## 3 Materials and Methods
### Development of Numerical Model  
#### 3.3.1 Governing Equations  
The governing equations are written for the fluid and particle phases as follow [45]:  
Fluid Phase  
(a) The mass conservation equation  
The equation for conservation of mass, or continuity equation, can be written as follows:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\overset{\rightarrow}{v} \right)=S_{m} \tag{5}\]  
where \(\rho\) is the fluid density, \(v\) is the fluid velocity vector, \(t\) is the time domain, and \(S_{m}\) is a source term that represents the mass added from the particles/droplets. (b) The momentum conservation equation  
The conservation of momentum is described by:  
\[\frac{\partial}{\partial t}\left(\rho\overset{\rightarrow}{v}\right)+\
abla \cdot\left(\rho\overset{\rightarrow}{v}\overset{\rightarrow}{v}\right)=- \
abla\text{p}+\rho\overset{\rightarrow}{g}+\overset{\rightarrow}{F} \tag{6}\]  
where \(p\) is the static fluid pressure, \(\rho\overset{\rightarrow}{g}\) is the gravitational body force, and \(\overset{\rightarrow}{F}\) is the external body force by the action of the droplets/particles. (c) The turbulence transport equations  
The transport equations for the RNG-k-\(\varepsilon\) turbulent model are given as:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}\left(\sigma_{k}\mu_{eff}\frac{\partial k }{\partial x_{j}}\right)+G_{k}+G_{b}-\rho\varepsilon-Y_{M}+S_{k} \tag{7}\]  
\[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}}( \rho\varepsilon u_{i})=\frac{\partial}{\partial x_{j}}\left(\sigma_{\varepsilon }\mu_{eff}\frac{\partial\varepsilon}{\partial x_{j}}\right)+C_{1\varepsilon} \frac{\varepsilon}{\bar{k}}(G_{k}+C_{3\varepsilon}G_{b})-C_{2\varepsilon} \rho\frac{\varepsilon^{2}}{k}-R_{\varepsilon}+S_{\varepsilon} \tag{8}\]  
where \(k\) is the turbulence kinetic energy, \(\varepsilon\) is the turbulence dissipation rate, and \(G_{k},G_{b}\) represent the generation of turbulence kinetic energy due to the mean velocity gradients and buoyancy, respectively. \(Y_{M}\) represents the effect of compressibility on turbulence, and \(S_{k}\) and \(S_{\varepsilon}\) are source terms for \(k\) and \(\varepsilon\) equations, respectively. \(\sigma_{k}\) and \(\sigma_{\varepsilon}\) are the turbulent Prandtl numbers for \(k\) and \(\varepsilon\), respectively. \(C_{1\varepsilon}\), \(C_{2\varepsilon}\), and \(C_{3\varepsilon}\) are constants. \(\mu_{eff}\) is the effective eddy viscosity. Particle Phase  
The trajectory of a discrete particle is described by the following equation  
\[\frac{d\overset{\rightarrow}{u}_{p}}{dt}=\frac{\overset{\rightarrow}{u}- \overset{\rightarrow}{u}_{p}}{\tau_{t}}+\frac{\overset{\rightarrow}{g} \left(\rho_{p}-\rho\right)}{\rho_{p}}+\overset{\rightarrow}{F} \tag{9}\]  
where the Saffman's lift force \(\overset{\rightarrow}{F}\) is calculated using the following equation  
\[\overset{\rightarrow}{F}=\frac{2 Kv^{1/2}\rho d_{ij}}{\rho_{p}d_{lp}(d_{lk}d_{ kl})^{1/4}}(\overset{\rightarrow}{u}-\overset{\rightarrow}{u}_{p}) \tag{10}\]  
where \(d_{p}\) is the particle size, \(\rho_{p}\) is the density of the particles, \(d_{ij}\) is the deformation tensor, \(\overset{\rightarrow}{u}_{p}\), is the velocity vector of the particles, and \(K\), is a constant. In this study, the FLUENT-ANSYS package was used to simulate the airflow and the transmission of the airborne particles for the two validation experiments and the classroom case study. 3-D unsteady state airflow models were developed. The models generallyresolve the details of eddies structures of the turbulent airflow field by using several two-equation turbulence models, such as the standard k- epsilon model, RNG k-epsilon model, and k-omega model. The Discrete Phase Model (DPM), which follows a Eulerian-Lagrangian approach, was adopted to track the airborne particles' transmission and the bioaerosols via the airflow field.