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2 International Journal of Heat an Mass Transfer 5 (009) Contents lists available at ciencedirect International Journal of Heat an Mass Transfer journal homeage: Analytical etermination of viscous ermeability of fibrous orous meia A. Tamayol *, M. Bahrami Mechatronic ystems Engineering, chool of Engineering cience, imon Fraser University, urrey, BC, Canaa V3T0A3 article info abstract Article history: Receive Aril 008 Receive in revise form 11 etember 008 Available online 1 January 009 Keywors: Fibrous meia Permeability Integral technique Unit cell Analytical solution Moeling In this stuy, the ermeability of orere fibrous meia towars normal an arallel flow is etermine analytically. In this aroach, orous material is reresente by a unit cell which is assume to be reeate throughout the meia. everal fiber arrangements incluing: touching an non-touching arrays are consiere. Moeling 1D touching fibers as a combination of channel-like conuits, a comact relationshi is roose to reict ermeability. Furthermore, emloying an integral technique an assuming a arabolic velocity rofile within the unit cells, analytical relationshis are eveloe for ressure ro an ermeability of rectangular arrangements. The eveloe moels are successfully verifie with existing exerimental ata collecte by others for square arrangement over a wie range of orosity. Due to the ranom nature of the orous micro structures, etermination of exact ermeability of real fibrous meia is imossible. However, the analyses eveloe for orere unit cells enable one to reict the trens observe in exerimental ata. The effects of unit cell asect ratio an fibers iameter on the ermeability are also investigate. It is note that with an increase in the asect ratio the normal ermeability ecreases while, the arallel ermeability remains constant. It is also shown that the ermeability of fibrous meia is relate to the iameter of fibers square. Ó 008 Elsevier Lt. All rights reserve. 1. Introuction Fibrous orous materials have alications in several engineering areas incluing: filtration an searation of articles, hysiological systems, comosite fabrication, heat exchangers, thermal insulations, an fuel cells [1 4]. Transort henomena in orous meia have been the focus of numerous stuies since 1940s which inicates the imortance of this toic. One of the most imortant roerties of orous structures is ermeability. Permeability is a measure of the ability of orous matrix to transort fluis. Preiction of velocity fiel lays a key role in estimating ermeability an analyzing the flow behavior in orous meia. This can be achieve by using Darcy s law which assumes a linear relationshi between volume-average suerficial flui velocity, U, an the ressure graient: rp ¼ l K U where l is the flui viscosity an K is the ermeability of the meium. Darcy s relationshi is emirical, convenient, an wiely accete. It can be shown that Darcy s equation hols while flui flowing through ores is in creeing regime [5]. To use Darcy s equation; however, we nee to know the ermeability of the meium beforehan. Permeability eens on several factors, incluing: * Corresoning author. aresses: ata4@sfu.ca (A. Tamayol), mbahrami@sfu.ca (M. Bahrami). ð1þ orosity, fibers size istribution, an arrangement; it is tyically foun using emirical correlations for most alications. Preiction of the ermeability of fibrous meia ates back to exerimental work of ullivan in 1940s [6] an theoretical works of Kuwabara [7], Hasimoto [8], Hael [9], an arrow an Loeffler [10] in 1950s. Kuwabara [7] reicte the ermeability of flow normal to ranomly arrange fibers for materials with high orosity. He solve the stream function an the vorticity transort equations aroun with limite bounary layer aroach. Hasimoto [8] an arrow an Loeffler [10] etermine the ermeability of normal an arallel flow to orere arrangement of cyliners, resectively. Hael [9] analytically solve the toke s equation for arallel an normal flow to a single cyliner with free surface moel (limite bounary layer). He also roose that the ermeability of ranom fibrous meia is relate to arallel an normal ermeability of 1D array of cyliners. Later, angani an Acrivos [11], erforme analytical an numerical stuies of viscous ermeability of square an staggere arrays of cyliners for the entire range of orosity, while their axes were erenicular to the flow irection. Their analytical moels were accurate for lower an higher limits of orosity. angani an Yao [1] reorte numerical results for the ermeability of ranom 1D fibers towars normal an arallel flows. ahraoui an Kaviany [13] inclue inertial effects an numerically etermine the ermeability of cyliners in normal flow an roose a correlation. Van er Westhuizen an Du Plessis [14] using numerical simulations roose a correlation for reiction of normal ermeability of 1D fibers. Analytical /$ - see front matter Ó 008 Elsevier Lt. All rights reserve. oi: /j.ijheatmasstransfer

