## 2 Materials and Methods
### Experimental Procedure
#### 2.1.2 Mathematical Analysis  
The velocity profile at entry is severe, falling from \(v_{\text{ (max)}}\) at the measuring points to zero at the internal wall surfaces. The mean velocity should be lowered if the wall surface velocity is greater than zero. The mean velocity (\(v_{\text{m}}\)) is required to provide a measure of \(F\) for individual mask layers. For this, the equation used to describe velocity profiles intersecting at \(v_{\text{ (max)}}\) is given by:  
\[v=v_{\text{max}}\left(1-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}\right) \tag{1}\]  
where \(a\) and \(b\) are the rectangular aperture half-lengths. Base coordinates \(x\) and \(y\) define the two profiles intersecting at \(v_{\text{max}}\). The volume beneath the surface defined by the two profiles is found from a double integral:  
\[v=v_{\text{max}}\int_{-a}^{a}\int_{-b}^{b}\left(1-\frac{x^{2}}{a^{2}}-\frac{y^ {2}}{b^{2}}\right)\text{d}x\text{d}y \tag{2}\]  
which leads to:  
\[v=(4/3)\;a\;b\;v_{\text{max}}\]  
Equating to a rectangular volume (\(2a\times 2b\times v_{\text{m}}\)) allows a mean velocity \(v_{\text{m}}\) to be found from:  
\[(4/3)\;a\;b\;v_{\text{max}}=2a\times 2b\times v_{\text{m}}\]  
This gives \(v_{\text{m}}=v_{\text{ (max)}}/3\), where \(v_{\text{ (max)}}\) is a measured entry velocity at the centre of the rectangular area. It will be seen that this simplification of the mean velocity can discriminate between the passage of air through (face velocity) ordered and randomly arranged fibers in wool, silk, and polypropylene materials. The exit velocity (\(v_{1}\)) is assumed to be uniform given the maximum does not exceed 10.32 (m/s) (\(v_{1}\)); in contrast, the entry velocities (\(v_{0}\)) lay in a narrow range across all masks of 37-39 (m/s). Assuming the parabolic profile in velocity across the central sections, it can be shown:  
\[\overline{v}_{\text{m}}=\frac{v\omega}{3}\]  
Figure 1: (**a**) Air flow duct in position within the vertical air flow instrument. (**b**) Air duct mask container. given that \(F\) (%) is a difference between the velocity at entry and exit and divided by that at entry, expressed as a percentage. Hence, \(F\) can be estimated as a percentage using the following formula:  
\[F\left(\%\right)=\left[\ \left(v_{0}/3-v_{1}\right)/\left(v_{0}/3\right)\right] \text{C}\times 100\]  
where C is an empirical constant that enables the conversion from velocity to filtration efficiency [\(F\)]% for a single layer to a whole mask (see the silk example below). Hence, the following formula gives the filtration efficiency of individual mask layers:  
\[F\left(\%\right)=\left[1-\left(3v_{1}\right)/v_{0}\right)\right]\text{C} \times 100\]  
To account for this, a conversion factor (C) has been introduced with the equation above. This assumes that filtration efficiency and velocity difference are proportional. For example, the average filtration efficiency of a single layer of silk was found to be 83%. Since the filtration efficiency of a whole silk mask is 88% [21], therefore, a conversion factor of C = 0.83 applies. Alternatively, as used here, \(F\) for the whole mask has been calculated as [23]:  
\[F\left(\text{total}\right)=1-\left(\text{F}_{\text{L1}}\times\text{F}_{\text{ L2}}...\right)\]  
where, \(\text{F}_{\text{L1}}\times\text{F}_{\text{L2}}\ldots\) is the product of F of individual layers. The usual procedure (ISO, ASTM) for calculating filtration efficiency is based upon the measurement of particle count at entry and exit. For example, the filtering efficiencies of the three surgical layers are 31, 89, and 50%, respectively (from Table 1); therefore, using the above formula to calculate the \(F\) (total) = 1 - (0.31 \(\times\) 0.89 \(\times\) 0.50) = 0.87 or 87%, which compares well with the experimental data (\(F\) whole mask = 88%).