An important vector for host-to-host infectious disease transmission is given by the transport of tiny pathogen-laden droplets. These are commonly exhaled by individuals while breathing, speaking, coughing, or sneezing. Depending on their size and ambient conditions, they may follow different paths, either settling on surfaces where the pathogen can be further transmitted by contact, or remaining airborne after evaporation where the pathogen can be inhaled. Our understanding of pathogen transmission from the fluid mechanics perspective is still somewhat limited, especially in quantitative terms. In the current work, starting from the fundamental laws of fluid mechanics and diffusion, a detailed analytical model of droplet transport and evaporation in humid air streams is presented and successfully validated against available data in the literature finding remarkable agreement. The model implements closed-form analytical solutions of the equations of transport, evaporation, and energy balance, and an algebraic model to account for the droplet chemical composition. It also features an analytical model of droplet transport within the buoyant exhaled breath cloud based on momentum conservation addressing both jet and puff phases and is able to handle periodic respiratory events. Turbulent dispersion is modeled with a discrete random walk approach. A simple inhalation model is also proposed. Such a model may help in better understanding droplets' fluid dynamic behavior and may be used to assess the risks associated with pathogen transmission under different scenarios for any type of respiratory event. Overall, the computational cost is relatively low, allowing extensive simulation campaigns to be performed easily.

The recent SARS-CoV-2 pandemic has drawn the attention of many researchers, not only in the medical field, in trying to understand how the virus infects the lungs, spreads among people, and how the diseased individuals may be treated. In fact, knowing how a pathogen operates and preventing its diffusion are key elements in reducing the pressure on healthcare systems, provide better treatment, and save lifes in the middle of a pandemic.

The well-known rules we all became acquainted with over the last few years such as frequent hand washing, social distancing, and face masks wearing, all go in the direction of prevention. While these remain important behaviors to be followed in a pandemic situation, their formulation is still based on an incomplete knowledge of the pathogen transmission routes.1,2 The understanding that diseases could spread from an infected person to a susceptible one dates back millennia, yet the idea that this might be due to microorganisms was scientifically proven only in late nineteenth century with the work of Pasteur3 and Koch.4 The interest was then drawn to understanding the possible transmission routes of these microorganisms. In the pioneering work of Wells5,6 on droplets fall and evaporation, it was shown that larger drops fall to the ground before evaporating, while smaller drops fully evaporate into droplet nuclei that remain suspended in the air. Thus, both fomite (i.e., contact with an infected surface) and airborne (i.e., inhalation of infected droplets) transmission routes are possible. A cutoff size between large and small droplets in the 97 172 μ m range was hypothesized by Wells, depending on the ambient relative humidity that governs the droplet evaporation rate. Nonetheless, the idea of airborne transmission was aversed as late as up to the '80s before being accepted,7 yet with often largely different cutoff thresholds in the 5 10 μ m range that are unjustified from fluid mechanics perspective.

Trying to resume the large literature on the topic and the differences in terminology that are found, five basic transmission routes can be assumed as reported by the World Health Organization (WHO):8 

  1. Direct contact, occurring through contact with an infected person, e.g., by hand-shaking;

  2. Indirect contact, or fomite, occurring through contact with an infected surface, e.g., touching a door knob;

  3. Direct inhalation, occurring by direct inhalation of droplets exhaled by an infected person, e.g., speaking face to face;

  4. Indirect inhalation, or airborne, occurring by inhalation of dried droplet nuclei suspended in the air, also referred to as aerosol route;

  5. Orofecal, occurring in case the pathogen can pass to the feces of an infected person, and from there infect other individuals, e.g., through the aerosol generated when flushing the toilet, or by contamination of water sources or food.

To some extent, the latter route can also be considered a sub-case of the others observing, for instance, that the aerosol generated when flushing the toilet can either be inhaled or deposit on surfaces. In any case, fundamental infection carriers are the droplets that, crossing the air or through the mediation of a surface, reach the airways of a susceptible individual. In addition, smaller droplets and droplet nuclei are more easily carried by the airflow and are thus able to penetrate deeper the lower airways into the lungs once they have been inhaled thanks to their small Stokes numbers.

The routes mentioned above are common to all infectious diseases, such as tuberculosis, measles, smallpox, chickenpox, influenza, rhinovirus, as well as MERS and SARS,9 even though the severity of the disease, the entity of the epidemic outbreaks, and the need for prevention differ from case to case. The various pathogens also have different stability depending on their type and on the environment. For instance, in Ref. 10, the infectious titer decay of SARS-CoV-2 was measured in 1.1 h half-life in air, 5.6 h on stainless steel, and 6.8 h on plastic. This means viable viruses can be found on certain surfaces even after days, while for airborne transmission, the time horizon is in the range of a daytime.

Differences in stability influence the relative importance of the transmission routes, even though this is an aspect difficult to be quantified. The reason is that host-to-host transmissibility of a pathogen also depends on many additional factors, equally difficult to be quantified, such as the virion shedding rate, the quantity and size distribution of the droplets exhaled, the environmental conditions,11 the infective dose needed to cause the infection in a susceptible host, the individual behavior, and the prevention measures adopted, alongside with the many medical and physiological considerations that could be raised. In addition, from a global perspective, the chance of infection also depends on the status of the epidemic outbreak and the potential for asymptomatic individuals to spread the pathogen unknowingly.

Infectious disease transmission from a fluid mechanics perspective has been dealt with rather extensively in the literature, in particular over the last few years following SARS-CoV-2 pandemic.

A large share of works in the literature addressed the problem of ventilation in confined spaces either experimentally or numerically. These, in fact, are among the most critical environments for disease transmission since the pathogen may accumulate in the air in the presence of an infected individual. To reduce the risk, it is necessary to dilute the pathogen by increasing the air changes per hour,12 yet this is in contrast with the thermal comfort and the building energy efficiency. To be also considered that dilution goes hand in hand with the uniform spreading of the pathogen throughout the room. Different ventilation approaches have been investigated, including natural, mixing, displacement, and personalized ventilation. Natural ventilation is very energy efficient but little controllable.13 Mixing ventilation is very common for its simplicity and consists in introducing air at relatively high velocity to promote mixing and dilution.14 In displacement, ventilation air is introduced at low velocity at the floor level and extracted close to the ceiling. The goal is to promote air stratification so that any airborne pathogen could be gradually pushed above the heads of the occupants and expelled. The presence of heat sources or people moving in the room may disrupt the stratification.15,16 Personalized ventilation consists of a diffused intake system where clean air is introduced in the breathing area of each individual. While being an energy efficient option,17 in most cases it is too complicated and expensive for practical application. The chance that an infected individual may spread the pathogens over the room calls for personalized exhaust terminals as well.18,19 While the studies above provide a qualitative idea of the pros and the cons of different ventilation systems, airflow in a closed environment depends on too many factors (e.g., type and layout of ventilation and heating systems, room shape, occupancy, etc.) so that any quantitative result cannot be generalized. Furthermore, it is worth mentioning that risk-free ventilation does not exist since a susceptible individual by chance may always find himself downstream of an infected person,20 this risk growing the closer the infected person to the intake grill and the susceptible individual to the exhaust grill.

A number of works in the literature also addressed pathogen transmission from other perspectives, focusing on different aspects.

First of all, to assess the chance of infectious disease transmission, it is important to know the quantity, the size distribution, and the velocity of the droplets exhaled under different circumstances (e.g., while breathing, speaking, coughing, or sneezing). Experimental measures of droplet size distribution were first presented in Ref. 21 for speaking, coughing, and sneezing. More recently, a large number of works addressed the same issue: e.g., Ref. 22 for speaking and coughing,23 for speaking, whispering, and breathing with nose or mouth,24 for breathing, speaking, and singing. However, at first sight, there is not much agreement among these studies. This may be due to a large variability of the exhalations among people, but also to the experimental method and the instruments employed since no single instrument can capture droplets size over a large range of dimensions. In fact, sizes from 0.1 μ m to 1 mm were experimentally detected. In Ref. 1, a review of these studies is presented. It is shown how usually lognormal distributions in the instrument operating range were proposed for the droplet size probability density function, while multi-modal distributions were attributable to composite instrumentation. Furthermore, it was noted that all these experimental data are quite superimposable if normalized over the mean diameter of each distribution. In view of this, a Pareto distribution of power 2 is proposed as a more credible option to model exhaled droplet size distribution.25 

Second, it is necessary to evaluate the viral content of exhaled droplets and estimate the average infectious dose that could potentially transmit the disease. Viral content in coughs was measured in Ref. 26 for influenza, while for SARS-CoV-2, a median viral load in saliva samples from infected individuals of 3.3 copies / nl was measured in Ref. 27, even though the viral load changes significantly as the disease progresses. Values as high as 120 copies / nl were found in hospitalized patients and up to 130 000 copies / nl in a patient who died.28 The infectious dose likely to cause the disease is not known, but from preliminary analyses,29 it is believed to be at least 50 100 virus copies: a similar value to that known for influenza.

Then, the droplet transport and evaporation should be addressed. Depending on their size and evaporation history, droplets may fall to the ground or remain airborne so that different relevance can be assumed by contact and inhalation routes. Many works address the problem by means of Computational Fluid Dynamics (CFD). In Ref. 30, the CFD analysis of a sneeze is presented where head motion and dynamic pressure variation during the sneeze are modeled. The CFD analysis of a sneeze is also presented in Ref. 31 where the influence of the ambient relative humidity on the droplet evaporation is analyzed: high relative humidity slows down droplet evaporation and reduces the fraction of airborne particles. A similar numerical study is presented in Ref. 32 for a cough and in Ref. 33 also for a cough but using the lattice Boltzmann method. Droplet distribution in a confined environment with different types of ventilation is investigated in Ref. 34 for coughing and breathing.

Other works propose analytical models implementing mass transfer and momentum equations to predict droplets transport and evaporation. In Ref. 35, the focus is on the role of relative humidity on evaporation, while the exhaled flow is modeled as a jet. A similar analysis is carried out in Refs. 36 and 37 for a cough. In Ref. 38, a physical model of the droplet behavior within a jet is proposed following a set of experimental data collected for a cough. In Ref. 25, it is underlined how the one or two meters of social distancing rule adopted in the recent pandemic is mostly based on the pioneering works of Wells5 where the droplet transport due to the exhaled flow was not considered. The idea that this rule, although reasonable, is not based on sound scientific evidence is also highlighted in Ref. 39. In addition,25 it underlines that exhalations cannot be modeled as continuous jets but rather as intermittent jets and puffs, since these have different space vs time evolutions. This is supported by Ref. 40 where it is also shown how the work of Wells overestimates actual evaporation times by at least one order of magnitude. Wells evaporation-falling curve is analytically revisited in Ref. 41 to account for the reduced evaporation within the exhalation jet. In the model proposed, an empirical formula is introduced to account for the jet buoyancy42 and Raoult's law is used to correct droplet evaporation predictions: the presence of solutes in the saliva, in fact, significantly slows down the last stage of the evaporation process where the water mole fraction vanishes and the solute mole fraction becomes dominant. Furthermore,43 it introduces a more elaborated multi-component model for saliva including proteins, lipids, carbohydrates, and salts. In Ref. 44, the droplet is modeled with a multiple shell approach to account for temperature and concentration gradients within the droplet itself, allowing finer predictions of the last evaporation stage. A few works also deal with short-range transmission, focusing on the interaction between the exhaled cloud and the inhaled air, and proposing exhalation–inhalation numerical models.45,46

Last, once the virion shedding rate is known and an estimate of the airborne virus fraction is made, statistical models can be drawn to predict the risk of airborne transmission in a closed environment in function of the virion concentration, as given by the balance between emission and removal rates in time, and the exposure time.29,47,48 The average infectious dose is typically referred to as quantum to decouple the mathematical formulation of the problem from difficult experimental assessments of virion emission rates and average infectious doses. These models are based on the well-known Wells–Riley equation49,50 with different enhancements proposed by various authors.

