**2. Porosity and porosity parameters**

Flat textile materials, e.g. fabrics, are porous materials which allow the transmission of energy and substances and are therefore interesting materials for different applications. In general, they are used for clothing, interior and wide range of technical applications. Fabric as porous barrier between the human body an environment should support heat and water vapour exchange between the body and environment in order to keep the body temperature within the homeostasis range. Besides thermo-physiological protection, fabrics also play an important role by heat protection due to the flames or convection heat, contact heat, radiant heat as well as due to the sparks and drops of molten metal, hot gases and vapours [11]. Fabrics protect users against micro-organisms, pesticides, chemicals, hazardous particles and radiations (radioactive particles, micro-meteorites, X-rays, micro-waves, UV radiation, etc.). They act very important role also by environmental protection as filters for air and water filtrations, sound absorption and isolation materials against noise pollution, adsorption materials for hazardous gas pollution, etc. [10, 12, 13]. By all mentioned applications dedicated to absorption, desorption, filtration, drainage, vapours transmission, etc., the essential constructional parameter that influences fabric efficiency to protect human or environment is porosity. The fabric in a dry state is a two-phase media which consists of the fibrous material – solid component and void spaces containing air – gas (void) component. The porosity of a material is one of the physical properties of the material and describes the fraction of void space in the material. The porosity (or void volume fraction) is expressed as coefficient ranging between 0 and 1 or as percentage ranging between 0% and 100% (by multiplying the coefficient by 100). Mathematically, the porosity is defined as the ratio of the total void space volume to the total (or bulk) body volume [14, 15]:

174 Genetic Programming – New Approaches and Successful Applications

centralized evolutionary processes [1, 8].

trials could be reduced.

algorithms (GA), respectively.

**2. Porosity and porosity parameters** 

real ones, but on the other hand they require numerous fabric samples data. The problem with extrapolation still remains. In general, when deterministic modelling is used, the obtained models are the results of strict mathematical rules and/or the models are set in advance. In this case the goal is to discover merely a set of numerical coefficients for a model whose form has been pre-specified. However, nowadays more and more processes and systems are modelled and optimized by the use of non-deterministic approaches. This is due to the high degree of complexity of the systems, and consequently, inability to study them successfully by the use of conventional methods only. In non-deterministic modelling of systems, no precise and strict mathematical rules are used [2, 3, 4, 5, 6, 7]. For example, in genetic programming, no assumptions about the form, size, and complexity of models are made in advance. They are left to the stochastic, self-organized, intelligent, and non-

Fabrics are porous materials having different porous structures as the consequence of different manufacturing techniques needed to interlace the fundamental structural elements, e.g. fibres, yarns or layers, into fibrous assembly. Fabric porosity strongly determines important physical, mechanical, sorptive, chemical, and thermal properties of the fabrics such as mechanical strength, thermal resistance, permeability (windproofness, breathability), absorption and adsorption properties (wicking, wetting), translucence, soiling propensity, UV light penetration, sound absorption ability, etc. [9, 10]. Knowledge about the fabric's porous structure is, therefore, an important step when characterising fabrics, in order to predict their behaviour under different end-usage conditions regarding a product. Hence, if porosity is estimated or predicted then when developing a new product the desired porosity parameters can be set in advance on the basis of selecting those fabric constructional factors that have an effect on porosity and, in this way sample production

This chapter gives some basic information about the porosity, porosity parameters of woven and nonwoven fabrics, and the results of the studies dealing with the prediction of porosity parameters of two types of fabrics, e.g. woven fabrics made from the 100% cotton staple yarns and needle-punched nonwovens made from the mixture of viscose/polyester fibres, using nondeterministic modelling tools, e.g. genetic programming (GP) and genetic

Flat textile materials, e.g. fabrics, are porous materials which allow the transmission of energy and substances and are therefore interesting materials for different applications. In general, they are used for clothing, interior and wide range of technical applications. Fabric as porous barrier between the human body an environment should support heat and water vapour exchange between the body and environment in order to keep the body temperature within the homeostasis range. Besides thermo-physiological protection, fabrics also play an important role by heat protection due to the flames or convection heat, contact heat, radiant heat as well as due to the sparks and drops of molten metal, hot gases and vapours [11].

