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An analytical model for the longitudinal permeability of aligned fibrous media
Composites Science and Technology 72 (2012) 1500–1507
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An analytical model for the longitudinal permeability of aligned fibrous media C. DeValve, R. Pitchumani ⇑ Advanced Materials and Technologies Laboratory, Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061-0238, USA
a r t i c l e
i n f o
Article history: Received 1 September 2011 Accepted 26 April 2012 Available online 5 June 2012 Keywords: A. Fibers B. Transport properties C. Modeling E. Resin transfer molding (RTM) Flow through porous media
a b s t r a c t This paper presents an analytical series solution of the longitudinal fluid flow in porous media consisting of aligned rigid fibers, which, in turn, is used to establish a relationship for the longitudinal permeability as a function of the fiber packing geometry, fiber volume fraction and the fiber radius. The analytical series solution is developed for rectangular and staggered packing arrangements of the fibers using the boundary collocation method where the constants in the series solution are solved for numerically. As the number of boundary collocation points is increased, the analytical solution is shown to converge with a finite element solution of the identical flow situation for all fiber volume fractions and packing arrangements. In addition, the permeability results are presented in a dimensionless form as a function of the fiber volume fraction and fiber packing arrangement for a general applicability and easy use of the results for predicting the longitudinal permeability of fiber tows consisting of aligned rigid fibers. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Longitudinal fluid flow through an arrangement of aligned rigid cylindrical fibers occurs in most composite processing techniques including liquid molding processes. Resin flow through fiber tows during the resin infiltration stage of composite fabrication is often described as flow through a porous medium characterized principally by the fiber volume fraction and a permeability tensor. The permeability tensor can be constructed based on the transverse and longitudinal permeability of the aligned fiber bundle and the local orientation of the fiber bundle within the preform. Therefore, the evaluation of the transverse and longitudinal permeability of aligned rigid fibers has been the focus of many studies in the literature. Fundamentally, the permeability of a porous medium is a geometric parameter and a measure of the resistance offered by the porous microstructure to the fluid flow as a function of the relative arrangement of the fibers, the fiber volume fraction, and the individual fiber radius. Sangani and Yao [1] developed an analytical solution for the transverse and longitudinal permeability of randomly packed fiber arrangements up to a fiber volume fraction of 0.7. Bruschke and Advani [2] used lubrication theory to predict transverse permeability values close to the fiber packing limit and a simplified cell model to predict the transverse permeability at low fiber volume fractions. A comprehensive review of the previous analytical and experimental work on transverse and longitudinal flow through aligned fibrous media published through ⇑ Corresponding author. Tel.: +1 540 231 1776; fax: +1 540 231 9100. E-mail address: [email protected] (R. Pitchumani). 0266-3538/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2012.04.019
1986 is summarized by Jackson and James [3], from which, a few of the reviewed analytical longitudinal permeability models are reiterated here. Happel [4] solved the longitudinal fluid flow problem by simplifying the fluid boundary of the unit cell domain from a square to a circle, and obtained an approximate solution for the flow. A solution for the pressure drop and the shear stress distribution along the fiber wall for longitudinal flow through aligned rigid fibers was developed by Sparrow and Loeffler [5] and was extended to determine permeability values in [3]. The method of distributed singularities was used by Drummond and Tahir [6] to arrive at an approximate solution in the form of a power series. Wang [7,8] solved for the longitudinal permeability by modifying the fiber geometry based on the fiber packing arrangement, i.e., approximating the circular fibers as square in cross section for square packing and hexagonal in cross section for hexagonal packing. Toll [9] proposed the superposition of three fundamental solutions, which approximately satisfy the boundary conditions of the longitudinal flow configuration. Fractal geometry was applied by Pitchumani and Ramakrishnan [10,11] to characterize the dualscale nature of flow through porous preforms in order to evaluate permeability values. Zhao and Povitsky [12] applied the method of fundamental solutions using a Stokeslet-based model and low Reynolds number flow to model the flow across an array of cylinders oriented transverse to the flow, where the results can be combined with Darcy’s law to predict permeability values. A three-dimensional finite element model of a simplified plain weave unit cell was used to simulate the resin flow through a preform by Ngo and Tamma [13], and an effective permeability based on the simulation results was reported. Zhou et al. [14] developed an analytical
C. DeValve, R. Pitchumani / Composites Science and Technology 72 (2012) 1500–1507
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Nomenclature a b C L N P r R Rb Re u
vf x y z
width of unit cell (m) height of unit cell (m) Eq. (12) constants length of unit cell (m) number of boundary collocation points pressure (Pa) radial-direction coordinate (m) fiber radius (m) radius to flow boundary (m) Reynolds number velocity vector (m/s) fiber volume fraction x-direction coordinate (m) y-direction coordinate (m) z-direction coordinate (m)
algorithm to predict the transverse permeability of fiber tows considering the unsaturated flow length in the fiber perform. A common method for experimentally determining the permeability of fibrous reinforcements is to physically force resin through a preform mat and monitor the pressure and flow rate throughout the mold, combining these results with Darcy’s law or an empirically fit equation to determine the permeability of the preform as a function of the fiber volume fraction (e.g. [15– 21]). A full review of these experimental methods for determining permeability was presented by Sharma and Siginer [22] in 2010, and this review also includes a short summary of the permeability prediction models documented in the literature. Experiments have demonstrated that the capillary forces within a fiber tow increase with an increase in fiber volume fraction (e.g. [23]), indicating the relative importance of including capillary effects in permeability calculations at high fiber volume fractions. It is evident from the foregoing review that the analytical approaches in the literature are based on approximations of either the geometry or the governing flow physics, which leads to permeability values that substantially deviate from the actual values, particularly for arrangements with high fiber volume fraction. This paper presents an analytical series solution for the problem of fluid flow through aligned rigid fibers and, in turn, evaluation of the longitudinal permeability. The governing equations for the fluid flow in a representative volume element are solved using the separation of variables technique combined with a boundary collocation method [24], in which the boundary conditions are satisfied along the outside edges of the fluid domain at discrete points. Using this approach, the analytical results are shown to be in excellent agreement with a numerical finite element solution for rectangular and staggered packing arrangements of the fibers and for all fiber volume fractions, vf. The paper is organized as follows: Section 2 explains the problem formulation and solution, the results are presented and discussed in Section 3, and the conclusions of the study are summarized in Section 4.
Greek symbols surface tension (N/m) d fiber packing angle (°) h angular-direction coordinate (°) j permeability (m2) l fluid viscosity (Pas) r contact angle (°)
c
Superscripts/subscripts dimensionless value hR rectangular fiber packing hS staggered fiber packing h2n index of constants in solution equation
different fiber packing arrangements are analyzed, described as rectangular-packed or staggered fiber tows, shown, respectively, in Fig. 1a and b. By virtue of the symmetry of the fiber packing arrangements, the geometry can be simplified as a representative unit cell of width, a, height, b, and length, L, for each packing configuration with an associated fiber radius, R, and packing angle, d, as shown in Fig. 1a and b. The flow is assumed to be laminar and the inertial forces are considered to be much less than the viscous forces, or equivalently, that the flow Reynolds number, Re 1, such that the fluid motion is described by Stokes flow [25]. Further, for a fully-developed flow in the representative volume element, the pressure gradient in the direction of the flow can be expressed as @P ¼ DLP, where Dp is the @z pressure difference between the inlet and the outlet planes of the unit cell, and L is the length of the unit cell in the flow (z) direction. The governing equation can then be written in cylindrical polar coordinates, in a nondimensional form, as
1 @ r @ u 1 @2u þ 2 2 1¼0 r @r @r r @h
ð1Þ
where referring to Fig. 1, all the unit cell parameters are normalized with respect to the fiber radius, R, as {a, b, x, y, r} = {a, b, x, y, r}/R, h is the angular coordinate and the dimensionless axial flow velocity, u,
2. Permeability model The goal of the modeling is to develop an analytical relationship between the longitudinal permeability, j, and the governing geometric parameters for a porous medium comprised of a periodic arrangement of aligned rigid fibers. To this end, the problem of a viscous fluid flowing longitudinally through a bundle of aligned rigid fibers, presented schematically in Fig. 1, is considered first. Two
Fig. 1. Schematic illustration of aligned rigid fibers arranged in a (a) rectangular array and (b) staggered array, with the associated representative unit cell configurations and geometric parameters.
