**4. Theoretical approach**

"Meltblowing is a process in which, usually, a thermoplastic fiber forming polymer is extruded through a linear die containing several hundred small orifices. Convergent streams of hot air (exiting from the top and bottom sides of the die nosepiece) rapidly attenuate the extruded polymer streams to form extremely fine diameter fibers (1–5 mm). The attenuated fibers are subsequently blown by high-velocity air onto a collector conveyor, forming a fine fibered selfbonded nonwoven meltblown **web** [1,4]." Figure 6 shows the schematic illustration of

**Figure 6.** Meltblowing process; primary and secondary air flow and web formation

meltblown nonwovens (Figure 7).

The main force that holds meltblown fibers together in a nonwoven structure is a combination of entanglement and cohesive sticking. Nonwoven produced by meltblown method have low to moderate strength. During the process, the fibers are drawn to their final diameters while still in the semi-molten state; there is no downstream method of drawing the fibers before they are deposited onto the collector, and this is the reason of moderate mechanical properties of

All the above-described processes are based on fibers and fibrous laps or webs. The charac‐ teristics of the web are determined by the mode of web formation which is related to web geometry. This web geometry takes into account fiber orientation (oriented or random), type

The knowledge of the web geometry is very important because physical and mechanical properties are directly related to it. For instance, as far as geotextile properties are concerned, separation, reinforcement, stabilization, filtration, and drainage are related to the mass per

of bonding, crimp, mass per unit area, and weight evenness and distribution.

unit area and the distribution of mass per unit area [5,23].

meltblown process.

60 Non-woven Fabrics

## **4.1. The ideal fibrous web**

In this chapter, the fiber web is split into several macro sample-elements; these elements have the same area *A* (Figure 8). The fiber web element area *A* is defined as the sum of fiber microstrand *da* with *A*=*n* ⋅*da*, where *n* is the number of micro-elements in the macro-fiber-sample. The width *L* of the macro-web-element is considered constant. Its length is *<sup>l</sup>* <sup>=</sup> *<sup>A</sup> <sup>L</sup>* called cut length. Then, we suppose that *da* = *L* ⋅*dt*, where *dt* is the length of the fibrous micro-element located at the distance *t* from the strand origin. The abscissa origin 0 is located in the middle of the fiber web [6,7,8,9,26].

We assume that *<sup>μ</sup>* =( *<sup>m</sup> da* ) as the fiber strand area density of the fiber web (web manufactured by the textile machines); *m* is defined as the micro-elementary fiber-strand-mass. In this theory, the two random variables related to the cutting length *l* are *μ* and *a*, where *μ* can be described with the help of the usual statistical parameters as the mean and the variance [10,11].

**Figure 8.** Scheme of an ideal fiber web

**Figure 8.** Scheme of an ideal fiber web

The unevenness of the area density of the fiber web can be characterized by the abovementioned dispersion parameters which give an approach of the overall irregularity. The overall variance is denoted by the following limit conditions:

$$\begin{aligned} A &\to \infty \qquad ∧ \qquad da \to 0, \\ \mathfrak{n} &\to \infty \end{aligned}$$

When the limits are taken, the variation coefficient of the fiber web and the overall variance can be written, respectively, as follows:

$$\begin{aligned} CV\left(0, \infty\right) &= CV(\infty) \\ V\left(0, \infty\right) &= V(\infty) \end{aligned}$$

Taking into account that these values are not directly measurable, an estimation of the values can be calculated by extrapolation. Based on previous studies, [12,13,14,15,27], the mean and overall variance can be estimated as follows:

$$\overline{\mu}\_{i} = \frac{\sum \mu\_{i}}{n} = \frac{M}{A} \tag{1}$$

where *M* is the web-element-mass

and

$$
\sigma\_{\mu\_i}^2 = \frac{\sum \left(\mu\_i - \overline{\mu}\_i\right)^2}{n} \tag{2}
$$

#### **4.2.** *B***(***A***) and** *CB***(***A***) functions**

The unevenness of the area density of the fiber web can be characterized by the abovementioned dispersion parameters which give an approach of the overall irregularity. The

**A**

**xi xi+1** 

**Process** 

**Figure 8.** Scheme of an ideal fiber web

**da x l**

*A and da* 0

When the limits are taken, the variation coefficient of the fiber web and the overall variance

0, ( ) 0, ( ) *CV CV V V*

Taking into account that these values are not directly measurable, an estimation of the values can be calculated by extrapolation. Based on previous studies, [12,13,14,15,27], the mean and

*i*

*M n A* m

( )<sup>2</sup>

m m

*i i n*

= = å (1)

**L**

**dt**


*i*

m

2 *i*

s

m

¥= ¥ ¥= ¥

( ) ( )

® ¥ ®

overall variance is denoted by the following limit conditions:

can be written, respectively, as follows:

**0**

**Figure 8.** Scheme of an ideal fiber web

**t**

overall variance can be estimated as follows:

where *M* is the web-element-mass

and

62 Non-woven Fabrics

*n*

® ¥

Let *B*(*A*) be the variance between areas density of the ith macro-fibrous-element. Also, we define the macro-fiber-area density *μ*(*x*, *A*), where *A* is the mean value of the surface element and *x* is the abscissa (location of the element) [14,15,28,29].

