**2.3 Porosity of the medical masks**

We have applied J-method to characterise the porosity of a medical mask. The walls of pores are defined by fibres. In contrast to a woven fabric, where pores are straight from one surface to the opposite one and where the length of pores is equal to thickness of the fabric, the pores in the non-woven fabric changes its direction and are thus much longer than the fabric's thickness. It is this property that makes them an excellent filtration media and at the same time, very difficult to characterise. Even though the viruses are much smaller than the hydraulic diameter of pores, the configuration of pores allows for high filtration efficiency.


Table 4. Comparison of chosen physical parameters of the fibres from which the mask is made. \* - not blocked with fibres pressed together

All three layers are made of polypropylene fibres. The area of the mask that allows the air flow is around 160cm2, table 4. The active surface of the outer layers is smaller than the active surface of the inner layer. Both outer layers are strengthened for their ability to retain shape 6 Will-be-set-by-IN-TECH

Fig. 1. Computer-aided reconstruction of a rotavirus based on several electron micrographs;

We have applied J-method to characterise the porosity of a medical mask. The walls of pores are defined by fibres. In contrast to a woven fabric, where pores are straight from one surface to the opposite one and where the length of pores is equal to thickness of the fabric, the pores in the non-woven fabric changes its direction and are thus much longer than the fabric's thickness. It is this property that makes them an excellent filtration media and at the same time, very difficult to characterise. Even though the viruses are much smaller than the hydraulic diameter of pores, the configuration of pores allows for high filtration efficiency.

> Thickness [*μ*m]

Outer 17.6 92 5 18 87 160 Inner 20.4 74 37 2 100 160 Outer on the subject face 19.1 120 7 18 75 160 Table 4. Comparison of chosen physical parameters of the fibres from which the mask is

All three layers are made of polypropylene fibres. The area of the mask that allows the air flow is around 160cm2, table 4. The active surface of the outer layers is smaller than the active surface of the inner layer. Both outer layers are strengthened for their ability to retain shape

Number of fibre layers

Thickness of fibres [*μ*m]

Active surface\* [%]

Total surface [cm2]

Wikipedia (2011)

**2.3 Porosity of the medical masks**

Non-woven layer Mass

made. \* - not blocked with fibres pressed together

[g/m2]

by pattern of dots where the fibres are partially pressed together. In this way, the parts of the other layers are transformed into a foil. Nevertheless, it is the open surface P that allows the airflow, table 5. The inner layer sub-layers are not pressed nor melted together and hence the air can flow through sub-layers and pores that are actually blocked by the outer layers areas transformed into the foil. Hence, 100% of active surface of the inner layer reported in table 4 holds.


Table 5. Parameters of porosity for all three non-woven layers of mask; the surface of samples: 1cm2; liquid in the pores: n-butanol

Fig. 2. The inner non-woven layer of the medical mask zoomed 200 times

The mask porosity parameters are presented in table 5. Number of the pores on 1cm2 is 45294. The maximal diameter of pores is 30*μ*m. The open area (free for air flow) is (8.42%. The coefficient *a* (regression equation 2 - flow air through dry sample), is 0.0921. The exponent *b*

Fig. 4. Model of the stream of air through the non-woven mask due to the respiration

the inner layer.

has smaller active surface due to their additional function to provide mechanical support for

Novel Theoretical Approach to the Filtration of Nano Particles Through Non-Woven Fabrics 213

The sample, which is undergoing the porosity test, is taken from the mask and placed in the apparatus head in exactly the same way. The area of the head is 1cm2. The layers in the sample are pressed together during the test to avoid side leakage. This boundary effect affects relatively small area and may be neglected. True difference comes with the pressure at which the test is carried out. The test pressure at test compresses all three layers of a mask together. This effect obstructs free air passage beneath the areas of the outer layers that are

Fig. 3. The inner non-woven layer of the medical mask zoomed 2200 times

(regression equation 2 - flow air through dry sample) is 0.7521. Mean hydraulic diameter of pores is 13.91*μ*m.

The nomenclature of table 5 is as following: *dmax* stands for the average pore diameter of the first interval (the largest pores), *dmin* stands for the average pore diameter of the last interval (the smallest pores), *dp* stands for the average pore diameter of the sample and *P* stands for the average open hydraulic flow area. The pore distribution is presented in table 6.