3 408 A. Tamayol, M. Bahrami / International Journal of Heat an Mass Transfer 5 (009) Nomenclature A ore cross-sectional area (normal to flow), m fiber iameter, m D h hyraulic iameter of the ore, m f Fanning friction coefficient I olar moment of inertia of ore cross-section, m 4 I imensionless olar moment of inertia of ore crosssection, I ¼ I =A K ermeability, m K * non-imensional ermeability, K ¼ K= L channel eth, m P ressure, N/m Q volumetric flow rate, m 3 /s istance between ajacent fibers in square arrangement, m x istance between ajacent fibers in rectangular unit cell in x-irection, m y u u s U w istance between ajacent fibers in rectangular unit cell in y-irection, m normal velocity, m/s velocity at the borer of unit cell, m/s volume-average suerficial velocity, m/s arallel velocity, m/s Greek symbols C erimeter of flow assages, m e orosity l viscosity, N s/m u soli fraction, u =1 e non-imensional arameter, u 0 = /4u u 0 ubscrit D-W Darcy Weisbach reiction of the ermeability of general triangular arrangement was resente by Hellou et al. [15]. They roose a correlation for etermination of ermeability of erioic triangular arrangements. Recently, obera an Kleijn [16] stuie the ermeability of ranom 1D an D fibrous meia both analytically an numerically. Their analytical moel was base on scale analysis an the roose relationshi was a function of fibers istance an a non-imensional ranomness number. A comarison of moel of [16] with numerical results showe that their moel was accurate in highly orous materials [16]. However, the ifference in low orosity was consierable. Tomaakis an otirchos [17] roose a moel which enables the reiction of anisotroic ermeability through 1D, D, an 3D ranom fibrous bes. This moel was roose for ranomly overlaing fibers. Although moel of [17] were meant to cover all tyes of ranom fibrous meia, in some cases the errors between the moel an exerimental ata were consierable. A thorough comarison of [17] moel with exerimental ata, is available in []. Avellanea an Torquato [18] using conuction-base techniques, roose an uer boun for the ermeability of generalize fibrous meia. Tomaakis an Robertson [] showe that this boun is violate by several ata oints available in the literature. Using exerimental ata, Tomaakis an Robertson [] state that the uer an lower bouns for fibrous meia with ranom orientation of fibers were normal an arallel ermeability of 1D arrangements. everal exerimental stuies have also been conucte for etermination of the ermeability of fibrous meia [19 5]. Goo reviews of exerimental works are available in Jackson an James [6], Astrom et al. [7], an Tomaakis an Robertson []. Porous meia have otential alications in comact heat exchangers an fuel cell technologies. Our literature review inicates that less attention has been ai to etermination of ermeability of orere acke fibrous materials. More imortantly, we observe that: (1) majority of the existing correlations for ermeability are base on curve-fitting of exerimental or numerical ata; () most of the analytical moels foun in the literature are not general an fail to reict ermeability over the entire range of orosity. In this stuy, the ermeability of touching an non-touching orere fibrous meia towars normal an arallel flow is stuie. A comact relationshi is resente that can be use for touching fibers an also acke bes. In aition, novel analytical moels are eveloe using the concet of unit cell an integral technique. Assuming a arabolic velocity rofile within the unit cells an integrating the continuity an momentum equations, comact analytical relationshis are erive for ressure ro an ermeability of consiere atterns. Using the roose moel, one only nees average fibers iameter an the meium orosity to reict the ermeability. More imortantly, the resent analysis oes not require any tuning arameters. Normal an arallel ermeability of 1D fibrous meia are stuie to establish bouns, as reviously ointe out by Tomaakis an Robertson []. It is shown that the roose normal flow ermeability of square unit cell reicts the trens observe in exerimental ata; an serves as a lower boun for the ermeability of fibrous meia. The roose moel is successfully valiate against exerimental ata collecte from several sources over a wie range of orosity, matrix materials, an fluis.. Moel eveloment Due to the ranom nature of orous micro structures, etermination of exact ermeability of real fibrous meia is highly unlikely. As a result, simlifying assumtions shoul be mae to moel the geometry of the microstructure. Fibrous meia can be categorize into three forms []: (1) One imensional (1D): fibers