## 4 Hydrodynamical Modeling of Direct Exposure  
In the previous sections, we infer the submicron characteristics and rate of emission of respiratory droplets expected to be carried by exhaled ECA. We need to estimate now how far can these respiratory droplets be carried to evaluate the distance for direct exposure of bystanders to pathogens potentially carried by these droplets  
Exhaled ECA is injected into surrounding air a given horizontal distance roughly in the direction of the exhaled flow. Since it involves a finite fluid mass of a SFF aerosol during a finite injection time (exhalation time), the appropriate dynamical model for it is a turbulent puff with a starting momentum dominated jet that lasts while the fluid injection is on (see [103; 104; 105; 106; 107; 108; 109]). A schematic description of this system is furnished by Figure 1. We are not concerned with the few larger ECA or respiratory particles (diameters \(d_{p}\), \begin{table}
\begin{tabular}{c c c c} \hline
**Study Authors** & **Droplet Numbers \(N_{r}\),** & **Subjects** & **Comments** \\
**and Reference** & **Density \(n_{r}\) and Diameters \(d_{r}\)** & & **and Technique** \\ \hline [PERSON] and [PERSON] & Mean \(N_{r}=12.5/\text{L}\), (\(<1\) μm) & 5 healthy & Table 2 \\
[89] & Mean \(N_{r}=1.9/\text{L}\), (\(>1\) μm) & & OPC, EM \\ \hline [PERSON] and [PERSON] & \(n_{r}<0.25/\text{cm}^{3}\) (\(V_{T}\)) & 17 healthy & Figures 3 and 7. \\
[93] & up to \(n_{r}=2.5/\text{cm}^{3}\) (deep) & ages 19–60 & BH decreases \\  & & & droplet numbers \\  & & APS \\ \hline [PERSON] et al. & Mean \(N_{r}=98/\text{L}\) & 15 healthy & nose inhalation \\
[94] & Mean \(d_{r}=0.8\) μm & ages \(<35\) & and mouth exhalation \\  & & & APS \\ \hline [PERSON] et al. & \(N_{r}=230/\text{L}\), (18–1000)/\(\text{L}\) & 10 healthy & Tidal Volume \\
[95] & \(d_{r}=0.3\text{--}0.4\) μm & ages 29–69 & Tables 2 and 3 \\  & 98\(\%\)\(d_{r}<1.0\) μm & & OPC \\ \hline [PERSON] et al. & Median \(n_{r}=3.1\) (0.6–82)/\(\text{cm}^{3}\) & 16 healthy & Tables 3 and 4 \\
[96] & \(d_{r}=0.07\) μm & & Two super emitters \\  & \(V_{n_{r}}=351\text{--}1701\) cm\({}^{3}\) & & SMPS \\ \hline [PERSON] et al. & \(N_{r}\approx 10\text{--}50/\text{exh}\) & 21 healthy & Close to \(V_{T}\) \\
[97] & Median \(d_{r}=0.28\) μm & (4 smokers) & \(V_{T}/V_{C}\approx 0.2\) \\  & & & CNC \\ \hline [PERSON] et al. & GMean \(N_{r}=7.4/\text{L}\) LE & 19 subjects & 4 HE \\
[90] & GMean \(N_{r}=3500/\text{L}\) HE & (7 asthmatic) & Table 1 \\  & 82\(\%\)\(d_{r}=0.3\text{--}0.5\) μm & & OPC \\ \hline [PERSON] et al. & Median \(N_{r}=38.3(3.3\text{--}1456)/\text{L}\) & 79 healthy & 4–19\% high emitters \\
[91] & 90\(\%\)\(N_{r}<150/\text{L}\), LE & (14 asthmatic) & follow up of subjects \\  & 99.9\(\%\)\(d_{r}<1.0\) μm & & OPC \\  & 75\(\%\)\(d_{r}<0.5\) μm & & \\ \hline [PERSON] et al. & \(N_{r}\approx 10/\text{exh}\) LE & 29 healthy & Figures 2 and 4 \\
[98] & up tp \(N_{r}\approx 1000/\text{exh}\) HE & (13 smokers) & Close to \(V_{T}\) \\  & Median \(d_{r}=0.3\) μm & 28 COPD & \(V_{T}/V_{C}\approx 0.2\) \\  & & 10 asthmatic & CNC \\ \hline [PERSON] et al. & \(N_{r}\approx 1/\text{s}\) & 48 healthy & Figure 5 \\
[92] & \(d_{r}=0.75\text{--}1.0\) μm & age 18–45 & much larger in speech \\  & \(n_{r}<0.1/\text{cm}^{3}\) & & than in breathing \\  & & 10 asthmatic & APS \\ \hline \end{tabular}
\end{table} 
Table 2: Droplet emissions for mouth breathing and tidal volume. The symbols \(N_{r}\) and \(n_{r}\) stand for respiratory particle (droplet or nuclei) number per exhalation and average number density (\(\text{cm}^{-3}\)). LE, HE, BH, L, and “exh” are Low Emitters, High Emitters, Breath Hold, L and exhalation, while OPC, EM, APS, SMPS and CNC are optical particle counter, electron microscopy, aerodynamic particle size, scanning mobility particle sizeer and condensation nucleus counter, respectively. Table and figure numbers (4 th column) correspond to articles cited in the Table. The subjects in all studies (save possibly the study by [PERSON] and [PERSON] [93]) breathed through a mouthpiece wearing a nose clip.