The COVID19 pandemic has spotlit the scientific field of fluid dynamics governing airborne transmission through virusladen mucosalsalivary droplets. In this work, a mathematical model for airborne droplet dispersion and viral transmission centered on evaporating droplets containing solid residue was proposed. Droplet dynamics are influenced by factors such as initial velocity, relative humidity (RH), and solid residue, in agreement with analytical and experimental results. Interestingly, the maximum droplet dispersion distance depends strongly on initial droplet size and RH, such as 0.8mmdiameter droplet at 0.3 RH, 1.0 mm at 0.6 RH, and 1.75 mm at 0.9 RH, but only weakly on initial projected velocity. Under realistic conditions, an evaporating sputum droplet can cover a dispersion distance at least three times than that of a pure water droplet. Based on Wells falling curves, the critical droplet size, the largest droplet that can remain suspended in air without settling due to gravity, ranges from 120 μm at 0.3 RH to 75 μm at 0.9 RH. Together, our results highlight the role of evaporation on droplet lifetime, dispersion distance, and transmission risks.
I. INTRODUCTION
The ongoing pandemic of coronavirus disease (COVID19) caused a large global crisis with a death toll in the millions.^{1} A reason for its infectiousness lies in the mechanism of spreading, which for the case of COVID19, is in the form of droplet transmission through the air route.^{2} During respiratory activities such as breathing, talking, coughing, and sneezing, mucosalsalivary fluid is expelled from an infected carrier in the form of aerosolized droplets. These droplets contain infectious pathogens and can be transferred to another person over a distance. Small droplets can be borne by air movements, inhaled, or evaporate into driedout solid residue that persists due to low settling rates.^{3} People who are in direct or indirect contact with the droplets emitted from the infected person are exposed to COVID19 infection risks.
The transmission pathways for COVID19 are broadly known by three routes, namely, inhalation of virusladen droplets, droplet deposition on a surface followed by contact due to face or mouth touching, and airborne transmission in the form of droplet nuclei.^{2,4} Violent expiratory activities such as coughing and sneezing can generate turbulent jets with a Reynolds number up to 10^{4} (Refs. 5 and 6) as the typical droplet expulsion velocity ranges from 10 m/s for coughing^{7} to 20 m/s for sneezing^{8} (assuming a typical mouth size of 2 cm). In addition to the spread of droplets, indirect transmission via surface contamination cannot be underestimated. The possibility of fomitemediated transmission of COVID19 has been supported by a Ministry of Health's report on a cluster in a church in Singapore, where a case was linked to two cases from Wuhan.^{9} Airborne transmission of COVID19 has also been widely recognized as it is supported by much epidemiological evidence during the pandemic.^{10–13}
Numerous studies have reported the droplet number, the size distribution, and the droplet velocity from different respiratory activities.^{14–16} These activities produce various amounts of droplets with widely varying sizes. Measured droplet counts range over several orders of magnitude O(1–1000), with size distributions over O(0.1–1000) μm, depending on droplets formation and the flow passages.^{17,18} Generally, respiratory droplets constitute of both water and nonvolatile solid components such as salt and pathogens in the form of protein and lipids.^{19} Water evaporates from the droplet after being released from its source and exposed to unsaturated air. Upon complete evaporation, the droplet dries into solid residue or droplet nuclei, which could contain virions.
The fate of a droplet is governed by evaporation, inertia, and gravity forces, where the droplet size plays an important role. Wells^{20} defined a critical diameter of 100 μm to differentiate between large and small droplets. The critical droplet size refers to the maximum size of a droplet that can remain suspended in air without settling due to gravity. Droplets larger than 100 μm typically settle to the ground by gravity before they evaporate completely.^{2} In contrast, small droplets less than 100 μm tend to evaporate out before settling and therefore persist airborne for a longer period. The evaporation of droplets depends on many factors such as the ambient temperature and humidity. Furthermore, the evaporation rates of pathogen laden droplets such as mucosalivary fluid are not well understood due to the presence of their complex nonvolatile composition. Therefore, it is important to understand the droplet evaporation to study COVID19 transmission.
