**2.1. Woven fabric's ideal geometric model of porous structure**

When a woven fabric is treated as a three dimensional formation, different types of pores are detected [22, 23, 24]: 1. inter-pores, e.g. the pores which are situated between warp and weft yarns (macropores, interyarn pores) and pores which are situated between fibres in the yarns (mesopores, interfiber/intrayarn pores), 2. intra-pores, e.g. the pores which are situated in the fibres (micropores, intrafiber pores). The structure and dimensions of the inter- or intrayarn pores are strongly affected by the yarn structure and the density of yarns in the woven structure [22]. As fibrous materials, woven fabrics have, with regard to knitted fabrics or nonwovens, the most exactly determined an ideal geometrical model of a macroporous structure in the form of a tube-like system, where each macropore has a cylindrical shape with a permanent cross-section over all its length (Figure 6) [25]. Because the warp density is usually greater than the weft density, the elliptical shape of the pore cross-section is used to represent the situation in Figure 6. Macropores are opened to the external surface and have the same cross-section area. They are separated by warp or weft yarns, and are uniformly distributed over the woven fabric area.

178 Genetic Programming – New Approaches and Successful Applications

right angles to each other,

porous structure will be presented.

fabrics.

fabrics

On the basis of an ideal geometrical model of porous structure, the pore size distribution which is also an important parameter of material porous structure can not assessed while the pores in geometrical model are usually assumed to be the same sizes. Such situation rarely occurs in the real fabrics. The further considerations of ideal geometrical models of material porous structures and porosity parameters will be focused on different types of

Fabrics are flat textile materials which are produced by different manufacturing techniques using different fibrous forms of input material (or structural element), and consequently having different porous structures. Following basic types of fabrics are known (Figure 5):

woven fabrics which are made by interlacing vertical warp and horizontal weft yarns at

knitted fabrics which are made by forming the yarn into loops and their interlacing in

nonwoven fabrics which are produced from the staple fibres, filaments or yarns by

**Figure 5.** 2-D schematic presentations of woven-, knitted-, and nonwoven (made from staple fibres)

While this chapter is focused on the genetic methods in order to predict porosity of woven and nonwoven fabrics, only those types of fabrics and their ideal geometric models of

When a woven fabric is treated as a three dimensional formation, different types of pores are detected [22, 23, 24]: 1. inter-pores, e.g. the pores which are situated between warp and weft yarns (macropores, interyarn pores) and pores which are situated between fibres in the yarns (mesopores, interfiber/intrayarn pores), 2. intra-pores, e.g. the pores which are situated in the fibres (micropores, intrafiber pores). The structure and dimensions of the inter- or intrayarn pores are strongly affected by the yarn structure and the density of yarns in the woven structure [22]. As fibrous materials, woven fabrics have, with regard to knitted fabrics or nonwovens, the most exactly determined an ideal geometrical model of a macroporous structure in the form of a tube-like system, where each macropore has a cylindrical shape with a permanent cross-section over all its length (Figure 6) [25]. Because the warp density is usually greater than the weft density, the elliptical shape of the pore cross-section

vertical (warp-knitted fabrics) or horizontal (weft-knitted fabrics) direction,

different web-forming, bonding and finishing techniques.

**2.1. Woven fabric's ideal geometric model of porous structure** 

The primary constructional parameters of woven fabrics which alter the porous structure are:


When fibre properties (fibre density, dimension, and shape) are different, two woven fabrics with similar woven structures and geometrical configurations can have distinctly different porosity [22].

**Figure 6.** 2D and 3D presentations of an ideal model of the porous structure of a woven fabric [27, 28] (d – yarn thickness, p – yarn spacing, MP - macropore; 1, 2 indicates warp and weft yarns, respectively)

To compare woven fabrics with porosity, the following porosity parameters can be calculated on the basis of the woven fabric primary constructional parameters and the ideal model of porous structure in the form of a tube-like system:

 (total) porosity by using Equation 6 where the bulk density of the material is actually the woven fabric density and the density of solid component is the yarn density. If the fibre volume fraction (yarn packing factor) is exposed from the Equation 7 which represents the yarn diameter calculation, and then inserted in Equation 8 by assuming Equation 9 for woven fabric density at the same time, the porosity of woven fabrics can be then written in the form of Equation 10:

$$d = \sqrt{\frac{4T}{10^5 \pi \cdot \rho\_{ym}}} = \sqrt{\frac{4T}{1000 \cdot \pi \cdot \rho\_{fb} \cdot \beta\_{fb}}} \tag{7}$$

$$
\rho\_{yarn} = \rho\_{f\text{ib}} \cdot \beta\_{f\text{ib}} \tag{8}
$$

The Usage of Genetic Methods for Prediction of Fabric Porosity 181

*N gg <sup>p</sup>* 1 2 (14)

