Diagrams of distribution

## Abstract

This chapter reports on the statistical analysis of 2D fiber lap using variance analysis and autocorrelation function. It begins with a short overview of the nonwoven processes showing the importance of lap and web formation. It then proceeds to describe the theory of the ideal fiber web. The real defects are taken into account based on random irregularity, periodic irregularity, and compound irregularity. To conclude, the chapter highlights the efficiency of this theoretical approach and its application on 2D fibrous material.

### Keywords

- Fiber lap
- fiber web
- textile processes
- statistical analysis
- variance analysis
- autocorrelation

## 1. Introduction

Textile and nonwoven industries are fiber-based industries. They use both continuous fibers as filament yarns and/or short fibers as staple fibers. During the manufacturing processes, these fibers, very thin 1D elements, are transformed in oriented or random 2D laps or webs. Then, these products are converted in 1D end product as staple yarns or 2D end products as woven or knitted fabrics or nonwoven fabrics. The physical and mechanical properties of these end products strongly depend on the regularity of laps or webs. Therefore, an accurate knowledge of the web and lap formation as well as the associated theoretical approach are absolutely necessary.

## 2. Textile processes

The classical textile spinning process is based on Figure 1. First of all, staple fibers highly compacted in form of bales are opened with the help of a bale opening line. Moving through a coarse and a fine opener, the size of fiber tufts is reduced to be able to feed the carding machine chute. The chute delivers a lap of fibers to the carding machine feed rolls.

In a second time, the carding machine, as shown in Figure 2, individualizes the fibers tufts and delivers the fibers in the form of a web. The web is then condensed in a form of sliver.

The sliver goes through a drawing frame and a roving frame which produces slivers which are drawn and twisted on a spinning machine, that is, ring spinning frame. The quality of the final product strongly depends on the efficiency of the carding operation. It can be noticed that this efficiency is strongly correlated to the quality and the evenness of both the fed lap and delivered web. An exact knowledge of the quality of these 2D textiles is absolutely necessary [1,16,17].

## 3. Nonwoven processes

Nonwoven materials can be considered as “**sheet or web** structures bonded together by entangling fibers or filaments, by various mechanical, thermal and/or chemical processes. These are made directly from separate fibers or from molten plastic or plastic film [1]” or as “a manufactured sheet, **web or mat** of directionally or randomly orientated fibers, bonded by friction, and/or cohesion and/or adhesion, excluding paper and products which are woven, knitted, tufted, stitch-bonded incorporating binding yarns of filaments, or felted by wet-milling, whether or not additionally needled [2]”. As above defined, the web formation is a very important part of the nonwoven processes.

Nonwoven production systems are normally based on deposition or laying the fiber material or extruded thermoplastic polymers on a forming or conveying surface. The physical environment at this phase can be dry, quenched in air, wet, or molten – drylaid, wetlaid, or spun [3].

### 3.1. Nonwoven formation

In a general point of view, there are three main methods for nonwoven forming:

The drylaid system with carding or airlaying

The wetlaid system

The polymer-based system, which includes spunlaying (spunbonding) or specialized technologies like meltblown, or flashspun fabrics [4]

The following chapter presents a non-exhaustive overview of a nonwoven manufacturing process.

### 3.2. Drylaid nonwoven

The drylaid nonwoven process consists in fiber preparation, web formation, web bonding, and stabilization (Figure 4).

The fiber preparation is close to classical textile industry staple fiber preparation including bale opening, blending, coarse, and fine opening. Then the web forming machine is fed with the help of a chute in case of short staple fibers or with the help of a hopper in case of long staple fibers. The fibers fed by a chute or hopper condensed into the form of a lap are introduced into a carding machine. The carding machine separates fibers, starts the process of individualization, and delivers the fibers in the form of a web [24,25].

The web is then formed into the desired web structure by layering the webs extracted from the carding machine. Based on the final chosen weight and web structure, the layering can be completed by using longitudinal layering, cross layering, or perpendicular layering.

Web strengthening can be done mechanically with the help of needle punching which consists in bonding nonwoven web by mechanically interlocking the fibers through the web using barbed needles (Figure 4), with the help of stitch bonding which consists in consolidating fiber webs using knitting elements or with the help of hydro-entanglement which consists in locking the fibers together using fine high pressure water jets directed across the web.

Mechanical bonding can be replaced with web chemical bonding which consists in applying the chemical binder to the web and curing it.

In case of thermoplastic and thermostable fiber blend web, the strengthening can be achieved by thermal bonding.

### 3.3. Wetlaid nonwovens

This method relates to paper-based nonwoven fabrics which are manufactured by suspending short length fibers in water and pumping the suspension over a moving mesh in order to form a fibrous **web** [3].

Figure 5 is a schematic illustration of wetlaid technique. The blades of a very strong mixer transform fibers, wood pulp, and water to a perfect slurry suspension. Webformer deposes the suspension over the screen, and depending on the application of nonwoven, it will pass through a suitable bonding stage. Finally, nonwoven will be dried and wound up.

All fibers which are dispersible in fluids and do not dissolve can be transformed into a **web** form by using the wetlaid method. One of the main advantages of the wetlaid method is a very good product homogeneity due to the very good homogeneity of the **web**.

### 3.4. Meltblown nonwoven

As mentioned before, the meltblown process belongs to the general category of polymer-laid nonwoven material. It has been defined as below:

“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 self-bonded nonwoven meltblown **web** [1,4].” Figure 6 shows the schematic illustration of meltblown process.