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is given by u ¼ uLl=R2 DP, in which u is the axial velocity in the zdirection along the fibers, and l is the fluid viscosity. Of the four boundary conditions needed to solve Eq. (1) uniquely, three are identical for both the rectangular and the staggered fiber packing arrangements shown in Fig. 1, namely, (1) the fluid velocity at the fiber-fluid interface (r = R) must satisfy the no-slip condition and the fluid shear stress must be zero (i.e. symmetry lines) along both (2) the bottom edge (h = 0) and (3) the left edge (h = p/2) of the unit cell. These conditions are defined mathematically in dimensionless form as:
ðr ¼ 1; hÞ ¼ 0 u
ð2Þ
@u ðr ; h ¼ 0Þ ¼ 0 @h
ð3Þ
@u p r ; h ¼ ¼0 @h 2
ð4Þ
The fourth boundary condition is dependent on the fiber packing arrangement. For the rectangular-packed fiber arrangement shown in Fig. 1a, the velocity field is symmetric about the boundary, Rb ðhÞ, where Rb ðhÞ ¼ Rb ðhÞ=R denotes the dimensionless radial distance to the outer fluid boundary, Rb(h), of the rectangular unit as shown in Figure 2a, and y ¼ b, cell geometry defined by x¼a such that
@u @u ¼0 ; y Þ ¼ ¼ bÞ ðx ¼ a ðx; y @x @y
ð5Þ
For the staggered fiber arrangement presented in Fig. 1b, the velocity field is symmetric with respect to the center of the unit =2; b=2Þ. cell along any line drawn through the central point ða Additionally, the gradient of the velocity field exhibits an anti-sym =2; b=2Þ metry with respect to the central point ða along this same line. For simplicity, this line passing through the unit cell center, Rb ðhÞ, on which the two aforementioned boundary conditions are satisfied, is taken to be the diagonal line connecting points ð0; bÞ ; 0Þ, as shown by the dashed line in Fig. 2b. In this case, and ða the boundary conditions are expressed mathematically in a dimensionless form as:
ðRðh2 ÞÞ ðRb ðh1 ÞÞ ¼ u u
ð6Þ
@u @u ðRb ðh1 ÞÞ ¼ ðRb ðh2 ÞÞ @ x @ x
@u @u ðR ðh ÞÞ ¼ ðRb ðh2 ÞÞ b 1 @y @y
ð7Þ Fig. 2. Schematic of the flow domain used in the analytical solution for a (a) rectangular array, (b) staggered array at low fiber volume fraction, and (c) staggered array at high fiber volume fraction. Also illustrated are the boundary collocation points, shown as open circles on the outer boundary for each unit cell configuration.
where
Rb ðhi Þ ¼
ðsinðdÞÞ a ; sinðd þ hi Þ
> h1 > 0
i ¼ 1; 2; h2 ¼ tan1
2 tan ðdÞ for d tanðh1 Þ ð8Þ
Equivalently, the boundary conditions in Eqs. (6) and (7) can be expressed by considering any two points on the boundary line de =2; b=2Þ fined through ða which are on opposite sides of the center =2; b=2Þ ða and equidistant from this point, denoted by D in Fig. 2b and c, where, at these two points, the velocities are equal and the velocity gradients are equal in magnitude but opposite in sign. As the fiber volume fraction, vf, for the staggered unit cell is increased towards the fiber packing limit, the fiber will eventually intersect the diagonal dashed line defining the boundary in Fig. 2b. For this configuration, it is necessary to modify the boundary, Rb ðhÞ, as shown by the dashed lines in Fig. 2c, on which Eqs. (6) and (7) are satisfied in order to ensure a continuous fluid presence along the outer edge of the unit cell. This is accomplished by defining the upper boundary to be a line drawn at an angle of d from vertical, as indicated in Fig. 2c, which passes through the unit cell Þ; i ¼ 1; 2 and h are written as =2; b=2Þ, center, ða for which bðh i 2
Rb ðhi Þ ¼
a ; 2ðcosðdÞ cosðd hi ÞÞ
h2 ¼ 2d h1
i ¼ 1; 2; 2d p2 if d > p4 for d > h1 > 0 if d p4
ð9Þ
In addition, as a result of this boundary modification, a vertical becomes necessary to completely define the line segment at x¼a fluid domain boundary, on which the first part of Eq. (5) is applied to complete the boundary condition specifications. Expressing the dimensionless velocity as a superposition of two function, f ðr Þ and gðr ; hÞ; uðr ; hÞ ¼ f ðrÞ þ gðr ; hÞ, it follows from Eq. (1) and the associated boundary conditions, Eqs. (2)–(4), that
1 d r df ðr Þ þ 1 ¼ 0; r dr dr
f ðr ¼ 1Þ ¼ 0
ð10Þ
( gðr ¼ 1; hÞ ¼ 0 1 @ r@gðr ; hÞ 1 @ 2 gðr ; hÞ ¼ 0 @g þ 2 2 r @r r @r ðr ; h ¼ 0Þ ¼ @g ðr ; h ¼ p2Þ ¼ 0 @h @h @h ð11Þ
C. DeValve, R. Pitchumani / Composites Science and Technology 72 (2012) 1500–1507
Solving equation set (10) for f ðr Þ and using the method of separation of variables [26] to solve equation set (11) for gðr; hÞ the solution for uðr ; hÞ can be written as
ðr ; hÞ ¼ u
X1 1 C 2n ðr2n r 2n Þ cosð2nhÞ ð1 r 2 Þ þ C 0 ln r þ n¼1 4 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} f ðr Þ
ð12Þ
gðr;hÞ
where C0, C2, . . ., C2n are the coefficients to be determined using the boundary conditions, Eqs. (5)–(7), on the outer edge of the fluid domain, Rb ðhÞ, for each of the two fiber packing arrangements. The boundary conditions on Rb ðhÞ were enforced using the boundary collocation method in which Eqs. (5)–(7) were satisfied at a finite number of discrete points chosen along the boundary of each of the domains depicted in Fig. 2, resulting in a system of equations which were solved simultaneously to find the constants in Eq. (12). Fig. 2a–c illustrates this technique for rectangular and staggered arrangements of fibers, in which six boundary collocation and ða ; 0Þ points are shown by the open circles, excluding ð0; bÞ and equally spaced to minimize the error in the resulting solution. In general, choosing N discrete points to enforce the boundary conditions will result in N equations to be solved simultaneously for C0, . . ., C2(N1) in Eq. (12). For the rectangular-packed unit cell in Fig. 1a, using the relapffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 and h ¼ tan1 ðy = tionships r ¼ x2 þ y xÞ Eq. (5) is converted to cylindrical polar coordinates as:
@r @ u @h @u @u ; y Þ ¼ ðx ¼ a þ @ x @r @ x @h @ x x¼a;y ! 1 X a C R0 1 þ ¼a 2 2nC R2n 2 ½ðr 2n þ r 2n Þ cosð2nhÞ r r 2 n¼1 þ ðr2n r 2n Þ sinð2nhÞ tan h ¼ 0 @r @ u @h @u ¼ @u ¼ bÞ ðx; y þ @h @ y x;y¼b @y @r @ y ! R 1 X b C0 1 þ ¼b 2nC R2n 2 r 2 2 r n¼1 sinð2nhÞ 2n 2n ðr þ r Þcosð2nhÞ ðr 2n r 2n Þ ¼0 tanh
ð13Þ
ð14Þ
where the coefficients in Eqs. (13) and (14) are superscripted by R as C R0 and C R2n to denote the rectangular-packing geometry. Using the boundary collocation method, Eqs. (13) and (14) are evaluated at discrete points along the top and right faces of the unit cell in Fig. 1a for several values of the fiber volume fraction, vf, and fiber packing angle, d, and the values of the coefficients C R0 ; . . . ; C R18 are presented in Table 1. For the staggered fiber arrangement as in Fig. 1b and the definitions in Eqs. (8) and (9) mentioned above, the boundary conditions in Eqs. (6) and (7) may be expressed as:
! 1 X
1 2 Rb ðh1 Þ S 2 þ R ðh2 Þ Rb ðh1 Þ þ C 0 ln C s2n R2n b ðh1 Þ 4 b Rb ðh2 Þ n¼1 2n 2n R2n b ðh1 Þ cosð2nh1 Þ Rb ðh2 Þ Rb ðh2 Þ cosð2nh2 Þ ¼ 0 ð15Þ ! ! Rb ðh1 Þ C S0 Rb ðh2 Þ þ sin h2 2 2 Rb ðh1 Þ Rb ðh2 Þ " 1 X sin h1 2n 2nC s2n R2n þ b ðh1 Þ þ Rb ðh1 Þ cosð2nh1 Þ Rb ðh1 Þ n¼1 sinð2nh1 Þ sin h2 2n 2n þ ðh Þ R ðh Þ Rb ðh2 Þ R2n 1 1 b b tan h1 Rb ðh2 Þ 2n sinð2nh2 Þ 2n þR2n ¼ 0 ð16Þ b ðh2 Þ cosð2nh2 Þ Rb ðh2 Þ Rb ðh2 Þ tan h2
sin h1
C S0
1503
where the coefficients in Eqs. (15) and (16) are superscripted by S as C S0 and C S2n to denote the staggered-packing geometry. As the packing limit is approached and the diagonal line through the unit cell center is modified as discussed previously and illustrated in Fig. 2c, the additional boundary on the right face of the unit cell will be satisfied by applying Eq. (13). Table 2 presents where x¼a the values of the coefficients C S0 ; . . . ; C S18 calculated using the boundary collocation method with N = 10 discrete boundary points for several values of the fiber volume fraction, vf, and fiber packing angle, d. Using Darcy’s law [27], the longitudinal permeability can be related to the average velocity through the cross section of the fluid region in the unit cell, the fluid viscosity, and the pressure gradient as j = lLuavg/Dp, from which the dimensionless permeability, j ¼ j=R2 , is obtained as
