# Design and Evaluation of Face Mask Filtration: Mechanisms, Formulas, and Fluid Dynamics Simulations
## 3 Results and Discussions
### Prediction from Fluid Flow Mechanics
From the velocity profiles above, it emerges that the velocity profile distribution for the k-\(\varepsilon\) and k-\(\omega\) models differs from the one obtained for the present model. In particular, for the model presented in this paper, the velocity profile for a rectangular cross section has a parabolic shape, whereas the k-\(\varepsilon\) and k-\(\omega\) models have a similar Gaussian velocity distribution for a circular section. When required, the two models described above show a wider velocity distribution, which admits a more accurate prediction of the velocity profile far from the boundary (mannequin mouth). The k-\(\varepsilon\) model simulates the mean flow features for turbulent flow conditions, primarily for flows with relatively small pressure gradients and high Reynolds numbers. In contrast, the k-\(\omega\) model captures the effects of turbulent flow conditions and is used for flows with low Reynolds numbers, performing better under adverse pressure gradients. The advantage of  
Figure 4: CFD simulations of velocity profile contours and respective velocity profile distributions for (**a,d**) turbulent k-\(\varepsilon\) model, (**b,e**) k-\(\omega\) turbulent model, and (**c,f**) our model. For all models, the velocity profiles are plotted against the width of the source (distance from the mannequin mouth). using the k-\(\omega\) model over the k-\(\epsilon\) model lies in its greater accuracy in estimating the rate at which flows spread [43; 44], although it requires longer computational time to produce a solution. In summary, the k-\(\epsilon\) model is more suitable for simulations in indoor environments with low ventilation, while the k-\(\omega\) model is more reliable for simulating turbulent flow conditions, such as coughing or sneezing [45] in both ventilated indoor and outdoor environments [46]. However, although the k-\(\epsilon\) and k-\(\omega\) models allow a better geometry optimization of the source (mannequin mouth), the two models have been found to overestimate the air flow expansion [47]. On the other hand, the laminar model allows better prediction of the flow behaviour within the entire flow domain. In our previous study [5], we analyzed the air velocity profile distribution based on the k-\(\epsilon\) turbulence model for the six different face masks. In this study, we analyzed the air velocity profile distribution of these masks using the quadrant-based average velocity model (Equation (10)). We simulated the exhalation phase (t = 3 s) of the breathing cycle for modelling and visualizing the air velocity distribution through the face masks by using the mathematical model developed in this study. The pore size data of the masks was measured and used [5] to simulate the air flow resistance. The no-slip boundary condition was applied at the mouth, nose, and forehead of the face mask wearer. For the scope of this study, we did not focus on modeling the shear stress behavior of airflow at these boundaries. Instead, the velocity profile for laminar and turbulent flow were simulated along with temperature influences. An inlet velocity flow based on the quadrant-based average velocity model was considered a boundary condition at the mouth of the mannequin. The external or ambient temperature was set to 23 \({}^{\circ}\)C. The initial temperature inside the mask was set at 36.7 \({}^{\circ}\)C, and the humidity average was set to 84% based on the average data obtained from the 12 volunteers [9]. The domain was discretized using a grid of tetrahedral cells, with total of approximately 3 \(\times\) 10\({}^{6}\) cells. The skewness parameter was evaluated to determine the deviation from the optimal cell size, with a range 0 (ideal) to 1 (worst). In our simulation, the cells exhibited skewness values raging from a minimum of 0.001 to a maximum of 0.80. These simulations showed that the maximum and minimum velocity during the exhalation of six different masks were 2.35 m/s and 0.9 m/s, respectively.