## 2 Evaluation Study and Design
### Fluid-Dynamic Methods and Simulations
#### 2.3.1 Lagrangian Method  
In the Lagrangian approach, the motion and temperature fields within the environment are initially simulated under steady-state conditions. Subsequently, particle input conditions are imposed, with the particle trajectories derived from a temporal dynamic calculation (i.e., using a transient model) based on the balance of forces acting on the particles, specifically aerodynamic forces. This simulation method allows for a more accurate reproduction of the distribution and diffusion of the aerosol particles emitted by an infected individual through breathing, talking, and coughing. However, as the duration of the simulation increases, the computational power and time required grow exponentially.  
The particle distribution considered was characterized with the spectrum summarized in Table 4, as described in [30; 31].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Season** & **Characteristic** & **Value** \\ \hline \multirow{5}{*}{Summer} & T\_Human\_Head [K] & 310 \\ \cline{2-3}  & T\_Human\_Body [K] & 300 \\ \cline{2-3}  & Air\_Density [kg/m\({}^{3}\)] & 1.1762 \\ \cline{2-3}  & T\_Windows [K] & 302 \\ \cline{2-3}  & T\_Walls [K] & 300 \\ \cline{2-3}  & T\_Office [K] & 300 \\ \cline{2-3}  & T\_Air [K] & 298 \\ \cline{2-3}  & Volume\_Flow\_Rate\_X\_Fan [m\({}^{3}\)/s] & 0.0333–0.0666 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Boundary conditions for summer conditions.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Activity** & **Normal Speaking** & **Breathing** & **Normal Breathing** \\ \hline \multirow{2}{*}{Exhale Velocity [m/s]} & Sinusoidal (0–1) & \multirow{2}{*}{1} & Sinusoidal 0–1 \\  & and peak of 5 & & \\ \hline Duration [s] & 9.2 & 12.5 &
\begin{tabular}{c} 40 (20 top of the area) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 3: Infected individual’s emission conditions.  
Respiratory droplets are inherently complex, comprising mixtures of water, salts, proteins (e.g., mucin), and biological materials such as virions. Nevertheless, the modelling approach employed in this study simplifies droplets as single-component spherical particles to concentrate on their aerodynamic transport and dispersion behavior rather than their intricate internal composition or biological viability.  
The representation of respiratory droplets as spherical particles is widely accepted in the literature, particularly for the purposes of airflow interaction and transport, as demonstrated in previous studies [7; 32; 33; 34; 35]. These models have effectively captured the fundamental dynamics of droplet sedimentation and airborne suspension.  
Moreover, due to the rapid evaporation of the water content [32; 36], the residual droplet nuclei primarily consist of non-volatile materials and virions. Previous studies [5; 37] have shown that the trajectory and suspension time of these nuclei are predominantly governed by their aerodynamic size and environmental conditions rather than their detailed chemical composition. Once the water has evaporated, the particle's effective size stabilizes, and its subsequent motion can be reasonably approximated using a spherical model.  
This simplification may constrain the ability to predict virion viability or detailed droplet-phase behavior. However, the primary focus of this study is on modeling transport pathways and exposure potential. In this context, the assumption of spherical droplets offers a computationally efficient and physically reasonable representation.  
By employing the Lagrangian approach, the trajectory of emitted aerosol particles can be accurately tracked. This method necessitates transient simulations, which are constrained by the requirement for substantial computational power and extended processing times. In these transient models, the emission from the infected individual is represented using a sinusoidal function, meaning the digital dummy emits particles at intervals with modulated intensity.