R p=2 R Rb ðhÞ
j ¼ uav g ¼
0
uðr; hÞr dr dh 1 R p=2 R Rb ðhÞ r dr dh 0 1
ð17Þ
where the numerator denotes the dimensionless volumetric flow rate and the denominator is the dimensionless cross sectional area p=4Þ for the b of the fluid region of the unit cell equaling ða rectangular packing as in Fig. 1a and ðab=2 p=4Þ for the staggered fiber arrangement as in Fig. 1b. In Eq. (17) the velocity profile, uðr ; hÞ, is given by Eq. (12) and the associated coefficients tabulated in Tables 1 and 2. Eq. (17) was evaluated using a computational quadrature method to determine the dimensionless permeability as a function of the volume fraction, vf, and the fiber packing angle, d, for the two packing arrangements considered in this study. 3. Results and discussion The analytical model presented in the previous section was compared to three-dimensional numerical simulations of the fluid flow through identical rectangular and staggered unit cell geometries (as shown in Fig. 1) for a range of fiber volume fractions and fiber packing angles. The numerical solution was obtained with the commercially available finite element software COMSOLÓ using a tetrahedral mesh consisting of 2 104 elements, and the results were non-dimensionalized according to the parameters presented in Section 2 in order to compare the numerical results to the dimensionless analytical series solution. The results of the comparison are presented and discussed in this section. Fig. 3 presents contours of the dimensionless velocity profiles evaluated using the analytical solution in a rectangular unit cell geometry with a fiber packing angle of 45° (Fig. 3a) and a staggered unit cell geometry with a fiber packing angle of 60° (Fig. 3b), both with a fiber volume fraction of 0.25. The analytical solution is based on ten boundary collocation points in the evaluation of the boundary conditions, Eqs. (5)–(7), using the constants given in Tables 1 and 2 in Eq. (12). Also included in the figures are the dimensionless velocity profile values obtained from a numerical fluid dynamics simulation of the flow through the respective geometries, shown by the discrete triangular markers in the figures. Excellent agreement is noted between the analytical solution and the corresponding numerical solution. For the rectangular fiber arrangement in Fig. 3a, the maximum dimensionless velocity is approximately 0.27 and located at the upper right-hand corner of the fluid domain. At the same volume fraction, the staggeredpacked case in Fig. 3b shows a smaller maximum dimensionless velocity of approximately 0.19 to occur at the points (0, 1.16) and (1, 0.57) in the unit cell, as shown by the arrows in Fig. 3b. From Fig. 3, it is also evident that the maximum velocity locations for each fiber packing arrangement are situated on the fluid boundary at the furthest distance from the fiber surfaces in their respective
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Table 1 First ten coefficients in Eq. (12) for a rectangular fiber arrangement. d
vf
C R0
C R2 (101)
C R4 (102)
C R6 (103)
C R8 (104)
C R10 (105)
C R12 (106)
C R14 (107)
C R16 (108)
C R18 (109)
45°
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.79
0.637 0.637 0.637 0.637 0.636 0.636 0.636 0.638
0 0 0 0 0 0 0 0
3.13 3.13 3.10 3.02 2.86 2.62 2.29 1.94
0 0 0 0 0 0 0 0
13.2 13.0 12.2 10.1 6.08 0.277 8.29 15.3
0 0 0 0 0 0 0 0
49.4 46.5 34.0 1.44 62.4 161 270 321
0 0 0 0 0 0 0 0
175 154 63.2 175 643 1350 2030 1960
0 0 0 0 0 0 0 0
55°/35°
0.10 0.20 0.30 0.40 0.50 0.54
0.909 0.909 0.909 0.908 0.908 0.910
1.21 1.16 1.09 0.997 0.891 0.840
3.10 2.93 2.64 2.25 1.78 1.56
4.24 3.56 2.40 0.792 1.17 2.05
8.59 6.62 3.04 2.24 8.88 11.8
13.8 8.10 2.49 18.4 38.4 46.9
25.2 12.4 12.4 50.9 99.2 119
35.4 12.0 34.1 106 196 232
40.0 10.4 49.1 144 260 307
24.3 2.49 41.1 110 193 224
65°/25°
0.10 0.20 0.30 0.36
1.366 1.358 1.345 1.341
3.02 2.82 2.54 2.34
4.22 3.48 2.43 1.66
5.73 3.60 0.514 1.74
8.27 3.46 3.59 8.76
10.7 2.43 9.88 18.9
11.4 0.878 14.9 26.5
9.00 0.362 14.4 24.8
4.62 0.576 8.40 14.2
1.15 0.221 2.29 3.83
C R1 ; C R3 ; C R5 ; C R7 ; C R9 ; C R11 ; C R13 ; C R15 ; C R17 ; C R19 ¼ 0.
unit cell domains. It follows from the dimensionless velocity profiles that the dimensionless volumetric flow rate through the rectangular fiber packing geometry is greater than the staggered fiber arrangement at the same fiber volume fraction. Since the results are presented in dimensionless form, the velocity profiles are generally applicable to longitudinal flow situations through aligned rigid fibers without limitation on the fiber radius or fluid viscosity. In addition, the qualitative relative trends in the flow behavior for the two unit cell geometries are generally applicable throughout the entire range of fiber volume fractions and packing angles. The results in Fig. 3 were based on ten collocation points on the boundaries of the rectangular and staggered unit cells in Fig. 2a–c, which, in turn, governs the number of terms used in evaluating the series solution in Eq. (12). It is instructive to assess the influence of the number of boundary collocation points—and, in turn, the number of terms in Eq. (12)—on the accuracy of the analytical solution. To this end, Fig. 4 presents the magnitude of the error in the analytical solution of the volumetric flow rate for the rectangular fiber arrangement in comparison to the converged numerical solution of
the volumetric flow rate, expressed as a percentage of the numerical solution, as the number of coefficients determined in Eq. (12) is increased. The error magnitudes are presented at incremented values of the volume fraction, vf, and at four different values of the packing angle, d, (Fig. 4a–d) as noted in the figure caption. The maximum attainable volume fraction at the packing limit decreases as the packing angle is increased, which can be seen from inspection of the unit cell’s geometry and is reflected in the different values of the volume fraction upper limits as the packing angle is changed, presented in the legend of Fig. 4. Fig. 4 generally shows that as the number of terms in the expansion of Eq. (12) is increased, the accuracy of the analytical solution increases accordingly and the error with respect to the converged numerical solution approaches zero. It is apparent from Fig. 4a that for a rectangular packing angle of d ¼ 45 , the error remains less than about 2.8% when at least six terms are included in the expansion of Eq. (12) and further reduces to less than 1% when the number of terms is increased to eight. The relatively greater amount of error apparent in the case where vf = 0.79 and d ¼ 45
Table 2 First ten coefficients in Eq. (12) for a staggered fiber arrangement. d
vf
C S0
C S2 (104)
C S4 (104)
C S6 (103)
C S8 (105)
C S10 (105)
C S12 (106)
C S14 (107)
C S16 (108)
C S18 (109)
60°/30°
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
0.551 0.551 0.551 0.551 0.552 0.554 0.551 0.551 0.551
0.921 0.893 0.604 0.813 5.31 19.4 0.0696 0.0658 0.997
0.701 0.667 0.444 0.524 3.40 11.9 0.0301 0.0338 0.378
8.46 8.46 8.44 8.35 8.03 7.10 7.37 6.39 5.13
3.89 3.79 2.73 2.51 19.3 72.5 16.2 18.1 1.90
3.13 3.05 2.20 1.99 15.7 59.9 18.4 14. 5 1.95
89.2 88.6 81.3 44.5 76.8 454 149 327 452
105 102 76.2 53.9 481 1930 12.7 6.32 117
254 251 213 23.6 585 3230 61.4 27.4 542
815 813 785 639 191 2330 14,300 22,700 9690
70°/20°
0.10 0.20 0.30 0.40 0.50 0.57
0.875 0.874 0.871 0.868 0.862 0.875
1100 1070 1020 965 905 771
276 263 243 217 188 122
4.79 4.51 4.07 3.42 2.68 0.487
67.8 61.6 53.2 41.6 31.1 120
13.8 14.0 15.3 17.6 22.4 37.6
20.2 24.3 35.0 52.0 80.6 98.8
26.3 38.1 65.9 110 180 179
24.1 40.3 75.8 130 213 217
9.18 17.6 34.8 59.7 96.6 137
80°/10°
0.10 0.20 0.27
1.832 1.711 1.620
4490 4520 4412
447 431 388
4.32 3.96 3.24
36.3 31.9 24.9
2.28 1.93 1.57
0.982 0.803 76.4
0.267 0.209 0.266
0.0405 0.0294 0.0587
0.00258 0.00165 0.00628
C S1 ; C S3 ; C S5 ; C S7 ; C S9 ; C S11 ; C S13 ; C S15 ; C S17 ; C S19 ¼ 0. Rows with italicized numbers indicate solution for the modified boundary, as shown in Fig. 2c.
