Modeling and simulation of the potential indoor airborne transmission of SARS-CoV-2 virus through respiratory droplets

Respiratory viruses mainly spread through respiratory droplets released during breathing, talking, coughing, and sneezing.^1–3 During social interactions, such as group meetings, the viral infections can spread through the respiratory droplets,⁴ which get advected in a room due to air circulation. Therefore, after the initial outbreak of corona virus disease (COVID-19) caused by severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2) in 2019–2020, physical group meetings were discontinued worldwide. Many businesses and educational institutes started organizing meetings over the internet, which vastly controlled the spread of the virus. However, it has also reduced the pace of progress due to issues such as internet connectivity and limitations of online communication with the current technology. Also, the physical conference room meetings play a vital role in faster decision-making, which is difficult to recreate through virtual platforms. Therefore, soon after the descent of the first wave of COVID-19 infections, many businesses returned to physical group meetings. However, the resumption of social gatherings also resulted in multiple waves of the pandemic in many countries worldwide. Also, many variants of the virus have emerged over time, making development of effective vaccines a challenge. Therefore, to avoid further upsurge of the virus, it is essential to understand the transport mechanisms of respiratory droplets under various conditions as we return to normalcy.

Respiratory droplets provide a protective space for the viruses to remain undamaged as they travel from one person to another.⁵ These droplets of various sizes ranging from $0.3$ to $1500 μm$⁠,^6,7 are released by human beings during breathing, talking, coughing, and sneezing. A droplet's lifespan and path of travel are governed by its size and the surrounding air conditions, such as temperature, air velocity, and relative humidity. Larger droplets tend to quickly fall to the ground (or other surfaces) due to gravity and inertia. On the other hand, the smaller droplets are easily dragged further away in the room by the flowing air due to forced and natural convection. However, the smaller droplets have shorter lifespan as they are evaporated quickly due to their larger surface area to volume ratio. This phenomenon was studied experimentally at different relative humidities by Wells⁸ and studied again using modern techniques by Xie et al.⁹ In a room with stagnant air, the path followed by the respiratory droplets is dictated by the initial velocity of the droplets and the turbulence generated by the mouth.¹⁰ However, the distance traveled by the droplets significantly increases in the presence of ambient velocity, as per a recent study.¹¹ A more detailed study on the effects of air-conditioning systems in a hospital environment¹² showed that air-conditioning plays a vital role in determining the dispersion of respiratory droplets in the room. However, the study was limited to the coughing event while neglecting natural ventilation and face mask.

Face masks are passive control devices which form a physical barrier to trap the respiratory droplets. This has been the initial line of defense against the virus during COVID-19 pandemic. However, a significant amount of leakage of respiratory droplets was observed around the surgical marks in a recent numerical study,¹³ which led the authors to emphasize on the importance of social distancing. The filtration efficiency of homemade cloth masks was seen to vary widely in an experimental study,¹⁴ with efficiencies up to 90% for particles larger than $300 nm$ with multilayered masks. The same study also observed more than 60% reduction in filtration efficiency due to improper fit and leakage of exhaled air. Therefore, it becomes essential to consider the effect of face masks while studying the spread of respiratory droplets.

