**1. Introduction**

204 Woven Fabrics

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The motivation for this work comes from studying the composition of a medical face mask made of non woven fabrics and the mechanisms of filtrations inside the mask. Present medical mask filters passing air with the maximum efficiency of 99.97%, which might be seen as excellent. Main question is: "What presents this 0.03%?" If an aggressive virus is found in this tiny share, the effectiveness of the mask would be questionable.

The filtration mechanism in a non woven fabric is a complex process. It is obvious that air passing through a non woven fabric has to go through voids in the fabric. In order to understand the filtration mechanism, it is essential to be able to define fabrics porosity parameters, the nature of the air flow through voids in the fabric and the behaviour of the nano-particle caught inside the flow.

The classical theory of filtration is based on a single fibre in the fluid flow; Sharma (2000),Brown (1993), and Hutten (2007). It neglects neighbouring fibres and thus all porosity parameters. It was reasonable simplification due the fact that no easy method that would be reliable at the same time did exist until J-method for assessing the flat textiles porosity was introduced in Jakši´c & Jakši´c (2007) and Jakši´c & Jakši´c (2010). J-method enables us to determine all relevant porosity parameters, which makes this approach to filtration possible.

Medical face masks, which are made of textile fibres, are light, easy to use, relatively low-cost, and very efficient in combating air-borne infections. The mask design should provide tight fit to the skin of a face in order to ensure that air flows only through the mask. The design of the mask should also ensure filtration, not only dust particles, but also microbes and viruses. The masks are meant for a single use to avoid saturation of a mask. They are also suitable for use in a dust environment until saturation of a mask.

The mask composition must be proper regarding its use. The filtering layer must be protected with additional layers at its side faced towards a subject's face and outside side. Voids (pores) in the inner layer, which is actually filter passing air, should not lead air from the outer side to the inner side of the layer directly, as a channel. The dust particles, microbes and viruses might penetrate a mask designed in that way by being trapped in the unobstructed air flow. Pores in the inner layer, that enables air flow through mask, should be small enough and winded like a channel frequently changing its direction, in order to ensure maximum filtering efficiency.

where *Vi* stands for the air volume flow rate through the sample at the air pressure pi, *A* for a regression coefficient when fitting equation (2) to the measured dry data, *P* for the open surface, *vi* for the linear air flow velocity, *a* for the coefficient and *b* for the exponent. The parameters *a* and *P* are unknown and they have to be estimated as well. The solution of the problem is enabled by equation (3) by putting the velocity *vi* in the relationship with the air pressure *pi*. The value for the exponent *b* is bounded between 0.5 and 1.0. The last part of equation (3) holds in the ideal circumstances, when all of the energy dissipation mechanisms

Novel Theoretical Approach to the Filtration of Nano Particles Through Non-Woven Fabrics 207

*<sup>i</sup>* <sup>=</sup> 1.28 *<sup>p</sup>*0.5

<sup>−</sup> *Vi*−<sup>1</sup> *pb i*−1

*<sup>a</sup>*<sup>∗</sup> ; *<sup>a</sup>*<sup>∗</sup> <sup>=</sup> *nt*

*ncj*

The selective squeezing out the fluid from pores enables us to compute the number of pores at each interval defined by the incremental pressure growth. The number of pores in the *i*

> *Vi pb i*

The presumption of the equal regime of the air flow through the wet sample's open area and the dry one at the same pressure is taken into account. Small values of the Reynolds number in the extreme causes (maximal hydraulic diameter of pore), support that presumption. The air flow is either laminar through the open pores in the wet sample and through all pores in the dry sample, or the type of the air flow is same. This is the criterion for using the exponent *b*, which is estimated when equation (2) is fitted to the measured dry data, in the process of determining the pore distribution from the measured wet data. The corresponding coefficient

<sup>=</sup> 1.28

where *nt* stands for the true number of pores, *ncj* for the computed number of pores and *aj* for the corrected *a* in equation (4). The values of theoretical limits, for exponent *b* (*b*<sup>0</sup> = 0.5) and coefficient *a* (*a*<sup>0</sup> = 1.28), that are used in the second procedure are shown in the last part

Four different samples were used for the method's testing, which practically encompasses all the fabric types that the method is suitable for. The basic design parameters of the woven fabrics are presented in table 1. They are made of monofilament, multifilament and cotton yarn. The measured average pore's hydraulic diameters of the textiles are in the interval of 18

The results of the textile's porosity tests are presented in table 2. The first procedure is used for all four samples. The second procedure was used for porosity parameters estimation of