$$\int\_{y\_0}^{y\_0 \ast L} \int\_{x\_0}^{x\_0 \ast \xi} \mathcal{N}\_{\perp}(x, y) dx dy = m \{l, L\} \tag{3}$$

We designate *E*(*μ*) as the average of the random variable *μ*(*x*, *y*) ; where *E* is the symbol of the mathematical expectation. *E*(*μ*)= *E*(*μ*(*x*, *y*)) can also be written as follows:

$$E\left(\mu\left(\mathbf{x},\boldsymbol{y}\right)\right) = E\left[\int\_{\mathbf{x}\_{0}}^{\mathbf{x}\_{0}+\mathbf{A}} \Big| \int\_{\mathbf{y}\_{0}}^{\mathbf{y}\_{0}+\mathbf{L}} \mu\left(\mathbf{x},\boldsymbol{y}\right) d\mathbf{x} d\boldsymbol{y}\right] \tag{4}$$

and based on the properties of the *E* operator:

$$E\left[m\left(l,L\right)\right] = \int\_{x\_0}^{x\_0+l} \int\_{y\_0}^{y\_0+l} E\left[\mu\left(x,y\right)\right] dx dy = l \cdot L \cdot E\left[\mu\left(x,y\right)\right] = A \cdot E\left[\mu\left(x,y\right)\right] \tag{5}$$

In the framework of this theory, the *B*(*A*) variance between areas density is described by the second order moment as follows:

$$E\left[m\left(l,L\right)\right] = E\left[m\left(l,L\right) - E\left(\mu\left(l,L\right)\right)^2\right] \tag{6}$$

When Equation (5) is substituted into Equation (6), we obtain the following relationship:

$$E\left[m\left(l,L\right)\right] = E\left[\int\_{x\_0}^{x\_0+\frac{\omega}{\sqrt{L}}} \int\_{y\_0}^{y\_0+L} \left(\mu\left(\mathbf{x},y\right) - E\left(\mu\left(\mathbf{x},y\right)\right)\right) d\mathbf{x} dy\right]^2 = E\left[\int\_{x\_0}^{x\_0+\frac{\omega}{\sqrt{L}}} \int\_{y\_0}^{y\_0+L} \mu\_c\left(\mathbf{x},y\right) d\mathbf{x} dy\right]^2\tag{7}$$

where *μc* (*x*, *y*)=*μ*(*x*, *y*)−*E*(*μ*(*x*, *y*))

Then we set *u* =*ξ* − *x* and *v* =*η* − *y*. The *B*(*A*) function can be converted into a double integral assuming independence of position variables *u* and *v*. So, the *B*(*A*) function can be written as follows:

$$B(A) = E\left[\int\_{x\_0}^{x\_0 \circ \mathcal{H}} \int\_{y\_0}^{y\_0 \circ L} \int\_{x\_0}^{x\_0 \circ \mathcal{H}} \mathcal{N}\int\_{y\_0}^{y\_0 \circ L} \mu\_c(x, y)\,\mu\_c(\xi, \eta)\,dxdydd\xi d\eta\right] \tag{8}$$

If we consider the two random functions *μ*(*x*, *y*)−*E*(*μ*(*x*, *y*)) and *μ*(*ξ*, *η*)−*E*(*μ*(*ξ*, *η*)) , the Γ covariance function can be defined as follows:

$$\Gamma\left(\mu,\upsilon\right) = \mathbb{E}\left[\left(\mu\left(\mathbf{x},\boldsymbol{y}\right) - \mathbb{E}\left(\mu\left(\mathbf{x},\boldsymbol{y}\right)\right)\right) \cdot \left(\mu\_c\left(\mathbf{x}+\boldsymbol{u},\boldsymbol{y}+\upsilon\right) - \mathbb{E}\left(\mu\_c\left(\mathbf{x}+\boldsymbol{u},\boldsymbol{y}+\upsilon\right)\right)\right)\right] \tag{9}$$

Hence Γ(0, 0)= *E μ*(*x*, *y*)− *E*(*μ*(*x*, *y*)) <sup>2</sup> =*V μ*(*x*, *y*) .

This allows showing that the covariance of the random variable *μ*(*x*, *y*) is a stationary random variable of second order [18,26]. In this case, the covariance is only depending on the difference of positions (*v* + *x*) and on (*w* + *x*):

$$(\upsilon + \mathbf{x}) - (\upsilon + \mathbf{x}) = \upsilon - \mathbf{z}\upsilon \tag{10}$$

Moreover, this covariance is an even function having the following property:

$$
\Gamma\left(\upsilon - w\right) = \Gamma\left(w - \upsilon\right) \tag{11}
$$

So, the new *B*(*A*) equation can be defined as follows:

$$B\left[m\left(l,L\right)\right] = \int\_{x=0}^{\xi\_L^L} \int\_{y=0}^{L} \int\_0^{\xi\_L^L} \int\_0^L \Gamma\left(u,v\right)dvdv\tag{12}$$