C. DeValve, R. Pitchumani / Composites Science and Technology 72 (2012) 1500–1507
Fig. 3. Contour plots of the dimensionless velocity in unit cells of a (a) rectangular fiber arrangement with vf = 0.25 and d = 60°/30°.
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vf = 0.25 and d = 45° and (b) staggered fiber arrangement with
(Fig. 4a) when compared to the other cases is due to the presence of narrow fluid regions at either side of the unit cell which occur as a result of the unit cell geometry approaching the fiber packing limit. Fig. 4b–d indicates that for packing angles other than 45°, the error is less than 0.9% and 0.5% with the inclusion of six and eight terms, respectively. In general, the error variations in Fig. 4 demonstrate that by including ten terms in the expansion of Eq. (12), a highly accurate solution for the flow rate through the unit cell can be found throughout the ranges of the geometrical parameters involved, including conditions approaching and equal to the fiber packing limits. The error in the solution of Eq. (12) using a staggered packing arrangement of the fibers is similar to the trends presented in Fig. 4 and is omitted for brevity. The variation of the dimensionless longitudinal permeability, j, with the fiber volume fraction, vf, for rectangular fiber arrangements with different packing angles is depicted in Fig. 5. The figure
shows the analytical solution obtained from Eq. (17) with ten boundary collocation points and using the coefficients in Table 1, as solid lines, which is compared with the numerical finite element solution presented as the discrete triangular symbols. The designation of two angles for the different lines in Fig. 5 other than d ¼ 45 represents the fact that either packing angle results in the same dimensionless fluid flow rate and therefore the same dimensionless permeability value. In general, it is noted that the dimensionless permeability decreases with increasing fiber volume fraction, owing to the resulting increase in the flow resistance. Furthermore, for any given fiber volume fraction, the permeability is the lowest for the packing angle of 45°, again signifying the greatest flow resistance for this packing configuration. It is seen in Fig. 5 that the packing angle of 45° allows the maximum fiber volume fraction of 0.785 to be attained, which results in the least dimensionless permeability of approximately 2.31 102.
Fig. 4. Variation of the error between the numerical and analytical solutions for flow rate through a rectangular fiber array at different fiber volume fractions as the number of terms in Eq. (12) is varied; for (a) d = 45°, (b) d = 50°/40°, (c) d = 55°/35°, and (d) d = 60°/30°.
Fig. 5. Variation of the dimensionless permeability with the fiber volume fraction and fiber packing angle for a rectangular arrangement of fibers.
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Fig. 6. Variation of the dimensionless permeability with the fiber volume fraction and fiber packing angle for a staggered arrangement of fibers.
Similar trends in the dimensionless permeability variation with fiber volume fraction and fiber packing angle were found for the staggered fiber arrangement, as presented in Fig. 6. As in Fig. 5, the discrete triangular markers denote the numerical finite element solution and the solid lines represent the analytical solution from Eq. (17) based on 10 boundary collocation points using the coefficients in Table 2. Two different packing angles are shown for each line to signify that the identical dimensionless flow rates are found by applying either of the packing angle values. For the staggered fiber arrangement, the highest volume fraction attainable is 0.907 at a packing angle of 60 =30 , where the corresponding dimensionless permeability value is approximately 3.26 103, which is approximately an order of magnitude lower than the minimum permeability noted for the rectangular packing in Fig. 5. In both Figs. 5 and 6, excellent agreement is seen between the analytical results and the numerical values. It is evident from a comparison of the permeability values in Figs. 5 and 6 that at any fiber volume fraction, the rectangular fiber packing offers less resistance to the fluid flow than the staggered fiber arrangement, which is reflected in the higher dimensionless permeability value for the rectangular fiber arrangement. This difference is particularly pronounced at higher fiber volume fractions and reduces as the fiber volume fraction decreases, where the smallest tested fiber volume fraction of 1e-3 resulted in the dimensionless permeability values converging to the same value of approximately 450 for all cases of the different relative fiber arrangements. The longitudinal permeability values presented in Figs. 5 and 6 can be compared to experimental results reported in the literature to evaluate the accuracy of the analytical results compared to physical measurements. An experiment measuring the longitudinal permeability of an array of aligned cylinders was conducted by Brennan et al. [20], who reported a dimensionless permeability on the order of 102 for a staggered fiber arrangement with a volume fraction of 0.6. Considering the results presented in Fig. 6 for a staggered fiber arrangement, it is evident that at vf = 0.6 the analytical solution also predicts a permeability on the order of 102 and therefore exhibits good agreement with the experimentally measured value in [20]. In addition, Li et al. [23] studied a fiber tow made up of glass filaments with a diameter of 12 lm and experimentally measured the dimensionless longitudinal permeability to be on the order of 102 for a fiber volume fraction of 0.6, which is on the same order of magnitude as the measurements in [20] and the analytical results presented in Figs. 5 and 6.
As the fiber volume fraction of the fiber tow is increased, the effects of capillary pressure will become increasingly significant in permeability measurements and predictions (e.g. [23,28]). According to [28], the capillary pressure, Pcap, in the longitudinal direction of aligned rigid fibers can be expressed in terms of the fiber volume fraction, vf, fiber radius, R, surface tension, c, and contact angle, r, as: P cap ¼ f2v f c cosðrÞg=fRð1 v f Þg. For typical materials used in the liquid composite molding process, the surface tension, contact angle and the fiber radius are on the order of 102 N/m, 10°, and 105 m, respectively. Using the above expression and considering a fiber tow with a volume fraction of 0.8 results in a capillary pressure of approximately 8 kPa. The applied pressure in a liquid composite molding application is typically on the order of 100 kPa, which is more than an order of magnitude higher than the calculated capillary pressure. Therefore, reasonable accuracy should be expected of the permeability values presented in Figs. 5 and 6 even in the presence of capillary effects in practical liquid molding conditions. The analytical permeability model presented in this article may be extended to include the capillary effect in the governing physics in a future work.
4. Conclusions An analytical series solution was developed for the longitudinal fluid flow through aligned rigid fibers which was used to establish a relationship between the dimensionless permeability, j, the fiber volume fraction, vf, the relative fiber arrangement, and the fiber packing angle. The results are valid for all fiber volume fractions, vf including the fiber packing limit. The analytical model together with the coefficients in Tables 1 and 2 serve to obtain the values of the longitudinal flow permeability in simulation of resin flow through aligned fibrous media, providing a method for predicting the local longitudinal permeability of fiber tows in preform materials used in composites manufacturing.
Acknowledgments This research is funded in part by the National Science Foundation with Grant No. CBET-0934008, and the U.S. Department of Education through a GAANN fellowship to Caleb DeValve through Award No. P200A060289. Their support is gratefully acknowledged.
References [1] Sangani AS, Yao C. Transport processes in random arrays of cylinders II. Viscous flow. Phys Fluids 1988;31(9):2435–44. [2] Bruschke MV, Advani SG. Flow of generalized Newtonian fluids across a periodic array of cylinders. J Rheol 1993;37(3):479–98. [3] Jackson GW, James DF. The permeability of fibrous porous media. Can J Chem Eng 1986;64:364–74. [4] Happel J. Viscous flow relative to arrays of cylinders. AIChE J 1959;5(2):174–7. [5] Sparrow EM, Loeffler AL. Longitudinal laminar flow between cylinders arranged in regular array. AIChE J 1959;5(3):325–30. [6] Drummond JE, Tahir MI. Laminar viscous flow through regular arrays of parallel solid cylinders. Int J Multiphase Flow 1984;10(5):515–40. [7] Wang CY. Longitudinal flow through an array of rectangular cylinders. IMA J Appl Math 1997;58:111–20. [8] Wang CY. Longitudinal flow past cylinders arranged in a triangular array. Appl Math Mod 1999;23:219–30. [9] Toll S. A solution technique for longitudinal Stokes flow around multiple aligned cylinders. J Fluid Mech 2001;439:199–216. [10] Pitchumani R, Ramakrishnan B. A fractal geometry model for evaluating permeabilities of porous preforms used in liquid composite molding. Int J Heat Mass Trans 1999;42:2219–32. [11] Pitchumani R, Ramakrishnan B. Fractal permeation characteristics of preforms used in liquid composite molding. Polym Compos 2000;21(2):281–96. [12] Zhao S, Povitsky A. Method of fundamental solutions for partial-slip fibrous filtration flows. Int J Num Meth Fluids 2009;61:255–74. [13] Ngo ND, Tamma KK. Microscale permeability predictions of porous fibrous media. Int J Heat Mass Trans 2001;44:3135–45.