Fluid dynamics plays an important role in almost every aspect of the COVID19 pandemic, from droplet formation to droplet dispersion. Computational fluid dynamics (CFD) has emerged as a popular tool to address droplet dispersion under different indoor and outdoor environments.^{21–29} These CFD studies can numerically simulate the flight trajectories and travel distances of droplets produced from expiratory activities, which are affected by many factors such as the environmental relative humidity (RH), wind speed, and temperature. However, it is still challenging to perform direct fullscale simulation of droplet dispersion in real environments of COVID19 transmission. It can also be timeconsuming for CFD to evaluate the impact of every factor affecting the spread.
On the other hand, theoretical studies can provide additional avenues for rapid evaluation as well as a fundamental understanding of how a droplet travels under certain standardized scenarios to evaluate the airborne transmission risks.^{30–33} Even though it provides fewer details compared to numerical simulations, analytical or semianalytical solutions are far less computationally expensive and convenient compared to largescale numerical simulations in terms of gauging droplet dispersion distances. Most of the existing theoretical work is mainly applicable to nonevaporative droplets.^{32,33} Liu et al.^{34} and Xie et al.^{35} have considered saline solution to evaluate the effect of dried droplet nuclei size on the droplet dispersion. Their studies and results provide important information for developing nonpharmaceutical measures to control infectious diseases in the community. Wang et al.^{36} proposed a continuous random walk (CRW) algorithm to simulate the turbulent fluctuation during violent expiratory activities. The model has significant improvement for the prediction of the mediumsized droplet dispersion.
In general, respiratory droplets contain not only dissolved particles such as salt (0.89%) but also nondissolved particles including protein (2.3%), lipid (1.9%), carbohydrate (1.3%), and DNA (0.08%) based on the mass percentage.^{19} Understanding the composition and behavior of the solid residue in droplets is crucial toward determining potential transmission risks of respiratory diseases. In view of this, the current work proposed a simple and realistic physical model to quantify droplet evaporation and dispersion. Unlike existing homogeneous phase models in the literature, our model considers the solid residue separately, which realistically accounts for the effects of nondissolved constituents on droplet evaporation and dispersion. To the best of the authors' knowledge, this is the first time that the solid residue is handled separately for an evaporating droplet inflight. This would allow us to uncover an interesting phenomenon related to the droplet vertical velocity (see later).
Our proposed model couples mass and momentum transport to predict the evaporation rate, velocity, and travel distance of droplets produced by common expiratory activities. In contrast to computationally demanding computational fluid dynamics (CFD) studies, our model can be deployed rapidly and economically to predict trajectories of emitted droplets. Thus, it can serve as a valuable modeling tool for studying droplet dynamics in standardized environments.
II. THEORY
A. Model assumptions
The evolution of respiratory droplets in air is governed mainly by inertial, viscous, and gravitational forces, coupled with shrinkage due to evaporative loss. The evaporation rate depends on parameters such as temperature, relative humidity (RH), relative velocities of droplet and air, and the chemical composition within the droplet. Once expelled from the body, respiratory droplets evolve passively according to local conditions. The Weber number defined as the ratio of drag and cohesion forces, including surface tension, ranges from 1.7 × 10^{−3} to 5.1, for a 100 μm droplet assuming peak coughing velocity of 10 m/s.^{7} In this study, secondary effects due to droplet deformation and breakup, as well as angular momentum, are neglected.
As ambient temperatures are usually cooler than body temperatures, the warm droplet cools in air until stable wetbulb (adiabatic saturation) temperature is reached.^{37} This cooling is faster than evaporation at wetbulb temperature,^{37} and hence, the droplet can be assumed at its wetbulb temperature depending on the local air temperature and RH. The salt and polymer content in respiratory droplets is known to reduce water activity, which results in low evaporation rates.^{38} During evaporation, the concentrations of chemical components in droplets change with time. Using a rapid diffusion model, the density can be assumed to be homogeneously uniform.
