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

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 than those in other regions.

**Figure 8.** 2D and 3D presentations of an ideal model of the porous structure of a nonwoven fabric (with detail to define opening diameter of pore by 2D presentation)

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


182 Genetic Programming – New Approaches and Successful Applications

cross-section shape (Equation 19) [30].

same as open porosity:

*macro*

*p*

*p y*

*p y*

respectively;

1 1

*K*

10 *open*

1212

*open p p N A* (17)

(18)

(19)

(16)

11 22 10

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,

 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

> 2 4 *circle <sup>d</sup> A r*

> > 4 *<sup>p</sup>*

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;

 maximal an minimal macropore diameters which refer to the elliptical shape of macropore cross-section. In the case where warp density is greater than weft density the maximal diameter is equal to p2-d2, while minimal diameter is equal to p1-d1 (Figure 7); macroporosity which describes the portion of macropore volume in volume unit of woven fabric. In general, it is defined using Equation 20. In the case of the elliptical macropore cross-section shape, the macroporosity, defined with Equation 21, is the

*A*

*e*

*d*

2

12 12

(21)

(20)

 

*pp p*

11 22 11 22 1 2 1 2 ( )( ) ( )( )

 

4 4

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

*macro open*

*V V ppD p p*

*V pd pdD pd pd*

*V AD A V V ppD pp*

*ddgg dg dg*

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 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 nonwoven fabric density and the density of solid component is the fibre density. The nonwoven fabric density is calculated on the basis of primary nonwoven constructional parameters, e.g. fabric mass and thickness using Equation 9 where index fab in this case refers to the nonwoven fabric. Substituting Equation 9 into Equation 6, final Equation 22 of nonwoven porosity which refers to inter- (pores between fibres in nonwovens) and intra-pores (pores inside the fibres) is obtained:

$$\varepsilon = 1 - \frac{\rho\_b}{\rho\_s} = 1 - \frac{\rho\_{fib}}{\rho\_{fib}} = 1 - \frac{m\_{fib}}{D\_{fib} \cdot \rho\_{fib} \cdot 1000} \tag{22}$$

The Usage of Genetic Methods for Prediction of Fabric Porosity 185

Needle-punched nonwoven fabric is a sheet of fibres made by mechanical entanglement, penetrating barbed needles into a fibrous mat [31]. Needle-punched nonwovens represent the largest segment of filtration materials used as dust filters [32]. The geometrical model of three-dimensional needle-punched nonwoven fabric proposed by Mao & Rusell [33], is also known from the literature, and it is constructed on a two-dimensional fibre orientation within the fabric plane, with interconnecting fibres oriented in the z-direction (Figure 10). Such model relies on the following basic assumptions: 1. the fibres in the fabric have the same diameter, and a fraction of the fibres is distributed horizontally in the two-dimensional plane, the rest are aligned in the direction of the fabric thickness, 2. fibre distribution in both the fabric plane and the z-direction is homogeneous and uniform, 3. in each twodimensional plane, the number of fibres oriented in each direction is not the same, but obeys the function of the fibre orientation distribution Ω(α), where α is the fibre orientation angle, 4. the distance between fibres and the length of individual fibres is much greater than the fibre diameter. The basic porosity parameters which are based on the mentioned geometrical model of needle-punched nonwoven fabric are still difficult to define due to the fact that in each fabric planes fibres lie in different direction and in this way produce pores with different orientations, diameters, connectivity and accessibility to fluid flow (Figure 9).

The only porosity parameters that are calculated from such model are:

total porosity (Equation 22) and

g/cm3.

the following relation proposed by White [34]:

**Figure 9.** Geometrical models of needle-punched nonwoven fabric and porous structure [14, 34]

*p*

*d*

mean pore diameter which is deduced from the fibre radius and porosity according to

35.68 *fib*

*<sup>T</sup> <sup>d</sup>*

where, dp is the mean pore diameter in µm, ε is the nonwoven fabric porosity, dfib is the fibre diameter in µm, T is the fibre linear density in tex, and ρfib is the fibre density in

1 2

*fib*

*fib*

(27)

(28)