C. DeValve, R. Pitchumani / Composites Science and Technology 72 (2012) 1500–1507 [14] Zhou F, Kuentzer N, Simacek P, Advani SG, Walsh S. Analytic characterization of the permeability of dual-scale fibrous porous media. Compos Sci Tech 2006;66:2795–803. [15] Han KK, Lee CW, Rice BP. Measurements of the permeability of fiber preforms and applications. Compos Sci Tech 2000;60:2435–24441. [16] Shojaei AS, Trochu F, Ghaffarian SR, Karimian SMH, Lessard L. An experimental study of saturated and unsaturated permeabilities in resin transfer molding based on unidirectional flow measurements. J Reinf Plast Compos 2004;23(14):1515–36. [17] Rodriquez E, Giacomelli F, Vazquez A. Permeability–porosity relationship in RTM for different fiberglass and natural reinforcements. J Compos Mater 2004;38(3):259–68. [18] Endruweit A, Matthys KS, Peiro J, Long AC. Effect of differential compression on in-plane permeability tensor of heterogeneous multi-layer carbon fibre preforms. Plast Rubb Compos 2009;38(1):1–9. [19] Brennan KP, Walrath DE. Macroscale simulation of fiber bed impregnation utilizing microscale material properties. J Reinf Plast Compos 2010;29(8): 1248–66. [20] Brennan KP, Straka B, Walrath DE. Investigation of a simple tensorial permeameter. Polym Compos 2010;31(3):434–42.
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[21] Antonucci V, Esposito M, Ricciardi MR, Raffone M, Zarrelli M, Giordano M. Permeability characterization of stitched carbon fiber preforms by fiber optic sensors. Polym Lett 2011;5(12):1075–84. [22] Sharma S, Siginer DA. Permeability measurement methods in porous media of fiber reinforced composites. Appl Mech Rev 2010;63:1–19. [23] Li M, Wang S, Gu Y, Zhang Z, Li Y, Potter K. Dynamic capillary impact on longitudinal micro-flow in vacuum assisted impregnation and the unsaturated permeability of inner fiber tows. Compos Sci Tech 2010;70:1628–36. [24] Han LS, Cosner AA. Effective thermal conductivities of fibrous composites. J Heat Trans 1980;103:387–92. [25] Munson BR, Young DF, Okiishi TH. Fundamentals of fluid mechanics. 5th ed. New Jersey: Wiley; 2006. [26] Kreyszig E. Advanced engineering mathematics. 9th ed. New Jersey: Wiley; 2006. [27] Advani SG. Flow and rheology in polymer composites manufacturing. Amsterdam: Elsevier; 1994. [28] Young W. The effect of surface tension on tow impregnation of unidirectional fibrous preforms in resin transfer molding. J Compos Mater 1996;30(11): 1191–209.
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Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
An analytical model for the longitudinal permeability of aligned fibrous media C. DeValve, R. Pitchumani ⇑ Advanced Materials and Technologies Laboratory, Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061-0238, USA
a r t i c l e
i n f o
Article history: Received 1 September 2011 Accepted 26 April 2012 Available online 5 June 2012 Keywords: A. Fibers B. Transport properties C. Modeling E. Resin transfer molding (RTM) Flow through porous media
a b s t r a c t This paper presents an analytical series solution of the longitudinal fluid flow in porous media consisting of aligned rigid fibers, which, in turn, is used to establish a relationship for the longitudinal permeability as a function of the fiber packing geometry, fiber volume fraction and the fiber radius. The analytical series solution is developed for rectangular and staggered packing arrangements of the fibers using the boundary collocation method where the constants in the series solution are solved for numerically. As the number of boundary collocation points is increased, the analytical solution is shown to converge with a finite element solution of the identical flow situation for all fiber volume fractions and packing arrangements. In addition, the permeability results are presented in a dimensionless form as a function of the fiber volume fraction and fiber packing arrangement for a general applicability and easy use of the results for predicting the longitudinal permeability of fiber tows consisting of aligned rigid fibers. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Longitudinal fluid flow through an arrangement of aligned rigid cylindrical fibers occurs in most composite processing techniques including liquid molding processes. Resin flow through fiber tows during the resin infiltration stage of composite fabrication is often described as flow through a porous medium characterized principally by the fiber volume fraction and a permeability tensor. The permeability tensor can be constructed based on the transverse and longitudinal permeability of the aligned fiber bundle and the local orientation of the fiber bundle within the preform. Therefore, the evaluation of the transverse and longitudinal permeability of aligned rigid fibers has been the focus of many studies in the literature. Fundamentally, the permeability of a porous medium is a geometric parameter and a measure of the resistance offered by the porous microstructure to the fluid flow as a function of the relative arrangement of the fibers, the fiber volume fraction, and the individual fiber radius. Sangani and Yao [1] developed an analytical solution for the transverse and longitudinal permeability of randomly packed fiber arrangements up to a fiber volume fraction of 0.7. Bruschke and Advani [2] used lubrication theory to predict transverse permeability values close to the fiber packing limit and a simplified cell model to predict the transverse permeability at low fiber volume fractions. A comprehensive review of the previous analytical and experimental work on transverse and longitudinal flow through aligned fibrous media published through ⇑ Corresponding author. Tel.: +1 540 231 1776; fax: +1 540 231 9100. E-mail address: [email protected] (R. Pitchumani). 0266-3538/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compscitech.2012.04.019
1986 is summarized by Jackson and James [3], from which, a few of the reviewed analytical longitudinal permeability models are reiterated here. Happel [4] solved the longitudinal fluid flow problem by simplifying the fluid boundary of the unit cell domain from a square to a circle, and obtained an approximate solution for the flow. A solution for the pressure drop and the shear stress distribution along the fiber wall for longitudinal flow through aligned rigid fibers was developed by Sparrow and Loeffler [5] and was extended to determine permeability values in [3]. The method of distributed singularities was used by Drummond and Tahir [6] to arrive at an approximate solution in the form of a power series. Wang [7,8] solved for the longitudinal permeability by modifying the fiber geometry based on the fiber packing arrangement, i.e., approximating the circular fibers as square in cross section for square packing and hexagonal in cross section for hexagonal packing. Toll [9] proposed the superposition of three fundamental solutions, which approximately satisfy the boundary conditions of the longitudinal flow configuration. Fractal geometry was applied by Pitchumani and Ramakrishnan [10,11] to characterize the dualscale nature of flow through porous preforms in order to evaluate permeability values. Zhao and Povitsky [12] applied the method of fundamental solutions using a Stokeslet-based model and low Reynolds number flow to model the flow across an array of cylinders oriented transverse to the flow, where the results can be combined with Darcy’s law to predict permeability values. A three-dimensional finite element model of a simplified plain weave unit cell was used to simulate the resin flow through a preform by Ngo and Tamma [13], and an effective permeability based on the simulation results was reported. Zhou et al. [14] developed an analytical
C. DeValve, R. Pitchumani / Composites Science and Technology 72 (2012) 1500–1507
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Nomenclature a b C L N P r R Rb Re u
vf x y z
width of unit cell (m) height of unit cell (m) Eq. (12) constants length of unit cell (m) number of boundary collocation points pressure (Pa) radial-direction coordinate (m) fiber radius (m) radius to flow boundary (m) Reynolds number velocity vector (m/s) fiber volume fraction x-direction coordinate (m) y-direction coordinate (m) z-direction coordinate (m)
algorithm to predict the transverse permeability of fiber tows considering the unsaturated flow length in the fiber perform. A common method for experimentally determining the permeability of fibrous reinforcements is to physically force resin through a preform mat and monitor the pressure and flow rate throughout the mold, combining these results with Darcy’s law or an empirically fit equation to determine the permeability of the preform as a function of the fiber volume fraction (e.g. [15– 21]). A full review of these experimental methods for determining permeability was presented by Sharma and Siginer [22] in 2010, and this review also includes a short summary of the permeability prediction models documented in the literature. Experiments have demonstrated that the capillary forces within a fiber tow increase with an increase in fiber volume fraction (e.g. [23]), indicating the relative importance of including capillary effects in permeability calculations at high fiber volume fractions. It is evident from the foregoing review that the analytical approaches in the literature are based on approximations of either the geometry or the governing flow physics, which leads to permeability values that substantially deviate from the actual values, particularly for arrangements with high fiber volume fraction. This paper presents an analytical series solution for the problem of fluid flow through aligned rigid fibers and, in turn, evaluation of the longitudinal permeability. The governing equations for the fluid flow in a representative volume element are solved using the separation of variables technique combined with a boundary collocation method [24], in which the boundary conditions are satisfied along the outside edges of the fluid domain at discrete points. Using this approach, the analytical results are shown to be in excellent agreement with a numerical finite element solution for rectangular and staggered packing arrangements of the fibers and for all fiber volume fractions, vf. The paper is organized as follows: Section 2 explains the problem formulation and solution, the results are presented and discussed in Section 3, and the conclusions of the study are summarized in Section 4.