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 meltblown nonwovens (Figure 7).

All the above-described processes are based on fibers and fibrous laps or webs. The characteristics 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 of bonding, crimp, mass per unit area, and weight evenness and distribution.

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

The following paragraphs will present a theoretical approach of an ideal fiber web. This theoretical approach will simulate the real faults of the fiber-web forming-step during the industrial process, random irregularity, periodic irregularity, and compound irregularity.

## 4. Theoretical approach

### 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

We assume that

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

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

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:

where

and

### 4.2. B ( A ) and C B ( A ) functions

Let ^{th} macro-fibrous-element. Also, we define the macro-fiber-area density

We designate

and based on the properties of the

In the framework of this theory, the

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

where

Then we set

If we consider the two random functions

Hence

This allows showing that the covariance of the random variable

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

So, the new

The covariance is only a function of

Let us now introduce the autocorrelation function [21]:

Hence, a new form of the variance between areas density

Then, let us use the initial function

In the interval

Finally,

We designate

and

## 5. Discussion

### 5.1. Random irregularity

The fiber flocks length distribution is always considered of a great importance for textile laps and web processing. Nowadays, it is still a source of statistical interpretations more or less empirical, such as cumulative frequency diagram.

In the following part, the cumulative frequency function

Based on textile sciences, these diagrams are usually represented by permuting the coordinated axes, that is, by plotting the value of the fiber length

The usual empirical criterion for the quality estimation of the fiber web is based on the shape of the diagram. If all fiber flocks have the same length

Based on the shape of the diagram of cumulative frequency, the function

where

It can be highlighted that the autocorrelation function is a double integration of the histogram (frequency distribution function) of a fiber web numerical sample.

The cumulative frequency function

While, if

Then

with

For a fiber web having the **isoprobable distribution** of the fiber flocks, the normalized between-area-density variance

The difference between a “short” and a “long” fiber web element is shown by the diagrams of

Otherwise, for the **equiprobable distribution**, the corresponding diagram has a triangular shape (see Table 1). By a similar calculation to the previous we obtain two expressions for

It can be noticed that the graph of

Finally, the **uniform distribution** that has a trapezoidal shape (Table 1) has the mean and the variation coefficient respectively expressed [22] as follows:

It can be noticed that if

In this case of uniform distribution, the cumulative frequency function and the autocorrelation function have, respectively, the following expressions:

After integration

According to the values of

As shown in Figure 12, if

Considering the three above-described distributions, the random unevenness varies with the variation of the fibrous web length and as shown is Figures 9 and 10, if

This can be explained by a greater possibility of compensating local irregularities in case of longer surfaces of fiber web. Normally, the

### 5.2. Periodic unevenness

We suppose that the area density of the fibrous web is defined as follows:

where

We have seen that

with

Hence by replacing

and finally after integration:

Hence we deduce the corresponding variation coefficient:

where

The

### 5.3. Compound unevenness

In the above-mentioned development, we have intentionally unheeded the random component and assumed the presence of only one type of periodic sinusoidal irregularity. In fact, the random component is always present in the 2D textile fibrous structures and is existing; the additional components are of several sinusoidal types generated by the manufacturing process. That way, the periodic irregularity has to be split according to Fourier series [17] and the composition of the

Firstly, only one type of sinusoidal periodic irregularity is taken in account as follows:

where

Based on the law of additive covariances composition [26], we deduce the expression of the autocorrelation function

Let

with

The figure [28] shows that the standardized form of the variance composition can be written as follows:

The extension of the above results to the more general case of the superposition of n sinusoidal type irregularities can be written as follows:

where

Figures 15 and 16 show some simulation examples of compositions of one or two sinusoidal irregularities with a random irregularity (isoprobable distribution). The interpretation of the curves

## 6. Conclusion

Nonwoven industries are fiber-based industries. Based on their physical and mechanical properties, the nonwoven applications are increasing inducing a strong growth of the nonwoven production. The physical and mechanical properties of these products strongly depend on the evenness of laps or webs. Thus, the nonwoven manufacturers absolutely need tools to evaluate these properties. Hence, an accurate knowledge of the web and lap formation as well as the associated theoretical approach is absolutely necessary. In this paper we propose a theoretical approach simulating the real faults of the fiber-web forming-step during the industrial process. Thanks to this theory, we were able to calculate from measurements both the random and periodic components of the real defects for all textile types as fibrous webs and nonwovens. The periodic component depends on the production machine and the random component depends mainly on the characteristics of the fibers. In this study, we are not interested in uniformity of the visual appearance (2-D uniformity) of the fiber web but we assess the 3-D uniformity from the area density. This uniformity is determined from small surface-elements and not from image analysis [16,24]. We did not analyze the correlation between the irregularity of the yarns and the woven fabric [20] but measured directly surface and mass irregularities thereof. This involves the use of specific sensors (such as capacitive, ultrasonic, and radioactive sensors). To highlight these irregularities (random or periodic), the fiber web is divided into several elements that have the same surface. Then the between-area-density variance function is defined with the help of the autocorrelation function. The common distribution functions of the fiber flocks are used to carry out the random component of the equation. Finally, according to the law of additive covariances composition, periodic defects have been added to the random component in order to determine the theoretical equation. This information is fundamental to the production manager in order to detect the earliest possible manufacturing breakdown and to optimize the machine settings and textile production quality.