$$
\varepsilon = \frac{V\_v}{V} \tag{1}
$$

where, ε is the porosity expressed as coefficient, Vv is the volume of the total void space in cm3, and V is the total or bulk body volume in cm3. The total volume of the body consists of the volumes of the solid and void components as follows:

$$V = V\_v + V\_s \tag{2}$$

where, V is the total volume of the body in cm3, Vv is the volume of void component in cm3, and VS is the volume of solid component in cm3. If the volume of void component is exposed from the Equation 2, the Equation 1 can be further written as follows:

$$\varepsilon = \frac{V\_v}{V} = \frac{V - V\_s}{V} = 1 - \frac{V\_s}{V} = 1 - \beta \tag{3}$$

$$
\mathcal{B} = \frac{V\_s}{V} \tag{4}
$$

where, β is the fulfilment (or solid volume fraction) which describes the fraction of solid component volume in the material expressed as coefficient ranging between 0 and 1 or as percentage. If we take into account the common equation for material density (Equation 5), and assume that the mass of the material is actually the mass of solid component (ms=mb), the Equation 3 could be further written in the form of Equation 6:

$$
\rho = \frac{m}{V} \tag{5}
$$

$$\varepsilon = 1 - \frac{V\_s}{V} = 1 - \frac{m\_s \rho\_b}{\rho\_s m\_b} = 1 - \frac{\rho\_b}{\rho\_s} \tag{6}$$

where, ε is the porosity expressed as coefficient, Vs is the volume of solid component in cm3, V is the volume of the body (or bulk volume) in cm3, ms is the mass of solid component in g, mb is the mass of the body (or bulk mass) in g, ρb is the bulk density in g/cm3, and ρs is the density of solid component in g/cm3.

The Usage of Genetic Methods for Prediction of Fabric Porosity 177

4. according to the pore shape into (Figure 2):

**Figure 2.** Types of pores according to the pore shape [19]

**Figure 3.** Different shapes of pore cross-sections [20]

a. pores with a permanent cross-section,

5. according to the geometry of pore-cross section into (Figure 3): a. pores with geometrically regular cross-sectional shape and b. pores with geometrically irregular cross-sectional shape,

6. according to the uniformity of pore cross-section over the pore length into (Figure 4):

a b c d

(the most constricted, the largest, the mean pore diameters).

**Figure 4.** Pores with permanent (a) and non-permanent (b) cross-sections over their length

Four groups of pore descriptors, e.g. size, shape, orientation, and placement, are defined as important parameters [21]. Pores can be mathematically assessed on the basis of known model of pores geometry and constructional parameters of the material with the following parameters: the number of pores, pore size, pore volume, pore surface area, pore length, etc.

a a b a

b. pores with a different cross-sections and for which different diameters are defined

a. cylindrical pores, b. slit-shape pores, c. cone-shape pores, and d. ink bottle pores;

In this way exactly defined porosity of the material is useful parameter, only, when materials with the same porous structure are compared, and gives an indication which material possesses more void space in the bulk volume. It does not give any information about the porous structure of the material, so it is an insufficient parameter for describing fibre assembly characteristics [16]. Namely, the materials with the same porosity could have very different porous structure and consequently, in the case of fabrics, different protection, filtration, sound absorption, etc., properties; so the need to define porous structure and some other porosity parameters is essential. From the theoretical point of view, the porosity parameters could be easily determined on the basis of an ideal geometrical model of the material porous structure. The simpler models consider that all pores, whatever their shape, are the same and regularly arranged in a fibre assembly [16, 17]. Ideal models are based also on some other simplifying assumptions depending on the fibre assembly type. Porosity parameters calculated on the basis of ideal models of porous structures are usually not in a good correlation with the real porosity parameters. Real porous media generally have rather complex structures that are relatively difficult to define. But the advantage of ideal geometric models of porous structures is the possibility to understand the influence of porous structure on some end-usage properties of the material, which is crucial by a new product planning.