Some studies have been recently undertaken to understand the flow physics of respiratory droplets and spread of viruses in different environments.^15–20 The studies in the literature, on the spread of respiratory droplets, are mostly confined to the regions close to a person. Also, these studies generally consider flow patterns due to coughing, while neglecting breathing and talking. Moreover, the effects of air-conditioning and masks are not considered together in many of these studies. Even though such focused studies are necessary for understanding the physics of local flows, they fail to display the complex flow patterns which arise in practical scenarios. Therefore, it is crucial to conduct large-scale studies to understand the advection patterns of respiratory droplets in a room involving multiple people with complex inhalation–exhalation patterns and viral spread to the neighboring people. It is not possible to carry out such large-scale studies experimentally, due to the limitations of visualization techniques. On the other hand, it is challenging to perform such a numerical study as the numerous parameters involved in a realistic simulation are not readily available. In this paper, an effort is made to carry out a realistic simulation with sufficient simulation parameters reported in the literature and deriving the ones which are not available. The dispersion of respiratory droplets is studied in a typical conference room with an ongoing meeting involving ten people. We consider a two-minute scenario with the people in the room breathing normally, and intermittently coughing and talking during the meeting. The fluid properties of the exhaled air and ambient room air are derived using fundamental gas laws and air compositions from the literature. The distribution of droplet diameters in the simulation is calculated based on previous experimental studies by fitting the available data using a least-square error fit. Also, the equation for packing fraction, which is not available in the literature for multilayered homemade cloth masks, is derived for woven fabric. The derived equation for packing fraction is necessary for calculating the viscous and inertial coefficients to model the porous media of the mask. The results obtained in this work display the room-wide effects and spread of respiratory droplets during meetings. The parameters and equations derived in this work can also be used in future studies for analyzing different social gatherings in confined spaces during and after the pandemic.

The paper is organized as follows. In Sec. II, we discuss the model and methodology which consists of five subsections. Subsection II A discusses the dimensions of the conference room and its occupants, Subsec. II B describes the governing equations and the mathematical models used in the simulation. Subsection II C describes the constitutive equations used for deriving air properties such as density and viscosity of the inhaled and exhaled air. Section II D describes the simulation parameters and their derivations, which include droplet size distribution, mask modeling, initial, and boundary conditions. The last subsection under methods and models is Subsec. II E, which discusses the grid convergence for a transient simulation. Section III on the results and discussion displays various snapshots during the simulation and discusses the flow of respiratory droplets under various conditions. Finally, Sec. IV concludes the paper with discussion on work done and directions and scope for future work.

A typical conference room occupied by ten people is modeled, with eight of them sitting and two standing as shown in Fig. 1. The three-dimensional (3D) geometric model is constructed using Autodesk Fusion 360 software. The transient flow in the room is simulated by solving Reynolds-averaged Navier–Stokes (RANS) equations using the Ansys Fluent software. The two-phase mixture model is used to simulate mixing of the exhaled air with ambient room air. The air conditioning and the outlet duct ensure air circulation in the room. A scenario involving breathing, talking, and coughing by the occupants is created to study the transport of respiratory droplets introduced and their spread in the room over time.

A sequence of events is constructed to resemble an ongoing meeting in a conference room for the study. The ten people in the room are labeled P1 through P10 as shown in Fig. 1(a). The meeting scenario lasts for 120 s where every person is breathing continuously while talking in turns and coughing/sneezing intermittently. The sequence of events are as follows. The conversation is started by P1 who talks for the first 10 s and also P3 coughs right at the beginning. After a silence of 10 s, the conversation is continued by P5 from the other side of the table, who talks between 20 and 30 s. The conversation is then continued by P8, who talks between 40 and 50 s. However, unlike P1 and P5, P8 coughs at the end of the talk. Just before P8's completion, P4 begins talking for about 10 s. The conversation is then continued by P2, P7, and P3 sequentially. All the people cough intermittently during the meeting. The final inhalation and exhalation mass flow rate is a function of time, which is computed by superimposing the flow rates produced by breathing, talking, and coughing events described above. The resulting mass flow rates are plotted in Figs. 6 and 7. Sections II A–II E provide a detailed description of the design of the room, governing equations, air properties, simulation parameters, and grid convergence study conducted for the present work.