Based on the above considerations, the following assumptions are made:

The droplet does not rotate or vibrate (oscillate).

The droplet and its residue remain spherical all the time.

The droplet remains at its wetbulb temperature.^{37}

The density of the droplet is uniform (rapid diffusion and mixing).

Air properties surrounding the droplet, including temperature and RH, keep unchanged.

The evaporation rate per unit area is uniform on the surface of the droplet.
B. Droplet evaporation
C. Equation of motion
D. Numerical method
The fourthorder Runge–Kutta method is implemented in Fortran to numerically solve the coupled nonlinear ordinary differential equations [Eqs. (12), (13), (16), and (17)]. However, Eq. (16) contains terms proportional to 1/r^{2}, 1/r, and 1/r^{3}, respectively, which are singular for small r. To ensure the stability and accuracy of the numerical solution, the following iterative scheme is proposed.
E. Model validation
The present method is verified against two known solutions for two problems involving pure water droplet, namely, droplet motion without evaporation and droplet motion with a constant evaporation rate. In both cases, analytical solutions are available for comparison.
 The first scenario corresponds to the stationary freefall condition of $\alpha =0,\u2009 \u2009 \delta s=0,$ and horizontal velocity $u t=0$. The droplet radius remains constant as evaporation is not considered. The vertical component of the droplet acceleration in Eq. (16) is$ d v d t=\u2212 \beta 1 r 0 2v\u2212 \beta 2 r 0 v 2+ \beta 3.$Equation (25) is a nonlinear ordinary differential equation with constant coefficients. The corresponding analytical solution is$v t= v T\u2212 \beta 12 + 2 v T v T \u2212 v 0 \beta 12 + v T + v 0 e \beta 1 r 0 2 + 2 \beta 2 r 0 v T t + v T \u2212 v 0,$where $ \beta 12= \beta 1/( r 0 \beta 2)$ and v_{T} is the terminal velocity expressed as$ v T= \beta 12 2 1 + 4 \beta 2 \beta 3 \beta 1 \beta 1 r 0 3 \u2212 1.$
 The second scenario corresponds to the situation of $ \delta s=0$ and $ V t = 0= 0 , \u2009 v 0$ at constant $\alpha $. The droplet size and motion are described by$ r 2= r 0 2\u22122\alpha t,$$ d v d t=\u2212 \beta 1 r 2v\u2212 \beta 2 r v 2+ \beta 3.$Equation (29) is similar to Eq. (25), except that the coefficients are time dependent since the droplet radius changes with time. The corresponding analytical solution as derived in Sec. 2 in the Appendix is$v r *= G r * F r *+ v 0 F 2 1 \u2212 G 1 F 1 1 + \beta 2 * v 0 F 2 1 \u2212 G 1 F 1 \eta 3 r *\xb7 r * \u2212 \beta 1 * F 2 r *,$where r^{*} = r/r_{0} is the nondimensional radius with initial value of r^{*}(0) = 1 and$vF= r * \u2212 \beta 1 * \u2211 k = 0 \u221e a k r * 3 k,\u2003G=\u2212 \beta 3 * r 2 \u2212 \beta 1 \u2032 \u2211 k = 0 \u221e a k r * 3 k 3 k + 1 \u2212 \beta 11,$$ a k= \beta 3 * \beta 2 * 9 k k \u2212 \beta 11 a k \u2212 1,\u2003 \eta 3 r *= \u222b r * 1 \chi \u2212 \beta 1 * F 2 \chi d \chi ,$$ \beta 11= 1 + \beta 1 * 3,\u2003 \beta 1 *= \beta 1 \alpha ,\u2003 \beta 2 *= \beta 2 r 0 \alpha ,\u2003 \beta 3 *= \beta 3 r 0 2 \alpha .$
In the first scenario, evaporation is not considered and droplet diameter is constant. C_{1} in Eq. (15) is chosen to be 1. The droplet diameter is chosen to be 500 μm, and its initial vertical velocity is set as 0.2 m/s. Note that the positive direction of the y axis is pointing in the direction of the gravity acceleration vector. The stationary air temperature and RH are 24 °C and 0.6, respectively. For ease of discussion, droplet diameter is referred to instead of radius henceforth. Figure 1(a) shows excellent agreement between our proposed iterative solution [Eq. (26)] and analytical solution [Eq. (27)].
