# Design and Evaluation of Face Mask Filtration: Mechanisms, Formulas, and Fluid Dynamics Simulations
## 3 Results and Discussions
### Prediction from Fluid Flow Mechanics  
Navier-Stokes (N-S) equations allow us to predict and compare the behaviour of fluid flow. The main equations are mass and momentum conservation (Supplementary Materials S2: Navier-Stokes equations) [38; 39]. Both laminar and turbulent fluid behaviors can be explained by using the N-S equations. However, the phenomenon of turbulence is very complex, and therefore, in order to describe it more simply, turbulence models such as the k-\(\epsilon\) and the k-\(\omega\) are coupled with the N-S equations. These models use turbulent energy dissipation and turbulent kinetic energy factors to provide the time-averaged behaviour of a turbulent fluid [40]. Furthermore, to predict the pressure drop of a fluid passing through porous media (e.g., facemask fabrics) and to simulate the flow resistance through the mask, the following equation was used [28]:  
\[\frac{\Delta p}{d_{f}}=-D\ \mu_{f}U_{f}-0.5\ I\ \rho_{f}\ \Big{|}U_{f}\Big{|}^{2} \tag{17}\]  
where (D) is the Darcy viscous coefficient, (I) is the inertial coefficient, and \(\mu_{f},\rho_{f}\) and \(\text{U}_{f}\) are the viscosity, density, and velocity of the fluid, respectively. The Darcy and I coefficients can be estimated according to previous studies [41; 42] (Supplementary Materials S3: turbulent flow resistance of porous media). Due to both the Darcy and inertial coefficients, Equation (17) not only takes into account the fluid features (e.g., viscosity, velocity, and density) but also different face mask characteristics such as the fiber diameters, the packing fiber density, and the particle diameter, thus allowing for better prediction accuracy. \begin{table}
\begin{tabular}{c c} \hline \hline
**Time (s)** & **\(\Delta\)x (m)** \\ \hline
60 & \(3\times 10^{-5}\) \\
900 & \(1\times 10^{-4}\) \\
1800 & \(1.5\times 10^{-4}\) \\
2700 & \(1.8\times 10^{-4}\) \\
3600 & \(2.1\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Particle displacement at different time durations. These properties change the velocity counters for each mask (e.g., in the manner shown in Figure 5). In this study, we designed and developed a model describing a velocity profile with maximum velocity at the centre (the air duct centre) and a velocity that falls to zero at the inner air duct surfaces. A representation of this model is based on a modulated ellipsoidal function containing a floating term to represent reflective velocity effects. This has the advantage of a mathematical analysis of the average air flow velocity in connection with filtration efficiency determination. COMSOL Multiphysics was used to perform CFD simulations to obtain an overview of the different velocity profiles and contours for the different models discussed above. Again, the initial velocity flow was set to 38 m/s based on the average data in Table 1, the ambient temperature was set to 23 \({}^{\circ}\)C, and the temperature at the face of the mannequin was set to 32.1 \({}^{\circ}\)C based on the average data obtained from the 12 volunteers [9]. The physical domain is described by the mannequin head, while the external computational domain was reduced to reduce the processing time. From the simulations above, it emerges that our model (see Figure 4c) generates a larger region with higher velocity (the red region) compared to the k-\(\omega\) and k-\(\varepsilon\) models (see Figure 4a,b). Moreover, the model developed in this paper has a high velocity in the centre and a low velocity moving with the x axis from the mannequin mouth (source). As mentioned above, the physical domain included the mannequin head. The externally computational domain (see Figure 4a-c) was reduced to decrease the processing time. In order to have a better visualisation of the velocity distributions for the three different models, the velocity profiles have been plotted against the width (distance from the mannequin mouth) (see Figure 4d,f).