In the present work, an analytical model of the exhaled droplets transport and evaporation is presented and discussed. The model is based on the fundamental laws of fluid dynamics and on convection–diffusion equation, and features a model of droplet transport within the exhaled breath cloud including turbulent dispersion. The analytical model can be easily implemented in a numerical tool allowing the evaluation of spatial and time distribution of droplets of any size under different initial and ambient boundary conditions within seconds of CPU time per droplet. The computational cost varies largely depending on the droplet size, or better on whether the droplet remains trapped in the breath cloud or not: typical simulation time ranges from < 0.1 s for large droplets, to 60 s for smaller ones on an ordinary laptop running in single core. The model described has been implemented in Python language and validated against additional numerical and experimental data in the literature finding good agreement. With respect to infectious disease transmission, with the due boundary conditions, such a model is rather versatile and can be used to assess the risks associated with fomite, direct inhalation, and indirect inhalation routes for any type of respiratory event.

A comparison with similar analytical works in the literature is given in Table I in terms of the modeling assumptions made, even though the meaning of some of them may become clearer to the reader only further in the text. Obviously, most works include models for the droplet transport, evaporation, energy balance, and chemical composition in terms of nonvolatile components. Yet, several works limit their interest to the case of Stokes' flow without including drag correction terms due to non-continuum effects, formally restricting the model validity to droplets diameters in the range from a few micrometers to a few tens of micrometers. The majority of the works in the literature also include a breath cloud model, at times also addressing its buoyancy, even though always with an empirical approach. Nonetheless, except from the present work and the one in Ref. 25, respiratory events are always modeled as steady continuous jets, thus neglecting their time limited or intermittent nature, besides their unsteadiness. Turbulent dispersion, whenever addressed, is always based on empirical profiles of the turbulent quantities derived from steady jets.

TABLE I.

Synoptic table of the modeling assumptions made in analytical works on exhaled droplet transport and evaporation. The works are listed in chronological order.

Authors Transport Evaporation Energy Non-volatiles Jet breath cloud Puff breath cloud Cloud buoyancy Turbulent dispersion Periodic events Randomization Inhalation
Wang et al.35   ✓  ✓  ✓  ✓  ✓             
Xie et al.41   ✓  ✓  ✓  ✓  ✓    ✓         
Parienta et al.44   ✓  ✓  ✓  ✓  ✓             
Redrow et al.43   a,b  ✓  ✓  ✓  ✓    ✓ 

c

 
     
Wei and Li36   ✓  ✓  ✓ 

d

 
✓    ✓ 

e

 
     
Balachandar et al.25   a 

f

 
✓  d 

g

 
g  ✓ 

h

 
     
Chen et al.45   a  ✓  ✓  d  ✓    ✓        ✓ 
Wang et al.38   ✓  ✓      g             
Wang et al.37   a  ✓  ✓  d  ✓      e       
Present work  b  ✓  ✓  ✓  ✓  ✓  ✓  e  ✓  ✓  ✓ 
Authors Transport Evaporation Energy Non-volatiles Jet breath cloud Puff breath cloud Cloud buoyancy Turbulent dispersion Periodic events Randomization Inhalation
Wang et al.35   ✓  ✓  ✓  ✓  ✓             
Xie et al.41   ✓  ✓  ✓  ✓  ✓    ✓         
Parienta et al.44   ✓  ✓  ✓  ✓  ✓             
Redrow et al.43   a,b  ✓  ✓  ✓  ✓    ✓ 

c

 
     
Wei and Li36   ✓  ✓  ✓ 

d

 
✓    ✓ 

e

 
     
Balachandar et al.25   a 

f

 
✓  d 

g

 
g  ✓ 

h

 
     
Chen et al.45   a  ✓  ✓  d  ✓    ✓        ✓ 
Wang et al.38   ✓  ✓      g             
Wang et al.37   a  ✓  ✓  d  ✓      e       
Present work  b  ✓  ✓  ✓  ✓  ✓  ✓  e  ✓  ✓  ✓ 
a

Limited to the case of Stokes' flow.

b

Including Cunningham correction factor for drag.

c

Based on additional 3D CFD simulations.

d

Raoult correction for the evaporation rate missing.

e

Based on empirical profiles for the turbulent quantities in jets.

f

Constant evaporation rate assumed.

g

Uniform velocity of the whole cloud assumed.

h

Limited to the late stage where the cloud is dispersed in the ambient.

The breath cloud model proposed springs from momentum conservation equation to which a momentum decay factor is added. It can handle both the jet and the puff phases of a respiratory event, being able to address both isolated and periodic events. The jet/puff velocity profile is non-uniform in space and unsteady. To the authors' knowledge, this is the first time these two characteristics are featured together in a similar analytical model. The cloud buoyancy and the droplet turbulent dispersion are addressed through empirical models as done in some previous work in the literature. In particular, turbulent dispersion is implemented with a discrete random walk model. Furthermore, the droplet emission can be randomized both in space (within the mouth cross section) and time (along the jet phase): this is another aspect never addressed in previous works. These last features are most relevant as they enable the model to be used for statistical analyses on the spatial distribution of droplet trajectories, a characteristic strictly related to the risk of disease transmission through different routes. A simple inhalation model is also proposed in the end.

In this section, the analytical model is presented quite extensively, beginning from the governing equations of the droplet transport and evaporation, and continuing with the rather elaborated cloud model proposed.

While the first part of the analysis derives from rather well-established physical models, in the authors' opinion and for the sake of completeness, it is still worth focusing briefly on these notions, enriching the discussion with examples that may help the reader in gaining a deeper insight into the subject. This, also considering that erroneous equations are occasionally found in the literature.

To model the transport of the exhaled droplet, without loss of generality, it is assumed its shape is spherical. The flow past a sphere is a well-known fluid dynamic scenario, that in the case of low Reynolds numbers is known as Stokes' flow. The basics of this flow are briefly recalled in a general form. In the following, it is assumed that thermophysical properties are uniform within the droplet and vary with the temperature.

The forces acting on the droplet as it moves through the air are its weight due to gravity (acting along z direction) and drag (acting in the direction opposite to the relative velocity of the droplet with respect to the surrounding air). Other forces can be assumed negligible as demonstrated in Ref. 51. Including buoyancy, weight force in the vertical direction is
W z = ( ρ p ρ m , s ) g V p = 1 6 ( ρ p ρ m , s ) π g D p 3 ,
(1)
where g is the gravitational acceleration, ρ is the density, V is the volume, and D is the diameter. The subscripts “p” and “m,s” refer to the droplet and to the humid air mixture by the droplet surface, respectively. In case the droplet is evaporating, saturated air can be assumed at the droplet surface; otherwise, local ambient hygrometric conditions apply. Drag force in modulus is, by definition,
D = 1 2 C d ρ m , s A p u r 2 = 1 8 C d ρ m , s π D p 2 u r 2 ,
(2)
where C d is the drag coefficient and u r is the modulus of the relative velocity of the droplet with respect to air ( u r = u p u m). Assuming the droplet relative velocity forms an angle θ with the horizontal, velocity and drag components are
u r , x = u r cos θ ; u r , z = u r sin θ
(3)
and
{ D x = D cos θ = 1 8 C d ρ m , s π D p 2 u r u r , x D z = D sin θ = 1 8 C d ρ m , s π D p 2 u r u r , z .
(4)
The droplet Reynolds number is computed in terms of relative velocity and local air thermophysical properties
Re = ρ m , s D p u r μ m , s = D p u r ν m , s .
(5)
According to Stokes, if Re 1, the drag coefficient tends to
C d = 24 Re .
(6)
This allows Eq. (2) to be simplified into
D = 3 μ m , s π D p u r ,
(7)
known as Stokes' law. Let us consider a sphere freely falling in still air ( u m = 0) from a given initial velocity u p , 0 under the assumption of Stokes' flow. The sphere will accelerate or decelerate up to when gravity and drag forces are in equilibrium. By solving the equality W + D = 0, the terminal velocity of the sphere along z is found
u r , t = ( ρ p ρ m , s ) g D p 2 18 μ m , s ,
(8)
while any horizontal velocity component will be gradually zeroed by drag. By substituting Eq. (8) into Eq. (5), it is found that Re t D p 3. If air at normal temperature and pressure (NTP) conditions with 50% relative humidity and a non-evaporating particle in thermal equilibrium with the environment are assumed (i.e., T = 20 ° C, p = 1 atm, R H = 50 %, ρ m = 1.199 kg / m 3, μ m = 1.814 × 10 5 kg / m s, ρ p = 998.2 kg / m 3), it is easily shown that for Reynolds number to be < 0.1 it must be D p < 37.0 μ m. More in general, from horizontal and vertical forces balance according to Newton's second law of motion
{ u ̇ r , x = D x m p = u r , x τ d , u ̇ r , z = W z + D z m p = u r , t τ d u r , z τ d ,
(9)
where m p is the droplet mass and the characteristic time associated with drag
τ d = m p u r , z D z = ρ p D p 2 18 μ m , s
(10)
is introduced. To be noted that τ d does not depend on the droplet relative velocity and that
u r , t = ( 1 ρ m , s ρ p ) g τ d .
(11)
The solution of the system of equations in Eq. (9) for initial conditions u r , 0 = ( u r , x , 0 , u r , z , 0 ) is
{ u r , x ( t ) = u r , x , 0 e t / τ d u r , z ( t ) = u r , z , 0 e t / τ d + u r , t ( 1 e t / τ d )
(12)
from which the absolute droplet velocity is simply found by adding the local air velocity to the result
{ u p , x ( t ) = u m , x + u r , x , 0 e t / τ d u p , z ( t ) = u m , z + u r , z , 0 e t / τ d + u r , t ( 1 e t / τ d ) .
(13)
From Eq. (12), the physical meaning of characteristic time is apparent. The droplet velocity is given at any time by a weighted average of initial and terminal velocities: after a time t = τ d has passed the latter weighs ( 1 e 1 ) = 63 % over the total, when t = 7 τ d this weight grows > 99.9 % and the initial condition has been essentially washed away. Equation (13) clearly shows that in the case of small terminal velocity, after a brief initial transient, the droplet essentially moves with the local air and might be transported arbitrary far from where it was generated. By further integration with initial conditions s p , 0 = ( s p , x , 0 , s p , z , 0 ), the distance traveled by the droplet can be written as a function of time
{ s p , x ( t ) = s p , x , 0 + u m , x t + u r , x , 0 τ d ( 1 e t / τ d ) s p , z ( t ) = s p , z , 0 + ( u m , z + u r , t ) t + ( u r , z , 0 u r , t ) τ d ( 1 e t / τ d ) .
(14)
Let us now consider a generic flow past a sphere whose Reynolds number is not necessarily small so that Eqs. (6) and (7) do not hold. This means Eqs. (8)–(14) must be rewritten without these assumptions. From experiments, different flow regimes and drag coefficient behaviors are encountered for growing Reynolds numbers.52 For what concerns the case of exhaled droplets, it is sufficient to remind that Stokes' regime can only be assumed for Re < 0.1 and that drag coefficient gradually departs from Eq. (6) for larger Reynolds numbers reaching a minimum value of 0.4 for Re = 3 000. Several empirical correlations have been given in the literature providing the drag coefficient as a function of the Reynolds number either for spherical53 or non-spherical52 particles. In the following, the one from Clift and Gauvin,54 valid for spherical particles up to Re = 3 × 10 5, is adopted due to its accuracy and ease of calculation
C d = 24 Re ( 1 + 0.152 Re 0.677 ) + 0.417 1 + 5070 Re 0.94 .
(15)
The result of this correlation is compared to that derived from Stokes' law in Fig. 1. Furthermore, in the case of small particles, the drag coefficient computed from empirical correlations must be corrected to account for non-continuum effects, i.e., the slip flow at the particle surface that reduces the drag, dividing it by the Cunningham slip correction factor55 
C c = 1 + 2 Kn ( 1.257 + 0.4 e 0.55 / Kn ) ,
(16)
where the Knudsen number Kn is evaluated at the liquid film condition and, for a gas, is
Kn = μ m , s ρ m , s D p π 2 R m , s T s .
(17)
FIG. 1.

Drag coefficient as a function of the Reynolds number: Clift–Gauvin correlation vs Stokes' law.

FIG. 1.

Drag coefficient as a function of the Reynolds number: Clift–Gauvin correlation vs Stokes' law.