The conference room consists of a meeting table at the center of the room with equally spaced chairs around the table. The people in the room have a height of $1.73 m$⁠. The length, width, and height of the room are $10.2, 6$⁠, and $3 m$⁠, respectively. The table has a length and width of $6$ and $2 m$⁠, respectively, with a $0.5 m$-wide central slot. The chairs have a sitting height of $0.42 m$ and total height of $0.98 m$⁠. Two wall-mounted split-type air conditioning units are installed on side walls. The overall dimensions of the air conditioning units are based on SystemAir's design taken from their catalog. An outlet duct of height $0.1 m$ is designed on the walls of the room, close to the ceiling. A typical ratio of the body parts of the humans is used according to their height. An ellipse-shaped mouth of the humans is modeled with its horizontal major axis of $40 mm$ and minor axis of $16 mm$⁠. The depth of the mouth is $40 mm$ as shown in Fig. 2(a). The face mask is designed to cover the mouth, as shown in Fig. 2(b), with a gap of $2 mm$ between face and the mask to account for peripheral air leakage. The top view and the isometric view of the room are shown in Figs. 1(a) and 1(b), respectively.

The air is considered as a mixture of two fluids, i.e., the ambient air in the room (at temperature $Tr=25 °C$⁠) and the air exhaled by the people (at temperature $Te=37 °C$⁠). The air flow is governed by the Navier–Stokes equations for a mixture of gases,

and mass conservation equation,

where t is the time, $v→$ is the mixture velocity, and $g→$ is the acceleration due to gravity. The mixture density, ρ, and mixture dynamic viscosity, μ, are computed using volume fraction, α_r, as follows:

where subscripts r and e are used for room air properties and exhaled air properties, respectively, which are computed in Sec. II C. The drift velocity is computed as

Finally, the volume fraction is evolved for the primary phase (room air in this case) by solving the following equation:

The respiratory droplets are tracked as a discrete phase in a Lagrangian frame of reference. The gravitational and inertial forces on the droplets are computed using the density of the saliva and droplet diameter. The hydrodynamic drag force on the droplets is computed assuming a spherical shape using the drag coefficient given by.²¹ The forces imparted by the respiratory droplets on the surrounding air and the interactions between the droplets are assumed to be negligible. This approach is commonly known as one-way coupling.

The air in the room (inhaled air) and the exhaled air have different compositions,²² and therefore have different density and viscosity. The composition of these two gases, in percentage concentration by volume, are given in Table I adjusted for a total of 100%. The air in the room is assumed to be maintained at a temperature of $25 °C$ by the air conditioning system. The exhaled air is assumed to be at human body temperature of $37 °C$⁠.

The density of air in the room and the exhaled air is calculated using the ideal gas law,

where ρ is the density of the gas measured in $kg/m3$⁠, p is the absolute pressure measured in $Pa$⁠, T is the temperature of the gas measured in $K, m̂$ is the molar mass of the gas measured in $g/mol$⁠, and R_u is the universal gas constant measured in $J/K kmol$⁠. The pressure in the room and exhaled air is taken as $p=101 325 Pa$ and $Ru=8314 J/K kmol$⁠.

The combined molar mass of the air can be computed using a linear combination of each of the molar masses as follows:

The molar mass of the air in the room can be calculated as $28.89 g/mol$⁠, and the molar mass of exhaled air can be calculated as $29.305 g/mol$⁠. The density of the room air and the exhaled air can be computed using Eq. (1) with respective values of molar mass and temperature. Therefore, the air density in the room can be computed to be $ρr=1.180 92 kg/m3$⁠, and the density of the exhaled air can be computed to be $ρe=1.151 53 kg/m3$⁠.

The dynamic viscosity, μ, of the fluids at different temperatures is computed using Sutherland's formula,²³ which can be written as follows:

where $μref$ is the reference viscosity defined at the reference temperature $Tref$⁠, and S is known as Sutherland temperature. These parameters for individual constituents are obtained from Ansys Fluent database and tabulated in Table II, along with calculated viscosity at room temperature and human body temperature.

The combined viscosity is computed using the equation of Herning and Zipperer²⁴ where a linear combination of viscosity of individual gases is weighted by the molar fraction and square root of molar mass of each gas in the air. The viscosity of the air in the room can be computed to be $μr=1.84×10−5 Pa s$⁠, and the viscosity of the exhaled air can be computed to be $μe=1.86×10−5 Pa s$⁠.

