**3. Calculation of the infected droplets' trajectory and viral load**

To calculate droplets' trajectory and viral load variation, the model proposed by [15] was considered. However, a premise must be highlighted. Only part of the volume of the infected droplets is occupied by saliva, as virions occupy the remaining part. For example, considering the voidage ratio (i.e., the difference between the droplet's volume and the sum of the virions' volumes, divided by the volume of the infected droplets) in analogy with similar physical problems, only a part of the infected droplets' volume consists of a liquid that can evaporate. However, to date, the literature has no answer to RQ3: What effect the virions can induce on the droplet evaporation process? Therefore, it cannot be excluded that an infected droplet evaporates faster than a pure water droplet and that the evaporation time is proportional to the voidage ratio. On the other hand, it is not known, for example, if the virions hinder the mass transfer phenomenon by retaining water in the droplet. Similarly, virions' heat capacity is unknown, i.e., how long the virions take to reach thermal equilibrium with the water surrounding them. Therefore, due to the existing research gaps, the following assumption was made: the reduction of the infected droplet's evaporation time due to the lower volume of aqueous solution contained is counterbalanced by the potential obstacle the virions could cause during the evaporation mechanism. Therefore, the mass variation due to evaporation is calculated as

$$\frac{dm\_G}{dt} = \frac{2\pi p D\_G M\_V D\_\infty \mathcal{C}}{R\_0 T\_\infty} \ln\left(\frac{p - p\_{\rm ua}}{p - p\_{v,\infty}}\right). \tag{7}$$

*Perspective Chapter: Analysis of SARS-CoV-2 Indirect Spreading Routes and Possible… DOI: http://dx.doi.org/10.5772/intechopen.105914*

where *mG* is the mass of the droplet [kg], *DG* is the diameter of the droplet [m], *D*<sup>∞</sup> is the vapor diffusion coefficient in the air surrounding the droplet [m2 /s], *Mv* is the molar mass of the vapor in [kg/kmol], *R*<sup>0</sup> is the universal gas constant in [J/kmolK], *pva* and *pv*,<sup>∞</sup> are, respectively, the partial pressure of the vapor on the surface of the droplet and far away in [Pa], and *p* and *T*<sup>∞</sup> are the air pressure [Pa] and temperature [K], respectively. *C* is a corrective coefficient that takes into account the presence of other constituents in human saliva different from pure water. The temperature variation *TG* on droplet surface [K] due to the evaporation phenomenon is calculated as

$$m\_{G}c\_{L}\frac{dT\_{G}}{dt} = 2\pi D\_{G}^{2}K\_{\xi}\frac{T\_{\circ\circ}-T\_{G}}{D\_{G}} + r\frac{dm\_{G}}{dt} - \pi\Gamma\left(T\_{G}^{4}-T\_{\circ\circ}^{4}\right). \tag{8}$$

where *cL* is the specific heat in [kJ/kg�K], *Kg* is the thermal conductivity of the gas in [kJ/s�m�K], *r* is the latent heat of vaporization in [kJ/kg], Γ is the Stefan-Boltzmann constant in [kW/m<sup>2</sup> �K4 ]. In the second-member heat balance, therefore, there are three contributions, namely, heat conduction (first term), heat convection (second term), and heat radiation (third term). To calculate droplets' trajectory, the following equation was solved:

$$m\_G \overrightarrow{a\_G} = m\_G \overrightarrow{\mathbf{g}} \left( \mathbf{1} - \frac{\rho\_a}{\rho\_G} \right) - \mathbf{C}\_w A\_G \rho\_g \frac{\left| \boldsymbol{v\_{rel}} \right|^2}{2} \times \frac{\overrightarrow{\boldsymbol{v\_{rel}}}}{\left| \boldsymbol{v\_{rel}} \right|} \tag{9}$$

where *aG* is the droplet's acceleration [m/s<sup>2</sup> ], *g* is the gravity acceleration [m/s<sup>2</sup> ], *ρ<sup>a</sup>* and *ρ<sup>G</sup>* are, respectively, the air and the droplet's densities [kg/m<sup>3</sup> ], *Cw* is the Stokes coefficient [�], *AG* is the cross-sectional area of the droplet, and *v*rel is the relative velocity between the drop and the surrounding air. Since the droplets' diameter along the horizontal distance is known, the droplet viral load [RNA copies/mL] is calculated as

$$\begin{cases} \lambda(\mathbf{x}) = \frac{\lambda\_0 D\_{G,0}^3}{D\_G^{-3}(\mathbf{x})}. \\\ \lambda(\mathbf{x}) = \lambda\_{\text{max}} \end{cases} \tag{10}$$

In Eq. (10), λmax is calculated as the ratio between the number of virions and the volume of the minimum droplet in which they can be contained. Therefore, since virion volume is assumed to not change during droplets' evaporation, the voidage ratio decreases to a minimum, which corresponds to the maximum allowable concentration λmax. To date, no data about this maximum concentration exist. To cover the gap, the most conservative assumption is made. Specifically, the minimum voidage ratio of a bed of sphere, i.e., 39%, from [16] is considered. Therefore, the maximum viral load is equal to

$$\lambda\_{\text{max}} = \frac{\sum N\_{\text{virions}}}{V\_G} = \frac{(1 - \rho) \times \frac{V\_G}{V\_{\text{virions}}}}{V\_G} = \frac{(1 - \rho)}{V\_{\text{virions}}} = \mathbf{1.165} \times \mathbf{10}^{15} \left[\frac{\text{RNAcopies}}{\text{mL}}\right]. \tag{11}$$

where *N*virions is the number of virions contained in the droplet [#], *VG* is the droplet's volume [mL], φ is the minimum void ratio (assumed equal to 39%), and *V*virion is the mean virions' volume [mL]. For the analysis, an average virion diameter of 100 nm is assumed [17].