In the second scenario, Fig. 1(b) shows excellent agreement between our proposed iterative solution [Eq. (29)] and analytical solution [Eq. (30)] with only $\alpha $ set as 5.7 × 10^{−9} m^{2}/s. The droplet vertical velocity increases to a maximum at 1–2 s and then starts to decrease, in contrast to the earlier case without evaporation. This is a quite interesting phenomenon observed in the current work given the consideration of the solid density in the droplet. For a large droplet size of 500 μm, the droplet initially accelerates under gravity, but this is counteracted by evaporation causing the droplet to lose mass over time, leading to deceleration and a reduction of its vertical velocity.
The good agreements with analytical solutions validate our proposed method. Furthermore, validations were made between our model with experimental results for a pure droplet and a droplet with salt as solid residue. The droplets are levitated in air at temperature of 30 °C and RH = 0.5. Figure 2(a) shows the predicated results from the current model and the experimental data reported by Chaudhuri et al.^{25} The chemical coefficient C is calculated based on Eq. (11). The results from the current proposed model agree well with reported data, which serves as experimental validation. Furthermore, validation is made for droplet evaporation under air velocity of 2 m/s and air temperature of 30 °C and RH = 0.84 as shown in Fig. 2(b). It is clear that the current results are in good agreement with the numerical results reported by Ref. 26.
III. RESULTS AND DISCUSSION
A. Effects of droplet diameter on droplet velocity
Emitted droplets can be differentiated by size into “small droplet” and “large droplet.” The term, large droplets, was initially coined by Wells,^{20} who proposed 100 μm as a critical size to distinguish massive droplets which settle quickly from smaller ones which tend to remain airborne. The critical size, specifically the minimal size of droplets which settle quickly, depends on many parameters including RH, temperature, and air flow, to name a few. For example, Xie et al.^{35} found the critical droplet size to be roughly 65 μm at RH = 0.7, air temperature of 20 °C, and droplet temperature of 33 °C. In this study, Wells' size classification of 100 μm as the droplet critical size to distinguish between small and large droplets is adopted.
For small droplets (<100 μm), Fig. 3 shows the time evolution of droplet diameter, horizontal and vertical droplet velocities, respectively, with u_{0} = 10 m/s and v_{0} = 0 m/s. The stationary air temperature and RH are 24 °C and 0.6, respectively; under such conditions, the wetbulb temperature is 18.6 °C. Droplet content of 99.1% water and 0.9% salt as solid residue in terms of the mass fraction is assumed.^{35} Figure 3(a) shows that the evaporation time, or the time for droplet to completely evaporate, increases with increasing initial droplet diameter. Small droplets evaporate within a seconds leaving solid residue suspended in air. Figure 3(b) shows that the horizontal velocity of all small droplets decreases to zero within 1 s. A larger droplet carries greater momentum compared to a smaller one resulting in a longer time to slow down, i.e., a longer relaxation time. As for vertical velocity, Fig. 3(c) shows that it increases initially up to terminal velocity balancing drag force and gravity forces. The initial droplet acceleration is due to stronger gravitational force compared to drag force, but as the droplet evaporates and loses mass, gravitational force weakens relative to the drag force and droplet decelerates. In general, the droplets, even the droplet size of 100 mm, can evaporate within a few seconds and reach its terminal velocity within a few seconds. These droplets can suspend in air for sufficient long time and pose potential risks to susceptible persons at long distance.