The simulations are performed on a dual-socket NUMA architecture machine using 40 Intel Xeon Gold processors and one NVIDIA Quadro P1000 Graphics Processor. The transient Reynolds-averaged Navier–Stokes (RANS) equations are solved using the pressure-based model. The source term of acceleration due to gravity of $−9.81 m/s2$ in the vertical direction is used, so that the effects of natural convection and trajectories of respiratory droplets are captured correctly. The mixing of the two fluids (i.e., room air and exhaled air) is computed using mixture model of multiphase flow with an implicit body force formulation. The RANS equations are closed by using the $k−ω$ shear stress transport turbulence model, as it is known to accurately capture separating incompressible flows.²⁵ The PISO algorithm is used with the second-order approximation for pressure and momentum. The gradients are computed using least-square approximation. The transient solution is obtained using the second-order implicit formulation.

The transport of respiratory droplets in the room is studied by using the discrete phase model (DPM). Droplets of varying diameters are released from the mouth surface at a depth of $40 mm$ as shown in Fig. 2. In the experimental work of Xie et al.,⁷ the size of the droplets released from mouth during coughing was measured and corrected for evaporation using a simple evaporation model. The fraction of droplets greater than a given diameter d_p is calculated based on the experimental data⁷ and tabulated in Table III. We have fitted the data by the cumulative Rosin–Rammler distribution function,^26,27

as shown in Fig. 3, optimized using the least-square error method. This results in a mean diameter of $d¯p≈120 μm$ and a spread parameter of $n≈1.8$⁠, with an L₂-error of about 6.5%. The equivalent frequency function is given as $−∂Yd/∂dp$⁠. The range of droplet diameters from $2$ to $1000 μm$ is divided into ten discrete intervals (on log scale) for injecting 10 000 droplets per second (50 droplets are injected per mouth every 0.05 s) as shown in Fig. 4. The respiratory droplets, which is mostly saliva during coughing and talking, are assumed to have density similar to water.^28,29 The average mass of saliva droplets during twenty cough cycles was observed in a controlled experimental study⁷ to be about $85 mg$⁠. Assuming a duration of $120 s$ for twenty coughs, the mass flow rate of saliva can be calculated to be about $7.0833×10−4 kg/s$⁠, for use in DPM for droplet injection. The drag force on the droplets is computed assuming the droplets to be of spherical shape.

The droplets evaporate over time as they travel through air and/or settle on the table and other surrounding surfaces. Once the droplet has evaporated, the protection for the viruses cease to exist.⁵ The time required for evaporation of the droplet is dependent on the droplet diameter, ambient temperature, and relative humidity in the room. In general, smaller droplets in air with lower relative humidity evaporate sooner^8,9 and vice versa. Once the droplet settles on a surface the lifetime of the droplet also depends on the material of the surface. For example, on porous surfaces, such as paper or cloth, the droplet survives for a shorter period compared to impermeable surfaces, such as glass.¹⁶ In this work, the droplets are tracked through air and the distances spanned on the table are marked in the results. Droplets smaller than $40 μm$ are removed from the simulation after a residence time of 15 s to account for evaporation.

The mask covering the face of the people is modeled using a porous media formulation. A homogeneous porous medium is considered by adding source terms to the momentum equations. The gradient of pressure within the porous medium is given by

where $v→$ is the fluid velocity and $|v→|$ is the magnitude of the velocity. The two unknown coefficients, D and I, are used to appropriately scale the viscous (Darcy) and inertial losses, respectively. These coefficients depend on the mask material and design. The pressure drop for Stokes flow (modeled by Darcy's law³⁰) is estimated using a large set of experimental data^31,32 to fit the empirical relation,

where c is the packing fraction and d_f is the fiber (thread) diameter of the mask material. The Darcy coefficient can, therefore, be calculated as