The nomenclature in table 2 - *b* stands for the exponent in equation (2), *h* [*μ*m] for the width of the interval of the pore distribution, *m* for the number of the distribution intervals, *nt* for the true number of pores between the threads of the warp and the weft per cm2, *n* for the computed number of hydraulic pores between the threads of the warp and the weft per cm2, when the true number of pores (or number of hydraulic pores) is unknown (second procedure), *d* for the average hydraulic diameter of pores, *dt* for the optically measured average hydraulic pore diameter - for samples (b), (c) and (d); the pores are ill-defined in

*<sup>i</sup>* (3)

*th*

(4)

(5)

*vi* = *a*<sup>0</sup> *p<sup>b</sup>*

*ni* <sup>=</sup> *<sup>p</sup>*<sup>2</sup> *i* 4*πaα*<sup>2</sup>

*aj* = 1.28

up to 200 micrometers. The wide assortment of textiles is thus covered.

samples (a) and (b) due to large value of the parameter *a*0.

*ncj nt*

are neglected.

interval as

of equation (3).

*aj* are determined by equation (5)

The criteria can only be met with a non-woven fabric made of microfibers. The diameter of microfibers is normally between 1.5*μ*m and 2.5*μ*m. For comparison, the diameter of normal chemical fibres is normally around 20*μ*m.

The medical face mask analysed here is made of three non-woven fabrics layers of different quality. The outer and inner fabric are intended for the partial filtration only and primarily for the protection of the filtration fabric in the middle. Fibres in all three fabrics are placed in the random pattern enhancing air-flow filtration capability.

The arguments focused on the nature of the air flow through a fabric and the behaviour of the nano-particle caught inside the flow were made based on analytical and numerical analysis computational fluid dynamics (CFD).

The novel approach to the filtration of nano-particles through non-woven fabrics is based on a fabric porosity and consequently on the nature of the fluid flow through the fabric. It is thus essential to quickly recapitulate the method for estimating the porosity of flat textiles (J-method); Jakši´c & Jakši´c (2007) and Jakši´c & Jakši´c (2010). The design and the porosity parameters of the mask under consideration are presented next followed by description of the mechanisms of filtration in the mask.

#### **1.1 J-method for determining porosity of flat textiles**

The method is based on selectively squeezing the fluid in the pores out of the wet fabrics by air pressure and on the presumption that a pore is approximated with a cylinder. The selectivity is assured by the fact that the fluid is squeezed out of the pores with a certain hydraulic diameter providing that the precise value of the air pressure is applied. The air pressure is inversely proportional to the hydraulic pore diameter. Latter is important, while the process of squeezing out the fluid contained in the pores of the wet fabrics is under examination. There is always a small amount of the fluid that remains at the edges of pores if such edges exist.

We worked under two presumptions:


The pressure difference *pi* between the opposite surfaces of the flat textile, equation (1), results in squeezing the fluid out of the pores, which diameter is equal or larger than *di*. The fluid is characterized by the surface stress *α*.

$$d\_i \ge \frac{4\alpha}{p\_i} \tag{1}$$

The fluid is first squeezed out from pores, which have the largest hydraulic diameter. The flow of air will establish itself through these pores that are now empty. The volume flow rate of air through the flat textile can be described by equation (2)

$$\mathbf{V}\_{\mathbf{i}} = A \; p\_{\mathbf{i}} = P \; a \; p\_{\mathbf{i}}^{b} = P \; v\_{\mathbf{i}} \tag{2}$$

2 Will-be-set-by-IN-TECH

The criteria can only be met with a non-woven fabric made of microfibers. The diameter of microfibers is normally between 1.5*μ*m and 2.5*μ*m. For comparison, the diameter of normal

The medical face mask analysed here is made of three non-woven fabrics layers of different quality. The outer and inner fabric are intended for the partial filtration only and primarily for the protection of the filtration fabric in the middle. Fibres in all three fabrics are placed in

The arguments focused on the nature of the air flow through a fabric and the behaviour of the nano-particle caught inside the flow were made based on analytical and numerical analysis -

The novel approach to the filtration of nano-particles through non-woven fabrics is based on a fabric porosity and consequently on the nature of the fluid flow through the fabric. It is thus essential to quickly recapitulate the method for estimating the porosity of flat textiles (J-method); Jakši´c & Jakši´c (2007) and Jakši´c & Jakši´c (2010). The design and the porosity parameters of the mask under consideration are presented next followed by description of the

