key: cord-0965321-t0btm49z
authors: Srinivasan, Anand; Krishan, Jayant; Bathula, Sreekanth; Mayya, Yelia S.
title: Modeling the viral load dependence of residence times of virus‐laden droplets from COVID‐19‐infected subjects in indoor environments
date: 2021-06-12
journal: Indoor Air
DOI: 10.1111/ina.12868
sha: 42888b632bd1ba6636fc862db3c39eb47b2f369c
doc_id: 965321
cord_uid: t0btm49z

In the ongoing COVID‐19 pandemic situation, exposure assessment and control strategies for aerosol transmission path are feebly understood. A recent study pointed out that Poissonian fluctuations in viral loading of airborne droplets significantly modifies the size spectrum of the virus‐laden droplets (termed as “virusol”) (Anand and Mayya, 2020). Herein we develop the theory of residence time of the virusols, as contrasted with complete droplet system in indoor air using a comprehensive “Falling‐to‐Mixing‐Plate‐out” model that considers all the important processes namely, indoor dispersion of the emitted puff, droplet evaporation, gravitational settling, and plate out mechanisms at indoor surfaces. This model fills the existing gap between Wells falling drop model (Wells, 1934) and the stirred chamber models (Lai and Nazarofff, 2000). The analytical solutions are obtained for both 1‐D and 3‐D problems for non‐evaporating falling droplets, used mainly for benchmarking the numerical formulation. The effect of various parameters is examined in detail. Significantly, the mean residence time of virusols is found to increase nonlinearly with the viral load in the ejecta, ranging from about 100 to 150 s at low viral loads (<10(4)/ml) to about 1100–1250 s at high viral loads (>10(11)/ml). The implications are discussed.

The outbreak of COVID-19 disease at superspreading events intensifies the scientific discussion on the airborne infection transfer mechanism responsible for the global spread of this disease [1] [2] [3] [4] ; most of all for combating strategies and then to the issue of rebooting economic activity. Limited information is available on the formation of virus-laden droplets/aerosols, dispersion, and evaporation of expiratory droplet cloud in the indoor environment 5, 6 . Furthermore, there is a wide variation in the input parameters such as droplet size distribution, concentration, volume of expiratory fluid, release conditions, etc. Accordingly, a considerable amount of uncertainty is associated with modeling practices used for inhalation exposure assessment.

Although there are few methods of exposure assessment available in the literature [7] [8] [9] , it is necessary to address larger scientific questions which form the basis of these calculations. In this study, we wish to focus on the issue of residence time of cough droplet and droplet nuclei, ejected from the expiratory activity of infected subjects as localized puff to start with, in an indoor space. The residence times are not only important for exposure assessment, but are also important for assessing the performance of air cleaners, or for deciding the minimum level of Clean Air Delivery Rate (CADR) expected of them, as effective intervention technologies for minimizing the infection transfer risks in confined environments.

Generally, the mean residence time is derived from the reciprocal of the removal rate, which is a concept based on uniformly mixed chamber models 10 . This concept is best suited for fine aerosol particles of about 1 μm or less, predominantly occurring in the context of indoor pollution. On the other extreme is the models of residence time for falling-evaporating droplets, as described by Wells 11 long ago and revisited with considerable mathematical precision by Xie et al. 12 , which are best suited to describe the fate of droplets of 20 μm or more. There is a wide gap in modeling the residence time for droplets in the intervening size range (say, 5-20 μm), which due to natural convective mixing in a room, will be undergoing dispersal in the air currents during their fall under gravity, and cannot be described by either of the above models. With increasing concern expressed over the yet to be understood role of aerosol route of infection spread in the COVID-19 scenario, the assessment of the impact of size range is of considerable urgency. The first motivating factor for this paper is to formulate a model that smoothly transits from the falling droplet to a uniformly mixed chamber model (Fallingto-Mixing-Plate-out) across the entire size range of polydisperse droplets released in a cough.

A second and more important motivation for this study stems from the following consideration. In a recent paper 13 , we pointed out that the size distribution metrics of corona virus-laden polydisperse droplets expirated from infected subjects, could be very different from the original droplet distributions, due to the well-known Poissonian fluctuations [13] [14] [15] [16] [17] in virus incorporation propensities into droplets at the time of their formation. To focus on this viral loaddependent distinctiveness, a terminology "virusol" was proposed for the ensemble of virus-laden droplets. From both exposure evaluation and control technology point of view, the residence time of the virusolized fraction is a matter of concern, and not that of the entire droplet population. The Poissonian theory shifts focus on larger size fraction effectively rendering the residence times sensitive to the viral load in the ejecta of subjects. Examining this dependence is an important aspect of this study.