For large droplets (>100 μm), Fig. 4 shows the time evolution of droplet diameter, horizontal, and vertical droplet velocities, respectively, with same environmental conditions as for the small droplet case. As before, large droplets carry greater momentum compared to small ones. For example, the relaxation times for 1000 and 200 μm droplets are approximately 5 and 1 s, respectively. Larger droplets can be dispersed over longer distances compared to smaller ones, as shown in Fig. 4(a).
The maximum droplet dispersion distance refers to the ground distance covered by a droplet during its lifetime. Unlike the small droplets, the maximum dispersion distance for large droplet is dominated by inertial and gravitational forces.
On one hand, the inertial force for droplets with larger diameters is higher than that of droplets with smaller diameters. This leads to a slower reduction in the horizontal velocity as observed in Fig. 4(b). On the other hand, the lifetime for droplets with larger diameters is shorter than that of the droplets with smaller diameters, resulting in a shorter lifetime in air. The travel distance, or the ground distance covered by droplet, for large droplets depends on the competing effects between inertial and gravitational forces.
B. Effects of droplet initial velocity on droplet diameter and dispersion distance
Figure 5 shows that the evolution of the droplet diameter is insensitive to initial velocity at an air temperature of 24 °C and RH = 0.6. The droplet diameter is chosen to be 100 μm. This suggests that the second term in the bracket of Eq. (13) on the righthand side of the equation is small ( $0.3 R e 1 / 2 S c 1 / 3\u226a1$) and can be neglected.
Figure 6 shows how droplet travel distance increases with initial velocity for respiratory droplets up to 1000 μm and even larger droplets up to 3000 μm. Here, travel distance refers to the ground distance covered by a droplet released from a height of 2 m to the location where it settles. The droplet initial velocities ranging from 1 to 10 m/s, which covers typical velocities of droplets emitted due to respiratory activities including talking,^{15} singing,^{44} coughing,^{14} and sneezing^{5} are tested. Although the droplet lifetime tends to reduce with increasing droplet diameter, the greater initial momentum propels the droplet further resulting in a longer dispersion distance, an observation consistent with Xie et al.^{35} By inspection, a 1000 μm droplet with initial velocity u_{d} = 20 m/s can travel as far as 3.5 m under, also consistent with Pendar and Páscoa.^{23} This shows that our proposed model, while simplistic, can yield useful dispersion solutions for practical applications including advising guidelines for social distancing.
C. Effects of RH on droplet diameter and dispersion distance
Figure 7 shows how the time evolution of droplet diameter, initially 100 μm, depends on RH. Under high RH humid conditions, droplets evaporate slowly leading to long evaporation times.
Figure 8 shows the dispersion distance for droplet sizes larger than 100 μm. It is found that RH does not have significant effect on the droplet dispersion distance for droplet size smaller than 150 μm. For droplet diameter larger than 150 μm, RH reduces dispersion distance significantly. For example, a 2000 μm droplet can travel distances greater than 3.3 m at RH = 0.0 compared to 0.9 m at RH = 0.90, an almost 3.7 times difference. Note here that the viability of virus under different ambient temperature and humidity conditions is not considered in the current work.
D. Effects of sputum on droplet diameter and dispersion distance
Droplets from respiratory activities generally constitute not only salt but also high polymers such as lipid, protein, and DNA.^{19,45} These are called sputum droplets, with an estimated composition of 93.5% water and 6.5% solid residue in mass fraction.^{19} The lipid monolayer formed on the surface of the droplet can reduce the droplet evaporation rate significantly.^{46} Following Redrow et al.,^{19} a 50% reduction in evaporation rate of sputum due to the presence of high polymers is assumed here.