The method is based on selectively squeezing the fluid in the pores out of the wet fabrics by air pressure and on the presumption that a pore is approximated with a cylinder. The selectivity is assured by the fact that the fluid is squeezed out of the pores with a certain hydraulic diameter providing that the precise value of the air pressure is applied. The air pressure is inversely proportional to the hydraulic pore diameter. Latter is important, while the process of squeezing out the fluid contained in the pores of the wet fabrics is under examination. There is always a small amount of the fluid that remains at the edges of pores if such edges

• The regime of the air flow through the dry and the wet sample is the same at same pressure

• The number of the hydraulic pores is not the same as number of pores between threads of the warp and weft if the ratio of the rectangular sides, which represents real pore's

The pressure difference *pi* between the opposite surfaces of the flat textile, equation (1), results in squeezing the fluid out of the pores, which diameter is equal or larger than *di*. The fluid is

> *di* ≥ 4*α pi*

*Vi* = *A pi* = *Pap<sup>b</sup>*

The fluid is first squeezed out from pores, which have the largest hydraulic diameter. The flow of air will establish itself through these pores that are now empty. The volume flow rate

(1)

*<sup>i</sup>* = *P vi* (2)

difference regardless of the size of the open area of the wet sample.

of air through the flat textile can be described by equation (2)

chemical fibres is normally around 20*μ*m.

computational fluid dynamics (CFD).

mechanisms of filtration in the mask.

We worked under two presumptions:

cross-section, is at least 3:1.

characterized by the surface stress *α*.

exist.

**1.1 J-method for determining porosity of flat textiles**

the random pattern enhancing air-flow filtration capability.

where *Vi* stands for the air volume flow rate through the sample at the air pressure pi, *A* for a regression coefficient when fitting equation (2) to the measured dry data, *P* for the open surface, *vi* for the linear air flow velocity, *a* for the coefficient and *b* for the exponent. The parameters *a* and *P* are unknown and they have to be estimated as well. The solution of the problem is enabled by equation (3) by putting the velocity *vi* in the relationship with the air pressure *pi*. The value for the exponent *b* is bounded between 0.5 and 1.0. The last part of equation (3) holds in the ideal circumstances, when all of the energy dissipation mechanisms are neglected.

$$w\_i = a\_0 \, p\_i^b = 1.28 \, p\_i^{0.5} \tag{3}$$

The selective squeezing out the fluid from pores enables us to compute the number of pores at each interval defined by the incremental pressure growth. The number of pores in the *i th* interval as

$$m\_i = \frac{p\_i^2}{4\pi a a^2} \left(\frac{V\_i}{p\_i^b} - \frac{V\_{i-1}}{p\_{i-1}^b}\right) \tag{4}$$

The presumption of the equal regime of the air flow through the wet sample's open area and the dry one at the same pressure is taken into account. Small values of the Reynolds number in the extreme causes (maximal hydraulic diameter of pore), support that presumption. The air flow is either laminar through the open pores in the wet sample and through all pores in the dry sample, or the type of the air flow is same. This is the criterion for using the exponent *b*, which is estimated when equation (2) is fitted to the measured dry data, in the process of determining the pore distribution from the measured wet data. The corresponding coefficient *aj* are determined by equation (5)

$$a\_{\circ} = 1.28 \frac{n\_{c\circ}}{n\_{\text{f}}} = \frac{1.28}{a^\*} \quad ; \quad a^\* = \frac{n\_{\text{f}}}{n\_{c\circ}} \tag{5}$$

where *nt* stands for the true number of pores, *ncj* for the computed number of pores and *aj* for the corrected *a* in equation (4). The values of theoretical limits, for exponent *b* (*b*<sup>0</sup> = 0.5) and coefficient *a* (*a*<sup>0</sup> = 1.28), that are used in the second procedure are shown in the last part of equation (3).

Four different samples were used for the method's testing, which practically encompasses all the fabric types that the method is suitable for. The basic design parameters of the woven fabrics are presented in table 1. They are made of monofilament, multifilament and cotton yarn. The measured average pore's hydraulic diameters of the textiles are in the interval of 18 up to 200 micrometers. The wide assortment of textiles is thus covered.

The results of the textile's porosity tests are presented in table 2. The first procedure is used for all four samples. The second procedure was used for porosity parameters estimation of samples (a) and (b) due to large value of the parameter *a*0.

