# Multi-Disciplinary Optimization of UV-C Filter for Air Disinfection
## 2 Filter Modeling and Simulation
### CFD Simulation Model  
Each updated CAD configuration is exported in a proper format (.CATPart file) to ANSYS Workbench, which generates the mesh and performs the CFD analysis. The mass flow rate entering the filter is one of the design parameters, and the selected range of variation is [\(2.5\times 10^{-4}\): \(16\times 10^{-4}\)] kg/s, corresponding to [\(12.5\cdot 78.4\)] L/min. To save computational time, an axial-symmetry model, corresponding to a quarter of the total volume, was defined (Figure 4). The inlet of the air flow is defined on the quarter circular surface at the left of Figure 4, while the outlet is defined at the end of the pipe (right, out of picture). The mesh is tetrahedral with prismatic layers defined on the wall (Figure 5), for a total of about 1.5 M elements. The Reynolds number of the flow is in the order of \(1\times 10^{4}\), which leads to the definition of a wall distance (minimum mesh size) of 0.5 mm, to yield a \(Y+\) value \(<5\). For this mesh size, mesh independence studies revealed a proper convergence of the monitored quantity (maximum pressure and velocity on the median section of the filter; the errors for the selected mesh are lower than 1-2% with respect to a mesh of 3 M elements). In the first stage, Navier-Stokes equations (Equation (2)) are defined for the stationary CFD analysis (using the solver Fluent of ANSYS 2024R1), implementing an _SST k-\(\omega\)_ turbulence model (Figure 6). \[\frac{\partial}{\partial t}(\rho\mathbf{u})+\
abla(\rho\mathbf{u}\cdot\mathbf{u})=-\
abla p +\
abla\mathbf{\tau}+\rho g \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter (Reference ID)** & **Lower Bound** & **Higher Bound** \\ \hline Blade\_angle (x1) & 45\({}^{\circ}\) & 75\({}^{\circ}\) \\ Blade\_height (x2) & 15 mm & 25 mm \\ Inner\_radius (x3) & 10 mm & 20 mm \\ Z\_length (x4) & 25 mm & 80 mm \\ X\_inner (x5) & 7.5 mm & 12.5 mm \\ X\_outer (x6) & 0.1 mm & 3 mm \\ Y\_inner (x7) & 22.5 mm & 27.5 mm \\ Y\_outer (x8) & 0.1 mm & 3 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters and range of variation. Figure 4: Air flow across the filter: mesh view. In the second stage, on the basis of the flow field results across the filter, a particle tracking analysis was performed to understand how any pathogenic particle is transported by the air flow across the filter. Various methodologies are available for particle tracking [22]; we adopted the one available in ANSYS Fluent, based on [23] the Basset-Boussinesq-Oseen (BBO) equation (Equation (3)). \[\frac{\pi}{6}\rho_{P}d_{P}^{3}\frac{d\mathbf{U_{P}}}{dt}=3\pi\mu d_{P}(\mathbf{U_{f}}- \mathbf{U_{P}})-\frac{\pi}{6}d_{P}^{3}\
abla p+\frac{\pi}{12}\rho_{f}d_{P}^{3} \frac{d}{dt}\big{(}\mathbf{U_{f}}-\mathbf{U_{P}}\big{)}+\rho_{P}d_{P}^{3}\mathbf{\delta} \tag{3}\]  
In Equation (3), the left term reports the rate of change in the particle momentum (\(\rho\), \(d\), and \(\mathbf{U}\) are, respectively, the density, diameter, and velocity of the particle; the subscript \(P\) stands for particle). The right term reports the sum of the forces acting on the particle; the first term is the Stokes' drag (function of the fluid viscosity \(\mu\) and velocity difference between particle, \(\mathbf{U_{p}}\), and fluid, \(\mathbf{U_{f}}\)); the second term is the pressure gradient in the undisturbed flow; the third term is the added mass, which is an inertia forced induced by the accelerating particle moving a volume of surrounding fluid as it crosses through it; and the last term is the gravitational force. Figure 5: Air flow across the filter: mesh detail (the \(z\)-axis of the pipe is vertical). Figure 6: Air flow across the filter: velocity magnitude (**above**) and velocity vector (**below**) field. In the original formulation of the BBO equation, an additional force, the Basset term, is also introduced to evaluate the viscous effects of the temporal delay in boundary layer development as the relative velocity changes with time.