An analytical model for through-thickness permeability of woven fabric

Abstract

Woven fabric permeability is relevant to many applications, such as airbags, textile composites processing, paper making and air and water filtration. This paper proposes an analytical model to predict through-thickness fabric permeability based on viscous and incompressible Hagen–Poiseuille flow. The flow is modeled through a unit cell of fabric with a smooth fluid channel at the center with slowly varying cross-section. The channel geometry is determined by yarn spacing, yarn cross-section and fabric thickness. The shape of the channel is approximated by a parabolic function. Volumetric flow rate (Q) is formulated as a function of pressure drop and flow channel geometry for woven fabric. The permeability (K) is calculated thereafter according to Darcy’s law. The air permeability of nine different fabrics has been measured to verify the analytical model. A sensitivity study was carried out to understand how geometric parameters influence the fabric permeability. The analytical model shows very close agreement with the experimental data: within 5% for most fabrics. The sensitivity study on permeability indicates the importance of flow channel geometry in obtaining accurate predictions.

Keywords

analytical model Hagen–Poiseuille flow permeability woven fabric

Permeability is a measure of the ability of a porous material to transmit fluids. It is one of the primary properties for technical textiles used in fluid-related applications, such as airbags, textile composite processing, paper making and air and water filtration. The regular interwoven structure of woven fabric gives rise to arrays of fluid channels. These channels are in theory identical and repetitive. The geometry of each individual channel depends mainly on weave density, yarn shape and weave style. Fabric permeability is governed by these geometric parameters. A predictive model of permeability as a function of fabric structure is a desirable tool for optimum material design.¹

Research on the air permeability of textile materials began at the end of the 19th century when experimental methods for estimating the hygienic properties of materials for clothing began to be used.² As a result of studying flow through porous media, Darcy³ established the linear dependence of velocity on pressure drop. The first studies of the air permeability for fabrics conducted by Rubner⁴ were based on this theory. Darcy's law is presented in the following equation:

(1)

where μ is the fluid viscosity, L is the flow length, υ is the fluid velocity and

Δ P

is the pressure drop. Bruschke and Advani⁵ reported excellent agreement using experimental and numerical simulation for a Newtonian fluid through a fiber network compared with Darcy’s law. However, Darcy’s law does not relate the structural parameters of porous materials to the value of permeability.

Kozeny and Carman⁶ found a similar relation between pressure drop and fluid flow as in Darcy’s law separately, suggesting that the permeability is independently determined by material variables, that is, porosity (Ф) and specific surface area (S):

(2)

where k is the Kozeny coefficient. This classical permeability-porosity equation was originally developed for granular beds consisting of ellipsoids. Further modification of the equation subsequently made it applicable in various fields,^7,8 such as fouled fibrous filters and adsorbents, and other composite porous media. However, one weakness of the Kozeny–Carman equation (pointed out by Gutowski et al.⁹) when applied to composite manufacturing is in that the model parameters are only coupled to the geometry through the variables on the radius of the fiber and that the detailed geometry dependence is lumped together into the model parameter, which has to be determined experimentally. Published results also indicated that the parameter k may vary with the fiber volume fraction for a given fabric.¹⁰

Gebart¹¹ developed an analytical model for predicting the permeability of fiber bundles, which simulated the two-dimensional (2D) flow of a Newtonian fluid perpendicular to and parallel with unidirectional filaments. It can be considered analogous to flow within yarns in a fabric structure. Gebart¹¹ looked at two types of packing, array-quadratic and hexagonal, as follows:

(3)

(4)

where

K_{∥}

and

K_{⊥}

are permeability along and perpendicular to the fibers, respectively, and

V_{f max}

is the possible maximum fiber volume fraction, which depends on the arrangement of fibers. Values for

C_{1}, V_{f max}

and c were

16 / 9 π \sqrt{2}

π / 4

and 57, respectively, when the fiber arrangement was quadratic and

16 / 9 π \sqrt{2}

π / 2 \sqrt{3}

and 53, respectively, when it was hexagonal. This model was based on the assumption of circular fiber cross-section and the specified fiber arrangements. The predicted permeability had excellent agreement with experimental results only when the fiber radius in the analytical model was adjusted to match the permeability prediction to the experimental data for both flow along and across the fibers. Based on Gebart’s¹¹ work, Ngo and Tamma¹² and Hakanson et al.¹³ both made micro-scale permeability predictions for fibrous porous medium. The former applied computational approximations, while the latter mainly focused on analytical determination, both showing good agreement with experimental data for fiber bundle permeability prediction.