Close modal
The terminal velocity becomes
u r , t = 4 ( ρ p ρ m , s ) g D p 3 C d , t ρ m , s ,
(18)
where an iterative solution is now needed for its evaluation since drag coefficient depends on the Reynolds number, which in turns depends on the relative (terminal) velocity itself: subscript “t” is added to drag coefficient to highlight this. Drag characteristic time now depends on the droplet relative velocity and thus changes during its flight
τ d = 4 ρ p D p 3 C d ρ m , s u r .
(19)
When the terminal condition is attained,
τ d , t = 4 ρ p 2 D p 3 C d , t ( ρ p ρ m , s ) ρ m , s g ,
(20)
and Eq. (11) still holds given that τ t , d is used in place of τ d. Horizontal and vertical forces balance can now be written as
{ u ̇ r , x = D x m p = u r , x τ d u ̇ r , z = W z + D z m p = u r , t τ d , t u r , z τ d .
(21)
In theory, this system of equations cannot be solved easily since the characteristic time τ d is no longer constant, but rather depends on the relative velocity through the term C d u r. Yet, this dependency is weak, and if a numerical solution is sought over small time steps it can well be neglected resulting in a solution similar to the one in Eqs. (13) and (14)
{ u p , x ( t ) = u m , x + u r , x , 0 e t / τ d u p , z ( t ) = u m , z + u r , z , 0 e t / τ d + u r , t τ d τ d , t ( 1 e t / τ d ) ,
(22)
{ s p , x ( t ) = s p , x , 0 + u m , x t + u r , x , 0 τ d ( 1 e t / τ d ) s p , z ( t ) = s p , z , 0 + ( u m , z + u r , t τ d τ d , t ) t + ( u r , z , 0 u r , t τ d τ d , t ) τ d ( 1 e t / τ d ) .
(23)
Equations (13) and (14) are special cases of Eqs. (22) and (23) for τ d = τ d , t. Table II shows particle and terminal flow data for non-evaporating spherical water droplets of various size in the range of interest for infectious disease transmission. The droplets are assumed in thermal equilibrium with the environment, and falling in still air from zero initial velocity under NTP ambient conditions with 50% relative humidity. The thermophysical properties needed for the calculations are taken from CoolProp library.56 The last column of the table reports the time needed to reach the ground from an initial height of 1 m and is derived from time-stepped simulations of the droplet fall. To be noted the wide range of scales encountered for many quantities of interest. To correctly capture the transient droplet behavior from its emission to its reaching of the terminal velocity, the time step used in a numerical model must be sufficiently small compared to τ d , t. On the other hand, if the falling time is much larger than the terminal characteristic time ( t fl 7 τ d , t), the transient becomes negligible as the droplet spends most of its time falling at terminal velocity. Figure 2 shows the terminal values of the Reynolds number, drag coefficient, velocity, and characteristic time for droplets of different size for the same conditions assumed in Table II. Dashed lines are derived under the hypothesis of Stokes' law: deviations in the upper diameters range are to be attributed to non-Stokes flow ( Re > 1), those in the lower range to non-continuum effects ( C c > 1).
TABLE II.

Droplet and terminal data for the flow past a water sphere falling in still air from zero initial velocity.

D p ( mm) m p ( kg) A p ( m 2) W z ( N) Kn C c u t ( m / s) Re t C d , t τ d , t ( s) t 1 m ( s)
1 × 10+00  5.227 × 10−07  7.854 × 10−07  5.126 × 10−06  6.522 × 10−05  1.000  3.900 × 10+00  2.578 × 10+02  7.147 × 10−01  3.982 × 10−01  5.167 × 10−01 
3 × 10−01  1.411 × 10−08  7.069 × 10−08  1.384 × 10−07  2.174 × 10−04  1.001  1.179 × 10+00  2.339 × 10+01  2.344 × 10+00  1.204 × 10−01  9.486 × 10−01 
1 × 10−01  5.227 × 10−10  7.854 × 10−09  5.126 × 10−09  6.522 × 10−04  1.002  2.474 × 10−01  1.635 × 10+00  1.776 × 10+01  2.526 × 10−02  4.066 × 10+00 
3 × 10−02  1.411 × 10−11  7.069 × 10−10  1.384 × 10−10  2.174 × 10−03  1.005  2.654 × 10−02  5.264 × 10−02  4.629 × 10+02  2.710 × 10−03  3.768 × 10+01 
1 × 10−02  5.227 × 10−13  7.854 × 10−11  5.126 × 10−12  6.522 × 10−03  1.016  3.036 × 10−03  2.007 × 10−03  1.179 × 10+04  3.100 × 10−04  3.294 × 10+02 
3 × 10−03  1.411 × 10−14  7.069 × 10−12  1.384 × 10−13  2.174 × 10−02  1.055  2.841 × 10−04  5.634 × 10−05  4.040 × 10+05  2.901 × 10−05  3.520 × 10+03 
1 × 10−03  5.227 × 10−16  7.854 × 10−13  5.126 × 10−15  6.522 × 10−02  1.164  3.485 × 10−05  2.304 × 10−06  8.951 × 10+06  3.558 × 10−06  2.870 × 10+04 
3 × 10−04  1.411 × 10−17  7.069 × 10−14  1.384 × 10−16  2.174 × 10−01  1.560  4.204 × 10−06  8.338 × 10−08  1.845 × 10+08  4.292 × 10−07  2.378 × 10+05 
1 × 10−04  5.227 × 10−20  7.854 × 10−15  5.126 × 10−18  6.522 × 10−01  2.864  8.574 × 10−07  5.668 × 10−09  1.478 × 10+09  8.754 × 10−08  1.166 × 10+06 
D p ( mm) m p ( kg) A p ( m 2) W z ( N) Kn C c u t ( m / s) Re t C d , t τ d , t ( s) t 1 m ( s)
1 × 10+00  5.227 × 10−07  7.854 × 10−07  5.126 × 10−06  6.522 × 10−05  1.000  3.900 × 10+00  2.578 × 10+02  7.147 × 10−01  3.982 × 10−01  5.167 × 10−01 
3 × 10−01  1.411 × 10−08  7.069 × 10−08  1.384 × 10−07  2.174 × 10−04  1.001  1.179 × 10+00  2.339 × 10+01  2.344 × 10+00  1.204 × 10−01  9.486 × 10−01 
1 × 10−01  5.227 × 10−10  7.854 × 10−09  5.126 × 10−09  6.522 × 10−04  1.002  2.474 × 10−01  1.635 × 10+00  1.776 × 10+01  2.526 × 10−02  4.066 × 10+00 
3 × 10−02  1.411 × 10−11  7.069 × 10−10  1.384 × 10−10  2.174 × 10−03  1.005  2.654 × 10−02  5.264 × 10−02  4.629 × 10+02  2.710 × 10−03  3.768 × 10+01 
1 × 10−02  5.227 × 10−13  7.854 × 10−11  5.126 × 10−12  6.522 × 10−03  1.016  3.036 × 10−03  2.007 × 10−03  1.179 × 10+04  3.100 × 10−04  3.294 × 10+02 
3 × 10−03  1.411 × 10−14  7.069 × 10−12  1.384 × 10−13  2.174 × 10−02  1.055  2.841 × 10−04  5.634 × 10−05  4.040 × 10+05  2.901 × 10−05  3.520 × 10+03 
1 × 10−03  5.227 × 10−16  7.854 × 10−13  5.126 × 10−15  6.522 × 10−02  1.164  3.485 × 10−05  2.304 × 10−06  8.951 × 10+06  3.558 × 10−06  2.870 × 10+04 
3 × 10−04  1.411 × 10−17  7.069 × 10−14  1.384 × 10−16  2.174 × 10−01  1.560  4.204 × 10−06  8.338 × 10−08  1.845 × 10+08  4.292 × 10−07  2.378 × 10+05 
1 × 10−04  5.227 × 10−20  7.854 × 10−15  5.126 × 10−18  6.522 × 10−01  2.864  8.574 × 10−07  5.668 × 10−09  1.478 × 10+09  8.754 × 10−08  1.166 × 10+06 
FIG. 2.

Terminal characteristics for the flow past a spherical water droplet falling in still air as functions of the droplet diameter. Solid lines refer to actual values, and dashed lines are derived from Stokes' law.

FIG. 2.

Terminal characteristics for the flow past a spherical water droplet falling in still air as functions of the droplet diameter. Solid lines refer to actual values, and dashed lines are derived from Stokes' law.

Close modal
It is known that for Re > 100, the droplets tend to deviate from the spherical shape57 with an associated increase in drag. As the terminal Reynolds number of the largest droplet in Table II is slightly above this value, no such drag correction is applied in the present work. Furthermore, it must be considered that droplets will shatter if a critical Weber number of 13 is reached where
We = ρ m , s u r 2 D p σ ,
(24)
σ being water surface tension. Considering the droplet dimensions and velocities at stake this is an unlikely event even in the case of a fast respiratory event like sneezing. As such, droplet breakup is not addressed in the current work, as neither it is in previous similar works in the literature. Similarly, also droplet interaction and coalescence are not taken into consideration as in Ref. 22, the droplet number concentration has been estimated to be at most 5 cm 3 even very close to the mouth in the case of coughing.
Mass transfer from the droplet to the environment due to evaporation follows the well-known convection–diffusion equation. In the following, the cross-diffusion of air in water is neglected being much smaller than its counterpart. In stationary form, without source or sink terms, and in terms of water vapor molar concentration C v, the equation takes the form
n v = u C v D v C v ,
(25)
where u is the molar average velocity of the air–water mixture, D v is the mass diffusivity or binary diffusion coefficient of water in air, and n is the water vapor molar flux composed of an advective and a diffusive term as clearly shown in the equation. Discarding the advective term, Eq. (25) becomes Fick's first law of diffusion. The equation can be written indifferently in molar or mass flux forms. Considering that the only relevant direction for the droplet mass transfer case is the radial
n v = n v y v D v d C v d r = D v 1 y v d C v d r = C m D v 1 y v d y v d r ,
(26a)
m v = m v ω v D v d ρ v d r = D v 1 ω v d ρ v d r = ρ m D v 1 ω v d ω v d r ,
(26b)
where y v and ω v are the water vapor molar and mass fraction, and where the equalities y v = C v / C m and ω v = ρ v / ρ m holding for ideal gas mixtures have been exploited. The subscript “m” denotes the mixture. Let us consider for now the mass flux form in Eq. (26b), by further multiplying by the radial surface area 4 π r 2 the evaporating mass flow rate m ̇ is found
m ̇ = 4 ρ m π r 2 D v 1 ω v d ω v d r .
(27)
Separating the variables and integrating between the droplet surface and the ambient
r p r m ̇ 4 π r 2 d r = ω s ω ρ m D v 1 ω v d ω v ,
(28)
and hence,
m ̇ = 2 ρ m π D p D v ln ( 1 + B M ) ,
(29)
where B M = ( ω v , ω v , s ) / ( ω v , s 1 ) is the Spalding mass transfer number. Similar considerations for heat transfer along the radial direction lead to
m ̇ = 2 λ m π D p c p , v ln ( 1 + B T ) ,
(30)
where B T = m ̇ c p , v ( T s T ) / Q ̇ cv is the Spalding heat transfer number, λ m is the mixture thermal conductivity, c p , v is the water vapor specific heat at constant pressure, and Q ̇ cv is the convective heat transfer rate. From Eqs. (29) and (30), it follows that58 
ln ( 1 + B M ) = ln ( 1 + B T ) Le c p , m c p , v ,
(31)
where
Le = α m D v = λ m ρ m c p , m D v = Sc Pr
(32)
is the Lewis number.
In the following, the subscript “v” denotes water vapor and “a” dry air, while the subscript “m” used so far to indicate the mixture is dropped for brevity. From Fourier's law and considering Newton's law of cooling, the heat flux at the droplet surface can be written as
q s = λ s ( d T d r ) s = h ( T s T ) ,
(33)
where h is the convective heat transfer coefficient. Introducing the Nusselt number, defined using the droplet diameter as characteristic length, with a few algebraic passages, it can be shown that
Nu D p = h λ s = 1 T s T ( d T d r ) s = 2 ln ( 1 + B T ) D p B T .
(34)
Considering the analogy between Fourier's law and Fick's law of diffusion, mass transfer is usually treated in a similar fashion so that, for the case of pure diffusion,
n s = D v , s ( d C v d r ) s = k ( C v , s C v , ) ,
(35a)
m s = D v , s ( d ρ v d r ) s = k ( ρ v , s ρ v , ) ,
(35b)
where k is the convective mass transfer coefficient. Introducing the Sherwood number, it can be shown that
Sh D p = k D v , s = 1 ρ v , s ρ v , ( d ρ v d r ) s = 2 ln ( 1 + B M ) D p B M .
(36)
From the mole flux forms in Eqs. (26a) and (35a), and considering that for ideal gas mixtures molar fractions can be substituted by partial pressures, mole flux, mass flux, and mass flow rate are written as
n = D v , s 1 y v , s ( d C v d r ) s = k C s y v , s y v , 1 y v , s ,
(37a)
m = n M v = k ρ s M v M s p v , s p v , p a , s ,
(37b)
m ̇ = m S = k ρ s π D p 2 M v M s p v , s p v , p a , s ,
(37c)
where M denotes the molar mass. By differentiating the droplet mass and recalling the definition of Sherwood number, Eq. (37c) can be recasted in a more practical form in terms of droplet diameter reduction rate
d D p d t = 2 ρ s D v , s Sh ρ p D p M v M s p v , s p v , p a , s = 2 D v , s Sh ρ p D p R v T s p p a , s ( p v , s p v , ) .
(38)
Under the hypothesis of small vapor partial pressure in the droplet film ( p a , s p), Eq. (38) is simplified to a form commonly found in the specialized literature.59 Nusselt and Sherwood numbers, by similitude, can be evaluated from the well-known Ranz–Marshall empirical correlation for the flow past a sphere60,61
Nu = h D p λ s = 2 + 0.6 Re 1 / 2 Pr 1 / 3 ,
(39a)
Sh = k D p D v , s = 2 + 0.6 Re 1 / 2 Sc 1 / 3 ,
(39b)
where Re , Pr, and Sc are Reynolds, Prandtl, and Schmidt numbers, respectively,
Re = u r D p ν s ; Pr = ν s α s ; Sc = ν s D v , s ,
(40)
and where the subscript “s” indicates that the thermal properties of the mixture should be evaluated at the gas film by the droplet surface, where the temperature is T s = T p and air can be regarded as saturated if the droplet is evaporating. The evaporation rate K is defined as
K = d D p 2 d t = 2 D p d D p d t = 4 D v , s Sh ρ p R v T s p p a , s ( p v , s p v , ) .
(41)
Separating the variables and integrating between the initial time and a generic time t for an initial condition D p ( 0 ) = D p , 0,
D p ( t ) = D p , 0 2 K t ,
(42)
so that the droplet evaporation time can be estimated by equating to zero the right-hand side term
t ev = D p , 0 2 K .
(43)
This is just an estimate since the evaporation rate is not constant during the evaporation but depends on the droplet diameter through the Sherwood number. Nonetheless, for small diameters, K tends to a constant value, function of the ambient conditions, and the particle density alone.