The inertial coefficient is estimated using the Ergun equation,³³ which produces

where ψ is the correction for deviation from spherical shape, which can be calculated as 1.5 for long threads with circular cross section. It is a common practice, as the pandemic has spread over the world, for the people to use masks of woven fabric material. The woven masks made from natural materials are also encouraged for public use, from the environmental standpoint,³⁴ compared to mass produced non-woven masks, which are necessary for medical use. For a woven mask, the packing fraction, c, is a geometric parameter specifying the space occupied by the threads in unit volume of the mask material. The packing fraction can be calculated for a uniform multilayered woven cloth with perpendicular threads of circular cross section, with diameter d_f and pitch a, as follows:

We have derived this equation assuming the cloth thickness to be twice of the thread diameter, due to overlapping perpendicular threads, as depicted in Fig. 5. In the equation, Z is the number of layers of the cloth and δ is the average air gap between the layers. In the experimental work of Konda et al.,¹⁴ different types of homemade masks were studied with various thread diameters and pitch. Here, we choose the material for the mask to be cotton quilt (90% cotton, 5% polyester, 5% other fibers) with two layers (Z = 2) with an average gap of $δ=1 mm$ between the layers. From the literature,¹⁴ for the chosen material, $df=227 μm$ and $a≈350 μm$⁠. The packing fraction can be calculated using Eq. (9) to be c = 0.2424. Using Eqs. (7) and (8), the Darcy and inertial coefficients, which are required for simulating the flow in porous media (i.e., mask), can be calculated to be $D=2.664 529 7×108 m−2$ and $I=5.730 119×103 m−1$⁠, respectively. The gaps in between threads in such a porous material is equal to $(a−df)≈123 μm$⁠; hence, droplets larger than $125 μm$ are removed from the simulation when they come in contact with the mask surface, in simulations where masks are worn by the people.

The complete domain is initialized with room air properties as calculated in Sec. II C. The two air conditioners blow-in fresh room air at a constant mass flow rate of $11 kg/s$ (when switched on), which corresponds to the low speed setting of the wall mounted split-type air conditioner from SystemAir designed for similar application. A pressure outlet boundary condition is applied at the outlet vent, which is placed around the room wall close to the ceiling.

Air inhalation and exhalation patterns of the ten people in the room are very crucial as it governs the release of respiratory droplets from their mouths. The droplets placed deep within the mouth (40 mm inside) are dragged by the exhaled air depending on the instantaneous flow rate and droplet diameters. Each person's breathing, intermittent coughing, and talking are modeled as time-varying mass flow rates as described in this section. A negative value of mass flow rate implies inhalation and a positive value means exhalation. The measured breathing patterns are complicated and vary from person to person.³⁵ In this work, the mass flow rate of normal breathing is modeled using a sinusoidal function, as plotted in Fig. 6(a),

where t is the simulation time in seconds and T is the period for one complete breath (i.e., inhalation and exhalation). At a rate of 12 breaths per minute,³⁵ $T=5 s$⁠. The average volume flow rate during normal breathing was experimentally measured and reported to be approximately $4.5 l/m$⁠.^35,36 Therefore, the peak mass flow rate can be calculated to be $ṁbrmax=1.356 61×10−4 kg/s$ (equivalently peak volume flow rate is $1.1781×10−4 m3/s$ and peak velocity is $0.23438 m/s$⁠) during normal breathing. The phase of breathing for each person is shifted by t_p. The phase shifts are $tp=0 s$ for P1, $tp=1 s$ for P2, $tp=2 s$ for P3, and so on.

The flow rate during coughing is modeled using two Gaussian functions, as shown in Fig. 6(b), i.e., a deep inhalation followed by sudden exhalation,

where t_s is start time of inhalation, $te=ts+1.5$ is the end time of cough, and $tm=ts+1$ is the time of transition from inhalation to sudden exhalation. The average volume flow rate (Q) during coughing is measured in an experimental study,³⁷ where the Reynolds number is obtained to be about 10 000 using the formula $Re=ρ Q/(d μ)$ with $d=8 mm$ as the characteristic diameter. Using the findings of Scharfman et al.,³⁷ the average mass flow rate during coughing can be calculated as $1.487 79×10−3 kg/s$⁠, which is also within the range of values used by Zhao et al.³⁶ The peak flow rate can be calculated as $ṁcomax=3.381 34×10−3 kg/s$ (equivalently peak volume flow rate is $2.9364×10−3 m3/s$ and peak velocity is $5.8418 m/s$⁠).