Phelan and Wise¹⁴ studied transverse Stokes flow through an array of elliptical cylinders to determine the macroscopic permeability of unidirectional fabric. Each cylinder represents one yarn, which can be treated as a solid or porous material. From first principles a semi-analytical model based on lubrication analysis¹¹ was developed:

(5)

where L_x and L_y are half the length and width of the unit cell, respectively, and f is an explicit function of the fluid domain geometry, tow porosity and tow permeability. The fluid domain geometry is governed by tow packing arrangement as well as tow cross-sectional shape. The model showed that tow cross-sectional shape played an important role in the overall fabric permeability. A more elliptical tow shape had higher flow resistance along the major axis. One main assumption from this model was that the flow was unidirectional along the fluid channel axis by ignoring transverse flow velocity components. Using computational fluid dynamics (CFD) analysis, the author indicated that the assumption was valid, since the pressure gradient was highest near the narrowest regions of the flow channel where the walls were almost parallel. The model improves the permeability prediction by considering specific geometries.¹⁵ However, it is not directly applicable to woven fabric where the flow domain geometry is more complex.

Kulichenko¹⁶ developed an analytical model for the through-thickness permeability of woven fabric, based on the Poiseuille and Weisbach–Darcy equations, by simplifying the geometry of channels (pores) in a fabric as a system of parallel capillaries, such as the straight tube shown in Figure 1(a). After analysis of the fabric geometry and fitting with experimental data, the permeability is

(6)

where d_h is the hydraulic diameter of the pore, Φ is the porosity of fabric, which should be calculated by

Φ = A_{p} / A_{u}, A_{p}

is the area of the pore, while A_u is the area of a unit cell. Three methods were suggested for measuring porosity and hydraulic diameter for Equation (6), but no method could predict the permeability accurately in comparison to experiments. The other problems in verification are that the samples used were non-woven materials with no periodic unit cell.

Figure 1.

Unit cell of fabric and three-dimensional simplified channel geometry.

This paper takes into account the exact three-dimensional (3D) geometry of the fluid channels formed by the interlaced yarns, and an analytical model based on Poiseuille flow is developed for predicting the permeability of woven fabric in the following section. Experimental verification with nine fabrics (plain and twill series, as shown in Table 1) is presented in the third and fourth sections, and compared with experimental results and the Kulichenko model in the fourth section. This section also includes a parametric study for the model. The final section gives concluding remarks.

Table 1.

Fabrics specifications

Fabrics	Composition and structure	Average R_f10⁻⁶ m	Yarn V_f value	T 10⁻³ m	Yarn spacing 10⁻³ m		Yarn width 10⁻³ m
Fabrics	Composition and structure	Average R_f10⁻⁶ m	Yarn V_f value	T 10⁻³ m	Warp S_j	Weft S_w	Warp D_j	Weft D_w
U₁	100%Cotton plain	4.3	0.56	0.323	0.470	0.410	0.405	0.279
U₂	65/35 PET/cotton plain	5.4	0.58	0.229	0.235	0.365	0.195	0.255
C₁	67 PET/33 cotton Desized, scoured, bleached and mercerized 2/1 twill	5.9	0.56	0.419	0.340	0.480	0.310	0.310
C₂	67 PET/33 cotton Dyed but not finished 2/1 twill	5.9	0.63	0.425	0.330	0.532	0.300	0.430
C₃	67 PET/33 cotton Finished 2/1 twill	5.5	0.57	0.427	0.300	0.510	0.270	0.330
C₇	67 PET/33 cotton Desized, scoured, bleached and mercerized 2/1 twill	5.7	0.60	0.452	0.340	0.450	0.300	0.400
C₈	67 PET/33 cotton Finished 2/1 twill	5.7	0.61	0.455	0.340	0.430	0.300	0.350
C₉	60 cotton/40 PET Same process as C1 BUT 2/2 twill	5.6	0.67	0.560	0.356	0.520	0.332	0.450
C₁₀	60 cotton/40 PET finished 2/2twill	5.7	0.56	0.610	0.342	0.446	0.313	0.380

PET: polyethylene terephthalate

Development of the model

Hypothesis

Fluid

The Newtonian fluid (liquid or gas) considered in the model is assumed to be incompressible, with a constant viscosity and density due to the low flow rates.