The fundamental building elements of the material porous structure are pores (also capillaries, channels, holes, free volume) [15, 18]. Pores are void spaces within the material which are separated between each other, and are classified [19, 20]:

	- a. inter-pores, e.g. pores which lie between the structural elements of the material,
	- b. intra-pores, e.g. pores which lie within the structural element of the material;
	- a. macropores whose pore-width is greater than 50 nm,
	- b. mesopores whose pore-width lies in the range between 2 and 50 nm, and
	- c. micropores with the pore-width lower than 2 nm;
	- a. closed pores being inaccessible for fluid flow or surroundings,
	- b. blind pores which are accessible for fluid but terminate inside the material and prevent fluid flow, and
	- c. open (or through) pores which are open to external surface and permit fluid flow;

**Figure 1.** Types of pores according to the fluid accessibility

	- a. cylindrical pores,

density of solid component in g/cm3.

where, ε is the porosity expressed as coefficient, Vs is the volume of solid component in cm3, V is the volume of the body (or bulk volume) in cm3, ms is the mass of solid component in g, mb is the mass of the body (or bulk mass) in g, ρb is the bulk density in g/cm3, and ρs is the

In this way exactly defined porosity of the material is useful parameter, only, when materials with the same porous structure are compared, and gives an indication which material possesses more void space in the bulk volume. It does not give any information about the porous structure of the material, so it is an insufficient parameter for describing fibre assembly characteristics [16]. Namely, the materials with the same porosity could have very different porous structure and consequently, in the case of fabrics, different protection, filtration, sound absorption, etc., properties; so the need to define porous structure and some other porosity parameters is essential. From the theoretical point of view, the porosity parameters could be easily determined on the basis of an ideal geometrical model of the material porous structure. The simpler models consider that all pores, whatever their shape, are the same and regularly arranged in a fibre assembly [16, 17]. Ideal models are based also on some other simplifying assumptions depending on the fibre assembly type. Porosity parameters calculated on the basis of ideal models of porous structures are usually not in a good correlation with the real porosity parameters. Real porous media generally have rather complex structures that are relatively difficult to define. But the advantage of ideal geometric models of porous structures is the possibility to understand the influence of porous structure on some end-usage properties of the material, which is crucial by a new product planning.

The fundamental building elements of the material porous structure are pores (also capillaries, channels, holes, free volume) [15, 18]. Pores are void spaces within the material

a. inter-pores, e.g. pores which lie between the structural elements of the material, b. intra-pores, e.g. pores which lie within the structural element of the material;

b. blind pores which are accessible for fluid but terminate inside the material and

c. open (or through) pores which are open to external surface and permit fluid flow;

a b c

b. mesopores whose pore-width lies in the range between 2 and 50 nm, and

a. closed pores being inaccessible for fluid flow or surroundings,

which are separated between each other, and are classified [19, 20]:

2. according to the pore width (the shortest pore diameter) into: a. macropores whose pore-width is greater than 50 nm,

c. micropores with the pore-width lower than 2 nm;

3. according to the fluid accessibility into (Figure 1):

**Figure 1.** Types of pores according to the fluid accessibility

prevent fluid flow, and

1. according to the position in the material into:


**Figure 2.** Types of pores according to the pore shape [19]

	- a. pores with geometrically regular cross-sectional shape and
	- b. pores with geometrically irregular cross-sectional shape,

**Figure 3.** Different shapes of pore cross-sections [20]

	- a. pores with a permanent cross-section,
	- b. pores with a different cross-sections and for which different diameters are defined (the most constricted, the largest, the mean pore diameters).

**Figure 4.** Pores with permanent (a) and non-permanent (b) cross-sections over their length

Four groups of pore descriptors, e.g. size, shape, orientation, and placement, are defined as important parameters [21]. Pores can be mathematically assessed on the basis of known model of pores geometry and constructional parameters of the material with the following parameters: the number of pores, pore size, pore volume, pore surface area, pore length, etc.

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.

The Usage of Genetic Methods for Prediction of Fabric Porosity 179

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

The primary constructional parameters of woven fabrics which alter the porous structure are: yarn fineness, e.g. the mass of 1000 meter of yarn from which the yarn diameter can be

type of weave, e.g. the manner how the yarns are interlaced. It has an effect on the pore

the number of yarns in length unit (warp and weft densities), which directly alters the

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

**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

 (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

4 4

*yarn fib fib*

 (7)

10 1000

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

uniformly distributed over the woven fabric area.

size as well as on the shape of pore cross-section [26],

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

be then written in the form of Equation 10:

5

 

calculated,

pore size.

porosity [22].

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):


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

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 porous structure will be presented.