When a droplet is released, its evaporation starts immediately and is characterized by two phases, often kind of improperly referred to as transient and steady. During the initial transient period, the drop is cooled down (or warmed up) from its initial temperature to ambient wet bulb temperature, and it will then continue to shrink at constant temperature during the so-called steady phase. In case the droplet is composed of some nonvolatile fraction, at the end of the evaporation, it is still possible to distinguish a further transient phase in which the droplet warms to the ambient temperature, and an additional steady one where the droplet remains in thermal equilibrium with the environment. Due to the size of the droplet, transient phases are very short compared to the overall evaporation time.

From energy balance, it is required that the sensible thermal power accumulated by the droplet is balanced by convective heating and by evaporative cooling
Q ̇ sn = Q ̇ cv + Q ̇ ev ,
(44)
where
Q ̇ sn = c p , p m p d T p d t = c p , p ρ p π D p 3 6 d T s d t ,
(45a)
Q ̇ cv = h S ( T T s ) = λ s π D p Nu ( T T s ) ,
(45b)
Q ̇ ev = h lt d m p d t = h lt ρ p π D p 2 2 d D p d t = h lt ρ p π D p 4 K ,
(45c)
which results in
d T s d t = 6 λ s Nu ( T T s ) c p , p ρ p D p 2 3 h lt K 2 c p , p D p 2 ,
(46)
where h lt is the water latent heat of vaporization. By neglecting the dependency of K upon T s (the droplet temperature change is usually small and occurs only during the transient phases), Eq. (46) is a differential equation of the same form of Eq. (9) whose solution, for initial droplet temperature of T s , 0, is
T s ( t ) = T s , 0 e t / τ ev + T s , t ( 1 e t / τ ev ) ,
(47)
where
τ ev = c p , p ρ p D p 2 6 λ s Nu
(48)
is the characteristic time associated with evaporation, and
T s , t = T h lt ρ p K 4 λ s Nu = T + h lt ρ p D p 2 λ s Nu d D p d t
(49)
the droplet terminal temperature, i.e., the ambient wet bulb temperature. From the most general forms in Eqs. (41) and (46) that solve the transient evaporation problem in terms of diameter and temperature rate of change, solutions for the special cases outlined above are easily derived: steady evaporating droplet ( d T s / d t = 0, and thus, T s = T s , t), transient non-evaporating droplet ( d D p / d t = 0, and thus T s , t = T ), and steady non-evaporating droplet ( d T s / d t = d D p / d t = 0, and thus T s = T ).

The equations of transport, evaporation, and energy balance are coupled through the D p and d D p / d t terms, i.e., ultimately through K. Although the coupling is rather weak (K is only moderately influenced by the Sherwood number), this influences the characteristic times and the terminal velocity that may now changeover time. A coupled closed-form analytical solution is not viable, yet the system of equation can be easily solved numerically integrating over time.

Table III reports heat and mass transfer data for evaporating spherical pure water droplets of various size. The droplets are assumed to be falling in still air at NTP conditions with 50% relative humidity ( T wb = 13.77 ° C, Pr = 0.710, Sc = 0.625, Le = 0.880) from a zero initial velocity and with initial temperature of 34 °C. The thermophysical properties needed for the calculations are taken from the CoolProp library, except for mass diffusivity of water in air, which is evaluated from the correlation proposed in Ref. 62 
D v = 21.2 × 10 6 m 2 s K [ 1 K + 0.0071 ( T 273.15 ) ]
(50)
valid for temperatures between 273 and 318 K. The Reynolds number and the drag characteristic time slightly differ from those in Table II. This is because film and ambient conditions do not coincide anymore since the droplet is now evaporating. With the exception of the last two columns, all the data in the table are computed for terminal flow conditions. A clarification is required on the meaning of this: an evaporating droplet continuously changes its diameter over time, this means a terminal flow condition, strictly speaking, is never attained. Data in the table are computed hypothesizing the droplet is in the steady evaporation phase and is moving at its terminal velocity for the given diameter. The last two columns show the time needed for the complete evaporation of the droplet and the time needed to reach the ground from an initial height of 1 m, and are derived from time-stepped simulations of the droplet fall and evaporation. Note that only the larger drops will reach the ground before evaporating completely when released from that height. Furthermore, 7 τ ev , t t ev, meaning most of the evaporation time is spent in the steady phase. For small droplets, K tends to a constant value: as a consequence, d D p / d t grows D p 1.
TABLE III.

Heat and mass transfer data for an evaporating spherical water droplet falling in still air from zero initial velocity.

D p ( mm) Re t Nu Sh Nu / Sh d D p / d t ( m / s) K ( m 2 / s) τ d , t ( s) τ ev , t ( s) t ev ( s) t 1 m ( s)
1 × 10+00  2.673 × 10+02  10.759  10.392  1.035  1.428 × 10−06  2.856 × 10−09  3.970 × 10−01  2.555 × 10+00  5.201 × 10+02  5.159 × 10−1 
3 × 10−01  2.441 × 10+01  4.647  4.536  1.024  2.063 × 10−06  1.238 × 10−09  1.208 × 10−01  5.325 × 10−01  9.412 × 10+01  9.556 × 10−1 
1 × 10−01  1.724 × 10+00  2.703  2.674  1.011  3.618 × 10−06  7.237 × 10−10  2.560 × 10−02  1.017 × 10−01  1.531 × 10+01  4.771 × 10+0 
3 × 10−02  5.580 × 10−02  2.127  2.121  1.003  9.516 × 10−06  5.710 × 10−10  2.761 × 10−03  1.164 × 10−02  1.591 × 10+00  ⋯ 
1 × 10−02  2.129 × 10−03  2.025  2.024  1.001  2.720 × 10−05  5.440 × 10−10  3.160 × 10−04  1.358 × 10−03  1.817 × 10−01  ⋯ 
3 × 10−03  5.971 × 10−05  2.004  2.004  1.000  8.976 × 10−05  5.386 × 10−10  2.954 × 10−05  1.235 × 10−04  1.645 × 10−02  ⋯ 
1 × 10−03  2.435 × 10−06  2.001  2.001  1.000  2.688 × 10−04  5.377 × 10−10  3.614 × 10−06  1.374 × 10−05  1.829 × 10−03  ⋯ 
3 × 10−04  8.751 × 10−08  2.000  2.000  1.000  8.958 × 10−04  5.375 × 10−10  4.330 × 10−07  1.237 × 10−06  1.647 × 10−04  ⋯ 
1 × 10−04  5.891 × 10−09  2.000  2.000  1.000  2.687 × 10−03  5.375 × 10−10  8.744 × 10−08  1.375 × 10−07  1.830 × 10−05  ⋯ 
D p ( mm) Re t Nu Sh Nu / Sh d D p / d t ( m / s) K ( m 2 / s) τ d , t ( s) τ ev , t ( s) t ev ( s) t 1 m ( s)
1 × 10+00  2.673 × 10+02  10.759  10.392  1.035  1.428 × 10−06  2.856 × 10−09  3.970 × 10−01  2.555 × 10+00  5.201 × 10+02  5.159 × 10−1 
3 × 10−01  2.441 × 10+01  4.647  4.536  1.024  2.063 × 10−06  1.238 × 10−09  1.208 × 10−01  5.325 × 10−01  9.412 × 10+01  9.556 × 10−1 
1 × 10−01  1.724 × 10+00  2.703  2.674  1.011  3.618 × 10−06  7.237 × 10−10  2.560 × 10−02  1.017 × 10−01  1.531 × 10+01  4.771 × 10+0 
3 × 10−02  5.580 × 10−02  2.127  2.121  1.003  9.516 × 10−06  5.710 × 10−10  2.761 × 10−03  1.164 × 10−02  1.591 × 10+00  ⋯ 
1 × 10−02  2.129 × 10−03  2.025  2.024  1.001  2.720 × 10−05  5.440 × 10−10  3.160 × 10−04  1.358 × 10−03  1.817 × 10−01  ⋯ 
3 × 10−03  5.971 × 10−05  2.004  2.004  1.000  8.976 × 10−05  5.386 × 10−10  2.954 × 10−05  1.235 × 10−04  1.645 × 10−02  ⋯ 
1 × 10−03  2.435 × 10−06  2.001  2.001  1.000  2.688 × 10−04  5.377 × 10−10  3.614 × 10−06  1.374 × 10−05  1.829 × 10−03  ⋯ 
3 × 10−04  8.751 × 10−08  2.000  2.000  1.000  8.958 × 10−04  5.375 × 10−10  4.330 × 10−07  1.237 × 10−06  1.647 × 10−04  ⋯ 
1 × 10−04  5.891 × 10−09  2.000  2.000  1.000  2.687 × 10−03  5.375 × 10−10  8.744 × 10−08  1.375 × 10−07  1.830 × 10−05  ⋯ 

Figure 3 compares evaporation and falling times of water droplets of different sizes in a similar fashion to the famous Wells' curve.5 Still ambient air and zero droplet initial velocity are assumed. Droplets are released from different heights (namely, 0.5, 1.5, and 5.0 m), lines are drawn for three ambient conditions: NTP with R H = 50 %, T = 40 ° C with R H = 20 %, and T = 0 ° C with R H = 80 %. The last two are extreme conditions representing a dry summer environment promoting quick evaporation, and a wet winter environment hampering evaporation. Among the two, the evaporation time changes almost 20-fold mostly due to the change in the ( p v , s p v , ) term in Eq. (41). Non-evaporating particles (see the dotted lines in Fig. 3) after a first acceleration phase fall at their terminal velocity, while for evaporating droplets, the terminal velocity is progressively reduced to zero as they shrink. As a result, for any ambient condition, a droplet that evaporates completely as it touches the ground ( t fl = t ev) has a double falling time compared to that of an equivalent non-evaporating droplet.