Flow conditions

At the inlet, fluid is injected at a constant pressure P₁. The flow front pressure P₂ is ambient.

Inertial terms and yarn motion are neglected.

The flow is laminar and the flow process is quasi-steady state.

The velocity of the fluid at the surface and inside of the yarns is assumed to be zero, while at the center-line of the channel it is maximum.

Fluid flow is considered in the direction perpendicular to the fabric. The transverse component of the velocity is negligible, since the highest pressure gradient is near the narrowest region, where the flow is almost parallel to the channel surface.¹⁷

Geometry of the unit cell

Figure 2 describes the unit cell geometry in a plain woven fabric. D_w and D_j are widths of the weft and warp yarns, respectively, while S_w and S_j are the spacings of the weft and warp yarns, respectively. One single flow channel in the unit cell is then simplified as a smooth fluid channel with slowly varying circular cross-sections. The radius of the narrowest cross-section (R) is calculated as half the diameter of the rectangular channel cross-section in the real fabric:¹⁸

(7)

R is the radius of the narrowest cross-section and a is the distance from the narrowest channel surface to the boundary of the unit cell, which is calculated from

(8)

Figure 2.

Cross-section of flow channel and curve fitting with parabola.

Description of the yarn cross-section

Figure 3(a) shows a side view of the flow channel formed by the yarns. The channel surface curvature can be represented by a parabolic function, as illustrated by Figure 3(b), where the parabolic function matches well near the narrowest channel cross-section. This is the region most relevant for permeability prediction, since the highest pressure drop occurs at this confined region. The parabolic function for yarn shape is assumed to be

(9)

where β is a parameter that determines the channel geometry. Figure 4 shows four kinds of yarn cross-section with four different β values. It is noted that the smaller the β value, the sharper the central part of the parabolic curve.

Figure 3.

Effect of β on yarn (and hence channel) shape.

Figure 4.

Laminar flow through (a) straight tube and (b) curved tube.

The parabolic function is chosen, as it is easy to integrate the derived analytical model in the next section. It is noted in Figure 3(b) that the real fabric thickness is smaller than the crossponding height of the parabolic curve formed at the boundary of the unit cell; hence, the porabolic curve is truncated to match the fabric thickness.

Analytical model

According to fluid dynamics theory and the assumptions above, the analytical model is from the Hagen–Poiseuille equation,¹⁹ which describes fluid flow through a long straight tube, as shown in Figure 1(a):

(10)

where r is the radius of the tube and

\frac{dP}{dx}

is the pressure gradient. Figure 1(b) shows the curved channel described in the section Description of the yarn cross-section. Here the length of the channel is equal to the fabric thickness (T). For the varied cross-section, the form of Equation (10) is changed by integrating for a finite length d_x with a radius

r (x)

(11)

The channel radius varies depending on the distance along the x axis:

(12)

Equation (11) is transformed into Equation (13) with the radius substituted from Equation (12):

(13)

(14)

Setting

t = \frac{x}{\sqrt{2 aR}}

, then the integration in Equation (14) has the following solution:

(15)

Therefore Equation (14) becomes

(16)

It is possible to simplify Equation (16). Figure 5 shows the value of the integral

\frac{1}{(1 + t^{2})^{4}}

with the limit value

\frac{T}{\sqrt{β aR}}

. If the limit value was 2, the integral value would be 0.9807; while if the limit was set to more than 3, the integration would be 0.9817. Therefore,

5 π / 16 \approx 0.98175

was used instead of the complicated expression (15).

Figure 5.

The bounded integral value in Equation (15).

Accordingly, Equation (17) has been simplified significantly:

(17)

From Equation (17), the velocity of fluid flow through fabric (

ν_{1}

) and the permeability of fabric (K) can be obtained as follows:

(18)

(19)

Verification

Experimental approach

Through-thickness air permeability was measured according to BS EN ISO 9237:1995.²⁰ The apparatus for the experiment is an air permeability tester FX 3300, as shown in Figure 6.²¹ The fabric is held by the clamp under a certain pressure. A suction fan forces the air to flow perpendicularly through the fabric and the flow is adjusted gradually until the required pressure drop is achieved across the test region. D is a transducer that can determine the volumetric flow rate (m³/s). This value divided by the specimen area (10 cm²) gives the velocity of air flow. The pressure drop in the experiment for all fabrics is set to 500 Pa, with an accuracy of at least 2%. Using the measured velocity, pressure drop and fabric thickness, permeability is calculated according to Darcy’s law.