are arallel to each other but, ranomly istribute in the volume. () Two imensional (D): fibers are locate in arallel lanes in which the fibers can have ranom orientation. (3) Three imensional (3D): fibers can have any orientations an locations in the sace. Among the above three tyes, the 1D moel is the most anisotroic tye an is consiere in this stuy. Following the aroach use successfully in several alications such as sherical acke bes [8] an in gas iffusion layer of fuel cells [9], an fibrous meia [7 11,30,31] a unit cell is consiere to analyze the geometry of the fibrous meia. The unit cell (or the basic cell) is the smallest volume which can reresent characteristics of the whole microstructure. Porous meia are assume to be erioic an the consiere unit cells reeat throughout the material. In the following sections several fiber arrangements incluing: touching an non-touching arrays are consiere. The flow is assume to be creeing, incomressible an steay state. Determination of the exact velocity rofile requires etail knowlege of the geometry of the meium which is not feasible in the case of orous meia. Moreover, even with secifie geometry an bounary conitions, fining exact analytical solution is

4 A. Tamayol, M. Bahrami / International Journal of Heat an Mass Transfer 5 (009) not guarantee an is a ifficult task for most cases. To overcome this roblem, an integral metho is emloye in this stuy. The integral metho rovies a owerful technique for obtaining accurate but aroximate solutions to rather comlex roblems with remarkable ease. The basic iea is that we assume a general shae of the velocity rofile. It must be note that we are not intereste in the recise shae of velocity rofile but rather nee to know the ressure ro over the basic cell to calculate ermeability. This can be accomlishe by satisfying conservation of mass an momentum in a lume fashion across the unit cell. As a result, an aroximate arabolic velocity rofile is consiere which satisfies the bounary conitions within the unit cell. The integral technique has been alie successfully to several classical roblems such as moving late an bounary layer [3]. Use this technique to moel orous meia, however, is a novel aroach. In the following sections, a combination of integral technique an asymtotic solution are unertaken to stuy the flow in a variety of fiber arrangements. 3. Permeability of touching fibers Two limits can be recognize for fibrous meia: (1) Touching cyliners, meia can be envisione as a acke be. () Highly orous materials with orosities near 1. Although acke fibrous bes have alication in comact heat exchangers, most of the roose moels an analytical solutions existing in the literature fail to reict the ermeability of these materials [4]. ince no flow coul ass erenicular to touching fibers (see figures in Table 1), normal ermeability is zero. Flui assing arallel to the axis of uniirectional fibers exeriences a channel-like flow; thus, the meia is treate as a combination of arallel constant cross-sectional conuits. Therefore, the ermeability can be relate to ressure ro in these channel flows. In this aroach, the cross-sectional area an the erimeter of the channel are require. Pressure ro can be calculate using Darcy Weisbach relation [3]: P z DP L ¼ f qu e D h where D h is the hyraulic iameter, L is the channel eth, f is the Fanning friction factor, an e an U reresent the orosity an the volume-average suerficial velocity, resectively. Using Eq. (1) the ermeability becomes: K D W ¼ f e ld h fqu Table 1 resents several touching fibers arrangements an the calculate ermeability using Eq. (3). Note that the Fanning friction factor must be known to calculate ermeability from Eq. (3). Numerical values of the Fanning coefficients reorte by hah an Lonon [33] an the resulte ermeabilities are also liste in Table 1. It shoul be note that all of these ossible arrangements cannot be consiere as a unit cell. Fig. 1a c shows how the triangular, the rectangular, an the hexagonal arrangements can ma a orous meium; As a result, they are unit cells. However, the octagonal arrangement in Table 1 oes not reresent, by itself, the characteristics of the fibrous meia since it must be combine by the rectangular arrangement. election of the characteristic length is an arbitrary choice an will not affect the final solution. However, a more aroriate length scale leas to more consistent results, esecially when ranom cross-sections are consiere such as in orous meia. A a ðþ ð3þ Table 1 Touching arallel fibers arameters. Porosity (e) Arrangement shae f Ref.[33] D h K D W /, Eq. (3) b c Normal flow ermeability = 0 for these arrangements. 1 is the fibers iameter. Fig. 1. Triangular, square, an hexagonal unit cells an combination of octagonal an square array of cyliners.