The nomenclature in table 2 - *b* stands for the exponent in equation (2), *h* [*μ*m] for the width of the interval of the pore distribution, *m* for the number of the distribution intervals, *nt* for the true number of pores between the threads of the warp and the weft per cm2, *n* for the computed number of hydraulic pores between the threads of the warp and the weft per cm2, when the true number of pores (or number of hydraulic pores) is unknown (second procedure), *d* for the average hydraulic diameter of pores, *dt* for the optically measured average hydraulic pore diameter - for samples (b), (c) and (d); the pores are ill-defined in

sample (a), *P* [%] for the average open hydraulic flow area, *Pt* [%] for the average open flow hydraulic area computed on the bases of the optical experiment, *a*<sup>0</sup> for the coefficient

Novel Theoretical Approach to the Filtration of Nano Particles Through Non-Woven Fabrics 209

When dealing with the sample (b), the average ratio *l*/*w* in table 3 is 66.06/20.13 = 3.23. Hence, the criterion of having more than one hydraulic pore in a pore between threads of the warp and weft is thus met. The maximum value of the ratio between the value of the longer rectangular side *l* and the hydraulic diameter of the same pore is 6.84 and the mean value is 2.5. Hence, the true number of the hydraulic pores of the specimen (b) is 2200 · 2.5 = 5500, see table 2. The estimated number of hydraulic pores is 6213, table 2, which is only 13% more

The porosity of the non-woven flat fabrics is extremely difficult to characterize due to their irregular structure. The structure makes them better, more effective filtration media in comparison to the woven fabrics. Hence, the challenge is to estimate their porosity parameters. The experimental results presented in table 2 and 3, especially sample (b), proved

A medical mask fabric is composed of three layers. The outer layer and the layer suite on the face of a subject are composed of fibres which diameter is 18*μ*m. The inner layer is composed from microfibers which diameter is 2*μ*m, see figures 2 and 3. In this layer the fibres are

When discussing the mechanisms of filtration the initial state of the mask before usage must

2. The mask is placed over a subject's face in a way that it covers respiratory organs and air

5. A mask user is moderately active; volume flow: 40-60 l/min, breathing rate: 12-20 breaths

7. The fibres for the inner mask layer are randomly orientated and spread over the whole

8. There are no fibres in the direction perpendicular to the mask face, which means that the

Viruses are smaller than dust particles and microbes and thus the most difficult to filter out. They come in different shapes: spherical, see figure 1, helical, icosahedra, etc. and sizes, which varies between 10nm and 300nm. The average density of a virus is approximated to v

*a*, equation (2), at presumption that exponent *b* has minimal value (*b* = 0.5).

that the porosity of the non-woven flat textiles can be estimated by J-method.

than the true number of hydraulic pores in specimen (b).

**2. Medical mask**

be defined:

arranged in 37 sub-layers.

3. The mask assures 100

area, see figures 2 and 3.

non-woven fabric was not needled.

per minute.

**2.2 Virus**

**2.1 Initial state of a medical mask**

can pass only through the mask.

1. The mask is new and in the original package.

4. The laminar flow of air through mask is observed.

6. The mask thickness is 285 *μ*m and specific mass is 57.5g/m2.

= 1200kg/m3. We will focus on the spherical or near spherical viruses.


Table 1. Samples used in the testing of J-method


Table 2. Parameters of porosity estimated with J-method for all four samples. \* - the number corresponds to the product of the warp and weft. \*\* - corresponds to the 452 measured pores - between the threads of warp and weft only one typical pore was measured in each void between the threads of warp and weft.


Table 3. Results of the scanning-electron microscope pore's shape and open area measurement on 50 pores of the sample (b).

sample (a), *P* [%] for the average open hydraulic flow area, *Pt* [%] for the average open flow hydraulic area computed on the bases of the optical experiment, *a*<sup>0</sup> for the coefficient *a*, equation (2), at presumption that exponent *b* has minimal value (*b* = 0.5).

When dealing with the sample (b), the average ratio *l*/*w* in table 3 is 66.06/20.13 = 3.23. Hence, the criterion of having more than one hydraulic pore in a pore between threads of the warp and weft is thus met. The maximum value of the ratio between the value of the longer rectangular side *l* and the hydraulic diameter of the same pore is 6.84 and the mean value is 2.5. Hence, the true number of the hydraulic pores of the specimen (b) is 2200 · 2.5 = 5500, see table 2. The estimated number of hydraulic pores is 6213, table 2, which is only 13% more than the true number of hydraulic pores in specimen (b).

The porosity of the non-woven flat fabrics is extremely difficult to characterize due to their irregular structure. The structure makes them better, more effective filtration media in comparison to the woven fabrics. Hence, the challenge is to estimate their porosity parameters. The experimental results presented in table 2 and 3, especially sample (b), proved that the porosity of the non-woven flat textiles can be estimated by J-method.