The covariance is only a function of *u* where *u* =*v* −*w*; thus *B*(*A*) can be reduced to a simple integral [27]. Thereby, the covariance function *B*(*A*) can be rewritten in the following form:

$$B\left(A\right) = 2 \cdot \frac{L^2}{A^2} \int\_0^{\bigvee} \int\_0^L \Gamma\left(u, v\right) \cdot \left(\bigvee\_L - u\right) du dv \tag{13}$$

Let us now introduce the autocorrelation function [21]:

$$\rho\left(\mu,\upsilon\right) = \frac{\Gamma\left(\mu,\upsilon\right)}{\Gamma\left(0,0\right)}\tag{14}$$

Hence, a new form of the variance between areas density *B*(*A*) can be expressed as follows:

$$B\left(A\right) = 2\,\Gamma\left(0\right) \cdot \frac{L^2}{A^2} \int\_0^{\lambda\_L'} \rho\left(u\right) \cdot \left(\bigwedge\_L'-u\right) du\tag{15}$$

Then, let us use the initial function *B*(*A*)=*E μ*(*x*, *A*)− *E*(*μ*(*x*, *A*)) <sup>2</sup> to estimate *B*(0).

In the interval *xi* , *xi*+1 shown in Figure 1, if *<sup>A</sup> <sup>L</sup>* <sup>→</sup>0, then *μ*(*x*, *<sup>A</sup>*)→*μ*(*t*). This means that the punctual area density is given by the following equation:

$$B\left(0\right) = E\left[\mu\left(t\right) - E\left(\mu\left(t\right)\right)\right]^2 = V\left[\mu\left(t\right)\right] = \sigma\_{\mu\_i}^2\tag{16}$$

*σμi* 2 representing the overall variance of the fiber-web-element.

Finally, *B*(*A*) can be written as follows:

If we consider the two random functions *μ*(*x*, *y*)−*E*(*μ*(*x*, *y*)) and *μ*(*ξ*, *η*)−*E*(*μ*(*ξ*, *η*)) , the Γ

This allows showing that the covariance of the random variable *μ*(*x*, *y*) is a stationary random variable of second order [18,26]. In this case, the covariance is only depending on the difference

 m

( )( ) *vx wx vw* + - + =- (10)

G - =G - ( )( ) *vw wv* (11)

é ù <sup>=</sup> <sup>G</sup> ë û òòòò (12)

*<sup>A</sup> <sup>L</sup>* =× G × - ò ò (13)

<sup>G</sup> <sup>=</sup> <sup>G</sup> (14)

ò (15)

ë û (9)

() () () *uv E xy E xy x uy v E x uy v* , ,, , , (

( )) ( *c c* ( )( ) ( )) G = - × + +- + + é ù

mm

Moreover, this covariance is an even function having the following property:

( ) ( ) 0 00 0 , , *S S L L L L x y B mlL u v dvdw* = =

( ) ( ) ( ) <sup>2</sup> <sup>2</sup> 0 0 2 , *A L B A <sup>L</sup> <sup>L</sup> u v u dudv <sup>A</sup>*

( ) ( )

Hence, a new form of the variance between areas density *B*(*A*) can be expressed as follows:

( ) () ( ) ( ) <sup>2</sup>

*A B A <sup>L</sup> <sup>L</sup> u u du <sup>A</sup> <sup>A</sup> <sup>L</sup>* =G × × r

r*u v*

<sup>2</sup> <sup>0</sup> 2 0

( ) , , 0,0 *u v*

The covariance is only a function of *u* where *u* =*v* −*w*; thus *B*(*A*) can be reduced to a simple integral [27]. Thereby, the covariance function *B*(*A*) can be rewritten in the following form:

covariance function can be defined as follows:

Hence Γ(0, 0)= *E μ*(*x*, *y*)− *E*(*μ*(*x*, *y*)) <sup>2</sup> =*V μ*(*x*, *y*) .

So, the new *B*(*A*) equation can be defined as follows:

Let us now introduce the autocorrelation function [21]:

m

of positions (*v* + *x*) and on (*w* + *x*):

64 Non-woven Fabrics

$$B\left(A\right) = 2\left.B\left(0\right) \cdot \frac{L^2}{S^2} \int\_0^{\xi\_L^{\prime}} \left(\bigvee\_L -u\right) \rho\left(u\right) du\right.\tag{17}$$

We designate *CB*(*A*) as the variation coefficient associated to the between-area-density variance *B*(*A*). Therefore:

$$\text{LCB}^2\left(A\right) = \frac{\text{B}\left(A\right)}{\text{E}\left(\mu\right)^2} = \frac{2 \cdot \text{CV}\_{\mu}^2}{\text{A}\int\_{\text{L}}^{\text{M}} \int\_{0}^{\text{M}} \left(\text{A}\,\left\langle\frac{A}{L}\right\rangle - u\right) \rho\left(u\right) du} \tag{18}$$

and *CVμ* 2 = *σμi* 2 *<sup>E</sup>*(*μ*)<sup>2</sup> called the overall variation coefficient of the fiber web.