5 410 A. Tamayol, M. Bahrami / International Journal of Heat an Mass Transfer 5 (009) circular uct is fully escribe with its iameter, thus the obvious length scale is the iameter (or raius). For non-circular cross-sections, the selection is not as clear; many textbooks an researchers have conventionally chosen the hyraulic iameter. ffiffiffi Yovanovich [34,35] introuce the square root of area ( A ) as a characteristic length scale for heat conuction an convection roblems. ffiffiffi Bahrami et al. [36,37] through analysis showe that A aears in the solution of fully-eveloe ffiffiffi flow in non-circular ucts. They ffiffiffi also comare both D h an A an observe that using A as the characteristic length scale results in similar trens in Poiseuille number for microchannels with a wie variety of cross-sections. Therefore, in this stuy, the Fanning friction coefficient is calculate emloying Bahrami et al. [36] moel which consiere A ffiffiffi as the length scale. They roose a general moel that reicts the ressure ro for arbitrary cross-sectional channels. In the moel of [36], ressure ro is relate to geometrical arameters of the cross-section: DP L ¼ 16 lu I Ae ; I ¼ I A where I an A are the olar moment of inertia an the area of the assage cross-section, resectively. Using Darcy s relationshi an moel of [36], the non-imensional ermeability of erioic touching fibrous meia can be foun as: K ¼ K ¼ Ae 16 I ð5þ This relationshi can be easily alie to any touching fibrous arrangements incluing; triangular, rectangular, hexagonal, an checker boaring. Table comares the values calculate from Eq. (5) an the exerimental ata reorte by ullivan [6] for air flowing through staggere an square arrangements of coer wires, resectively. The ifference between the reicte values by the roose moel an the exerimental ata is reasonably within the context of orous meia. 4. Normal ermeability of square arrangement Fig. shows the rectangular arrangement of cyliners an the velocity rofile between these fibers. The unit cell is selecte as the sace between arallel cyliners as shown in this figure. For convenience an without losing generality, the unit cell is assume to be square, i.e., x = y =. The same aroach can be followe for the rectangular unit cell. The orosity for this arrangement can be etermine from: e ¼ 1 ð6þ 4 The ermeability is relate to the total ressure ro through the unit cell; see Eq. (1). Assuming creeing flow an neglecting inertial terms, the x-momentum equation reuces to tokes equation: Table Parallel ermeability of touching fibers. Porosity (e) Unit cell K*, Eq. (5) K*, ata [6] Difference (%) ¼ 1 P l x Due to symmetry, the y-comonent of velocity on the unit cell borer line is zero. The x-comonent of velocity is not necessarily zero; however, for very acke materials the borer velocity is negligible. At first, no borer velocity assumtion is mae to simlify the analysis. Later, the effect of the borer velocity on orosity will be investigate. olving Eq. (7) an assuming no-sli conition leas to a arabolic velocity rofile: u ¼ 1 P l x y ð8þ where is the half thickness of the unit cell in y-irection. For the unit cell of the rectangular arrangement, Fig. b, is: 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >< ¼ x; 0 6 x 6 4 ð9þ >: ; 6 x 6 Total ressure ro of the unit cell is calculate emloying an integral technique solution. ubstituting velocity rofile of Eq. (8) into continuity equation an integrating the result, the unit cell ressure ro is calculate: DP unitcell ¼ 3lQ Z = 0 x 3 ð7þ ð10þ where Q ¼ U is the total volumetric flow rate assing through the unit cell. The ressure ro in the basic cell is calculate as: 8 9 >< 3 tan 1 ffiffiffiffiffiffiffiffiffi þ DP unitcell ¼ 6lQ ð Þ þ >= ð Þ 5 >: >; þ 1lQ 3 ð Þ ð11þ Using the total ressure ro over the length of the unit cell, one can rewrite Darcy s relation as: DP unitcell Fig.. Rectangular arrangement of cyliners an consiere unit cell. ¼ lq K ð1þ