(15)

**Figure 7.** Real and binary images of the pore cross-section shape and the number of pores in real woven fabrics (magnification of binary images: 20 x, magnification of real images: 80 x, yarn fineness: 36 tex,

plain, 21/16 threads/cm twill, 27/22 threads/cm satin, 30/24 threads/cm

 number of macropores in the area unit (pore density). It can be seen from Figure 7, that one macropore belongs to one warp yarn and one weft yarn, so the number of macropores can be calculated on the basis of warp and weft densities using Equation

where, Np is the pore density in pores/cm2, g1 is the warp density in threads/cm, and g2 is the

 open porosity (open area) which describes the fraction of macropore cross-section area in the area unit of woven fabric. If we assume elliptical macropore cross-section area

> *A A pp*

where, εopen is the open porosity, Ap is the macropore cross-section area in mm2, Ay is the projection area of warp and weft yarns, which refers to one macropore in mm2, p is the yarn spacing in mm, d is the yarn diameter in mm, and subscripts 1 and 2 indicate warp and weft yarns, respectively. Open porosity can be calculated also on the basis of cover factor

*p*

*p y*

11 22 1 2 ( )( )

4

*A pd pd*

(Figure 7), the open porosity is calculated as follows:

*open*

(Equation 16) or pore density (Equation 17) [26, 29]:

fabric relative density: 83 %)

weft density in threads/cm;

14:

$$
\rho\_{fab} = \frac{m}{D \cdot 1000} \tag{9}
$$

$$\varepsilon = 1 - \frac{\rho\_b}{\rho\_s} = 1 - \frac{\rho\_{\rm fab}}{\rho\_{\rm yarm}} = 1 - \frac{100 \cdot m \cdot d^2 \cdot \pi}{D \cdot 4 \cdot T} \tag{10}$$

where, d is the yarn diameter in cm, T is the yarn fineness in tex, ρyarn is the yarn bulk density in g/cm3, ρfib is the fibre density in g/cm3, βfib is the fibre volume fraction (or yarn packing factor), ρfab is the woven fabric bulk density in g/cm3, m is the woven fabric mass per unit area in g/m2, D is the woven fabric thickness in mm, ρb is the body bulk density in g/cm3, and ρs is the density of solid component in g/cm3. It is worth to mention that in this way defined porosity refers to all types of pores regarding their position in the woven fabric, e.g. inter- and intra-pores;

 area of pore cross-section which refers only on macropores in a woven fabric. The ideal model of woven fabric porous structure is based on the assumption that macropores have cylindrical shape with circular cross-section. In real woven fabrics, the macropore cross-section shape is more likely to be irregular rather regular (Figure 7) [26]. The shape of pore cross-section and consequently the area of pore cross-section depend on the type of yarns used. Woven fabrics made from filament yarns have pure macropores with rectangular cross-sections, whilst woven fabrics made from staple yarns have a small percentage of pure macropores, some of partly latticed macropores as well as fully latticed macropores (as the consequence of the phenomenon of latticed pores) with mostly irregular cross-sections. The area of pore cross-section also depends on the phenomenon of changing the position of warp threads according to the longitudinal fabric axis and the phenomenon of thread spacing irregularity [28]. For the theoretical calculations of the macropore cross-section area three types of regular pore crosssection shapes are taken into account, e.g. circular (Equation 11), rectangular (Equation 12) and elliptical (Equation 13) as follows:

$$A\_{p/circular} = \frac{\pi}{16}(p\_1 + p\_2 - d\_1 - d\_2)^2 = \frac{\pi}{16} \left(\frac{10}{g\_1} + \frac{10}{g\_2} - d\_1 - d\_2\right)^2 \tag{11}$$

$$A\_{p/rec\,\tan\,quark} = (p\_1 - d\_1) \cdot (p\_2 - d\_2) = \left(\frac{10}{g\_1} - d\_1\right) \cdot \left(\frac{10}{g\_2} - d\_2\right) \tag{12}$$

$$A\_{p/cell|initial} = \frac{\pi}{4} \cdot (p\_1 - d\_1) \cdot (p\_2 - d\_2) = \frac{\pi}{4} \left(\frac{10}{g\_1} - d\_1\right) \cdot \left(\frac{10}{g\_2} - d\_2\right) \tag{13}$$

where, Ap is the area of macropore cross-section in mm2, p is the yarn spacing in mm, d is the yarn diameter in mm, g is the number of yarns per unit length in threads/cm, and subscripts 1 and 2 indicate warp and weft yarns, respectively;

