Transverse Permeability of Fibrous Porous Media Ali Tamayol * and Majid Bahrami * PhD Candidate, Assistant Professor,

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1 Proceeings (CD) of the 3 r International Conference on Porous Meia an its Applications in Science an Engineering ICPM3 June 05, 010, Montecatini, Italy Transverse Permeability of Fibrous Porous Meia Ali Tamayol * an Maji Bahrami * PhD Caniate, ata4@sfu.ca, Assistant Professor, mbahrami@sfu.ca Mechatronic Systems Engineering, School of Engineering Science, Simon Fraser University, BC, Canaa, V3T0A3 Abstract In this stuy, the transverse permeability of fibrous porous meia is stuie both experimentally an theoretically. A scale analysis technique is employe for etermining the transverse permeability of fibrous meia with a variety of fibrous matrices incluing square, staggere, hexagonal uniirectional fiber arrangements, simple two irectional mats, an simple cubic structures. In this approach, the permeability is relate to the porosity, fiber iameter, an tortuosity of the meium. In aition, the pressure rop in several samples of tube banks of ifferent arrangements an metal foams are measure in the creeping flow regime. The results are then use to calculate the permeability of the samples. The evelope compact relationships are successfully verifie through comparison with the present experimental results an the ata reporte by others. Our results suggest that the fiber orientation has an important effect on the permeability; however, these effects are more pronounce in low porosities, i.e., < 0.7. Keywors: Transverse permeability; Scale analysis; Fibrous meia; Moeling; Creeping flow; Experimental 1. Introuction Stuy of flow in fibrous porous meia is important in many natural an inustrial processes such as: physiological transport phenomena [1], filtration [4], composite fabrication [5,6], compact heat exchangers [7,8], paper prouction [9], an fuel cell technology [10, 11]. As such, preiction of the flow properties of fibrous materials incluing permeability an inertial coefficient has rawn the attention of numerous researchers. Authors have employe various theoretical an experimental techniques to investigate the problem. Comprehensive reviews of the pertinent literature can be foun in Refs. [4,6,1]. Permeability, which can be interprete as the flow conuctance of the soli matrix, is relate to geometrical features of the soli matrix incluing particle size an shape, pore size, an pore istribution. Fibrous materials can be ivie into 1,, an 3 irectional meia. In oneirectional (1D) structures the axes of fibers are parallel to each other. In twoirectional (D) fibrous matrices the fibers axes are locate on planes parallel to each other with ranom 1 positions an orientations on these planes. The axes of fibers in threeirectional (3D) are ranomly positione an oriente in space. Unlike the previous configurations, 3D structures can be isotropic. Complex geometry of actual fibrous materials avoi from exact solutions for flow an permeability. However, in preliminary esign an optimization processes, approximate solutions often suffice. To estimate the permeability of the fibrous structures, several researchers have moele the complex microstructure of the porous meia with simplifie 1D unit cells [131]. The moels evelope theoretically for transverse permeability of 1D structures are plotte against experimental ata in Fig. 1. The comparison shows that most of these moels have a limite range of accuracy. Only, the moel of Tamayol an Bahrami [19] captures the trens of experimental ata over the entire range of porosity. However, the moel of Tamayol an Bahrami [19] is limite to square arrays of fibers. Theoretical stuies of permeability of D an 3D materials are not as frequent as 1D arrangements which is a result of geometrical complexity of these meia. A selection of the existing moels for D an 3D structures [, 4] are plotte in Figs. an 3, respectively. It can be seen that these moels are not accurate over the entire range of porosity. K/ 10 Berglin et al. (1950) Kirsch an Fuchs (1967) Saiq et al. (1995) 10 1 Khomami an Moreno (1997) Zhong et al. (006) Tamayol an Bahrami (009) Happel (1959) Sahraoui an Kaviani (199) Van er Westhuizen (1996) 10 1 Drummon an Tahir (1984) Gebart (199) Transverse flow 1D square arrays Figure 1: Comparison of the existing moels for square arrangements with experimental ata.