6 A. Tamayol, M. Bahrami / International Journal of Heat an Mass Transfer 5 (009) Combining Eqs. (11) an (1), one can observe that the ermeability is only a function of the geometrical arameters of the meia, which is in line with the creeing flow assumtion. Introucing soli fraction as / =1 e, the ermeability of square arrangement becomes: 8 ffiffiffiffi 1 / 0 1 K ¼ ffiffiffiffi þ 18 þ 1 18 ffiffiffiffi >< / 0 ð /0 1Þ ffiffiffiffi / 0 / 0 / 0 1 / 0 >: ð Þ þ tan 1 1 ð/ 0 1Þ 5 ffiffiffiffiffiffiffi / þ >= >; 1 ð13þ where K* = K/ an / 0 = /4/. Eq. (13) is base on the assumtion of no borer velocity, i.e., u b = 0. In general, the borer velocity is not zero. It is zero on the ege of the cyliners (no-sli conition) an reaches its maximum value at the half istance between cyliners in the x-irection. It is also execte that the maximum borer velocity be a function of the orosity [1], i.e., as the orosity increases the maximum borer velocity increases. For lower orosities the borer velocity is very small an for highly orous limits, aroaches to the Darcy velocity. In this stuy, the borer velocity increases linearly from the ege of the fibers (u b = 0) to its eak at the center of the unit cell: u b ¼ UgðeÞ x ; 6 x 6 ð14þ In Eq. (14) the maximum borer velocity is relate to orosity through g(e) which is assume to be a linear function of orosity; with g(0.15) = 0 for touching fibers an g(1) = 1 for high orosity limits: gðeþ ¼1:74e 0:74 ð15þ Using Eq. (14), our moel can be extene to account for the borer velocity. ince the no-sli bounary conition hols for the fibers surface, the velocity rofile in Eq. (8) is vali for 0 6 x 6 / range. But, for / 6 x 6 s/, the velocity rofile becomes: u ¼ 1 P l x y 4! þ gðeþ x U; 6 x 6 ð16þ Using continuity equation an the efinition of volumetric flow rate, one can calculate ressure graient: P x ¼ 1lQ ð Þ 1 gðeþ x ; 3 6 x 6 ð17þ The ressure ro in the basic cell is calculate as: 8 9 >< 3 tan 1 ffiffiffiffiffiffiffiffiffi þ DP unitcell ¼ 6lQ ð Þ þ >= 5 >: >; þ 1lQ gðeþ ð Þ 3 ð18þ Following the same aroach, the ermeability of the square arrangement can be etermine as: ( K ¼ 1ð ffiffiffi 0 1Þ gðeþ ffiffiffi / 0 0 þ 18 þ 1ð/0 1Þ ffiffiffi / 0 / ð1 / 0 Þ 18 ffiffiffi 0 / tan 1 ffiffi ) 1 / þ ð19þ ð/ 0 1Þ 5 In Fig. 3 reicte results from the resent moels, Eqs. (13) an (19) are comare with exerimental ata collecte from several sources. The ±15% bouns of the moel are also shown in the Fig. 3. Comarison of resent moels with exerimental ata. lot, to better emonstrate the agreement between the ata an the moel. The exeriments were conucte using ifferent fluis incluing: air, water, oil, an glycerol with a variety of orous materials such as metallic ros, glass wool, an carbon. As execte, the ifference ue to neglecting the borer velocity is only consierable in highly orous materials. More imortantly, the roose moel, Eq. (19), accurately reicts the normal ermeability of square arrangement of fibers over the entire range of orosity. A comarison between Eq. (19), exerimental ata, an other existing moels is also resente in Fig. 4. Although most of the moels reict similar trens for orosities near unity, they fail to reict the ata for low orosities. The roose moel, on the other han, is the only analytical-base moel that catures the trens of exerimental ata over the entire range of orosity; an oes not inclue any unknown constants. Accoring to Tomaakis an Robertson [], values for flow normal an arallel to 1D fiber arrangement resent the lower an uer bouns for the ermeability of fibrous meia. obera an Fig. 4. Comarison between roose moel, exerimental ata, an other existing moels.