*d*

where, ε is the nonwoven fabric porosity, ρb is the body bulk density in g/cm3, ρs is the density of solid component in g/cm3, ρfab is the nonwoven fabric density in g/cm3, ρfib is the fibre density in g/cm3, mfab is the nonwoven fabric mass per unit area in g/m2, and Dfab is the nonwoven fabric thickness in mm;

 opening diameter which is the diameter of the maximum circle that can fit in a pore (Figure 8). It is predicted on the basis of nonwoven fabric constructional parameters and refers to the heat-bonded nonwoven fabrics, as follows [17, 21]:

$$d\_o = \frac{1}{\sqrt{C} \cdot L} - d\_{fb} \tag{23}$$

$$C = \frac{D\_{fab}}{d\_{fib}}\tag{24}$$

$$L = \frac{8 \cdot m\_{\text{fib}}}{\pi \cdot D\_{\text{fib}} \cdot d\_{\text{fib}} \cdot \rho\_{\text{fib}}} \tag{25}$$

where, d0 is the opening diameter in µm, C is the thickness factor, L is the specific total length of fibres per nonwoven unit area in mm-1, dfib is the fibre diameter in µm, Dfab is the nonwoven thickness in mm, mfab is the nonwoven fabric mass per unit area in g/m2, and ρfib is the fibre density in g/cm3;

 average area of pore cross-section which is for un-needled fabrics (e.g. fabrics made of layers of randomly distributed fibres) predicted using Equation 26 [17]:

$$A\_p = \frac{\pi \cdot \varepsilon \cdot d\_{fb}^2}{(1 - \varepsilon)^2} \tag{26}$$

where, Ap is the average area of pore cross-section in mm2, ε is the porosity, and dfib is the fibre diameter in µm. On the basis of calculated average area of pore-cross-section, the equivalent pore diameter is then calculated using Equation 19.

Needle-punched nonwoven fabric is a sheet of fibres made by mechanical entanglement, penetrating barbed needles into a fibrous mat [31]. Needle-punched nonwovens represent the largest segment of filtration materials used as dust filters [32]. The geometrical model of three-dimensional needle-punched nonwoven fabric proposed by Mao & Rusell [33], is also known from the literature, and it is constructed on a two-dimensional fibre orientation within the fabric plane, with interconnecting fibres oriented in the z-direction (Figure 10). Such model relies on the following basic assumptions: 1. the fibres in the fabric have the same diameter, and a fraction of the fibres is distributed horizontally in the two-dimensional plane, the rest are aligned in the direction of the fabric thickness, 2. fibre distribution in both the fabric plane and the z-direction is homogeneous and uniform, 3. in each twodimensional plane, the number of fibres oriented in each direction is not the same, but obeys the function of the fibre orientation distribution Ω(α), where α is the fibre orientation angle, 4. the distance between fibres and the length of individual fibres is much greater than the fibre diameter. The basic porosity parameters which are based on the mentioned geometrical model of needle-punched nonwoven fabric are still difficult to define due to the fact that in each fabric planes fibres lie in different direction and in this way produce pores with different orientations, diameters, connectivity and accessibility to fluid flow (Figure 9). The only porosity parameters that are calculated from such model are:

**Figure 9.** Geometrical models of needle-punched nonwoven fabric and porous structure [14, 34]

total porosity (Equation 22) and

184 Genetic Programming – New Approaches and Successful Applications

nonwoven fabric thickness in mm;

is the fibre density in g/cm3;

 (total) porosity by using Equation 6 where the bulk density of the material is actually the nonwoven fabric density and the density of solid component is the fibre density. The nonwoven fabric density is calculated on the basis of primary nonwoven constructional parameters, e.g. fabric mass and thickness using Equation 9 where index fab in this case refers to the nonwoven fabric. Substituting Equation 9 into Equation 6, final Equation 22 of nonwoven porosity which refers to inter- (pores between fibres in

> *fab fab b s fib fab fib*

where, ε is the nonwoven fabric porosity, ρb is the body bulk density in g/cm3, ρs is the density of solid component in g/cm3, ρfab is the nonwoven fabric density in g/cm3, ρfib is the fibre density in g/cm3, mfab is the nonwoven fabric mass per unit area in g/m2, and Dfab is the

 opening diameter which is the diameter of the maximum circle that can fit in a pore (Figure 8). It is predicted on the basis of nonwoven fabric constructional parameters and

> 1 *<sup>o</sup> fib d d*

> > *fab fib*

8 *fab*

where, d0 is the opening diameter in µm, C is the thickness factor, L is the specific total length of fibres per nonwoven unit area in mm-1, dfib is the fibre diameter in µm, Dfab is the nonwoven thickness in mm, mfab is the nonwoven fabric mass per unit area in g/m2, and ρfib

average area of pore cross-section which is for un-needled fabrics (e.g. fabrics made of

  2

*d*

<sup>2</sup> (1 ) *fib*

where, Ap is the average area of pore cross-section in mm2, ε is the porosity, and dfib is the fibre diameter in µm. On the basis of calculated average area of pore-cross-section, the

layers of randomly distributed fibres) predicted using Equation 26 [17]:

*p*

*A*

equivalent pore diameter is then calculated using Equation 19.