Size distribution of expiratory droplets is one of the basic input parameters required to determine the lifetime of the virus-laden droplets in the ambient environment. There are several experimental studies on the size distribution and concentration of expiratory droplets 7, [18] [19] [20] . These studies show a large variation in the size distribution parameters of the droplets ejected from the infected subjects. Equally important is the data on viral load among patients and the variability across several findings have been summarized by Anand and Mayya 13 . The combined effect of size dispersity and viral load has a complex influence on the lifetime of the virus-laden droplets as we will see in this paper.

Computational fluid dynamics (CFD) is an increasingly adopted approach to address the transport of droplets along with their evaporative dynamic, from the point of emission to their settling on surfaces [21] [22] [23] [24] . In a strictly theoretical sense, this would be most accurate for modeling residence time distributions. However, a large number of parameters are required for carrying out simulation using fluid dynamics. Therefore, CFD approaches are highly input parameter intensive. For example, to handle turbulence in a room it would be necessary to specify all the processes responsible for creating turbulence. It should then include both mechanical and thermal sources, and this would make the CFD models quite useful in specific contexts, but lack generic predictive power. To have generality, CFD would require averaging over a large number of scenarios before arriving at the reasonable estimates of a representative droplet residence time. On the other hand, phenomenological model based on turbulent diffusion as the governing transport mechanism in addition to gravitational settling addresses the dispersive mechanisms directly through an assumed diffusion coefficient and hence will be more robust in providing generic estimates of the residence time distributions. Besides, treating droplet transport within the framework of an assumed bulk diffusion process obviates the need to examine the origin of its occurrence for the purposes of describing the migration dynamics of the droplets. Equally important, turbulent bulk diffusion models are amenable to analytical solutions for falling droplets which is useful for comparison purposes for providing insights into the role of individual processes.

In the phenomenological approach, the emitted droplet begins to disperse in bulk space as it settles under gravity simultaneously evaporating and undergoing wall removal processes. The removal rates are matched with the boundary conditions using Lai and Nazaroff model 10 . This phenomenological model integrates smoothly the wall removal as well as air exchange removal without having to concern about the points of air entry and exit for various realistic emission scenarios. We term it as a residence time distribution problem since the released droplets are not uniformly mixed in the indoor environment, and the inverse of the mean residence time is equivalent to the removal rate of uniformly mixed model. The dynamics of these evaporating and settling virusol puff is studied under different conditions by varying the important input parameters such as relative humidity, differential temperature between the droplet and ambient atmosphere, release height, bulk turbulent coefficient, friction

A recent study points out that due to Poissonian fluctuations, only a small fraction of droplets expirated from COVID-19-infected subjects would contain viruses. As 

Let us consider an expiratory event in which droplets of initial concentration (N 0 ) are released from an infected person at a height (z 0 ) in a closed environment (Figure 1 ). The convective diffusive expansion of the puff containing airborne droplets is modeled using the mixing transition model, which combines homogeneous mixed room model with falling droplet model seamlessly. Many researchers 6, 25, 26 argue that the cut-off diameter for the aerosol definition in the present context should be a single discrete number (e.g., <5 μm or <20 μm).

However, in the present study, a higher cut-off diameter (100 μm) is considered so that the transition to inhalable aerosol size from the evaporation of continuous size spectrum of ejected droplets can be modeled seamlessly. in the vertical direction (z) due to turbulence, gravitational settling, and inactivation of viruses in the droplets is given by where D is the bulk diffusion coefficient along z-direction due to turbulence, V g is the gravitational settling velocity from Equation (6) 

F I G U R E 1 Schematic diagram capturing various physical processes during a typical expiratory event in the indoor environment where V + D and V − D are the deposition velocities on floor and roof respectively, obtained from boundary layer models 10 

The integration of solution of Equations (1-2) over the spatial domain gives the droplet/aerosol volume concentration survive in air at any time t. If N 0 is the total number of particles released at time t = 0, then the mean residence time of the expiratory droplets in air is prescribed as,

The definition of residence time is more general compared to the lifetime definition used in single exponential decay expressions.

In the latter case, the mean residence time refers to time required to decrease 1/e th of original concentration, equivalent to ~63% removal in the single exponential decay process. Equation (3) is a measure of multi-exponential decay due to various processes acting on the removal/survival of the expiratory droplets.