FIG. 3.

Pure water droplet evaporation vs falling time in still air from different heights (see the colors) and under different thermohygrometric conditions (see the line types) as functions of the droplet diameter. Dotted lines refer to the falling time of a non-evaporating particle having the same characteristics of water.

FIG. 3.

Pure water droplet evaporation vs falling time in still air from different heights (see the colors) and under different thermohygrometric conditions (see the line types) as functions of the droplet diameter. Dotted lines refer to the falling time of a non-evaporating particle having the same characteristics of water.

Close modal

Figure 4 shows the time evolution of several quantities of interest, such as velocity, diameter, evaporation rate, temperature, Nusselt, and Sherwood numbers. In Fig. 4(a), it is seen how the droplet at first accelerates to its terminal velocity before slowing down hand in hand with the diameter reduction [Fig. 4(b)] that, following Eq. (42), at first appears almost imperceptible and then becomes more and more evident with time. Similarly, the evaporation rate in the case of small droplets is reduced at the same pace of temperature [Fig. 4(c)] due to the reduction of the water vapor partial pressure at the droplet surface. For larger droplets, the trend is inverted during the acceleration phase due to the growth of the Sherwood number with the velocity [Fig. 4(d)]. To be noted how the droplet temperature is smoothly reduced to its terminal value [Fig. 4(c)], and how this occurs in a slightly larger lapse of time than that required to reach terminal velocity [Fig. 4(a)] since τ ev , t > τ d , t.

FIG. 4.

Time history of several quantities of interest for pure water droplets of different sizes freely falling in still air at NTP with R H = 50 % from zero initial velocity.

FIG. 4.

Time history of several quantities of interest for pure water droplets of different sizes freely falling in still air at NTP with R H = 50 % from zero initial velocity.

Close modal

Let us focus on the parameters that affect the droplet fall and evaporation the most by looking at Eqs. (18) and (43) and at Tables II and III. The crucial role of the droplet diameter is apparent as the evaporation time scales with D p 2, while the falling time with D p 2 in the case of Stokes flow, and with D p 1 otherwise (please consider that in first approximation C d D p 1). Thus, a small change in the droplet diameter may drastically change its fate. Second, as already mentioned, the role of the ambient temperature and of the relative humidity on the evaporation rate K, and thus on the evaporation time t ev, is extremely relevant. Furthermore, falling and evaporation times also depend on the ambient conditions through various terms in the equations such as densities, dynamic viscosities, and the mass diffusivity. As these change only marginally in the range of temperatures of interest, their role is mostly negligible. Initial droplet conditions are also negligible as long as t ev τ ev and t fl τ d, i.e., unless the falling time of large inertia-driven droplets is of concern. For these reasons, a large portion of the older literature on the topic mention a droplet critical diameter to stress how the fate of a droplet (e.g., reaching the ground or evaporating before) is mostly a consequence of its size. While this may seem a result following from a limited view over the issue, the role of size is indeed unquestionable.

In case the droplet is a solution composed of an evaporating aqueous liquid fraction and a nonvolatile solid fraction (e.g., salts, pollutants, etc.), the evaporation will continue up to when the liquid is evaporated leaving a dry solid particle of smaller diameter. Thus, the density and the specific heat of the droplet will change during evaporation as a result of the change in its average chemical composition. Given a two-component solution with solid and liquid volume fractions φ sl and φ lq = 1 φ sl, the droplet density can be written as
ρ p = ρ sl φ sl + ρ lq ( 1 φ sl ) ,
(51)
where ρ sl and ρ lq are the densities of the solid and the liquid fractions taken separately, and are not to be intended in terms of mass concentration in the mixture. Since the volume fraction changes during the evaporation but the solid mass does not, it is practical to evaluate it from the initial conditions and the current diameter as
φ sl ( t ) = φ sl , 0 ( D p , 0 D p ( t ) ) 3 ,
(52)
while the droplet solid core diameter is easily computed as D sl = φ sl , 0 1 / 3 D p , 0. The droplet specific heat is
c p , p ( t ) = c p , sl ω sl + c p , lq ( 1 ω sl ) = ρ sl ρ p φ sl c p , sl + ρ lq ρ p ( 1 φ sl ) c p , sl .
(53)
From Raoult's law,63 the presence of the solute reduces the vapor pressure at the droplet surface (i.e., the p v , s term in the equations of Sec. II B) proportionally to the water molar fraction in the droplet
p v , s = y lq p v , sat ( T s ) .
(54)
This slows down the evaporation as the liquid fraction is reduced, stopping it slightly before it is completely dried out. Molar and volume fractions are linked by
y lq = φ lq M p M lq ρ lq ρ p .
(55)
With a few algebraic passages, the water molar fraction can be written as a function of the thermophysical properties (density and molar mass) of the droplet components, and their volume fractions
y lq = ( 1 + M lq M sl ρ sl ρ lq φ sl 1 φ sl ) 1 .
(56)
As a consequence of the evaporation rate reduction, convective heat transfer exceeds evaporative cooling so that, after the initial drop in Fig. 4(c), the droplet temperature approaches ambient temperature again while still evaporating. The evaporation terminates when p v , s = p v , , i.e., when y lq = R H and the nonvolatile solid fraction has become
φ sl = ( 1 + M lq M sl ρ sl ρ lq R H 1 R H ) 1 .
(57)

Let us consider a droplet of saliva. To esteem the impact that nonvolatile components may have on the droplet evaporation, its chemical composition must be considered. In addition to water, saliva is mainly composed of electrolytes (e.g., mostly dissociated KCl and NaCl salts), mucus, and enzymes, among other components. On the typical composition and the quantity of nonvolatiles, the literature does not agree, probably also due to some variability between individuals. Nonetheless, the volume fraction of the nonvolatiles is often assumed 1 %,41 of which one-third in mass being dissociated salts.43,64 Thermophysical properties of sodium, potassium, and chlorine can be found on any chemical engineering textbook, and the properties of any mixture of these components estimated with rather simple calculations. Data on the thermophysical properties of the organic compounds are not available in detail, but their molar mass is at least on the order of the tens of thousands, thus much larger than that of the salt ions. If these components have double mass fraction compared to the dissociated salts, this results in a triple molar mass of the nonvolatile fraction compared to salt ions. After these considerations, it is assumed that nonvolatile components in the droplet have an initial volume fraction of 1%, an average density of 1200 kg / m 3, a specific heat of 1100 J / kg K, and a molar mass of 0.1 kg / mol. This means the droplet solid core diameter is 21.5% of the initial diameter, and that for an ambient relative humidity of 50% evaporation terminates when the diameter is reduced to 23.0%, having a solid volume fraction of 82.2%.

Figure 5 shows diameter and temperature time histories of saliva droplets under the same conditions of Fig. 4. Data in the two figures differ only in the last portion of the evaporation where the evaporation rate decreases rapidly, the droplet diameter gradually reaches the terminal value [Fig. 5(a)], and the temperature rises from wet bulb to ambient value [Fig. 5(b)].

FIG. 5.

Time history of diameter and temperature for saliva droplets of different sizes freely falling in still air at NTP with R H = 50 % from zero initial velocity.

FIG. 5.

Time history of diameter and temperature for saliva droplets of different sizes freely falling in still air at NTP with R H = 50 % from zero initial velocity.

Close modal

To predict the trajectory and the evaporation of droplets, it is clear how local ambient conditions in terms of air velocity, direction, temperature, and relative humidity play a fundamental role. It must then be considered the case where the droplets are transported by a particular air flow. Essentially, the goal is to determine the value of the terms u m , x , u m , z , p v , , and T to be used in Eqs. (23), (41), and (49).

When dealing with pathogens spreading with the droplets emitted while breathing, coughing, or sneezing in a confined environment, for instance, it is important to address not only the thermohygrometric conditions and the ventilation characteristics of the room, but also the thermofluid dynamics of the cloud of exhaled air, which forms the environment into which the droplets are carried, at least during the first phase of their journey. Close to the mouth this cloud is generally characterized by higher fluid velocity, temperature, and relative humidity compared to the surroundings. This allows the droplets to be transported by drag over long distances while being mostly protected from evaporation. Furthermore, the cloud gradually mixes with the ambient air by entrainment up to when it will be no longer perceived.

A simplified model of the dynamic of such a cloud can be derived from momentum conservation equation. In the following, a periodic intermittent one-dimensional example is given. The model presented is inspired by the works in Refs. 25 and 65, other cloud models in view of pathogens transport are proposed in Refs. 41 and 45. The cloud model is successively extended to account for generic non-uniform and unsteady jet velocity profiles, and for the effects of momentum dissipation by friction over the cloud. In the following sections, models for the cloud buoyancy and the droplet turbulent dispersion will also be introduced.

It is known from observations that the exhaled cloud expands linearly in space, so that the jet generated has the shape of a truncated cone (or better, of a truncated spherical sector) whose radius r at a certain location is proportional to the distance s from the ideal cone tip O (see Fig. 6) so that
r = s tan ψ = β s ,
(58)
where ψ is the semi-aperture angle of the cloud cone. β is known as entrainment coefficient and from the literature is typically 0.1.66 In three dimensions, this corresponds to a cone solid angle
Ω = S r 2 = 2 π ( 1 cos ψ ) .
(59)
FIG. 6.

Scheme of the breath cloud cone.

FIG. 6.

Scheme of the breath cloud cone.

Close modal
β = 0.1 implies Ω = 0.0312 sr. If the mouth opening is assumed as a circle of radius r 0 , the distance r0 from the ideal cone tip and its subtended area along the spherical surface are
r 0 = r 0 sin ψ ; S 0 = r 0 2 Ω .
(60)
Assuming the cloud front has reached a distance r fr > r 0, its volume is
V ( ξ ) = r 0 r fr r 2 Ω d r = 1 ξ fr r 0 3 ξ 2 Ω d ξ = r 0 3 Ω 3 ( ξ 3 1 ) ,
(61)
where the non-dimensional distance ξ = r / r 0 is introduced.

It must be considered that the cloud does not strictly behave as a jet due to the intermittent nature of respiratory events. It rather evolves through two phases:65 a first jet phase in which a momentum flux is injected in the ambient through the mouth or nose, and a second puff phase in which the momentum emission stops and the cloud keeps expanding. The two phases are cyclically repeated in the case of breathing. The first can be modeled, assuming that a constant and uniform momentum flux P ̇ is introduced through the mouth and is conserved at successive radii as the cloud expands. The second, instead, can be modeled imposing the conservation of the momentum P previously introduced. Previous studies25 demonstrated how the momentum share associated with liquid droplets in the jet is negligible as their size and concentration is however small. As such, only humid air is considered here. For the same reason, droplets in a breath cloud can be assumed as isolated, and droplet to droplet interaction neglected, when studying their transport. Due to entrainment, the velocity of the air is reduced with the distance from the momentum source up to when it becomes of comparable size with respect to that of the surrounding ambient and its effect thus vanishes. The momentum conservation hypothesis is a simplification that allows the creation of a simple and effective analytical cloud dynamics model.