*m*

*fab fib fib*

*D C*

*D*

1000

*C L* (23)

*<sup>d</sup>* (24)

(25)

(26)

*m*

 (22)

nonwovens) and intra-pores (pores inside the fibres) is obtained:

11 1

refers to the heat-bonded nonwoven fabrics, as follows [17, 21]:

*L D d*

 mean pore diameter which is deduced from the fibre radius and porosity according to the following relation proposed by White [34]:

$$d\_p = \frac{\varepsilon}{1-\varepsilon} \cdot \frac{d\_{fb}}{2} \tag{27}$$

$$d\_{fib} = 35.68 \sqrt{\frac{T}{\rho\_{fib}}} \tag{28}$$

where, dp is the mean pore diameter in µm, ε is the nonwoven fabric porosity, dfib is the fibre diameter in µm, T is the fibre linear density in tex, and ρfib is the fibre density in g/cm3.

Three kinds of pores may be present in needle-punched nonwoven fabrics, namely, closed pores, open pores, and blind pores. The important pore structure characteristics of needlepunched nonwoven fabrics as filter media are the most constricted open pore diameter (smallest detected pore diameter), the largest pore diameter (bubble point pore diameter), and mean pore diameter (mean flow pore diameter) [35].

The Usage of Genetic Methods for Prediction of Fabric Porosity 187

generative method for the initial random population was *ramped half-and-half*. The method of selection was tournament selection with a group size of 7. For the purpose of this research 100 independent genetic programming runs were executed. Only the results of the best runs (i.e., the models with the smallest error between the measurements and predictions) are

Our experiments involved woven fabrics made from staple yarns with two restrictions: first, only fabrics made from 100% cotton yarns (made by a combing and carding procedure on a ring spinning machine) were used in this research; second, fabrics were measured in the grey state to eliminate the influence of finishing processes. We believe that it is very hard, perhaps even impossible, to include all woven fabrics types to predict individual macroporosity parameters precisely enough, and so we focused our research on unfinished staple yarn cotton fabrics. We would like to show that genetic programming can be used to establish the many relations between woven fabric constructional parameters and particular fabric properties, and that the results are more useful for fabric engineering than ideal theoretical models. The cotton fabrics varied according to yarn fineness (14 tex, 25 tex, and 36 tex), weave type (weave value), fabric tightness (55% - 65%, 65% - 75%, 75% - 85%), and denting. The constructional parameters of woven fabric samples are collected in Table 1. They were woven on a Picanol weaving machine under the same technological conditions. The weave values of plain (0.904), twill (1.188), and satin (1.379) fabrics, as well as fabric

We used an optical method to measure porosity parameters of woven fabrics, since it is the most accurate technique for macro-pores with diameters of more than 10 m. For each fabric specimen, we observed between 50 and 100 macro-pores using a Nikon SMZ-2T computeraided stereomicroscope with special software. We measured the following macro-porosity parameters: area of macro-pore cross-section, pore density, and equivalent macro-pore

Equations 29 and 30 present predictive models of the area of macro-pore cross-section Ap and macro-pore density Np, respectively [37]. Here *V* is the weave factor, *T* is the yarn linear density in tex, *t* is the fabric tightness in %, and *D* is the denting in ends per reed dent. The open porosity and equivalent diameter are calculated using Equations 17 and 19, respectively, where for Ap and Np the predicted values are taken into account. Because the model of the area of macro-pore cross-section is more complex, the functions f1, f2,…f10 are not presented here but are written in the appendix. When calculating the values of models, the following rules have to be taken into account: the protected division function returns to 1 if denominator is 0; otherwise, it returns to the normal quotient. The protected power function raises the absolute value of the first argument to the power specified by its second argument.

tightness, were determined according to Kienbaum's setting theory [36].

**3.2. Predictive models of woven fabric porosity parameters** 

presented in the paper.

diameters.

**3.1. Materials and porosity measurements** 