Table 6. Parameters of porosity for mask (for all three non-woven layers) - number of pores according to the pore size intervals

#### **2.4 Influence of the mask layered structure on the filtration**

There are some differences in the functioning of the mask layers during its usage and during the porosity parameter measurement that had to be noted. As already noted, both outer layers 8 Will-be-set-by-IN-TECH

Fig. 3. The inner non-woven layer of the medical mask zoomed 2200 times

pores is 13.91*μ*m.

Limits of the classes [*μ*m]

according to the pore size intervals

Hydraulic diameter of pores [*μ*m]

**2.4 Influence of the mask layered structure on the filtration**

(regression equation 2 - flow air through dry sample) is 0.7521. Mean hydraulic diameter of

The nomenclature of table 5 is as following: *dmax* stands for the average pore diameter of the first interval (the largest pores), *dmin* stands for the average pore diameter of the last interval (the smallest pores), *dp* stands for the average pore diameter of the sample and *P* stands for

25 – 27 26 3815 5.556 230 0.51 23 – 25 24 4133 13.100 329 0.73 21 – 23 22 4509 16.704 139 0.31 19 – 21 20 4959 66.931 2814 6.21 17 – 19 18 5510 120.306 3135 6.92 15 – 17 16 6199 233.431 6979 15.41 13 – 15 14 7089 323.569 6880 15.15 11 – 13 12 8266 516.463 16637 36.73 9 – 11 10 9919 652.269 8171 18.04 Table 6. Parameters of porosity for mask (for all three non-woven layers) - number of pores

There are some differences in the functioning of the mask layers during its usage and during the porosity parameter measurement that had to be noted. As already noted, both outer layers

Volume flow

[m3/s]·10−<sup>6</sup>

Number of pores

Portion of pores [%]

the average open hydraulic flow area. The pore distribution is presented in table 6.

Pressure [Pa]

has smaller active surface due to their additional function to provide mechanical support for the inner layer.

The sample, which is undergoing the porosity test, is taken from the mask and placed in the apparatus head in exactly the same way. The area of the head is 1cm2. The layers in the sample are pressed together during the test to avoid side leakage. This boundary effect affects relatively small area and may be neglected. True difference comes with the pressure at which the test is carried out. The test pressure at test compresses all three layers of a mask together. This effect obstructs free air passage beneath the areas of the outer layers that are

**3. Classical filtration theory**

4

1

filtration. They are described in the following list:

comparison to fluid induced forces.

microbes and viruses.

consideration.

2 3

5

Novel Theoretical Approach to the Filtration of Nano Particles Through Non-Woven Fabrics 215

7



Fig. 6. Classical theory of the particles filtration on one fibre in the case of the laminar air flow: 1 - filtration of particles because of gravitation; 2 - filtration of the smaller particles on the surface of the fibres when they change their position from central part of the air flow to the virtual surface of the flow; 3 - electro filtration of the particles; 4 - filtration particles because of their steadiness; 5 - filtration very small particles by diffusion; 6 - air flow; 7 - fibre. The classical filtration theory setting is presented in figure 6 – Sharma (2000), Brown (1993) and Hutten (2007), where a single fibre is put in the laminar fluid flow. The different sized particles are depicted with their supposed paths in order to depict different mechanisms of

1. The filtration due to gravitational field is applicable to largest and heaviest particles. This filtration mechanism can be observed in the outer fabric of the mask and does not apply to

2. The caption of a particle on a surface of the fibre due to its the collision with the fibre is generally applicable to the particles of all sizes. Unfortunately, the smaller the particle, the larger is the probability for the fluid flow to carry the particle pass the fibre due to the laminar nature of the air flow. The inertia of the particle in this case is negligible in

3. The electro filtration is a valid concept. However it is not applicable to the mask under



+



6

Fig. 5. Pore and a hydraulic pore of non-woven fabric. 1 - fibres that are narrowing the pore, 2 - fibre, 3 - hydraulic pore,4-a part of the pore where liquid has not been squeezed out during experiment of the J-method.

pressed together, which is not the case when a mask is used on a subjects face. In latter case all three mask layers are relatively loosely positioned together. The pressure caused by human breathing at supposed velocity of the air flow (1m/s) is so low (20Pa) that it does not affect the position of a mask layer in respect to the others. As a consequence, the experimentally estimated open area may be underestimated.

