## Abstract

Three-dimensional (3D) woven textiles, including orthogonal and angle-interlock woven fabrics, exhibit high inter-laminar strength in addition to good in-plane mechanical properties and are particularly suitable for lightweight structural applications. Resin transfer moulding (RTM) is a cost-effective manufacturing process for composites with 3D-woven reinforcement. With increasing preform thickness, the influence of through-thickness permeability on RTM processing of composites becomes increasingly significant. This study proposes an analytical model for prediction of the through-thickness permeability, based on Poiseuille’s law for hydraulic ducts approximating realistic flow channel geometries in woven fabrics. The model is applied to four 3D-woven fabrics and three 2D-woven fabrics. The geometrical parameters of the fabrics were characterized by employing optical microscopy. For validation, the through-thickness permeability was determined experimentally. The equivalent permeability of inter-yarn gaps was found to account for approximately 90 % of the through-thickness permeability for the analysed fabrics. The analytical predictions agree well with the experimental data of the seven fabrics.

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## Acknowledgements

The work was supported in part by the projects: RGC No.: 5158/13E and NSFC funding Grant No. 51373147 and Project code: JC201104210132A.

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## Appendix

### Appendix

The frictional pressure loss in flow along a duct with arbitrary cross section, e.g. the duct formed by interwoven yarns, is usually expressed in terms of a friction factor \( \xi \) (also called a resistance coefficient) which is defined as [27]

where \( \Delta P \) and *L* are the pressure loss and the length of flow channel, *D*
_{h} is the hydraulic diameter as defined below, *ρ* is the density of the fluid and *V* is the mean velocity over the duct cross section. The hydraulic diameter is defined as four times the duct cross-sectional area *A*′ divided by the wetted perimeter *O*

For a circular tube, *D*
_{h} is equivalent to its geometrical diameter. The friction factor can be derived analytically for many cross sections (circular, triangular, quadratic, etc.) in laminar flows [18, 28] and can be expressed as

where *c’* is a dimensionless shape factor and *μ* is the fluid viscosity. Then Eqs. a1 and a3 give

Comparing Eq. 1 with Eq. a4 gives

The Hagen–Poiseuille equation describes a laminar fluid flow along a circular tube (diameter *D*
_{h}), which has a relationship of pressure gradient and flow velocity

Comparison of Eqs. a6 and 1 gives the equivalent permeability of a circular tube

This implies that the value of *c’* is 64.

When converting ducts with arbitrary rectangular cross section to virtual ducts with circular cross section, friction constants reported in the literature [29] for rectangular ducts with different width/length ratios, *α*, were divided by *c’* to obtain *c* as listed in Table 3. These values can be fitted with a polynomial (coefficient of correlation *R*
^{2} = 1):

According to Eq. a8, the value of *c* can be obtained for calculation of *K*
_{g} for arbitrary gap length and width ratios, as demonstrated for the seven fabrics in Table 3.

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Xiao, X., Endruweit, A., Zeng, X. *et al.* Through-thickness permeability study of orthogonal and angle-interlock woven fabrics.
*J Mater Sci* **50**, 1257–1266 (2015). https://doi.org/10.1007/s10853-014-8683-4

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DOI: https://doi.org/10.1007/s10853-014-8683-4