## 2 Materials and Methods  
### Numerical Model  
The Lagrangian and Eulerian particle models are the two primary types of CFD particle transport models. The former is typified by the discrete phase model (DPM), which sees the particles as discrete entities with individual Lagrangian equations of motion. DPM resolves inertial effects and forces including drag, buoyancy, and gravity, as well as the motion of individual particles with varying weights and sizes. Instead, Eulerian models follow a finite volume element along the fluid flow path, treating the particles as a single phase. These approaches, in contrast to Lagrangian models, simulate the average behavior of the particles as opposed to that of individual particles. The concentration of massless particles is taken into account by Eulerian methods as a function of both space and time. This type of model is called a passive scalar model when there are no interactions between the fluid and particle phases [32]. Although Lagrangian approaches are more descriptive, they also require a lot more computing power, since good predictions necessitate simulating a large number of discrete particles [32]. Selecting the appropriate airflow model is equally crucial. More comprehensive discussions and comparisons of the various models for indoor particle simulations can be found in the references [31; 32; 33]. In this study, a combination of these two methods was used, as the Euler-Lagrange approach was employed. Since the diameters of the droplets considered in the present study were 1 \(\upmu\)m and 10 \(\upmu\)m, for which the Stokes number is very small, and the volume fraction of the particles was very small (<<10%), their influence on the mainstream air flow could be negligible. Therefore, the air velocity and temperature distributions in a room with an air cleaner were calculated with the Eulerian approach without considering the existence of the particles. The age of the air was also calculated at the same time. Then, by using these calculated velocity and temperature data in the room, the Lagrangian particle tracking method was employed to determine the trajectories of the particles in order to derive the transport characteristic of indoor particles. Then, the performance parameters of the ACD were calculated. The ANSYS Fluent (Release R19.0) software was used to solve the three-dimensional steady-state Reynolds averaged Navier-Stokes (RANS) and energy equations [34]. Under the Eulerian framework, the governing equations explaining the airflow in the solution domain can be written as given in Equations (1)-(3), obtained from the conservation of mass, momentum, and energy, respectively. Conservation of mass:  
\[\frac{\partial(\rho u_{i})}{\partial x_{i}}=0 \tag{1}\]Conservation of momentum:  
\[\frac{\partial}{\partial x_{i}}\big{(}\rho u_{i}u_{j}\big{)}=-\frac{\partial p}{ \partial x_{i}}+\frac{\partial}{\partial x_{i}}\bigg{[}(\mu+\mu_{t})\frac{ \partial u_{j}}{\partial x_{i}}\bigg{]}+S_{i} \tag{2}\]  
Conservation of energy:  
\[\frac{\partial}{\partial x_{i}}\big{(}\rho u_{i}c_{p}T\big{)}=\frac{\partial}{ \partial x_{i}}\bigg{[}\lambda\ \frac{\partial T}{\partial x_{i}}\bigg{]}+S_{k} \tag{3}\]  
where x\({}_{i}\) (m) is the coordinate component, u\({}_{i}\) (m/s) is the fluid velocity component, T (K) is the temperature, \(\rho\) (kg/m\({}^{3}\)) is the fluid density, c\({}_{p}\) (kJ/kgK) is the specific heat at the constant pressure, \(\mu\) (kg/ms) is the dynamic viscosity, \(\mu_{t}\) (kg/ms) is the turbulent viscosity, p (Pa) is the pressure, \(\lambda\) (W/mK) is the heat transfer coefficient of the fluid, and S is the volumetric source term. The SIMPLE scheme was chosen for velocity-pressure coupling, and the standard method was used for pressure discretization while the second order discretization was applied to other quantities [34]. Turbulence properties were calculated using the RNG k-\(\epsilon\) model, which was used in several research works dealing with air movement in enclosed spaces [22; 35; 36; 37].