AUTORES
ESTUDIO
Persistencia del aerosol en relación con la posible transmisión del SARSCoV2
ABSTRACT
The evaporation of a spherical droplet in an environment with a known relative humidity (RH) can be evaluated using the diffusion model presented and validated in Ref. 30. The rate of change in the mass of the droplet, m_{d}(t), is given by
∂md(t)∂t=4πR2(t)Dva∂C(r,t)∂r∣∣∣r=R(t),∂mdt∂t=4πR2tDva∂Cr,t∂rr=Rt,

(1) 
where R(t) is the radius of the droplet, D_{va} is the diffusivity of water vapor in air, and C(r, t) is the water vapor concentration along direction r. Assuming that the droplets are sufficiently spaced and that the relative humidity of the air in which they are falling through does not change, the final term can be written as
∂C(r,t)∂r∣∣∣r=R(t)=(C(r=∞)−C(R(t),t)(1R(t)+1πDt‾‾‾‾√).∂C(r,t)∂rr=R(t)=Cr=∞−CRt,t1R(t)+1πDt.

(2) 
The water vapor concentration at the surface of the droplet [i.e., r = R(t)] is given by the equilibrium vapor pressure, ρ_{vap}, of the environment and, very far away from the droplet surface [i.e., r ≫ R(t)], is given by the product of the RH of the environment and ρ_{vap}, resulting in
∂md(t)∂t=4πR2(t)Dvaρvap(RH−1)(1R(t)+1πDt‾‾‾‾√).∂md(t)∂t=4πR2tDvaρvapRH−11R(t)+1πDt.

(3) 
Assuming that the solids (salt, proteins, and possibly virus particles) constitute a “spherical core” of the droplet, the mass, m_{d}, of the droplet at any time is given by
md(t)=4π3R30ρs+4π3(R3(t)−R30)ρw,mdt=4π3R03ρs+4π3R3t−R03ρw,

(4) 
where ρ_{s} is the density of the solid found in human mucus/saliva (i.e., 1500 kg/m^{3}) from Ref. 31 and ρ_{w} is the density of liquid water. Differentiating Eq. (4) with respect to time and combining the result with Eq. (3) give a nonseparable differential equation for the evolution in size of the droplet due to evaporation, where evaporation stops when the droplet is completely composed of the solid fraction or when R(t) = R_{0},
∂R(t)∂t=ρvapDvaρw(RH−1)(1R(t)+1πDvat‾‾‾‾‾√).∂R(t)∂t=ρvapDvaρwRH−11R(t)+1πDvat.

(5) 
For the purpose of the following calculations, it is taken that the solid core R_{0} of each droplet is half of the initial size R(t = 0) and corresponds to an initial density of ∼1080 kg/m^{3} for the water–solute mixture. Figure 3 displays solutions to Eq. (5) for the largest microdroplet sizes and shows the influence of the RH on the evaporation kinetics of a 10 µm droplet. Within 1 s, the evaporation of the small microdroplets is complete, resulting in a solid core.
Due to the fact that the evaporation occurs quickly, the dominant mode of decline in suspended droplets is sedimentation. As we will show below, the exponential decay in the number of drops that we observe can be quantitatively accounted for by taking only the sedimentation of already evaporated droplets into account. At all times, the droplets are assumed to be vertically falling at their terminal velocities described by Stokes flow,
∂h(R(t),t)∂t=2(ρd(t))9ηgR2(t).∂h(Rt,t)∂t=2ρdt9ηgR2t.

(6) 
This describes the rate of change in the height, h(R(t), t), through which the droplet has fallen where ρ_{a} is the density of air and g is the acceleration due to gravity. By solving Eqs. (5) and (6) numerically, the progressive evaporation and sedimentation of the droplets are coupled and comparable to models presented in Refs. 5 and 32. For the framework presented herein, how the number of droplets in a given volume evolves can be predicted, allowing the persistence calculations in Fig. 4 to be made. For this calculation, it is assumed that the droplets of each size class have a uniform random initial height in the volume in which they progressively sediment. From the particle size distribution, the total number of particles of each size class, N(R(t = 0)), initially in the volume can be obtained. The evolution in the total number of particles for each size class is then directly given by N(R(t),t)=N(R(t=0))h(R(t),t)hsysNRt,t=NRt=0h(Rt,t)hsys, taking h(R(t), t) to always be smaller than the system height h_{sys} in which dispersion experiments are made and the computational domain height in which the sedimentation and evaporation of the droplets are calculated. The total number of droplets in the system, N_{total}, at any time t is then the discrete summation of this number over all particle sizes, n, for which h(R(t), t) < h_{sys}.
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