Figure 6.

Sketch of air permeability tester FX3300.

Nine fabrics with different weave styles and materials were investigated, as listed in Table 1. Each test was repeated three times with a fresh sample. Yarn cross-sections were cut using a laser beam razor blade and cross-section images were obtained using a TM – 1000 tabletop microscope. The images were used to measure the yarn spacing and yarn widths, as well as the fiber radius and yarn fiber volume fraction. The fabric thickness was measured using the Kawabata Evaluation System for Fabrics (KES-F) at a pressure of 0.05 kPa.²²

Curve fitting for channel geometry

The exact flow channel geometry in a woven fabric can be obtained by measurement from microscopic images of fabric cross-sections, as shown in Figure 7.

Figure 7.

Determination of yarn cross-section. (a) A cross-section of fabric C₁₀. (b) Channel formed by yarns and its mathematical description.

Coordinates of the yarn cross-section were measured using image analysis software, and these were approximated by a second-order polynomial using least-square analysis. This allowed the β value in Equation (9) to be determined directly.

Results and discussion

Comparison of analytical and experimental results

Table 2 presents the values of geometric parameters for unit cells of nine fabrics based on the experimental measurements and Equations (7)–(9). With the calculated values in Table 2, the error between the exact integral in Equation (15) and the approximate value

5 π / 16

was less than 2% for all fabrics, apart from Fabrics U₁ and U₂. For Fabric U₁ the error was 13.11%, while for U₂ it was 5.05%. The reason might be the structure of the unit cell. The channel radius of Fabric U₁ is much larger than the other fabrics, while the thickness smaller. This indicates that the simplified Equation (17) might not be suitable for loose and thin fabrics.

Table 2.

Geometric parameter values of geometries for nine fabrics

Fabrics	R 10⁻³ m	α 10⁻³ m	R² parabolic	β calculated
U₁	0.0434	0.1755	0.9906	6.284
U₂	0.0293	0.1136	0.9441	4.755
C₁	0.0255	0.1735	0.9841	4.509
C₂	0.0232	0.1805	0.9569	7.950
C₃	0.0257	0.1632	0.9730	8.004
C₇	0.0222	0.1714	0.9857	6.920
C₈	0.0267	0.1632	0.9945	8.253
C₉	0.0179	0.1935	0.9729	1.274
C₁₀	0.0201	0.1734	0.9336	4.100

Based on the geometric measurements for all the nine fabrics, the permeability for each fabric was calculated. Table 3 compares fabric permeability from the new analytical model (Equation (19)) and the Kulichenko straight flow channel model (Equation (6)) with the experimental measurements. The second column in the table is the air velocity through the narrowest cross-section where the maximum velocity occurs, according to continuity theory.²³ The velocity is calculated by Equation (20) and it is used to check the state of the fluid according to the Reynolds number based on Equation (21):

(20)

(21)

Table 3.

Prediction of permeability (Equation (19)) compared against experimental data and the Kulichenko model

Fabrics	υ m/s	Re	k Expt 10⁻¹² m² (±SD)	k Eq. (19) 10⁻¹² m²	k Kuli 10⁻¹² m²	Difference Eq.(19) vs. Expt	Difference Kuli vs. Expt
U₁	5.18	28.7	12.135(±0.0470)	13.956	3.714	13.05%	69.39%
U₂	4.67	17.5	8.031(±0.0200)	8.390	1.811	4.28%	77.45%
C₁	2.03	6.6	3.988(±0.0047)	4.034	0.534	1.13%	86.62%
C₂	1.13	3.3	2.018(±0.0060)	2.067	0.348	2.36%	82.75%
C₃	1.70	5.6	3.609(±0.0055)	3.637	0.613	0.78%	83.02%
C₇	1.23	3.5	2.331(±0.0033)	2.306	0.325	−1.10%	86.05%
C₈	1.79	6.1	4.191(±0.0075)	4.290	0.701	2.31%	83.27%
C₉	1.43	3.3	2.485(±0.0008)	2.456	0.114	−1.19%	95.40%
C₁₀	1.25	3.2	2.904(±0.0026)	2.855	0.220	−1.71%	92.43%

This shows that the Reynolds number for the flow through all fabrics is well below the critical value of 2300, where flow turns from laminar to turbulent. This validates the assumption of laminar flow for the analytical model.