The fluid flow rate, open area and fluid velocity are linked by equation (2) and the pressure (pressure difference between both sides of a mask) and air flow velocity are linked by equation (3). It is clear from these relations that if the same fluid flow rate is achieved in reality and at experiment, larger pressure inducing larger velocities would be needed at experiment, due to smaller open area, which is the consequence of mask layers being pressed together during the experiment.

The open surface was estimated to be 8.42% of the total active area. It has to be stressed that the estimate refers on the hydraulic area, defined by hydraulically active section of pores, see figure 5, in interval down to pore hydraulic diameter of 8*μ*m. The limitation is due to the maximal pressure that can be achieved during experimentation and the capacity of the volume flow. The pore distribution presented in table 6 indicates that there may be a lot of pores smaller than 8*μ*m. These pores play an important role in filtration process due to relative proximity of the pore wall to the microbe or virus on one hand and they also let air through the mask. Due to latter we can assume that the open area is somewhat larger than estimated.

### **3. Classical filtration theory**

10 Will-be-set-by-IN-TECH

Fig. 5. Pore and a hydraulic pore of non-woven fabric. 1 - fibres that are narrowing the pore, 2 - fibre, 3 - hydraulic pore,4-a part of the pore where liquid has not been squeezed out

pressed together, which is not the case when a mask is used on a subjects face. In latter case all three mask layers are relatively loosely positioned together. The pressure caused by human breathing at supposed velocity of the air flow (1m/s) is so low (20Pa) that it does not affect the position of a mask layer in respect to the others. As a consequence, the experimentally

The fluid flow rate, open area and fluid velocity are linked by equation (2) and the pressure (pressure difference between both sides of a mask) and air flow velocity are linked by equation (3). It is clear from these relations that if the same fluid flow rate is achieved in reality and at experiment, larger pressure inducing larger velocities would be needed at experiment, due to smaller open area, which is the consequence of mask layers being pressed together during the

The open surface was estimated to be 8.42% of the total active area. It has to be stressed that the estimate refers on the hydraulic area, defined by hydraulically active section of pores, see figure 5, in interval down to pore hydraulic diameter of 8*μ*m. The limitation is due to the maximal pressure that can be achieved during experimentation and the capacity of the volume flow. The pore distribution presented in table 6 indicates that there may be a lot of pores smaller than 8*μ*m. These pores play an important role in filtration process due to relative proximity of the pore wall to the microbe or virus on one hand and they also let air through the mask. Due to latter we can assume that the open area is somewhat larger than estimated.

during experiment of the J-method.

experiment.

estimated open area may be underestimated.

1

2

3

1

4

Fig. 6. Classical theory of the particles filtration on one fibre in the case of the laminar air flow: 1 - filtration of particles because of gravitation; 2 - filtration of the smaller particles on the surface of the fibres when they change their position from central part of the air flow to the virtual surface of the flow; 3 - electro filtration of the particles; 4 - filtration particles because of their steadiness; 5 - filtration very small particles by diffusion; 6 - air flow; 7 - fibre.

The classical filtration theory setting is presented in figure 6 – Sharma (2000), Brown (1993) and Hutten (2007), where a single fibre is put in the laminar fluid flow. The different sized particles are depicted with their supposed paths in order to depict different mechanisms of filtration. They are described in the following list:


is computed with maximal air velocity, even though that the virus is picked up by the air flow, to stay at conservative approach. In this case the *Re* = 0.02 *<<* 0.1, the limit for the laminar flow around the object in the fluid flow is much stricter than the one for the flow in a tube. We can conclude that the filtration in the inner mask layer is done in the laminar air flow.

Novel Theoretical Approach to the Filtration of Nano Particles Through Non-Woven Fabrics 217

The process of filtration that is carrying out in the medical masks is at the micro and nano scale. The size of the pores of the mask inner layer is at the micrometer scale and the size of viruses is at the nanometer scale. The Kundsen number (*Kn*) is used to determine whether the classical mechanics of continuum is still valid approach. Kundsen number is defined as

*Kn* <sup>=</sup> *kBT*

where *kB* stands for the Boltzmann constant, *T* for the thermodynamic temperature, *σ* for the particle hard shell diameter and *p* for the total pressure and finally *L* for the representative physical length scale. The value of *Kn* for maximal pore and virus size is *Kn* <sup>≈</sup> <sup>10</sup>−9, which is