Figure 9 shows the change of droplet diameter for salty, sputum, and pure water droplet with initial velocity of 10 m/s at T_{a} = 24 °C and RH = 0.6. The salty droplet is identical to droplets modeled in Secs. III B and III C. It is necessary to mention that the pure water droplet will eventually evaporate completely. The sputum droplet has a higher residue content of high polymers in the droplet compared to droplet with salt, which reduces water activity and slow evaporation.^{19}
Figure 10 shows the large droplet dispersion distance for pure water, salty water, and sputum droplets at T_{a} = 24 °C, RH = 0.6 with initial velocity of 10 m/s. The reduction of evaporation rate for sputum droplet leads to a slow droplet size shrinkage, resulting in a shorter lifetimes for large droplet. For droplet diameters less than 500 μm, dispersion distances between salty droplet and sputum droplet are identical. The dispersion distance for sputum droplet deviates from salty droplet for larger droplet diameters, due to shorter lifetimes. It is interesting to find that the dispersion distance for pure water droplet is shorter than both salty and sputum droplets, highlighting the importance of evaporation for large droplet dispersion. Failure to account for droplet evaporation due to chemical impurity may lead to an underestimate of environmental transmission risks of infectious airborne diseases.
E. Wells evaporationfalling curves
The current mathematical models and solutions can also be used to predict the famous Wells evaporationfalling curve.^{20} The falling time refers the time which large droplet settles down to the ground from air. The height of 2 m is considered in the current work. The effect of RH on small droplet evaporation and the lifetime for large droplet is considered. Figure 11 plots the Wells evaporationfalling curves under different RH. Since droplets evaporate faster at low RH, the evaporation curve persists downstream leading to a larger critical size than expected. Since evaporation does not play an important role for large droplets, the falling lines overlap for large droplets under different RHs.
IV. CONCLUSION
A simple theoretical model for the free dispersion of a single evaporating droplet in stationary air was proposed. Considering nonvolatile solid residue, the dispersion distance for the droplet nuclei related to the airborne transmission can be estimated under controlled environmental conditions. Our model results are in excellent agreement with analytical and experimental reports of droplet dynamics during evaporation.
Highlights of the current study are as follows:

Droplets in freefall accelerate due to gravitational force up to peak settling velocity offset by the evaporative mass loss of water, before drying out into solid residue with a lower terminal velocity (Figs. 3 and 4).

For evaporating respiratory droplets projected at velocities up to 10 m/s, dispersion distance is at maximum for 1000 μm droplets at 0.6 RH (Fig. 6), 1750 μm droplets at 0.9 RH, and 800 μm droplets at 0.3 RH (Fig. 8).

Evaporating sputum droplets projected at velocities up to 10 m/s can disperse at least 3 times further compared to pure water droplets at 0.6 RH (Fig. 10).

Critical droplet size based on falling times ranges from 120 μm at 0.3 RH to 75 μm at 0.9 RH (Fig. 11).
The current work showed how dispersion distance related to fomite transmission can be determined under different conditions including droplet initial velocity, RH, and chemical composition. The model is currently limited by the omission of droplet–droplet interactions and turbulent jet from the source. Future work can be extended to account for air velocity, droplet temperature variation, and turbulent jet from various respiratory activities. Viscoelasticity of the solid residue^{47} can also be considered as part of future work.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
LunSheng Pan: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Fong Yew Leong: Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). Evert Klaseboer: Formal analysis (equal); Writing – review & editing (equal). ChangWei Kang: Investigation (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Yun Ching Wang: Investigation (equal); Validation (equal). Keng Hui Lim: Resources (equal); Supervision (equal). George Xu: Investigation (equal); Supervision (equal). Cunlu Zhao: Formal analysis (equal); Investigation (equal); Writing – review & editing (equal). Zhizhao Che: Investigation (equal); Methodology (equal); Writing – review & editing (equal). Chinchun Ooi: Methodology (equal); Writing – review & editing (equal). Zhengwei Ge: Methodology (equal); Writing – review & editing (equal). Yit Fatt Yap: Supervision (equal); Writing – review & editing (equal). Hongying Li: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.