Greek symbols surface tension (N/m) d fiber packing angle (°) h angular-direction coordinate (°) j permeability (m2) l fluid viscosity (Pas) r contact angle (°)
c
Superscripts/subscripts dimensionless value hR rectangular fiber packing hS staggered fiber packing h2n index of constants in solution equation
different fiber packing arrangements are analyzed, described as rectangular-packed or staggered fiber tows, shown, respectively, in Fig. 1a and b. By virtue of the symmetry of the fiber packing arrangements, the geometry can be simplified as a representative unit cell of width, a, height, b, and length, L, for each packing configuration with an associated fiber radius, R, and packing angle, d, as shown in Fig. 1a and b. The flow is assumed to be laminar and the inertial forces are considered to be much less than the viscous forces, or equivalently, that the flow Reynolds number, Re 1, such that the fluid motion is described by Stokes flow [25]. Further, for a fully-developed flow in the representative volume element, the pressure gradient in the direction of the flow can be expressed as @P ¼ DLP, where Dp is the @z pressure difference between the inlet and the outlet planes of the unit cell, and L is the length of the unit cell in the flow (z) direction. The governing equation can then be written in cylindrical polar coordinates, in a nondimensional form, as
1 @ r @ u 1 @2u þ 2 2 1¼0 r @r @r r @h
ð1Þ
where referring to Fig. 1, all the unit cell parameters are normalized with respect to the fiber radius, R, as {a, b, x, y, r} = {a, b, x, y, r}/R, h is the angular coordinate and the dimensionless axial flow velocity, u,
2. Permeability model The goal of the modeling is to develop an analytical relationship between the longitudinal permeability, j, and the governing geometric parameters for a porous medium comprised of a periodic arrangement of aligned rigid fibers. To this end, the problem of a viscous fluid flowing longitudinally through a bundle of aligned rigid fibers, presented schematically in Fig. 1, is considered first. Two
Fig. 1. Schematic illustration of aligned rigid fibers arranged in a (a) rectangular array and (b) staggered array, with the associated representative unit cell configurations and geometric parameters.
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is given by u ¼ uLl=R2 DP, in which u is the axial velocity in the zdirection along the fibers, and l is the fluid viscosity. Of the four boundary conditions needed to solve Eq. (1) uniquely, three are identical for both the rectangular and the staggered fiber packing arrangements shown in Fig. 1, namely, (1) the fluid velocity at the fiber-fluid interface (r = R) must satisfy the no-slip condition and the fluid shear stress must be zero (i.e. symmetry lines) along both (2) the bottom edge (h = 0) and (3) the left edge (h = p/2) of the unit cell. These conditions are defined mathematically in dimensionless form as:
ðr ¼ 1; hÞ ¼ 0 u
ð2Þ
@u ðr ; h ¼ 0Þ ¼ 0 @h
ð3Þ
@u p r ; h ¼ ¼0 @h 2
ð4Þ
The fourth boundary condition is dependent on the fiber packing arrangement. For the rectangular-packed fiber arrangement shown in Fig. 1a, the velocity field is symmetric about the boundary, Rb ðhÞ, where Rb ðhÞ ¼ Rb ðhÞ=R denotes the dimensionless radial distance to the outer fluid boundary, Rb(h), of the rectangular unit as shown in Figure 2a, and y ¼ b, cell geometry defined by x¼a such that
@u @u ¼0 ; y Þ ¼ ¼ bÞ ðx ¼ a ðx; y @x @y
ð5Þ
For the staggered fiber arrangement presented in Fig. 1b, the velocity field is symmetric with respect to the center of the unit =2; b=2Þ. cell along any line drawn through the central point ða Additionally, the gradient of the velocity field exhibits an anti-sym =2; b=2Þ metry with respect to the central point ða along this same line. For simplicity, this line passing through the unit cell center, Rb ðhÞ, on which the two aforementioned boundary conditions are satisfied, is taken to be the diagonal line connecting points ð0; bÞ ; 0Þ, as shown by the dashed line in Fig. 2b. In this case, and ða the boundary conditions are expressed mathematically in a dimensionless form as:
ðRðh2 ÞÞ ðRb ðh1 ÞÞ ¼ u u
ð6Þ
@u @u ðRb ðh1 ÞÞ ¼ ðRb ðh2 ÞÞ @ x @ x
@u @u ðR ðh ÞÞ ¼ ðRb ðh2 ÞÞ b 1 @y @y
ð7Þ Fig. 2. Schematic of the flow domain used in the analytical solution for a (a) rectangular array, (b) staggered array at low fiber volume fraction, and (c) staggered array at high fiber volume fraction. Also illustrated are the boundary collocation points, shown as open circles on the outer boundary for each unit cell configuration.
where
Rb ðhi Þ ¼
ðsinðdÞÞ a ; sinðd þ hi Þ
> h1 > 0
i ¼ 1; 2; h2 ¼ tan1
2 tan ðdÞ for d tanðh1 Þ ð8Þ
Equivalently, the boundary conditions in Eqs. (6) and (7) can be expressed by considering any two points on the boundary line de =2; b=2Þ fined through ða which are on opposite sides of the center =2; b=2Þ ða and equidistant from this point, denoted by D in Fig. 2b and c, where, at these two points, the velocities are equal and the velocity gradients are equal in magnitude but opposite in sign. As the fiber volume fraction, vf, for the staggered unit cell is increased towards the fiber packing limit, the fiber will eventually intersect the diagonal dashed line defining the boundary in Fig. 2b. For this configuration, it is necessary to modify the boundary, Rb ðhÞ, as shown by the dashed lines in Fig. 2c, on which Eqs. (6) and (7) are satisfied in order to ensure a continuous fluid presence along the outer edge of the unit cell. This is accomplished by defining the upper boundary to be a line drawn at an angle of d from vertical, as indicated in Fig. 2c, which passes through the unit cell Þ; i ¼ 1; 2 and h are written as =2; b=2Þ, center, ða for which bðh i 2
Rb ðhi Þ ¼
a ; 2ðcosðdÞ cosðd hi ÞÞ
h2 ¼ 2d h1
i ¼ 1; 2; 2d p2 if d > p4 for d > h1 > 0 if d p4
ð9Þ
In addition, as a result of this boundary modification, a vertical becomes necessary to completely define the line segment at x¼a fluid domain boundary, on which the first part of Eq. (5) is applied to complete the boundary condition specifications. Expressing the dimensionless velocity as a superposition of two function, f ðr Þ and gðr ; hÞ; uðr ; hÞ ¼ f ðrÞ þ gðr ; hÞ, it follows from Eq. (1) and the associated boundary conditions, Eqs. (2)–(4), that
1 d r df ðr Þ þ 1 ¼ 0; r dr dr
f ðr ¼ 1Þ ¼ 0
ð10Þ
( gðr ¼ 1; hÞ ¼ 0 1 @ r@gðr ; hÞ 1 @ 2 gðr ; hÞ ¼ 0 @g þ 2 2 r @r r @r ðr ; h ¼ 0Þ ¼ @g ðr ; h ¼ p2Þ ¼ 0 @h @h @h ð11Þ
C. DeValve, R. Pitchumani / Composites Science and Technology 72 (2012) 1500–1507
Solving equation set (10) for f ðr Þ and using the method of separation of variables [26] to solve equation set (11) for gðr; hÞ the solution for uðr ; hÞ can be written as
ðr ; hÞ ¼ u
X1 1 C 2n ðr2n r 2n Þ cosð2nhÞ ð1 r 2 Þ þ C 0 ln r þ n¼1 4 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} f ðr Þ
ð12Þ
gðr;hÞ