In addition to the mechanisms of filtration described by the classical filtration theory the filtration caused by laminar flow through pores is described. A pore in the mask inner layer is approximated by a tube or a channel. The inner layer is formed in complicated way - it is composed of 37 sub-layers. Each of these sub-layers is composed of fibres randomly placed in random direction, making a pore meandering through all sub-layers a complex path for air to

1. A pore changes its direction 35 times, when leaving on sub-layer and enters the other.

3. At the direction change the pore may split. Nevertheless, during the experiment for determining porosity parameters, we have actually taken into account these newly formed

4. The length of a pore is at least twice the width of the inner layer, which means that the

5. Estimated pore diameter in the experiment is the smallest diameter of the pore over a

6. The particle can be filtered on the obstacle (classical filtering theory) or in a void after the

Based on the presumptions about shape of the pore the simple numerical model was built in order to show different possibilities of filtration and to asses some postulates of the classical

A simple 2D computational fluid dynamics (CFD) model was used for the simulation purposes of the virus behaviour in the laminar fluid flow in straight tube as a first

length of the pore is at least 570 *μ*m, together with both outer layers.

filtration theory. The shape of a virus is idealised with the sphere/circle.

whole width of the mask. Hence, the pore is acting as a tube or a channel.

<sup>√</sup><sup>2</sup> *πσ*<sup>2</sup> *pL* (7)

**4.2 Determination of the physical domain of the problem**

much less than 1. Hence, the mechanics of continuum is applicable.

**4.3 Numerical modelling of the virus behaviour in the straight pore**

take, see figure 4. We can suppose that:

2. The change of the direction is random.

obstacle or at the pore's wall.

pores.


#### **4. Novel approach to the filtration**

The classic theory of filtration is based on the flow around a single fibre. We argue that air is channelled through pores in the inner layer of the mask and hence, the modelling of filtration as a fluid flowing through a tube is justified. The justification is supported by the measurement of porosity parameters of the mask, which presumes the laminar flow through a sample Jakši´c & Jakši´c (2007) and Jakši´c & Jakši´c (2010).

#### **4.1 Nature of the air flow through a mask**

The nature of the fluid flow is described by Reynolds number, equation (6).

$$Re = \frac{\rho v d}{\mu} \tag{6}$$

where *ρ* stands for the fluid density, *v* is the mean velocity of the object relative to the fluid, or vice versa, *d* for the characteristic linear dimension (pore hydraulic diameter or particle diameter or fibre diameter) and *μ* for the dynamic viscosity of the fluid.

The value of the air density is *<sup>ρ</sup>* <sup>=</sup> 1.2kg/m3 and its dynamic viscosity *<sup>μ</sup>* <sup>=</sup> <sup>18</sup>*μ*Pa·s. The characteristic linear dimension is defined according to the object of interest. If the flow through a pore is studied, the dimension is its hydraulic diameter. On the other hand, if the flow around stationary particle is in question, the dimension is its diameter.

The velocity of the fluid (air) through a mask can be deduced from mask's porosity parameters and human physiology. The air velocity at inhaling or exhaling depends on breathing intensity. Latter depends on an activity and the intensity of the activity of the mask user. The exhaled air is normally denser than inhaled one due to its increased humidity. Let's suppose that a subject inhales from 12 litres of air per minute at normal pressure and during light activity and 60 l/min during moderate exercise. Further on, suppose that the subject inhales 12 times per minute. The size of the active surface of the mask inner layer is 160cm2, and the open area of the layer is 8.42% of the active surface, table 4. The open area is thus 13.5cm2. The air flow velocity through mask inner layer is thus approximately 0.17m/s during light activity and 0.86m/s during moderate exercise Zuurbier et al. (2009). As a conservative approach the value of 1m/s is taken in order to compute *Re* number.

The maximal pore hydraulic diameter is 30m, table 4. The Reynolds number for fluid flow through the largest pore is *Re* = 2 *<<* 2300, which ensures laminar air flow through the inner mask layer.

The mask is designed to filter all viruses, which come in different sizes: from 10nm to 300nm. When inhaling (or exhaling) the air starts to flow, but a virus is not following instantaneously. The Reynolds number at the moment when a virus is stationary and the air is already moving is computed with maximal air velocity, even though that the virus is picked up by the air flow, to stay at conservative approach. In this case the *Re* = 0.02 *<<* 0.1, the limit for the laminar flow around the object in the fluid flow is much stricter than the one for the flow in a tube.

We can conclude that the filtration in the inner mask layer is done in the laminar air flow.