Let us consider the jet phase of a breath cloud. It is assumed that the exhaled cloud is made of humid air and that its chemical composition does not change because of breathing. The lung's tidal volume for normal breathing is typically 0.5 l, the breathing period 4 s, and the exhalation time amounts to 60% of the period.67 The mouth opening radius is around r mt = 1 cm.65 Assuming a constant volume flow rate exhalation, V ̇ 0 = V ex / t ex is found where the subscript “ex” refers to the exhalation phase. The average air velocity at the mouth is u 0 = V ̇ 0 / S 0. With the data above, it would result r 0 = 0.100 m, u 0 = 0.662 m / s, and t ex = 2.4 s. It is assumed, for now, that the velocity and all the fluid properties only vary with the distance from the mouth, while they are assumed uniform over the solid angle spanned. If the momentum flux injected by such a steady and uniform jet
P ̇ st = m ̇ 0 u 0 = ρ 0 u 0 2 r 0 2 Ω
(62)
is conserved at each downstream section, it follows that
u ( ξ ) = u 0 ξ ρ 0 ρ ,
(63)
i.e., the velocity scales with ξ 1 and is proportional to u0. From Eq. (60), ξ r mt 1, and if a given exhaled volume boundary condition is assumed, then u 0 r mt 2. This means that the local velocity and the distance traveled by the droplet while in the breath cloud are inversely proportional to the mouth opening. Furthermore, volume and mass flow rates grow linearly with the distance
V ̇ ( ξ ) V ̇ 0 = ξ ρ 0 ρ ; m ̇ ( ξ ) m ̇ 0 = ξ ρ ρ 0 .
(64)
These quantities can also be expressed as result of the adiabatic mixing of two streams: a constant exhaled flow rate, and an entrained flow rate growing with the distance from the source. The two streams have different characteristics, and the thermophysical properties at any location can be computed in terms of weighted averages as
ρ ( ξ ) = ρ 0 V ̇ 0 V ̇ + ρ ( 1 V ̇ 0 V ̇ ) ,
(65a)
h ( ξ ) = h 0 m ̇ 0 m ̇ + h ( 1 m ̇ 0 m ̇ ) ,
(65b)
A H ( ξ ) = [ A H 0 1 + A H 0 m ̇ 0 m ̇ + A H 1 + A H ( 1 m ̇ 0 m ̇ ) ] 1 [ A H 0 1 + A H 0 m ̇ 0 m ̇ + A H 1 + A H ( 1 m ̇ 0 m ̇ ) ] ,
(65c)
where h is the stream specific enthalpy and AH denotes its absolute humidity. Since in Eq. (65a), the density ρ appears on both sides of the equation, explicitly on the left-hand side, and through V ̇ 0 / V ̇ term on the right-hand side, the equation is better recasted as
ρ ( ξ ) = ρ [ 1 + Γ 2 ξ 2 ( 1 1 + 4 ξ 2 Γ ) ] ρ ( 1 Γ ξ ) ,
(66)
where
Γ = ( ρ ρ 0 + ρ 0 ρ 2 )
(67)
is a positive quantity introduced for convenience, and the approximation in Eq. (66) is derived noting that the term under square root is always 1 as long as ρ / ρ 0 ratio is close to unity. In the range of densities of interest, this is the case, and the error associated with this simplification is < 0.2 % at the mouth and decreases rapidly for larger distances. In fact, if the exhaled stream is assumed to be at T = 34 ° C and R H = 95 % and the ambient is at NTP with R H = 50 %, it follows that ρ 0 = 1.128 kg / m 3, ρ = 1.199 kg / m 3, and 4 / Γ > 10 3. Integrating Eq. (63) by separating the variables, and neglecting the weak dependency of the density upon distance
1 ξ fr ρ ρ 0 ξ d ξ = 0 t u 0 r 0 d t ρ fr ρ 0 ξ fr 2 = 1 + 2 u 0 r 0 t ,
(68)
i.e., the cloud front advances proportionally to the square root of time ( ξ fr t 1 / 2) as known from the literature on jets.40,68
Let us now consider the puff phase of a breath cloud. While Eqs. (65b) and (66) still hold, some hypothesis on the variation of the cloud velocity is now needed. It is assumed that the momentum is conserved as the cloud expands, and so, the momentum flux is no longer constant at each section. For simplicity, it is assumed that for a generic time t > t ex, the velocity profile still retains its original shape but is uniformly scaled by a factor γ,
u ( ξ ) = u 0 ξ ρ 0 ρ γ .
(69)
From momentum conservation, the cumulative momentum injected since the beginning of the respiratory event is
P ( t ) = 0 t P ̇ ( t ) d t = 1 ξ fr ρ ρ 0 ρ 0 u 0 γ r 0 3 Ω ξ d ξ
(70)
from which
γ ( ρ fr ρ 0 ξ fr 2 1 ) = 2 u 0 r 0 P ( t ) P ̇ st
(71)
is derived. Substituting Eq. (71) into Eq. (69) and integrating by separating the variables
γ ( t ) = P ( t ) P ̇ st 2 0 t P ( t ) P ̇ st d t
(72)
is obtained. Equation (72) follows from momentum conservation given the assumption in Eq. (69), while no assumption on P(t) has been made. As a consequence, the equation does not hold only for the puff phase of a breath cloud [i.e., when P(t) remains constant] but for any jet, either uniform or non-uniform, steady or unsteady, periodic or not. P ̇ st, as reported in Eq. (62), stands for the momentum flux of an equivalent steady and uniform jet releasing the same tidal volume V ex over the given exhalation time span t ex and mouth surface r 0 2 Ω. By substituting Eq. (72) into Eq. (71), it can be shown that for the puff phase, the cloud front advances proportionally to the fourth root of time ( ξ fr t 1 / 4) as known from the literature.40,68

In the case of a generic isolated respiratory event (e.g., coughing or sneezing), γ will gradually fade to zero with time after the event terminates due to the progressive growth of the denominator in Eq. (72). In the case of a train of intermittent periodic events (e.g., breathing), γ will oscillate between jet and puff phases tending to a constant value, function of the momentum flux impulse shape and its duration within the event period. This means it will still be ξ fr t 1 / 2 but with a lesser proportionality constant compared to an equivalent continuous event, as known from the literature.1,67

Assuming a generic non-uniform radial velocity profile at the mouth does not affect the result of Eq. (72) since, with regard to momentum flux, it would result in the same scaling factor to be applied to P(t) and to P ̇ st.

From the considerations above and from Eqs. (69) and (72), in the most general case the local jet velocity is obviously function of its distance from the momentum source ξ, the relative position within the cloud radius ζ = r / β ξ r 0, and time t through the γ factor. Regarding time, this means that particles of the cloud generated at different moments in time will be characterized by different velocity histories. Likewise, if it is assumed that droplet emission occurs throughout the duration of a respiratory event, droplets exhaled at different moments in time will be characterized by slightly different trajectories while inside the breath cloud boundaries. Focusing on a gas particle exhaled with velocity u0 after a time t dl since the respiratory event beginning and integrating Eq. (69), the distance traveled by the fluid can be written as a function of time as
ρ ρ 0 ξ 2 1 = 2 u 0 r 0 t dl t γ ( t ) d t .
(73)
Equation (70) can be improved to account for the effects of momentum dissipation on the breath cloud. By applying an exponential decay factor proportional to time to the momentum flux, the cloud momentum becomes
P ( t 1 ) = 0 t 1 P ̇ ( t ) exp ( t 1 t τ ds ) d t ,
(74)
where τ ds is a jet momentum characteristic dissipation time which for breathing was rated in 1.5 s after some preliminary CFD tests performed for calibration purpose. In these tests, a breathing mannequin in a large empty room was modeled, and the extent of the breath cloud compared to that obtained from the model using the same boundary conditions and different dissipation times. It is highlighted that this is only a preliminary estimate that would require further investigation for a better evaluation, and that the value of τ ds may change for different respiratory events.
Figure 7 shows the trends of the main parameters describing the exhaled cloud according to the model outlined above through an example in which a non-uniform and time-dependent velocity profile is assumed for both isolated and periodic breathing. The exhaled velocity profile is Gaussian in space as suggested in Refs. 36 and 37 and follows a piecewise defined sinusoidal function in time: respiratory events, in fact, are commonly characterized by a quick rise in the velocity followed by a slower decay. Here, it is assumed that peak velocity is reached after a time t pk = 0.6 s, while the assumptions made previously hold for the other breath characteristics. The inflow velocity at the mouth ( ξ = 1) over a period is given by
u ( ζ , t ) = { π 2 u 0 e ζ 2 sin ( π 2 t t pk ) 0 t < t pk π 2 u 0 e ζ 2 cos ( π 2 t t pk t ex t pk ) t pk t < t ex 0 t ex t < t pr ,
(75)
while at any location inside the cloud, the local velocity is computed as
u ( ξ , ζ , t ) = u 0 ξ ρ 0 ρ e ζ 2 γ ( t ) .
(76)
FIG. 7.

Results of the exhaled cloud model. The wording “train of events” refers to the case of cyclical breathing, “isolated event” to a single exhalation. The colors in the bottom figures refer to cloud sections generated at different moments in time, solid lines identify jet phases, and dashed lines puff phases.

FIG. 7.

Results of the exhaled cloud model. The wording “train of events” refers to the case of cyclical breathing, “isolated event” to a single exhalation. The colors in the bottom figures refer to cloud sections generated at different moments in time, solid lines identify jet phases, and dashed lines puff phases.

Close modal

Figure 7(a) reports the trend of γ according to Eq. (72) for cyclical and isolated respiratory events with momentum decay as in Eq. (74). The same trends are reported with dots for the case of time-limited steady and uniform exhalation without decay. The impact of momentum dissipation on the cloud motion is apparent. Furthermore, Fig. 7(b) shows the value of the local cloud density as a function of the distance from the mouth according to Eq. (66). Similar plots can be derived for temperature or enthalpy and humidity following Eqs. (65b) and (65c) but are omitted for brevity. Figure 7(c) shows how different sections of the cloud undergo different space vs time histories after Eq. (73) and how the distance traveled scales differently with respect to time between jet and puff phases, as highlighted above. Finally, Fig. 7(d) shows how different sections of the cloud also undergo different velocity histories and how in the case of cyclical exhalations higher velocities are kept up to further distances from the momentum source thanks to the larger value of γ. Both Figs. 7(c) and 7(d) refer to the case where cloud momentum dissipation is present.

As an additional exhaled breath model feature, the effect of the jet potential core can be accounted for by preventing the reduction of the velocity and the change of the jet thermophysical properties with ξ within a certain non-dimensional distance ξ pc from the mouth. Furthermore, these quantities are computed on a reduced non-dimensional distance ξ ξ pc. In line with the literature37,45 this distance is chosen equal to 6.2 mouth diameters, which in non-dimensional form results in ξ pc = 12.4 sin ( arctan ( β ) ) = 1.234.

The analytical cloud model outlined above, by taking hint from similar models from the recent literature, is able to move a step forward in the modeling capabilities, overcoming several limitations of previous models with the sole assumption given in Eq. (69). As mentioned, in the sparse literature on the topic exhaled breath is either treated as a non-uniform steady jet or, less often, as a uniform and piecewise-steady sequence of jets and puffs. The cloud momentum dissipation is never accounted for. The model proposed here can handle fully non-uniform, unsteady, and periodic velocity profiles, taking into account the cloud momentum dissipation, and is able to evaluate the different trajectories droplets may undergo depending on their exhalation time within the respiratory event time span. Furthermore, the model proposed is consistent with the cloud front advancement foreseen by the jets and puffs theory.

The exhaled cloud is subject to buoyancy due to its lesser density compared to the surrounding environment. The effect of buoyancy is that of curving the cloud trajectory in the vertical direction. In Ref. 42, a rather popular empirical formula is given for the vertical deviation of a circular jet centerline as a function of the distance Δ r from its source
Δ z S 0 = 0.0354 Ri ( Δ r S 0 ) 3 T 0 T ,
(77)
where Ri = Gr / Re 2 is the Richardson number, erroneously referred to as Archimedes number in Ref. 42, computed using S 0 as characteristic length. Equation (77) is formally derived from experiments on horizontal jets, but can be reasonably applied also to jets along different directions as long as they do not deviate excessively from the horizontal, this is the case for respiratory events. Expanding the terms and recasting Eq. (77) using the non-dimensional distance ξ introduced in Sec. II E results in
Δ z = 0.0354 g r 0 2 Ω u 0 2 T 0 T T ( ξ 1 ) 3 ,
(78)
where ξ is to be intended as the non-dimensional distance from the ideal cloud cone vertex of the corresponding non-buoyant jet.
Let us consider a jet exhaled along a direction forming an angle χ0 with the horizontal. The jet centerline coordinates at any location ξ 1 can be expressed in parametric form as
{ x c = x vr + ξ r 0 cos χ 0 z c = z vr + ξ r 0 sin χ 0 + Δ z ,
(79)
where ( x vr , z vr ) are the coordinates of the ideal cone tip. To compute the cloud characteristics such as temperature, humidity, and velocity using the equations given in Sec. II E, the non-dimensional distance should rather be computed along the curved centerline, so that ξ must be substituted by the non-dimensional curvilinear distance ξ c where
ξ c = 1 ξ ( d x c ( ξ ) d ξ ) 2 + ( d z c ( ξ ) d ξ ) 2 d ξ .
(80)
Equation (80) has no closed form solution, but can be evaluated numerically: due to the nature of the equation, a third-order polynomial approximation would suffice.
Last, to compute the local ambient conditions encountered by a generic droplet, it must be assessed whether it falls within the breath cloud, and if so derive the ξ c associated with its position. Deriving Eq. (79) the angle χ c formed by the jet centerline with respect to the horizontal can be written as
χ c = arctan ( d z c d x c ) = arctan ( 0.1062 g r 0 Ω u 0 2 T 0 T T ( ξ 1 ) 2 cos χ 0 + tan χ 0 ) .
(81)
Let us assume that at a generic distance ξ, the centerline curvilinear coordinate is ξ c (see Fig. 8). Imagining to straighten the jet curved centerline along the direction identified by the angle χ c, the cone tip would be translated to the coordinates
{ x vr = x c ξ c r 0 cos χ c z vr = z c ξ c r 0 sin χ c .
(82)
FIG. 8.