where C0, C2, . . ., C2n are the coefficients to be determined using the boundary conditions, Eqs. (5)–(7), on the outer edge of the fluid domain, Rb ðhÞ, for each of the two fiber packing arrangements. The boundary conditions on Rb ðhÞ were enforced using the boundary collocation method in which Eqs. (5)–(7) were satisfied at a finite number of discrete points chosen along the boundary of each of the domains depicted in Fig. 2, resulting in a system of equations which were solved simultaneously to find the constants in Eq. (12). Fig. 2a–c illustrates this technique for rectangular and staggered arrangements of fibers, in which six boundary collocation and ða ; 0Þ points are shown by the open circles, excluding ð0; bÞ and equally spaced to minimize the error in the resulting solution. In general, choosing N discrete points to enforce the boundary conditions will result in N equations to be solved simultaneously for C0, . . ., C2(N1) in Eq. (12). For the rectangular-packed unit cell in Fig. 1a, using the relapffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 and h ¼ tan1 ðy = tionships r ¼ x2 þ y xÞ Eq. (5) is converted to cylindrical polar coordinates as:
@r @ u @h @u @u ; y Þ ¼ ðx ¼ a þ @ x @r @ x @h @ x x¼a;y ! 1 X a C R0 1 þ ¼a 2 2nC R2n 2 ½ðr 2n þ r 2n Þ cosð2nhÞ r r 2 n¼1 þ ðr2n r 2n Þ sinð2nhÞ tan h ¼ 0 @r @ u @h @u ¼ @u ¼ bÞ ðx; y þ @h @ y x;y¼b @y @r @ y ! R 1 X b C0 1 þ ¼b 2nC R2n 2 r 2 2 r n¼1 sinð2nhÞ 2n 2n ðr þ r Þcosð2nhÞ ðr 2n r 2n Þ ¼0 tanh
ð13Þ
ð14Þ
where the coefficients in Eqs. (13) and (14) are superscripted by R as C R0 and C R2n to denote the rectangular-packing geometry. Using the boundary collocation method, Eqs. (13) and (14) are evaluated at discrete points along the top and right faces of the unit cell in Fig. 1a for several values of the fiber volume fraction, vf, and fiber packing angle, d, and the values of the coefficients C R0 ; . . . ; C R18 are presented in Table 1. For the staggered fiber arrangement as in Fig. 1b and the definitions in Eqs. (8) and (9) mentioned above, the boundary conditions in Eqs. (6) and (7) may be expressed as:
! 1 X
1 2 Rb ðh1 Þ S 2 þ R ðh2 Þ Rb ðh1 Þ þ C 0 ln C s2n R2n b ðh1 Þ 4 b Rb ðh2 Þ n¼1 2n 2n R2n b ðh1 Þ cosð2nh1 Þ Rb ðh2 Þ Rb ðh2 Þ cosð2nh2 Þ ¼ 0 ð15Þ ! ! Rb ðh1 Þ C S0 Rb ðh2 Þ þ sin h2 2 2 Rb ðh1 Þ Rb ðh2 Þ " 1 X sin h1 2n 2nC s2n R2n þ b ðh1 Þ þ Rb ðh1 Þ cosð2nh1 Þ Rb ðh1 Þ n¼1 sinð2nh1 Þ sin h2 2n 2n þ ðh Þ R ðh Þ Rb ðh2 Þ R2n 1 1 b b tan h1 Rb ðh2 Þ 2n sinð2nh2 Þ 2n þR2n ¼ 0 ð16Þ b ðh2 Þ cosð2nh2 Þ Rb ðh2 Þ Rb ðh2 Þ tan h2
sin h1
C S0
1503
where the coefficients in Eqs. (15) and (16) are superscripted by S as C S0 and C S2n to denote the staggered-packing geometry. As the packing limit is approached and the diagonal line through the unit cell center is modified as discussed previously and illustrated in Fig. 2c, the additional boundary on the right face of the unit cell will be satisfied by applying Eq. (13). Table 2 presents where x¼a the values of the coefficients C S0 ; . . . ; C S18 calculated using the boundary collocation method with N = 10 discrete boundary points for several values of the fiber volume fraction, vf, and fiber packing angle, d. Using Darcy’s law [27], the longitudinal permeability can be related to the average velocity through the cross section of the fluid region in the unit cell, the fluid viscosity, and the pressure gradient as j = lLuavg/Dp, from which the dimensionless permeability, j ¼ j=R2 , is obtained as
R p=2 R Rb ðhÞ
j ¼ uav g ¼
0
uðr; hÞr dr dh 1 R p=2 R Rb ðhÞ r dr dh 0 1
ð17Þ
where the numerator denotes the dimensionless volumetric flow rate and the denominator is the dimensionless cross sectional area p=4Þ for the b of the fluid region of the unit cell equaling ða rectangular packing as in Fig. 1a and ðab=2 p=4Þ for the staggered fiber arrangement as in Fig. 1b. In Eq. (17) the velocity profile, uðr ; hÞ, is given by Eq. (12) and the associated coefficients tabulated in Tables 1 and 2. Eq. (17) was evaluated using a computational quadrature method to determine the dimensionless permeability as a function of the volume fraction, vf, and the fiber packing angle, d, for the two packing arrangements considered in this study. 3. Results and discussion The analytical model presented in the previous section was compared to three-dimensional numerical simulations of the fluid flow through identical rectangular and staggered unit cell geometries (as shown in Fig. 1) for a range of fiber volume fractions and fiber packing angles. The numerical solution was obtained with the commercially available finite element software COMSOLÓ using a tetrahedral mesh consisting of 2 104 elements, and the results were non-dimensionalized according to the parameters presented in Section 2 in order to compare the numerical results to the dimensionless analytical series solution. The results of the comparison are presented and discussed in this section. Fig. 3 presents contours of the dimensionless velocity profiles evaluated using the analytical solution in a rectangular unit cell geometry with a fiber packing angle of 45° (Fig. 3a) and a staggered unit cell geometry with a fiber packing angle of 60° (Fig. 3b), both with a fiber volume fraction of 0.25. The analytical solution is based on ten boundary collocation points in the evaluation of the boundary conditions, Eqs. (5)–(7), using the constants given in Tables 1 and 2 in Eq. (12). Also included in the figures are the dimensionless velocity profile values obtained from a numerical fluid dynamics simulation of the flow through the respective geometries, shown by the discrete triangular markers in the figures. Excellent agreement is noted between the analytical solution and the corresponding numerical solution. For the rectangular fiber arrangement in Fig. 3a, the maximum dimensionless velocity is approximately 0.27 and located at the upper right-hand corner of the fluid domain. At the same volume fraction, the staggeredpacked case in Fig. 3b shows a smaller maximum dimensionless velocity of approximately 0.19 to occur at the points (0, 1.16) and (1, 0.57) in the unit cell, as shown by the arrows in Fig. 3b. From Fig. 3, it is also evident that the maximum velocity locations for each fiber packing arrangement are situated on the fluid boundary at the furthest distance from the fiber surfaces in their respective
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Table 1 First ten coefficients in Eq. (12) for a rectangular fiber arrangement. d
vf
C R0
C R2 (101)
C R4 (102)
C R6 (103)
C R8 (104)
C R10 (105)
C R12 (106)
C R14 (107)
C R16 (108)
C R18 (109)
45°
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.79
0.637 0.637 0.637 0.637 0.636 0.636 0.636 0.638
0 0 0 0 0 0 0 0
3.13 3.13 3.10 3.02 2.86 2.62 2.29 1.94
0 0 0 0 0 0 0 0
13.2 13.0 12.2 10.1 6.08 0.277 8.29 15.3
0 0 0 0 0 0 0 0
49.4 46.5 34.0 1.44 62.4 161 270 321
0 0 0 0 0 0 0 0
175 154 63.2 175 643 1350 2030 1960
0 0 0 0 0 0 0 0
55°/35°
0.10 0.20 0.30 0.40 0.50 0.54
0.909 0.909 0.909 0.908 0.908 0.910
1.21 1.16 1.09 0.997 0.891 0.840
3.10 2.93 2.64 2.25 1.78 1.56
4.24 3.56 2.40 0.792 1.17 2.05
8.59 6.62 3.04 2.24 8.88 11.8
13.8 8.10 2.49 18.4 38.4 46.9
25.2 12.4 12.4 50.9 99.2 119
35.4 12.0 34.1 106 196 232
40.0 10.4 49.1 144 260 307
24.3 2.49 41.1 110 193 224
65°/25°
0.10 0.20 0.30 0.36
1.366 1.358 1.345 1.341
3.02 2.82 2.54 2.34
4.22 3.48 2.43 1.66
5.73 3.60 0.514 1.74
8.27 3.46 3.59 8.76
10.7 2.43 9.88 18.9
11.4 0.878 14.9 26.5
9.00 0.362 14.4 24.8
4.62 0.576 8.40 14.2
1.15 0.221 2.29 3.83
C R1 ; C R3 ; C R5 ; C R7 ; C R9 ; C R11 ; C R13 ; C R15 ; C R17 ; C R19 ¼ 0.
