perpendicular and (b) parallel Darcy hydraulic perme-ability of 1D fibrous media with different Kn.

cylindrical fibers ordered in different cubic lattices. Tamayol andBahrami [14] tested permeability of metal foam as 3D fibrous sys-tem using glycerol. Wiggins et al. [31] measured the hydraulicpermeability of various 3D fibrous media with several differentkinds of test liquids. More recently, Rahli et al. [32] experimen-tally investigated the hydraulic permeability of randomly ori-ented chopped fibers of bronze and copper wires. It can be seenfrom Fig. 7, most experimental data fall in between the line forrandomly distributed 3D fibrous media and the line for regularlydistributed ones. We can also see that the numerical results ofhydraulic permeability of ordered 3D cylinders in Ref. [11] areequivalent to the calculated results using our model for regularlydistributed fibers, confirming the validity of our model. As shownin the insertion of Fig. 7, the differences between the 1D, 2D and3D models for regular structure are indeed very large, and the 3Dmodel does fit the numerical data of Ref. [11] better than the 1Dand 2D models. Furthermore, We also find the hydraulic perme-ability of more randomly distributed 3D fiber media is higherthan that of less randomly ones at identical porosity and fiber ra-dius, but the difference is less pronounced than that in 1D and 2Dfibrous media. The following equation is the compact model ofhydraulic permeability of common 3D randomly distributed fi-brous media:


K=r2¼ 1ð1�eÞ

2:667ð1þ2KnÞ

�0:64ln 1�e1þ2e

� �þ 1�e

1þ2e

� ��0:737�0:25 1�e

1þ2e

� �2

þ2Kn �0:64ln 1�e1þ2e

� ��0:097þ0:25 1�e

1þ2e

� �2� �

2664

3775

þ 0:667

�0:64lnð 1�e1þ2eÞþ 1�e

1þ2e

� ��0:737�0:25 1�e

1þ2e

� �2

þKn 1�e1þ2e

� �2�2 1�e

1þ2e

� �þ1:28

� �2664

3775

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;

�1

:

ð36Þ

To investigate the influence of the through-plane fiber orienta-tion on the hydraulic permeability for flow perpendicular to 3D fi-brous media, we generated a series of 3D structures with differentthrough-plane fiber orientations, by changing the fraction of fibersin z-direction /z in steps, from 0 (2D layered random media), to 1/3(3D isotropic medium), and then to 1 (1D fibers parallel to theflow), and maintaining the same fractions of fibers for x and y-directions (/x = /y). The calculated results plotted in Fig. 8 showthat the hydraulic permeability for the flow perpendicular to the3D fibrous media increases significantly with increase in thethrough-plane fiber orientation.

3.4. Effect of Knudsen number

This section specifically considers the viscous flow through par-allel superfine fibers, although the present model can also be read-ily applied to 2D and 3D cases by applying Eq. (34). Thedimensionless hydraulic permeability and Darcy hydraulic perme-ability of 1D fibrous media with different Kn are calculated usingour model and are shown in Figs. 9 and 10, respectively.

From Fig. 9a, it can be seen that partial slip (10�2 < Kn < 1) in-creases perpendicular dimensionless hydraulic permeability ascompared to the no-slip (Kn < 10�2) case. The effect of Kn onincreasing the longitudinal dimensionless hydraulic permeabilityis even greater, as shown in Fig. 9b. This is understandable as par-tial slip enhances flow through the fibrous media. However, asshown in Fig. 10a and b, the Darcy hydraulic permeability ofthe fibrous media with higher Kn (i.e. more partial slip) is stillmuch lower than that with lower Kn. This can be ascribed to thatthe Darcy hydraulic permeability is proportional to the square offiber radius, which suppresses the slip effect on increasing perme-ability. Therefore, the effect of Darcy hydraulic permeabilitycaused by fiber radius is much greater than the accompanied slipinfluence.

We can also see from Fig. 9 that, the slip effect on the hydraulicpermeability is less pronounced at higher porosity of the fibrousmedia. Viscous flow through porous media is known to be in be-tween the Poiseuille’s and diffusional type. Flow is more of diffu-sional type in extremely high porous media, while it is more ofPoiseuille’s type at relatively low porosity regime. Therefore, theslip effect caused by Poiseuille flow is less significant at relativelyhigh porosity.

4. Conclusions

In this paper, we extensively studied the hydraulic permeabilityfor viscous flow through fibrous media with high porosity (e > 0.7).Unit cell method, Voronoi Tessellation method and volume averag-ing method are applied in steps to develop an analytical model ofhydraulic permeability for simplified and real fibrous media. Themodel is validated with the available theoretical, experimentaland numerical results from literature.

With the new model, the effects of fiber orientation, degree ofrandomness, and Knudsen number on the hydraulic permeabilityof 1D fiber arrays, 2D and 3D fibrous media are investigated. It

can be concluded that hydraulic permeability increases withincreasing degree of randomness, although the effect is less sig-nificant for higher dimensionality, and hydraulic permeability offibrous media is little dependent of in-plane orientation, but in-creases with increasing through-plane orientation. The slip effecton the longitudinal hydraulic permeability is greater than theperpendicular hydraulic permeability for 1D fibrous media. Ourwork provides an in-depth understanding of viscous flowmechanisms in fibrous media and can serve as foundation forthe design and optimization of fibrous media for variousapplications.

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