Columns 5 and 6 in Table 3 display this model and the Kulichenko model predictions for the permeability of each fabric. This model, derived from Hagen–Poiseuille flow through the double curvature channel, predicts the fabric permeability very accurately compared with the Kulichenko model. One of the reasons behind the accurate prediction is that the Hagen–Poiseuille flow assumption is generally accurate for flow through woven fabric, where the velocity component of laminar flow is only considered parallel to the channel axis. More importantly, the geometry of the channel has a strong effect on the flow resistance. The model includes this geometric influence by explicit flow integration over the channel geometry, using a parabolic function fitted to each fabric via microscopic analysis. Hence, all parameters in the current analytical model can be determined from the fabric geometry.

The last column in Table 3 shows the error for the Kulichenko model (Equation (6)) in comparison with experimental results. For all sample fabrics, the prediction is more than 50% in error. The Kulichenko model simulates the channels as a series of parallel straight tubes with constant cross-section. A constant correction factor for flow through the real fabric was determined by Kulichenko by data fitting with some experimental permeability measurements. The results shown by our study indicate that fabric permeability is strongly influenced by the shape of the flow channel. It cannot be expressed by an empirical parameter to account for the geometry. Hence the Kulichenko model fails to predict permeability for fabrics in general.

Sensitivity of through-thickness permeability to fabric geometry

The current analytical model helps one to understand how the flow channel geometry influences the fabric permeability. There are four geometric parameters in the Equation (19) that directly relate to the prediction, that is, radius of flow channel (R), radius of yarn (a), shape factor of yarn cross-section (β) and thickness of fabric (T). The effects of each parameter on permeability are shown in Figure 8.

Figure 8.

Relationship of permeability to each parameter.

From Equation (19), the relationship between the permeability K value and each parameter should be as follows:

It is noted that permeability would decrease with increasing radius of yarn or β value, while the permeability increases with the other two parameters, radius of pore or thickness of fabric. An increase in the radius of yarn causes the size of the unit cell to increase, which means the number of flow channels per unit area would decrease; therefore, the permeability would decrease, as shown in Figure 8(a). Permeability would increase as the channel cross-section is enlarged with increasing R, shown in Figure 8(c). When β is increased, the increased channel curvature brings the surfaces closer and the channel becomes less open near the ends. The volume of the flow channel decreases, leading to less volumetric flow through the pore in unit time, as shown in Figure 8(b). By increasing fabric thickness, the volume of the flow channel and the cross-section of the flow inlet and outlet increase. As a result, the volume flow rate increases, as shown in Figure 8(d). The parametric relationship between fabric geometry and fabric through-thickness permeability helps one to understand the mechanics of flow resistance through fabric. It also provides a useful tool for fabric design in various applications.

Conclusions

A novel generic analytical model was proposed for predicting the through-thickness permeability of woven fabric. The key feature in the model is that a parabolic function was used within the Hagen–Poiseuille flow integration to capture the geometry of the flow channel formed by interwoven yarns. Different channel shapes in various fabrics can be well represented by this, with the parameter β obtained from microscopic measurement.

For nine different fabrics, the model gives good predictions compared with experimental permeability measurements within 5% errors for most fabrics. The Kulichenko model, where a straight flow channel is assumed, gives over 60% errors in permeability prediction. It is believed that the inclusion of the geometry of the flow channel makes the current analytical model more accurate than existing models. A parametric study shows that there are four independent geometric variables relevant to fabric permeability. The model would assist with understanding the factors affecting permeability and could help to improve fabric design.

Footnotes

Funding

Some of the work reported in this publication was carried out under the Technology Strategy Board's collaborative research and development through Project No: RC 2071 ‘Multi-Scale Integrated Modelling for High Performance Flexible Materials'. The authors would like to thank the Technology Strategy Board for financial support, and acknowledge the participation of the industrial and academic partners involved in this project: Unilever UK Central Resources, OCF PLC, Croda Chemicals Europe Ltd, University of Manchester, Heriot-Watt University, ScotCad Textiles Ltd, Carrington Career and Workwear Ltd, Moxon Ltd, Airbags International and Technitex Faraday Ltd.

Conflict of interest statement

None declared.

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