Method to assess if a generic point P in space falls within the breath cloud.

FIG. 8.

Method to assess if a generic point P in space falls within the breath cloud.

Close modal
A generic position ( x p , z p ) to belong to the same cloud section must be located at the same non-dimensional distance ξ c from ( x vr , z vr ), i.e.,
ξ c = ( x p x vr ) 2 + ( z p z vr ) 2 = ( x c x vr ) 2 + ( z c z vr ) 2 ,
(83)
and there must exist an angle χ p so that
{ x p = x vr + ξ c r 0 cos χ p z p = z vr + ξ c r 0 sin χ p .
(84)
For each point in the domain one and only one solution to Eq. (83) exists, i.e., there is one point belonging to the cone centerline that satisfies the equation. This point can be found numerically by solving the inverse problem with a gradient descent method. If the difference between the resulting χ p and χ c angles is within the cloud cone semi-aperture angle ± ψ
| χ p χ c | ψ ,
(85)
then the point ( x p , z p ) belongs to the breath cloud.

Figure 8 exemplifies this procedure. At a distance ξ from the vertex VR, the cone centerline has shifted to point C due to buoyancy. Let the distance between VR and C along the curved centerline be ξ c. Straightening the cone along the χ c direction, VR is shifted to VR' (the red line in the figure shows the locus of VR' points for different values of ξ c). A point P having the same distance from VR' as C falls within the breath cloud limits if the angle between P, VR', and C is less than the cone half-opening angle ψ. In the example given P 1 is located in the breath cloud, while P 2 is not.

Figure 9 gives an example of the results obtained from the buoyant cloud model proposed. The boundary conditions are the same used in previous examples. Droplets of saliva of different diameter and 1% nonvolatile volume fraction are released from a height of 1.5 m from the middle of a 2 cm-diameter mouth at time t pk from the beginning of a periodic breath whose jet phase lasts 2.4 s and whose period is 4 s. The exhalation is characterized by an entrainment coefficient β = 0.1. Breath cloud momentum dissipation and jet potential core are accounted for. Ambient conditions are NTP with 50% relative humidity, while the exhaled cloud is at 34 °C and 95% relative humidity and is released horizontally, its velocity following Eq. (75). The droplets are initially in thermal equilibrium with the exhaled cloud and are released at a velocity equal to 95% of that of the cloud. The case of a breath is chosen since it is subject to more evident buoyancy effects due to the slow flow velocity associated with the respiratory event.

FIG. 9.

Results obtained from the buoyant cloud model.

FIG. 9.

Results obtained from the buoyant cloud model.

Close modal

Figure 9(a) shows the droplet trajectory. Dashed lines denote the limits and the centerline of the buoyant cloud. Larger droplets are little affected by the cloud and exit its zone of influence almost immediately following a parabolic trajectory. As the dimension of the droplet is reduced, the trajectory is more and more influenced by the breath cloud motion. The 100 μ m droplet, for instance, is carried horizontally up to when it exits the cloud falling vertically to the ground just a few centimeters away from the mouth. The 30 μ m droplet is carried further by the cloud; its terminal velocity is still high enough to escape the cloud at some point, but slow enough to terminate evaporation long before reaching the ground, remaining airborne. The same fate befalls the smallest droplet which yet cannot escape from the cloud and is easily carried over long distances barely moving away from the cloud centerline from which it was assumed to be released. In Fig. 9(b), the horizontal vs vertical velocity component history is compared for the same droplets. All droplets start from null vertical velocity (see the dot on the right side of the figure), and at first accelerate toward the ground due to gravity. Larger droplets keep accelerating downward as they exit the cloud either attaining or not their terminal velocity. The 30 μ m droplet soon after being released is pushed upward by buoyancy temporarily inverting its route up to when it falls out of the breath cloud. The effect of buoyancy is most evident on the smallest droplet, whose vertical velocity changes sign soon after being exhaled and after an initial vertical acceleration closely follows the decelerating cloud. The marginal momentum impulse given by the second breathing period is apparent on the top-left portion of the plot. In Figs. 9(c) and 9(d), the droplet time histories of temperature, diameter, and evaporation rate are shown. Larger droplets reach the ground without terminating evaporation, either attaining or not the terminal temperature. A sudden increase in the evaporation rate is always found when droplets exit the cloud. Droplets are initially cooled to the ambient wet bulb temperature whether they are located in the jet potential core ( T wb 33.4 ° C), in the ambient ( T wb 13.8 ° C), or in the breath cloud ( T wb decreasing with the distance). On this behalf, it is interesting to note the time histories of the smaller droplets. The 10 μ m and the 30 μ m droplets very soon reach the local wet bulb temperature, which gradually decreases as the cloud is cooled by entrainment. The 30 μ m droplet then exits the cloud and is suddenly cooled to the ambient wet bulb temperature before reaching thermal equilibrium with the ambient at the end of the evaporation process, when K drops to zero. On the contrary, the 10 μ m droplet terminates the evaporation while still in the cloud, as its temperature rises to the local cloud temperature after 0.79 s from exhalation. This is a time lapse approximately 4 times larger than that in Table III, which shows how the breath cloud protects the droplet from evaporation. Yet, it is not fully correct to state that the droplet terminates evaporation at this stage: in Fig. 9(d) in fact the evaporation rate drops sharply but is not zeroed. The droplet in fact locally reaches the evaporation stopping criterion in Eq. (57), but the relative humidity keeps being reduced by entrainment while moving forward in the cloud. As a result, the droplet diameter keeps reducing even though at a much slower pace. In the model proposed, being located in the jet potential core, in the breath cloud, or outside of the cloud are on/off conditions according to which the evaluation of the local thermohygrometric conditions changes. This translates into gradient discontinuities that can be seen in Figs. 9(c) and 9(d) but would be smoother in reality.

Droplet turbulent dispersion is implemented using a discrete random walk model as also found in similar works in the literature.36,37 This model aims at mimicking the droplet interaction with the turbulent eddies in the breath cloud with a stochastic approach. Turbulence is assumed isotropic so that the local air velocity fluctuations along the three coordinate directions can be written as
u = η 2 k 3 ,
(86)
where k is the turbulent kinetic energy and η a normally distributed random number vector. The perturbation thus computed is to be added to the air velocity term u m in the droplet transport Eqs. (22) and (23), where the extension from two to three dimensions is trivial. The perturbation in Eq. (86) is updated periodically at intervals of time equal to the minimum between the estimates of the characteristic eddy lifetime t el and of the droplet eddy crossing time t ec
t el = 0.3 k ε ; t ec = τ d ln ( 1 l e τ d u r ) ,
(87)
where ε is the turbulent dissipation rate, and l e = 0.164 k 1.5 / ε the eddy length scale. To evaluate the velocity fluctuation and its update, time estimates of k and ε fields in the breath cloud are needed. Here, the equations for jets proposed in Ref. 37 are adopted
k = u c 2 c 1 ( e c 2 ( ζ c 3 ) 2 + e c 2 ( ζ + c 3 ) 2 ) ,
(88a)
ε = u c 3 r mt c 4 ( e c 5 ( ζ c 6 ) 2 + e c 5 ( ζ + c 6 ) 2 ) ,
(88b)
where u c is the jet centerline velocity and the coefficients are as follows: c 1 = 0.0667 , c 2 = 1.079 , c 3 = 0.6853 , c 4 = 0.017 , c 5 = 1.963 , c 6 = 0.6126.

The inclusion of a discrete random walk model allows statistical analyses of droplet distribution in space to be performed. This feature is further improved by the randomization of the droplet initial conditions in both space and time. When simulating a droplet, its initial position is randomly chosen on the mouth surface area S0 (see Fig. 6), and its initial velocity aligned with the vector between this position and the ideal cone tip O. The probability for a certain location to be chosen is proportional to the exhaled velocity at that point (e.g., if the velocity profile chosen is Gaussian, the probability distribution becomes binormal) and the initial velocity is chosen as a fraction (e.g., 95%) of the exhalation velocity at that position and time. In a similar fashion, the droplet exhalation time is chosen randomly over the [ 0 , t ex ] interval, the probability being proportional to the instantaneous mass flow rate.

To assess the chance of inhaling a suspended particle, an additional model is needed. A simple one is presented in a similar fashion to what proposed in Sec. II E for exhalation.

Let us assume a volume of V in = 0.5 l per breath is inhaled at constant flow rate through a r 0 = 1 cm radius orifice in t in = 1.6 s time, in line with previous examples. Let the problem have radial symmetry, i.e., the inhaled volume is a hemisphere ( Ω = 2 π) in front of an individual's face. For radial symmetry, it is assumed that the surface through which air is inhaled is also hemispherical ( S 0 = r 0 2 Ω). The inhaled volume is such that
V in = Ω 3 ( r in 3 r 0 3 ) ,
(89a)
r in = 3 V in Ω + r 0 3 3 .
(89b)
With the numbers above, for instance, r in = 6.21 cm and u 0 = V ̇ 0 / S 0 = 0.497 m / s. Due to symmetry, the volumetric flow rate is the same at each radial coordinate so that
V ̇ 0 = Ω r 0 2 u 0 = Ω r 2 u ,
(90a)
u ( ξ ) = u 0 ξ 2 ,
(90b)
where the non-dimensional distance ξ = r / r 0 is introduced. Any droplet that in its trajectory crosses the volume V in with an orthogonal velocity lower than u is likely deviated and inhaled.

Such a simple model may apply both to direct and indirect particle inhalation.

The analytical model proposed is validated against available data in the literature for what concerns the droplet transport, evaporation, and the exhaled cloud modeling. Unfortunately, the data in the literature are very sparse and the boundary conditions not always clearly stated. To the authors' knowledge, due to the difficulties in carrying out accurate experiments at such small scales, experimental results are only available for droplet evaporation, while the other aspects of modeling can only be compared to additional analytical or numerical works based on rather different assumptions.

In Fig. 10(a), the evaporation model is successfully validated against three sets of experimental results,61,69,70 and another numerical model41 based on similar equations:

  • in the experiment by Ranz and Marshall,61 a still 1.04 mm-diameter pure water droplet evaporates in a quiescent dry air environment. The droplet is initially at 9 ° C, while the air is kept at 25 ° C. The correspondence with the present work is excellent,

  • in the work by Smolik et al.,69 a suspended 1.2 mm-diameter pure water droplet evaporates in a constant velocity airstream. The droplet is initially at 14 °C, and the air is kept at 24 °C and 35% relative humidity, while the airstream velocity is 0.203 m / s. The experiment is carried out in a pressurized vessel at 3 atm. The agreement with the present work is again very good,

  • in the work by Chaudhuri et al.,70 a still 593 μ m-diameter droplet made of an aqueous solution with 1 % w / w dissociated NaCl content evaporates in stagnant air. The ambient is approximately at 30 °C and 50% relative humidity, and the droplet is initially in thermal equilibrium with the environment. It is pointed out that the thermohygrometric conditions are not controlled nor accurately known. For this reason, the agreement with the present work is only fair, but it would be enough to assume a 60% relative humidity [see the thinner line in Fig. 10(a)] for the results to match,

  • in the simulations by Xie et al.,41 a numerical model is tested against the previous experiments by Ranz and Marshall,61 and by Smolik et al.69 The agreement with experiments is acceptable in the former case but not in the latter. It is deemed that the authors omitted to set the correct pressure in their simulations: if atmospheric pressure is assumed, in fact, the correspondence with the current model is reasonably good.