The equations of evaporation, vertical motion, and temperature of individual droplets are solved independently, and they are coupled together with the solution of mixing transition model through gravitational settling velocity (V g ). The governing equations that describe the vertical motion of the evaporating and settling droplets are given by where N w is the number of water molecules in the airborne droplets, d p (t) is the droplet diameter, Sh(t) is the Sherwood number (ratio of convective mass transfer to the mass diffusivity), given by

is the total number of molecules (non-volatiles/residue + water)

in the droplet, and p d (t) is the equilibrium vapor pressure of water in the droplet at temperature T d (t). k g is the thermal conductivity of air,

is the Nusselt number used to correct the convective heat transfer rate, given by 

The total volume (number) and size distribution of droplets per unit volume of exhaled gas largely depend on the expiratory activities and it is an important input parameter to this model. The droplet size distribution functions are generally obtained by fitting the measurement data using two-parameter lognormal distribution characterized by median diameter and its corresponding geometric standard deviation. In the present study, the lognormal distribution of droplet number concentration is given by

where N 0 is the total number of droplets ejected, CMD is count median diameter, and g is the geometric standard deviation (GSD).

Since atomization is the method of droplet ejection from the infected subject, the probability of viral load in each droplet will depend on the virus concentration (v c ) in the biological fluid and droplet size 14, 28 

The dispersion of a puff containing virus-laden droplets and aerosols exhaled during a cough expiratory event by an infected subject is analyzed using 1-D model described in the Materials and Methods section.

The puff is released at a height of 1. 

The rate of evaporation of droplets and its composition determines the final size of airborne droplets/aerosols (virusols) which play a key role in governing their residence time in the indoor atmosphere. To study these effects, the virusols residence time is estimated at different RH, and compared with that of droplets without solute. The temporal evolution of single droplet diameter (14 μm is chosen since its CMD of the cough droplets considered in this study) at different RH and composition is shown in Figure 3A . At a lower RH (10%), Furthermore, the final size of all the droplets as a function of initial droplet size is presented in Figure 3B . The overall reduction in the final size is about ~55% for all particle sizes (i.e., droplet with a size of 100 µm reduces to ~45 µm) in the case of RH = 50% and for the given residue composition. This size reduction is more pronounced if solute content in the saliva is reduced as discussed above.

It is to be noted that the initial droplet may be large enough to settle down under the force of gravity but simultaneous evaporation will reduce the size significantly (depending upon the atmospheric conditions), thereby reducing the gravitational effects and leading to higher residence time in the air.

The effect of AER on virusol mean residence time is studied with two different relationships of AER vs D ((i) D as a function of AER given These results show that the virusol mean residence time decreases monotonically when the AER increases; however, the reduction is only 0.43 times when the AER is increased from 0 h −1 to 5 h −1 . Hence, a large AER is required to reduce the virusol exposure significantly.

The virusol mean residence time of droplets as a function of initial droplet size is compared for different heights of release (z 0 = 0. spectively. This shows that there is very low probability that smaller droplets will contain any virus in the case of low viral load (mild-tomoderate cases) in biological fluid (e.g., saliva). It is interesting to note that there is a larger difference in the concentrations at the lower side of the size spectrum between complete droplets and virusols as compared to that at larger sizes. This is due to the fact that the value of P (Equation (9)) tends to unity for large droplets (size >40 μm).

Comparison of number-size distribution at t = 100 s with other initial size spectra shows that most of the particles greater than ~70 μm are removed from the system mainly due to gravitational settling; also, the evaporation process plays an important role in shifting the droplet spectrum in the range of (22-70) μm significantly ( Figure 6 ).

We further study the effect of virus concentration on the residence time and exposure for a wide range of concentration from 10 3 to 10 13 RNA copies/ml, and the results are presented in Figure 7 for various RH. The virusol residence time varies from ~100 s (RH = 90%

for viral load <10 6 RNA copies/ml) to ~1200 s (RH = 10% and viral load >10 11 RNA copies/ml). The results further show that MRT of virusols in the severe cases (viral load >10 11 RNA copies/ml) is eight times to that of mild-to-moderate cases (<10 6 RNA copies/ml) for RH = 50%, and this ratio increases with RH. As RH increases from 10% to 90%, the evaporation rate reduces which leads to larger final Study results indicate that virusol system residence times could be far less than that for the complete droplet cloud system, and smaller droplets (<20 μm) are more likely to be blanks in mild-to-moderate cases. Virusol system will settle down faster than the complete droplet system and hence less likely to be airborne when the biological viral load is <10 8 RNA copies/ml. However, greater viral load in the biological fluid leads to longer residence time of virusol system. This has important implications from an air cleaning perspective aimed at COVID-19 risk mitigation in enclosed spaces. It may not be necessary to capture ultrafine particles as these would be generally harmless; one may relax filtration efficiency criteria limiting them to essentially larger particles, thereby opting for coarser filters and reducing pressure Riya Dey for the preparation of the artwork (Figure 1 ).

The authors declare no competing interests. 

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