Airborne virus transmission via respiratory droplets: Effects of droplet evaporation and sedimentation

Abstract

Airborne transmission is considered as an important route for the spread of infectious diseases, such as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), and is primarily determined by the droplet sedimentation time, that is, the time droplets spend in air before reaching the ground. Evaporation increases the sedimentation time by reducing the droplet mass. In fact, small droplets can, depending on their solute content, almost completely evaporate during their descent to the ground and remain airborne as so-called droplet nuclei for a long time. Considering that viruses possibly remain infectious in aerosols for hours, droplet nuclei formation can substantially increase the infectious viral air load. Accordingly, the physical-chemical factors that control droplet evaporation and sedimentation times and play important roles in determining the infection risk from airborne respiratory droplets are reviewed in this article.

Keywords: Airborne virus transmission; Droplet evaporation; Droplet nuclei; Droplet sedimentation; Wells model.

Graphical abstract

Figure 1

A main question regarding the airborne transmission of infection is how long human respiratory droplets stay floating in air. To answer this question, the evaporation and sedimentation processes of saliva droplets have to be characterized. If droplets are small enough to completely evaporate to so-called droplet nuclei before they hit the ground, they will remain airborne for hours. Larger droplets, however, fall to the ground in a few seconds. Sedimentation and evaporation times of droplets are controlled by various physical-chemical effects and relevant parameters, which are listed in the diagram.

Figure 2

Evaporation and sedimentation times of pure water droplets as a function of the initial droplet radius $R_0$ for an initial height of $z_0=1.5m$. Results are shown for different relative humidities. Solid and broken lines indicate the evaporation times (Eq. (2)) and the sedimentation times (Eq. (4)), respectively. In the limit of $RH=1$, no evaporation takes place, and Eq. (4) yields a sedimentation time that equals Eq. (1) (shown by a dotted line), which neglects the evaporation-induced variation of the droplet size. Droplets with initial radii below the critical radius $R_0^{crit}$ given by Eq. (5) (which is the initial radius at which the evaporation and sedimentation times are equal) completely evaporate before they hit the ground, and thus, their sedimentation time is infinity.

Figure 3

Variation of the droplet radius $R$ with time $t$ in the presence of nonvolatile solutes according to Eq. (7). The liquid solution is assumed ideal $\left(\gamma =1\right)$, and data are shown for initial droplet radius $R_0=50\mu m$, relative humidity $RH=0.5$, and two different initial solute volume fractions of ${\text{Φ}}_0={10}^{-3}$ (main figure) and ${\text{Φ}}_0={10}^{-2}$ (inset). The y-axis is rescaled by $R_{ev}$, the equilibrium radius of the droplet at the end of the evaporation process (Eq. (6)). Solid and dashed lines indicate the results considering and neglecting the effect of the solute-induced water vapor-pressure reduction (which is reflected by the logarithmic term in Eq. (7)), respectively.

Figure 4

Evaporation and sedimentation times as a function of the initial radius $R_0$ in the presence of nonvolatile solutes, for an initial height of $z_0=1.5m$. Panel (a) shows results for a fixed initial solute volume fraction ${\text{Φ}}_0=0.01$ and different relative humidities. Panel (b) shows results for a fixed relative humidity $RH=0.5$ and different initial solute volume fractions. Solid and broken lines indicate the evaporation and sedimentation times, which are obtained from Eqs. (8), (9), respectively. For ${\text{Φ}}_0=0.01$ and $RH=0.99$, and ${\text{Φ}}_0=0.5$ and $RH=0.5$, no evaporation takes place, and Eq. (9) recovers the result of Eq. (1) (shown by dotted lines).

Figure 5

Evaporation time as a function of the initial solute volume fraction ${\text{Φ}}_0$. The liquid solution is assumed ideal $\left(\gamma =1\right)$, and the initial droplet radius is $R_0=50\mu m$. Panel (a) shows results for different relative humidities. Solid lines indicate results from Eq. (11), which accounts for the solute-induced water vapor-pressure reduction, the presence of internal concentration and diffusivity profiles, and the solute-concentration dependence of evaporation cooling. Dashed lines show the results from Eq. (10), which neglects the presence of internal concentration and diffusivity profiles and the solute-concentration dependence of evaporation cooling, but accounts for the solute-induced water vapor-pressure reduction. Dotted lines indicate the results from Eq. (8), in which the water vapor-pressure reduction is also neglected. Panel (b) shows the evaporation times estimated from Eq. (11) (solid line) along with those obtained from numerical solutions of the heat-conduction and water-diffusion equations for fixed relative humidity $RH=0.75$. Open squares indicate results that account for the solute-induced water vapor-pressure reduction, the presence of internal concentration and diffusivity profiles, the solute-concentration dependence of evaporation cooling, and the dependence of the water diffusivity on the local solute concentration profile. Downward triangles are obtained for infinitely rapid water diffusion in the droplet $D_w^{sol}\to \infty$ (i.e. neglecting internal water concentration gradients), filled circles are obtained for a constant but finite water diffusivity inside the droplet, and upward triangles are obtained for a constant finite water diffusivity inside the droplet and additionally neglecting the solute-concentration dependence of the evaporation cooling effect by setting ${\theta }^{sol}=\theta$ (Eq. (3)) in numerical calculations.