#### The Usage of Genetic Methods for Prediction of Fabric Porosity 181

180 Genetic Programming – New Approaches and Successful Applications

12) and elliptical (Equation 13) as follows:

subscripts 1 and 2 indicate warp and weft yarns, respectively;

fabric, e.g. inter- and intra-pores;

*yarn fib fib*

1000 *fab m D*

<sup>2</sup> <sup>100</sup> 11 1

where, d is the yarn diameter in cm, T is the yarn fineness in tex, ρyarn is the yarn bulk density in g/cm3, ρfib is the fibre density in g/cm3, βfib is the fibre volume fraction (or yarn packing factor), ρfab is the woven fabric bulk density in g/cm3, m is the woven fabric mass per unit area in g/m2, D is the woven fabric thickness in mm, ρb is the body bulk density in g/cm3, and ρs is the density of solid component in g/cm3. It is worth to mention that in this way defined porosity refers to all types of pores regarding their position in the woven

 area of pore cross-section which refers only on macropores in a woven fabric. The ideal model of woven fabric porous structure is based on the assumption that macropores have cylindrical shape with circular cross-section. In real woven fabrics, the macropore cross-section shape is more likely to be irregular rather regular (Figure 7) [26]. The shape of pore cross-section and consequently the area of pore cross-section depend on the type of yarns used. Woven fabrics made from filament yarns have pure macropores with rectangular cross-sections, whilst woven fabrics made from staple yarns have a small percentage of pure macropores, some of partly latticed macropores as well as fully latticed macropores (as the consequence of the phenomenon of latticed pores) with mostly irregular cross-sections. The area of pore cross-section also depends on the phenomenon of changing the position of warp threads according to the longitudinal fabric axis and the phenomenon of thread spacing irregularity [28]. For the theoretical calculations of the macropore cross-section area three types of regular pore crosssection shapes are taken into account, e.g. circular (Equation 11), rectangular (Equation

> 2 / 1212 1 2

 

 

10 10 ( ) <sup>16</sup> <sup>16</sup> *A ppdd p circular d d*

/ tan 11 22 1 2

/ 11 22 1 2

where, Ap is the area of macropore cross-section in mm2, p is the yarn spacing in mm, d is the yarn diameter in mm, g is the number of yarns per unit length in threads/cm, and

10 10 ( )( ) 4 4 *Ap elliptical pd pd d d*

10 10 ( )( ) *Ap rec gular pd pd d d*

1 2

1 2

1 2

*g g*

*g g*

*g g*

(8)

2

(11)

(12)

(13)

(9)

4

*m d D T*

(10)

 

*fab b s yarn*

 

**Figure 7.** Real and binary images of the pore cross-section shape and the number of pores in real woven fabrics (magnification of binary images: 20 x, magnification of real images: 80 x, yarn fineness: 36 tex, fabric relative density: 83 %)

 number of macropores in the area unit (pore density). It can be seen from Figure 7, that one macropore belongs to one warp yarn and one weft yarn, so the number of macropores can be calculated on the basis of warp and weft densities using Equation 14:

$$N\_p = \mathbb{g}\_1 \cdot \mathbb{g}\_2\tag{14}$$

where, Np is the pore density in pores/cm2, g1 is the warp density in threads/cm, and g2 is the weft density in threads/cm;

 open porosity (open area) which describes the fraction of macropore cross-section area in the area unit of woven fabric. If we assume elliptical macropore cross-section area (Figure 7), the open porosity is calculated as follows:

$$\varepsilon\_{open} = \frac{A\_p}{A\_p + A\_y} = \frac{\pi (p\_1 - d\_1) \cdot (p\_2 - d\_2)}{4 \cdot p\_1 \cdot p\_2} \tag{15}$$

where, εopen is the open porosity, Ap is the macropore cross-section area in mm2, Ay is the projection area of warp and weft yarns, which refers to one macropore in mm2, p is the yarn spacing in mm, d is the yarn diameter in mm, and subscripts 1 and 2 indicate warp and weft yarns, respectively. Open porosity can be calculated also on the basis of cover factor (Equation 16) or pore density (Equation 17) [26, 29]:

$$\mathcal{E}\_{open} = 1 - K = 1 - \left\lfloor \frac{\left(d\_1 g\_1 + d\_2 g\_2 - \left\{\frac{d\_1 d\_2 g\_1 g\_2}{10}\right\}\right)}{10} \right\rfloor \tag{16}$$

$$
\boldsymbol{\varepsilon}\_{\text{open}} = \mathbf{N}\_p \cdot \boldsymbol{A}\_p \tag{17}
$$