2 K/ 10 Davies (195) Molnar et al. (1989) Kostornov an Shevchuk (1977) 10 1 Gostick et al. (006) Zobel et al. (007) VanDoormaal an Pharoah (008) Tomaakis an Sotirchos (1993) Transverse flow D structures Figure : Comparison of the existing moels for D structures with experimental ata. K/ 10 3 Jackson an James (198) Carman (1938) 10 Rahli (1997) Bhattacharya et al. (006) Higon an For (1996) 10 1 Jackson an James (1986) Tomaakis an Sotirchos (1993) Transverse flow 3D structures Figure 3: Comparison of the existing moels for 3D structures with experimental ata. Therefore, the objectives of the present work are to: Develop a theoretical approach that is applicable to 1D, D, an 3D fibrous matrices an accurately captures the trens observe in experimental ata. Investigate the effect of relevant geometrical parameters involve an ientify the controlling parameters. Perform inepenent experimental stuies on fibrous structures to verify the evelope moels. A scale analysis technique is employe to preict the permeability of a variety of unit cells incluing square, staggere, an hexagonal arrangements of 1D fibers, simple D mats an simple cubic structures. This metho that was originally applie by Clauge et al. [5] to fibrous meia, is moifie to improve its accuracy. Moreover, compact relationships are presente for etermining the permeability of each category as a function of porosity an fiber iameter. In aition, pressure rop is measure for creeping flow through several samples of tube banks an aluminum foams with 1D an 3D structures, respectively. The evelope solutions are successfully compare with experimental an numerical results for a wie range of geometries an materials.. Geometrical moeling Depening on the orientation of fibers in the meium, the fibrous matrix coul be ivie into 1D, D, or 3D structures. The simplest representation of 1D structures or generally fibrous meia is orere arrangements of uniirectional cyliners. In the present stuy, several orere structures incluing square, staggere, an hexagonal arrays of fibers are consiere, see Fig. 4. The soli volume fractions, ϕ, for the arrangements shown in Fig. 4 are relate to the istance between the centers of ajacent fibers, S, an the fibers iameter, : Square 4S ϕ = Staggere (1) 3 S Hexagonal 3 3 S To obtain results for woven textile structures with nonoverlapping fibers, the structure, shown in Fig. 5, is consiere here. The relationship between ϕ an other geometrical parameters can be foun from Fig. 5: ϕ = () 4S Figure 4: Consiere unit cells for orere 1D structures: a) square, b) staggere, an c) hexagonal arrays of cyliners.

3 S The bulk flow was calculate by weighting the collecte test flui over a specific perio of time. The permeability of the samples is then calculate using Darcy equation escribe in the following section. Figure 8 shows that the measure pressure graients for samples of tube bank with square fiber arrangement have linear relationships with the Reynols number. Figure 5: The D unit cell consiere in the present stuy. In 3D structures such as metal foams, fibers can have any arbitrary orientation in space, see Fig. 6a. Many researchers have propose simple cubic arrangements as the representing unit cell for these structures, see in Fig. 6b. The relationship between the soli volume fraction an other geometric parameters of SC arrangement consiere in the present stuy is [6]: 3 ϕ = (3) 3 4 S S 3. Experimental approach Experimental ata for creeping flow through fibrous structures of our interests are not abunant in the open literature. As such, several samples of tube banks with square an staggere fiber arrangements an metal foams are teste using glycerol. The properties of the sample are summarize in Table 1. 3 Sample Valve Tank Pressure transucer Pressure taps Scale (a) (b) S Figure 7: Schematic of the test setup. The permeability of the samples is then calculate using Darcy equation escribe in the following section. Figure 8 shows that the measure pressure graients for samples of tube bank with square fiber arrangement have linear relationships with the Reynols number = 0.8, K/ = =0.85,K/ = = 0.9, K/ = Figure 6: 3D structures; a) metal foam, b) simple cubic arrangement. The gravity riven test be, illustrate in Fig. 7, is consiste of an elevate reservoir, an entry section, sample holer, an an exit section with a ball valve. The reservoir crosssection of 300X300 mm is large enough to ensure that the variation of the riving pressure is negligible uring the experiment. The pressure rop through the samples was measure using a ifferential pressure transucer (PX 154) provie by BEC Controls. 3 p / x Re Figure 8: measure pressure graients for samples of tube bank with square fiber arrangement.