unit cell domains. It follows from the dimensionless velocity profiles that the dimensionless volumetric flow rate through the rectangular fiber packing geometry is greater than the staggered fiber arrangement at the same fiber volume fraction. Since the results are presented in dimensionless form, the velocity profiles are generally applicable to longitudinal flow situations through aligned rigid fibers without limitation on the fiber radius or fluid viscosity. In addition, the qualitative relative trends in the flow behavior for the two unit cell geometries are generally applicable throughout the entire range of fiber volume fractions and packing angles. The results in Fig. 3 were based on ten collocation points on the boundaries of the rectangular and staggered unit cells in Fig. 2a–c, which, in turn, governs the number of terms used in evaluating the series solution in Eq. (12). It is instructive to assess the influence of the number of boundary collocation points—and, in turn, the number of terms in Eq. (12)—on the accuracy of the analytical solution. To this end, Fig. 4 presents the magnitude of the error in the analytical solution of the volumetric flow rate for the rectangular fiber arrangement in comparison to the converged numerical solution of
the volumetric flow rate, expressed as a percentage of the numerical solution, as the number of coefficients determined in Eq. (12) is increased. The error magnitudes are presented at incremented values of the volume fraction, vf, and at four different values of the packing angle, d, (Fig. 4a–d) as noted in the figure caption. The maximum attainable volume fraction at the packing limit decreases as the packing angle is increased, which can be seen from inspection of the unit cell’s geometry and is reflected in the different values of the volume fraction upper limits as the packing angle is changed, presented in the legend of Fig. 4. Fig. 4 generally shows that as the number of terms in the expansion of Eq. (12) is increased, the accuracy of the analytical solution increases accordingly and the error with respect to the converged numerical solution approaches zero. It is apparent from Fig. 4a that for a rectangular packing angle of d ¼ 45 , the error remains less than about 2.8% when at least six terms are included in the expansion of Eq. (12) and further reduces to less than 1% when the number of terms is increased to eight. The relatively greater amount of error apparent in the case where vf = 0.79 and d ¼ 45
Table 2 First ten coefficients in Eq. (12) for a staggered fiber arrangement. d
vf
C S0
C S2 (104)
C S4 (104)
C S6 (103)
C S8 (105)
C S10 (105)
C S12 (106)
C S14 (107)
C S16 (108)
C S18 (109)
60°/30°
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
0.551 0.551 0.551 0.551 0.552 0.554 0.551 0.551 0.551
0.921 0.893 0.604 0.813 5.31 19.4 0.0696 0.0658 0.997
0.701 0.667 0.444 0.524 3.40 11.9 0.0301 0.0338 0.378
8.46 8.46 8.44 8.35 8.03 7.10 7.37 6.39 5.13
3.89 3.79 2.73 2.51 19.3 72.5 16.2 18.1 1.90
3.13 3.05 2.20 1.99 15.7 59.9 18.4 14. 5 1.95
89.2 88.6 81.3 44.5 76.8 454 149 327 452
105 102 76.2 53.9 481 1930 12.7 6.32 117
254 251 213 23.6 585 3230 61.4 27.4 542
815 813 785 639 191 2330 14,300 22,700 9690
70°/20°
0.10 0.20 0.30 0.40 0.50 0.57
0.875 0.874 0.871 0.868 0.862 0.875
1100 1070 1020 965 905 771
276 263 243 217 188 122
4.79 4.51 4.07 3.42 2.68 0.487
67.8 61.6 53.2 41.6 31.1 120
13.8 14.0 15.3 17.6 22.4 37.6
20.2 24.3 35.0 52.0 80.6 98.8
26.3 38.1 65.9 110 180 179
24.1 40.3 75.8 130 213 217
9.18 17.6 34.8 59.7 96.6 137
80°/10°
0.10 0.20 0.27
1.832 1.711 1.620
4490 4520 4412
447 431 388
4.32 3.96 3.24
36.3 31.9 24.9
2.28 1.93 1.57
0.982 0.803 76.4
0.267 0.209 0.266
0.0405 0.0294 0.0587
0.00258 0.00165 0.00628
C S1 ; C S3 ; C S5 ; C S7 ; C S9 ; C S11 ; C S13 ; C S15 ; C S17 ; C S19 ¼ 0. Rows with italicized numbers indicate solution for the modified boundary, as shown in Fig. 2c.
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Fig. 3. Contour plots of the dimensionless velocity in unit cells of a (a) rectangular fiber arrangement with vf = 0.25 and d = 60°/30°.
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vf = 0.25 and d = 45° and (b) staggered fiber arrangement with
(Fig. 4a) when compared to the other cases is due to the presence of narrow fluid regions at either side of the unit cell which occur as a result of the unit cell geometry approaching the fiber packing limit. Fig. 4b–d indicates that for packing angles other than 45°, the error is less than 0.9% and 0.5% with the inclusion of six and eight terms, respectively. In general, the error variations in Fig. 4 demonstrate that by including ten terms in the expansion of Eq. (12), a highly accurate solution for the flow rate through the unit cell can be found throughout the ranges of the geometrical parameters involved, including conditions approaching and equal to the fiber packing limits. The error in the solution of Eq. (12) using a staggered packing arrangement of the fibers is similar to the trends presented in Fig. 4 and is omitted for brevity. The variation of the dimensionless longitudinal permeability, j, with the fiber volume fraction, vf, for rectangular fiber arrangements with different packing angles is depicted in Fig. 5. The figure
shows the analytical solution obtained from Eq. (17) with ten boundary collocation points and using the coefficients in Table 1, as solid lines, which is compared with the numerical finite element solution presented as the discrete triangular symbols. The designation of two angles for the different lines in Fig. 5 other than d ¼ 45 represents the fact that either packing angle results in the same dimensionless fluid flow rate and therefore the same dimensionless permeability value. In general, it is noted that the dimensionless permeability decreases with increasing fiber volume fraction, owing to the resulting increase in the flow resistance. Furthermore, for any given fiber volume fraction, the permeability is the lowest for the packing angle of 45°, again signifying the greatest flow resistance for this packing configuration. It is seen in Fig. 5 that the packing angle of 45° allows the maximum fiber volume fraction of 0.785 to be attained, which results in the least dimensionless permeability of approximately 2.31 102.
Fig. 4. Variation of the error between the numerical and analytical solutions for flow rate through a rectangular fiber array at different fiber volume fractions as the number of terms in Eq. (12) is varied; for (a) d = 45°, (b) d = 50°/40°, (c) d = 55°/35°, and (d) d = 60°/30°.
Fig. 5. Variation of the dimensionless permeability with the fiber volume fraction and fiber packing angle for a rectangular arrangement of fibers.
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Fig. 6. Variation of the dimensionless permeability with the fiber volume fraction and fiber packing angle for a staggered arrangement of fibers.
Similar trends in the dimensionless permeability variation with fiber volume fraction and fiber packing angle were found for the staggered fiber arrangement, as presented in Fig. 6. As in Fig. 5, the discrete triangular markers denote the numerical finite element solution and the solid lines represent the analytical solution from Eq. (17) based on 10 boundary collocation points using the coefficients in Table 2. Two different packing angles are shown for each line to signify that the identical dimensionless flow rates are found by applying either of the packing angle values. For the staggered fiber arrangement, the highest volume fraction attainable is 0.907 at a packing angle of 60 =30 , where the corresponding dimensionless permeability value is approximately 3.26 103, which is approximately an order of magnitude lower than the minimum permeability noted for the rectangular packing in Fig. 5. In both Figs. 5 and 6, excellent agreement is seen between the analytical results and the numerical values. It is evident from a comparison of the permeability values in Figs. 5 and 6 that at any fiber volume fraction, the rectangular fiber packing offers less resistance to the fluid flow than the staggered fiber arrangement, which is reflected in the higher dimensionless permeability value for the rectangular fiber arrangement. This difference is particularly pronounced at higher fiber volume fractions and reduces as the fiber volume fraction decreases, where the smallest tested fiber volume fraction of 1e-3 resulted in the dimensionless permeability values converging to the same value of approximately 450 for all cases of the different relative fiber arrangements. The longitudinal permeability values presented in Figs. 5 and 6 can be compared to experimental results reported in the literature to evaluate the accuracy of the analytical results compared to physical measurements. An experiment measuring the longitudinal permeability of an array of aligned cylinders was conducted by Brennan et al. [20], who reported a dimensionless permeability on the order of 102 for a staggered fiber arrangement with a volume fraction of 0.6. Considering the results presented in Fig. 6 for a staggered fiber arrangement, it is evident that at vf = 0.6 the analytical solution also predicts a permeability on the order of 102 and therefore exhibits good agreement with the experimentally measured value in [20]. In addition, Li et al. [23] studied a fiber tow made up of glass filaments with a diameter of 12 lm and experimentally measured the dimensionless longitudinal permeability to be on the order of 102 for a fiber volume fraction of 0.6, which is on the same order of magnitude as the measurements in [20] and the analytical results presented in Figs. 5 and 6.
As the fiber volume fraction of the fiber tow is increased, the effects of capillary pressure will become increasingly significant in permeability measurements and predictions (e.g. [23,28]). According to [28], the capillary pressure, Pcap, in the longitudinal direction of aligned rigid fibers can be expressed in terms of the fiber volume fraction, vf, fiber radius, R, surface tension, c, and contact angle, r, as: P cap ¼ f2v f c cosðrÞg=fRð1 v f Þg. For typical materials used in the liquid composite molding process, the surface tension, contact angle and the fiber radius are on the order of 102 N/m, 10°, and 105 m, respectively. Using the above expression and considering a fiber tow with a volume fraction of 0.8 results in a capillary pressure of approximately 8 kPa. The applied pressure in a liquid composite molding application is typically on the order of 100 kPa, which is more than an order of magnitude higher than the calculated capillary pressure. Therefore, reasonable accuracy should be expected of the permeability values presented in Figs. 5 and 6 even in the presence of capillary effects in practical liquid molding conditions. The analytical permeability model presented in this article may be extended to include the capillary effect in the governing physics in a future work.
4. Conclusions An analytical series solution was developed for the longitudinal fluid flow through aligned rigid fibers which was used to establish a relationship between the dimensionless permeability, j, the fiber volume fraction, vf, the relative fiber arrangement, and the fiber packing angle. The results are valid for all fiber volume fractions, vf including the fiber packing limit. The analytical model together with the coefficients in Tables 1 and 2 serve to obtain the values of the longitudinal flow permeability in simulation of resin flow through aligned fibrous media, providing a method for predicting the local longitudinal permeability of fiber tows in preform materials used in composites manufacturing.
Acknowledgments This research is funded in part by the National Science Foundation with Grant No. CBET-0934008, and the U.S. Department of Education through a GAANN fellowship to Caleb DeValve through Award No. P200A060289. Their support is gratefully acknowledged.
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