The Usage of Genetic Methods for Prediction of Fabric Porosity 183

the yarn diameter in mm, D is the woven fabric thickness in mm, Ap is the macropore area in mm2, εopen is the open porosity, and subscripts 1 and 2 indicate warp and weft yarns,

The porous structure of nonwoven fabric is a result of nonwoven construction (the type and properties of fibres or yarns as input materials, fabric mass, fabric thickness, etc.) as well as technological phases, e.g. the type of web production, bonding methods and finishing treatments. According to several different methods to produce non-woven fabrics having consequently very different porous structure, the ideal geometric model of porous structure in the form of tube-like system is partially acceptable only by those nonwovens which are thin and translucence, e.g. light polymer–laid nonwovens and some thin spun-laced or heatbonded nonwovens (Figure 8). Such model is based on the assumptions that fibres having the same diameter are distributed only in the direction of fabric plane and the distance between fibres and the length of individual fibres is much greater than the fibre diameter. Xu [21] found out that in most nonwoven fabrics, pore shape is approximately polygonal and that pores appear more circular when the fabric density increases. Pore orientation to some extent relates to fibre orientation. If pores are elongated and predominantly oriented in one direction, fibres are likely to be oriented in that direction. The variation in pore size is inherently high. Some regions may contain more pores than others or may have larger pores

**Figure 8.** 2D and 3D presentations of an ideal model of the porous structure of a nonwoven fabric (with

The primary constructional parameters of nonwoven fabrics which alter the porous

fibre fineness, e.g. the mass of 1000 meter of fibre, from which the fibre diameter can be

To compare nonwoven fabrics with porosity, the following porosity parameters can be calculated on the basis of the nonwoven fabric primary constructional parameters and the

detail to define opening diameter of pore by 2D presentation)

ideal model of porous structure in the form of a tube-like system:

**2.2. Nonwoven fabric's ideal geometric model of porous structure** 

respectively.

than those in other regions.

structure are:

calculated,

web thicknesses.

web mass per unit area and

where, εopen is the open porosity, K is the woven fabric cover factor, d is the yarn diameter in mm, g is the warp/weft density in threads/cm, Np is the pore density in pores/cm2, Ap is the area of macropore cross-section in cm2, and subscripts 1 and 2 indicate warp and weft yarns, respectively;

 equivalent macropore-diameter. If we assume that macropore has cylindrical shape, then the area of macropore cross-section is equal to the area of circle with radius r (Equation 18). Equivalent macropore diameter is the diameter of macropore with circular cross-section whose area is the same as the area of the macropore with irregular cross-section shape (Equation 19) [30].

$$A\_{circle} = \pi \cdot r^2 = \frac{\pi \cdot d^2}{4} \tag{18}$$

$$d\_e = \sqrt{\frac{4 \cdot A\_p}{\pi}}\tag{19}$$

where, Acircle is the circular cross-section macropore area in mm2, r is the macropore radius in mm, d is the macropore diameter in mm, de is the equivalent macropore diameter in mm, and Ap is the macropore cross-section area of macropore with irregular shape in mm2;


$$
\varepsilon\_{macro} = \frac{V\_p}{V\_p + V\_y} = \frac{A\_p \cdot D}{p\_1 \cdot p\_2 \cdot D} = \frac{A\_p}{p\_1 \cdot p\_2} \tag{20}
$$

$$\varepsilon\_{macro} = \frac{V\_p}{V\_p + V\_y} = \frac{\pi (p\_1 - d\_1) \cdot (p\_2 - d\_2) \cdot D}{4 \cdot p\_1 \cdot p\_2 \cdot D} = \frac{\pi (p\_1 - d\_1) \cdot (p\_2 - d\_2)}{4 \cdot p\_1 \cdot p\_2} = \varepsilon\_{open} \tag{21}$$

where, εmacro is the macroporosity, Vp is the macropore volume in cm3, Vy is the volume of warp and weft yarns which refers to one macropore in cm3, p is the yarn spacing in mm, d is the yarn diameter in mm, D is the woven fabric thickness in mm, Ap is the macropore area in mm2, εopen is the open porosity, and subscripts 1 and 2 indicate warp and weft yarns, respectively.