4 Table 1: Summary of the properties of the teste samples; glycerol use as test flui. Sample type ( mm ) Orientation Permeability ( K,m ) Tube bank (square) D Tube bank (square) D Tube bank (square) D Tube bank (staggere) D Tube bank (staggere) D Metalfoam (PPI=10) D Metalfoam (PPI=0) D Metalfoam (PPI=40) D Moel evelopment Experimental observations have shown that a linear relationship exists between the volumeaverage superficial flui velocity, U D, an the pressure graient; this is calle Darcy s law [4]: P μ = U D (4) x K where, μ is the flui viscosity an K is the permeability of the meium. Darcy s relationship is empirical, convenient, an wiely accepte; this equation hols when flow is in creeping regime [5]. However, one shoul know the permeability beforehan to use the Darcy s equation. Permeability can be calculate through the porescale analysis of flow in the soli matrix. In the creeping regime, the porescale velocity, V r, is governe by Stokes equation:. V r = 0 (5) μ V r = P (6) A scale analysis is followe for etermining the resulting pressure rop. In this approach the scale or the range of variation of the parameters is substitute in governing equations, i.e., erivatives are approximate with ifferences [7]. Following Clauge et al. [5] an Sobera an Kleijn [8], half of the minimum opening between two ajacent cyliners, δ min, is selecte as the characteristic length scale over which rapi changes of the velocity occurs, see Fig. 4. Sobera an Klein [8] propose to use the average velocity in the section with minimum frontal area as the characteristic velocity scale. However, this assumption is only accurate for highly porous structures, > 0.8, an overpreicts the pressure rop in low porosities [8]. Carman [9] argue that a flui particle shoul travel in a tortuous path of length L e to path through a sample of size L. Therefore, it is expecte that the resulting velocity scale from applying a constant pressure ifference to be inversely relate to L e / L ; this ratio is calle tortuosity factor, τ. Thus, the porelevel velocity scale becomes: r U D β V (7) τ where β is the ratio of the minimum to the total frontal areas in the unit cell. Substituting from Eq. (7) for velocity scale an using δ min as the length scale, permeability can be calculate as: K = C β δ min τ (8) where C is a constant that shoul be etermine through comparison with ata. Therefore one nees to know the ratio between minimum to total frontal area, β, an tortuosity factor, τ, to be able to calculate the permeability Tortuosity factor The tortuosity factor is efine as the ratio of the average istance, L e, that a particle shoul travel to cover a irect istance of L. Due to its importance in mass, thermal an electrical iffusion, several theoretical an empirical relationship have been propose for tortuosity calculation in the literature; goo reviews can be foun elsewhere [30,31]. Any relationship propose for tortuosity shoul satisfy three conitions [30]: τ >1; lim 1τ = 1; lim 0τ. One of the most popular empirical moels for etermination of tortuosity, that satisfies all these conitions, is the Archie s law [3]: α α 1 1 τ = = (9) 1 ϕ where α is a constant an is the porosity. Boureau [30] showe that α = 0.5 provies a goo estimate for tortuosity in packe bes. Due to similarity of flow in packe bes an flow in 1D an D fibers α is assume to be equal to 0.5. The stuy of Tomaakis an Robertson [1] showe that 3D fibrous structures are less tortuous in comparison with 1D an D matrices. Therefore for 3D structures α is assume to be equal to Results an iscussions Equation (8) relates the permeability of fibrous meia to the minimum opening between ajacent fibers, δ min, the 4

