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]
$$ \xi = \frac{\Delta P}{L} \cdot \frac{{2D_{h} }}{{\rho V^{2} }}, $$
(a1)
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
$$ D_{\text{h}} = 4A'/O $$
(a2)
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
$$ \xi = c' \cdot \frac{\mu }{{\rho VD_{\text{h}} }}, $$
(a3)
where c’ is a dimensionless shape factor and μ is the fluid viscosity. Then Eqs. a1 and a3 give
$$ \frac{\Delta P}{L} = c' \cdot \frac{\mu V}{{2D_{\text{h}}^{2} }} $$
(a4)
Comparing Eq. 1 with Eq. a4 gives
$$ {\text{K}} = \frac{{2D_{\text{h}}^{2} }}{c'} $$
(a5)
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
$$ \frac{\Delta P}{L} = \frac{32\mu V}{{D_{\text{h}}^{2} }} $$
(a6)
Comparison of Eqs. a6 and 1 gives the equivalent permeability of a circular tube
$$ {\text{K}}_{t} = \frac{{D_{\text{h}}^{2} }}{32} $$
(a7)
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):
$$ c = 1.5 - 2.0364\alpha + 2.964\alpha^{2} - 2.724\alpha^{3} + 1.677\alpha^{4} - 0.491\alpha^{5} $$
(a8)
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.