FIG. 10.

Validation of the analytical model proposed against data in the literature.

FIG. 10.

Validation of the analytical model proposed against data in the literature.

Close modal

In Fig. 10(b), the predicted droplet trajectory is compared to other sets of numerical results38,41 either including or not an exhaled cloud model. Dashed lines denote the limits of the buoyant exhaled cloud.

  • In the simulations by Wang et al.,38 pure water droplets of various diameters are released horizontally in stagnant air from a height of 1.8 m with a velocity of 6 m / s. The ambient thermohygrometric conditions are 23 °C with 50% relative humidity, and the droplet is initially in thermal equilibrium with the environment. No breath cloud is modeled. For brevity, only the 200 μ m droplet results are reported. This is a small enough size for the droplet not to follow a parabolic trajectory, and a big enough size for the droplet to fall relatively far from the source. The agreement with the current model is very good, the minor differences being ascribable to the different correlation used for evaluating drag coefficient,

  • in the simulations by Xie et al.,41 water droplets of various diameters with a NaCl content of 0.9 % w / v are released horizontally in a stagnant air environment from a height of 2 m and with a velocity of 10 m / s, modeling a cough. Ambient air is at 20 °C and 50% relative humidity, while the droplet is initially at 33 °C. The breath cloud is modeled as a buoyant jet with a mouth diameter of 4 cm, an average velocity of 10 m / s and a temperature of 33 °C. The jet velocity profile and buoyancy are computed from empirical formulas resulting in a compact and narrow jet whose horizontal velocity fades quickly to zero and having large vertical velocity components out of the jet core. No puff phase is modeled. This configuration is replicated at best using the model proposed in the present work even though the jet velocity profile is assumed Gaussian. In view of this, also the stochastic features discussed in Sec. II G are switched off. The comparison is reported in Fig. 10(b) for droplets diameters of 200 and 40 μ m, the latter being a small enough size for the droplet to remain trapped inside the breath cloud. The two models agree well, at least qualitatively. Some discrepancies in the trajectory are found indeed due to differences in the breath cloud modeling.

This is a common issue in the sparse breath cloud models found in the literature: small differences in the characteristics chosen to shape the jet/puff cloud may have an appreciable impact over the prediction of the time the droplet spends inside the cloud being carried by the air stream. Luckily enough, these differences are mostly limited to droplets of intermediate size: large droplets, in fact, leave the cloud almost immediately and are mostly unaffected its presence, while small droplets remain trapped inside and simply move with the flow.

Last, critical diameters predicted by Ref. 41 for different ambient relative humidities are compared with those predicted by the current model in Table IV. By critical diameter, it is meant the diameter of the droplet whose falling time equals the evaporation time. The table refers to droplets freely falling in stagnant air at 20 °C from a 2 m height. The exact match between the two models is apparent, also considering that data in Ref. 41 is given with a ± 2.5 μ m accuracy.

TABLE IV.

Validation of the analytical model: critical diameters for different ambient relative humidities.

Relative humidity (%) Critical diameter ( μ m)
Xie et al.41  Present work
125  125.1 
50  100  99.2 
70  85  85.0 
90  60  62.4 
Relative humidity (%) Critical diameter ( μ m)
Xie et al.41  Present work
125  125.1 
50  100  99.2 
70  85  85.0 
90  60  62.4 

In the present work, an analytical model of the exhaled droplets transport and evaporation in view of infectious disease transmission modeling has been presented and critically investigated. The model is grounded on the fundamental laws of fluid dynamics, convection–diffusion equation, and conservation equations, and features physical models of the droplet transport, evaporation, energy balance, composition in terms of nonvolatile fraction, and of the droplet transport within a buoyant exhaled breath cloud. A simple model for the droplet inhalation is also suggested and discussed.

The model has been validated against data in the literature finding a very good agreement overall, in particular with the few experimental works available. The comparison with other analytical or numerical works is also good, in particular where the fundamental equations are involved. Minor differences can be found depending on the level of detail, on the way the thermophysical properties of the fluids or the droplet drag are computed, but the physics behind, although extensive, is relatively well established. Things may differ for what concerns the breath cloud model. Depending on the assumptions made, several different approaches to building such a model are possible, and these reflect into slightly different sets of results. The detailed modeling of an unsteady three-dimensional breath cloud probably goes beyond the scope of a relatively simple analytical model and calls for extensive experimental campaigns or expensive CFD simulations that unfortunately are not available in the literature, and whose knowledge would help in further improving the prediction capabilities of simpler models as well.

The model proposed has been implemented in Python language and has proven to be very informative, being able to provide a deep insight into many details of droplet physics, as shown in the examples given along the text. It is also a very versatile and useful tool that can be applied to different scenarios with the proper sets of boundary conditions, allowing extensive statistical analyses of droplet trajectory distribution to be performed quickly. At the same time, sensitivity analyses71,72 on different scenarios would be simple and, considering that the model actually depends on many parameters, very informative. Together with additional information on the droplet size distribution and their viral content, it would allow the assessment of the risks associated with the different routes of infectious disease transmission for any type of respiratory event and environmental conditions.

The authors have no conflicts to disclose.

Marco Cavazzuti: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (lead); Supervision (equal); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Paolo Tartarini: Project administration (lead); Resources (lead); Supervision (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

A

Cross-sectional area ( m 2)

AH

Absolute humidity ( kg vap / kg air)

B M

Spalding mass transfer number

B T

Spalding heat transfer number

c p

Specific heat at constant pressure ( J / kg K)

C

Molar concentration ( mol / m 3)

C c

Cunningham correction factor

C d

Drag coefficient

D

Diameter ( m)

D v

Mass diffusivity ( m 2 / s)

D

Drag force ( N)

g

Gravitational acceleration ( m 2 / s)

Gr

Grashof number

h

Convective heat transfer coefficient ( W / m 2 K)

h lt

Latent heat of vaporization ( J / kg)

h

Specific enthalpy ( J / kg)

k

Convective mass transfer coefficient ( m / s)

k

Turbulent kinetic energy ( m 2 / s 2)

K

Evaporation rate ( m 2 / s)

Kn

Knudsen number

l e

Eddy length scale

Le

Lewis number

m

Mass ( kg)

m

Mass flux ( kg / m 2 s)

m ̇

Mass flow rate ( kg / s)

M

Molar mass ( kg / mol)

n

Molar flux ( mol / m 2 s)

Nu

Nusselt number

p

Pressure ( Pa)

P

Momentum ( N s)

P ̇

Momentum flux ( N)

Pr

Prandtl number

q

Heat flux ( W / m 2)

Q ̇

Heat transfer rate ( W)

r

Radius or radial direction ( m)

R

Specific gas constant ( J / kg K)

Re

Reynolds number

Ri

Richardson number

RH

Air relative humidity

s

Distance ( m)

S

Surface area ( m 2)

Sc

Schmidt number

Sh

Sherwood number

t

Time ( s)

T

Temperature ( K)

u

Velocity ( m / s)

u

Velocity fluctuation ( m / s)

V

Volume ( m 3)

V ̇

Volumetric flow rate ( m 3 / s)

W

Weight force ( N)

We

Weber number

y

Molar fraction

Greek Symbols
α

Thermal diffusivity ( m 2 / s)

β

Entrainment coefficient

γ

Cloud velocity scaling factor

ε

Turbulent dissipation rate ( m 2 / s 3)

ζ

Dimensionless radial distance

η

Normally distributed random number

θ

Droplet direction angle ( rad)

λ

Thermal conductivity ( W / m K)

μ

Dynamic viscosity ( kg / m s)

ν

Kinematic viscosity ( m 2 / s)

ξ

Dimensionless distance

ρ

Density ( kg / m 3)

σ

Water surface tension ( N / m)

τ

Characteristic time ( s)

φ

Volume fraction

χ

Cloud centerline angle ( rad)

ψ

Cloud semi-aperture angle ( rad)

ω

Mass fraction

Ω

Solid angle ( sr)

Subscripts
a

Dry air

c

Curvilinear centerline

cv

Convection

d

Drag

dl

Delay

ds

Dissipation

ec

Eddy crossing

el

Eddy lifetime

ev

Evaporation

ex

Exhalation

fl

Fall

fr

Exhaled cloud front section

in

Inhalation

lq

Liquid

m

Humid air mixture

mt

Mouth

p

Particle or droplet

pc

Potential core

pk

Peak

pr

Period

r

Relative

s

Droplet or particle surface

sat

Saturation

sl

Solid

sn

Sensible heat

st

Steady-state

t

Terminal

v

Water vapor

vr

Cloud cone vertex

wb

Wet bulb

x

Horizontal direction

z

Vertical direction

0

Initial or inlet condition

Ambient condition

A

Cross-sectional area ( m 2)

AH

Absolute humidity ( kg vap / kg air)

B M

Spalding mass transfer number

B T

Spalding heat transfer number

c p

Specific heat at constant pressure ( J / kg K)

C

Molar concentration ( mol / m 3)

C c

Cunningham correction factor

C d

Drag coefficient

D

Diameter ( m)

D v

Mass diffusivity ( m 2 / s)

D

Drag force ( N)

g

Gravitational acceleration ( m 2 / s)

Gr

Grashof number

h

Convective heat transfer coefficient ( W / m 2 K)

h lt

Latent heat of vaporization ( J / kg)

h

Specific enthalpy ( J / kg)

k

Convective mass transfer coefficient ( m / s)

k

Turbulent kinetic energy ( m 2 / s 2)

K

Evaporation rate ( m 2 / s)

Kn

Knudsen number

l e

Eddy length scale

Le

Lewis number

m

Mass ( kg)

m

Mass flux ( kg / m 2 s)

m ̇

Mass flow rate ( kg / s)

M

Molar mass ( kg / mol)

n

Molar flux ( mol / m 2 s)

Nu

Nusselt number

p

Pressure ( Pa)

P

Momentum ( N s)

P ̇

Momentum flux ( N)

Pr

Prandtl number

q

Heat flux ( W / m 2)

Q ̇

Heat transfer rate ( W)

r

Radius or radial direction ( m)

R

Specific gas constant ( J / kg K)

Re

Reynolds number

Ri

Richardson number

RH

Air relative humidity

s

Distance ( m)

S

Surface area ( m 2)

Sc

Schmidt number

Sh

Sherwood number

t

Time ( s)

T

Temperature ( K)

u

Velocity ( m / s)

u

Velocity fluctuation ( m / s)

V

Volume ( m 3)

V ̇

Volumetric flow rate ( m 3 / s)

W

Weight force ( N)

We

Weber number

y

Molar fraction

Greek Symbols
α

Thermal diffusivity ( m 2 / s)

β

Entrainment coefficient

γ

Cloud velocity scaling factor

ε

Turbulent dissipation rate ( m 2 / s 3)

ζ

Dimensionless radial distance

η

Normally distributed random number

θ

Droplet direction angle ( rad)

λ

Thermal conductivity ( W / m K)

μ

Dynamic viscosity ( kg / m s)

ν

Kinematic viscosity ( m 2 / s)

ξ

Dimensionless distance

ρ

Density ( kg / m 3)

σ

Water surface tension ( N / m)

τ

Characteristic time ( s)

φ

Volume fraction

χ

Cloud centerline angle ( rad)

ψ

Cloud semi-aperture angle ( rad)

ω

Mass fraction

Ω

Solid angle ( sr)

Subscripts
a

Dry air

c

Curvilinear centerline

cv

Convection

d

Drag

dl

Delay

ds

Dissipation

ec

Eddy crossing

el

Eddy lifetime

ev

Evaporation

ex

Exhalation

fl

Fall

fr

Exhaled cloud front section

in

Inhalation

lq

Liquid

m

Humid air mixture

mt

Mouth

p

Particle or droplet

pc

Potential core

pk

Peak

pr

Period

r

Relative

s

Droplet or particle surface

sat

Saturation

sl

Solid

sn

Sensible heat

st

Steady-state

t

Terminal

v

Water vapor

vr

Cloud cone vertex

wb

Wet bulb

x

Horizontal direction

z

Vertical direction

0

Initial or inlet condition

Ambient condition

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