7 41 A. Tamayol, M. Bahrami / International Journal of Heat an Mass Transfer 5 (009) Kleijn [16] also showe that among all ossible 1D arrangements, reeating attern of Fig. has the minimum ermeability. Therefore, Eq. (19) can serve as a lower boun for the ermeability of ranom fibrous meia. 5. Parallel ermeability of square arrangement For this arrangement, the selecte unit cell is the sace between arallel cyliners as shown in Fig. 5. The flui flows erenicular to aer. Following the same aroach iscusse in the revious section, a arabolic velocity rofile is assume: w ¼ 1 P l z y ; 0 6 x 6 ð0þ x! y w ¼ 1 P l z þ ; 6 x 6 where w is the velocity arallel to fibers in the z-irection, an is efine by Eq. (9). Using the continuity equation, one can calculate the volumetric flow rate through the unit cell as: Z Z Q ¼ wyx ð1þ A unitcell which can be evaluate as: " Q ¼ 1 P 6l þ z 3 4 þ þ # ð Þ3 þ 4 ðþ Using Darcy s relationshi an substituting for Q from the above, one can fin the arallel ermeability: ( " rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ¼ 4/ þ # s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4/ 1 þ / 9 8 þ / ) 1 / ð3þ 8 The resent moel, Eq. (3), is comare with analytical moel of Hael [9], numerical results of angani an Yao [1], an exerimental ata reorte by ullivan [6] an kartsis an Karos [3] for flow arallel to square arrangements of fibers, in Fig. 6. The analytical moel of [9] is accurate for high orosities [5]. This figure inicates that the roose moel shows a better agreement with exerimental ata in low orosities. 6. Effects of unit cell asect ratio After successfully valiating the roose moel, we can now use it to investigate the effect of major arameters. Effects of Flow y y x x Aunit cell Fig. 5. Parallel flow through rectangular unit cell. Fig. 6. Comarison of resent moel for arallel ermeability of square arrangement with exerimental ata of [6] an [3] an analytical moel of Hael [9]. orosity have alreay been iscusse in the revious sections. Eqs. (19) an (3) inicate that ermeability is irectly relate to fibers iameter square for square arrangement. Although Eqs. (19) an (3) are resente for square arrangement, the same analysis can be followe to stuy the effects of unit cell asect ratio, x / y, variation on the non-imensional ermeability of normal flow through rectangular fibers arrangement. In Fig. 7a the non-imensional ermeability, K*, is lotte versus unit cell asect ratio for normal an arallel flow through rectangular cell. Porosity an fibers iameter are ket constant while asect ratio is varie. The range of asect ratio is etermine by the nonoverlaing constraint for fibers, i.e., x, y P. For examle, for e = 0.5 the range of unit cell asect ratio will be x / y Fig. 7a shows that normal ermeability ecreases as unit cell asect ratio is increase. This is a irect result of reuction of the istance between ajacent fibers normal to the flow irection. For relatively low orosities the variation of ermeability is consierable; however, for highly orous materials this variation is relatively small. The non-imensional ermeability is lotte against unit cell asect ratio in Fig. 7b for arallel flow in rectangular cell. The changes of arallel ermeability ue to the variation of asect ratio are not as significant as the values of normal flow. It shoul be note that 1D fibers arallel to the flow can be treate as caillaries with sli velocity bounary conition; therefore, the ermeability in this case is relate to the cross-section of the ores. ince the area of the unit cell is not changing while orosity is ket constant, the arallel ermeability is not consierably affecte by the variation of basic cell asect ratio. 7. ummary an conclusions The ermeability of orere fibrous meia towars normal an arallel flow is analyze. In this stuy, orous material is reresente by a unit cell which is assume to be reeate throughout the meia. everal fiber arrangements incluing: touching an non-touching arrays are consiere. Moeling arallel touching fibers as a combination of channel-like assages, a comact relationshi is roose for reiction of the ermeability. Analytical moels are also eveloe by using the concet of unit cell an

8 A. Tamayol, M. Bahrami / International Journal of Heat an Mass Transfer 5 (009) The resent analysis rovies an in-eth knowlege on the effects of geometrical an thermohysical arameters involve on the ermeability of fibrous meia. This information can be use as guielines an criteria to esign, select, an otimize engineering systems that inclue orous meia. Acknowlegments The authors gratefully acknowlege the financial suort of the Natural ciences an Engineering Research Council of Canaa, (NERC). References Fig. 7. Non-imensional ermeability versus unit cell asect ratio for (a) normal, (b) arallel flow through square arrangement. introucing an integral technique solution. Assuming a arabolic velocity rofile within the unit cells, analytical relationshis are eveloe for ressure ro an ermeability of consiere atterns. The roose moel only requires the fibers iameter an the meium orosity, with no constant or arameter, to reict the ermeability of fibrous meia. The eveloe moels have been successfully comare with exerimental ata collecte from ifferent sources as well as existing moels in the literature for square arrangement over a wie range of orosity. It is also shown that the roose normal flow ermeability of square unit cell serves as a lower boun for the ermeability of fibrous meia. A arametric stuy was conucte an highlights of the analysis were: The ermeability is a function of geometrical arameters such as: orosity, fiber iameter an unit cell asect ratio in case of rectangular fibers arrangement. Normal an arallel ermeability are irectly relate to the fiber iameter square. [1] D.. Clauge, R.J. Philis, A numerical calculation of the hyraulic ermeability of three-imensional isorere fibrous meia, Phys. Fluis 9 (6) (1997) [] M.M. Tomaakis, T. Robertson, Viscous ermeability of ranom fiber structures: comarison of electrical an iffusion estimates with exerimental an analytical results, J. Comos. Materials 39 (005) [3] J.T. Gostick, M.W. 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