5 ratio between minimum to total frontal area, β, an tortuosity factor, τ, that can be calculate from Eq. (9). In the following subsections, using geometrical properties of the consiere microstructures, compact moels will be evelope that relate the permeability to the soli volume fraction Uniirectional arrangements For the three ifferent orere 1D unit cells shown in Fig.1, it can be seen that β = ( S ) / S an δ min = ( S ). Therefore, Eq. (8) can be rewritten as: 3 ( S ) K = C (10) S 1 ϕ Substituting from Eq. (1) an comparing the moel with experimental ata, the imensionless permeability of the orere structures is: K = ϕ 3 3 3ϕ 3 3ϕ 3 4ϕ 1 ϕ 3 3ϕ 1 ϕ 3 3 3ϕ 1 ϕ 4ϕ 3ϕ 3 3ϕ Square Staggere Hexagonal (11) In Fig. 9, Eq. (11) is compare with the present experimental results an the ata collecte by others [33 39]. As one can see, the moel is in agreement with experimental ata over the entire range of porosity. These experiments were conucte using ifferent fluis incluing: air, water, oil, an glycerol with a variety of porous materials such as metallic ros, acrylic cyliners, an carbon fibers. In Fig. 10 the preicte results of Eq. (11) for staggere arrangement of fibers are compare with present experimental ata an numerical results of Higon an For [6]. It can be seen that the propose moel can accurately preict the numerical results in the entire range of porosity. 5.. Twoirectional structures The ratio of the minimum frontal to the total unit cell areas for the D structure shown in Fig. b is not exactly known. Therefore, similar to 1D structures an using the Forchheimer law that relates average porescale velocity to U D / [5], the porelevel velocity scale is estimate as: r S V U D 3 / S (1) K/ 10 Berglin et al. (1950) 10 1 Chemielewski an Jayaraman (199) Khomami an Moreno (1997) Kirsch an Fuchs (1967) Saiq et al. (1995) Skartsis et al. (199) Zhong et al. (006) Scale analysis Present experimental ata Figure 9: Comparison of the propose moel for square arrangements with experimental ata. K/ 10 1 Higon an For (1996) Scale analysis Present experimental ata Figure 10: Comparison of the propose moel with numerical results of Higon an For [6] for staggere arrangements. Substituting for geometrical parameters from Eq. (), the imensionless permeability becomes: K = Flow 4ϕ 3 3 ( ) 3 / 1 ϕ 4ϕ 4ϕ (13) The constant value in Eq. (13), i.e., 0.008, is foun through comparison with experimental ata collecte from ifferent sources, see Fig. 11. It can be seen that Eq. (13) captures the trens of the experimental ata collecte from ifferent sources over a wie range of porosity. The experiments were conucte on glass ros, glass wool, cotton wool, kapok with application in filtration [40], alloy fibers [41], fiber reinforcing mats with application in moling an composite fabrication [4,43], an gas iffusion layers [10]. Kostornov an shevchuk [41] performe experiments with several fluis an they observe that permeability was epenent on the working flui, i.e., water resulte in higher permeability than alcohol. Flow δ min S 5

6 K/ Davies (195) Molnar et al. (1989) Kostornov an Shevchuk (1977) Gostick et al. (006) Zobel et al. (007) Scale analysis K/ Jackson an James (198) 10 1 Carman (1938) Rahli (1997) Bhattacharya et al. (00) Higon an For (1996), numerical Scale analysis Present experimental ata Figure 11: Comparison of the present moel with experimental ata for D structures Three irectional arrangements For simple cubic arrangement that is consiere in this stuy as a simple representation of 3D fibrous materials, the ratio of the minimum frontal to the unit cell areas is β = ( S ) / S. Therefore, the permeability of 3D structures becomes: K ( S ) S = (14) where the ratio of S to is calculate from Eq. (3). The constant in Eq. (14) is foun to be 0.08 through comparison of this equation with the numerical ata reporte by Higon an For [6] for SC arrangements over a wie range of porosity. Figure 1 inclues the present moel, current experimental measurements, an experimental ata collecte from ifferent sources. The plotte ata are base on permeability results for polymer chain in solutions [44], glass wool ranomly packe, stainless steel crimps [9], metallic fibers [45], an aluminum metal foams [46]. It can be seen that the present moel is in agreement with the numerical results over the entire range of porosity. 6. Effect of fiber orientation of the permeability To see the effect of fibers arrangement on the permeability of the fibrous structures, the propose relationships for 1D, D, an 3D arrays are plotte in Fig. 13. It can be seen that the square arrangements an D structures have similar permeabilities an 3D structures is the most permeable microstructure; this is in agreement with the results reporte by Tomaakis an Robertson [1]. It is noteworthy that the effect of microstructure is more significant in low porosities, where < 0.7 an the eviations are reuce in higher porosities; this is in line with our previous observations for parallel flow through 1D fibers [47] Figure 1: Comparison of the propose moel for SC, experimental ata. K/ 1D square arrangement 1D staggere arrangement D strucutres 3D strucutres Figure 13: Effect of fiber orientation on the permeability of fibrous structures. 7. Conclusions Scale analysis technique was employe for analyzing pressure rop an permeability of fibrous meia. The fibrous materials were represente by unit cells which were assume to be repeate throughout the meium an the present approach was applie to a variety of fibrous matrices incluing square, staggere, hexagonal uniirectional fiber arrangements, simple irectional mats, an simple cubic structures. Moreover, compact relationships have been reporte for the consiere geometries. In aition, pressure rop in samples of tube banks of ifferent arrangements an metal foam were measure in the creeping flow regime. The evelope compact relationships were successfully verifie through comparison with experimental an numerical ata, over a wie range of porosity. The results show that the microstructure effects were more significant for low porosities. Moreover, 3D structures ha the highest permeability. Since the present moel relates permeability to the tortuosity, fiber size an istribution, it is capable to be extene to inclue ranomness effects.

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