The mass flow rate during talking is modeled using Perlin noise³⁸ library, in the Python programming language using 100 octaves. The peak flow rate during talking is $1.8×10−3 kg/s$⁠, which is in between the normal breathing and coughing mass flow rate. A sample pattern of mass flow rate during talking is shown in Fig. 6(c). The boundary condition for the total, time-dependent, mass flow rate at the mouth surface (at a depth of $40 mm$ as in Fig. 2) is constructed by combining the above three patterns of inhalation and exhalation. The scenario designed for simulation of people breathing, talking, and coughing in the conference room is shown in Fig. 7.

The boundary condition applied to the respiratory droplet depends on the surface it comes in contact with during exit from the domain. The droplets escape from the domain (i.e., removed from simulation) if they touch the inner surface of the mouth (40 mm inside the mouth) or the outlet vent. All other surfaces such as walls, furniture, glass, and the human body trap the droplets upon contact, and such droplets are marked as trapped.

Before beginning the simulations for all the situations and complete time duration, a mesh convergence study is performed by using three levels of meshes for a duration of 5 s. The coarsest mesh consists of about 1.5 × 10⁶ cells, the medium mesh consists of about 3.5 × 10⁶ cells, and the finest mesh consists of about 5.9 × 10⁶ finite volume cells. The mouth, human body, and the neighboring area are locally refined for each of the meshes. A time step of 0.05 s is found to be appropriate for capturing the time varying nature of the flow based on multiple simulations. The face masks are not included in the mesh convergence study for simplicity. Thus, the cell counts are higher in actual simulations when masks are included due to local mesh refinement near the masks. All the studies use second-order approximations in space and time. The transient flow is monitored for a duration of 5 s with the air conditioning turned off. The magnitude of velocity of air in front of P6's mouth (about 10 mm ahead of the face) is plotted in Fig. 8, for the three meshes as a function of time. It can be observed that the medium and fine meshes adequately capture the flow features and converge to very similar transient solutions.

The study is conducted for two situations, with and without air conditioning. In both situations, the flow in the conference room is studied with the people wearing and not-wearing masks. The scenarios considered under each situation are discussed here one by one. The distance traveled by the respiratory droplets in still air before they evaporate and the wetted area on the table is of great interest. Hence, at first, the situation without air conditioning is discussed in detail, covering the normal breathing, coughing and talking scenarios separately. After that, the simulation results with air conditioning are discussed and compared to that of still air. In order to clearly distinguish and monitor droplets of various sizes, they are scaled and colored based on their diameter. The left side figures show the situation without mask and the right side figures show the situation with mask. Each figure also contains an overview sketch, marking the sitting position of the person of interest in that particular figure. It is difficult to display the 3D nature of the flow in 2D projected snapshots. Hence, animations are also provided for each figure shown in the results below to provide a better perspective and to show the dynamic nature of these situations.

The person P5 is chosen to display the effects of normal breathing and the associated flow at 11.25 s. The corner position of P5 allows for clear visualization of the respiratory droplets, as there is no other person obstructing the view. Also, the neighbors P4 and P6 are breathing normally until this time without drastically disturbing the flow. The flow in the neighborhood of P5 is shown in Fig. 9 (Multimedia view) at 11.25 s, at the beginning of the next inhalation, after completing two breathing cycles. It can be observed that the warmer exhaled air flows upward due to natural convection dragging some of the droplets along with it. The smaller droplets moving upward evaporate within a distance of $0.3 m$⁠. Most of the smaller droplets move upward, while the larger droplets fall down to the body and chair of the person due to gravity. When the person is wearing a mask, in Fig. 9(b), the amount of droplets falling down increase considerably due to the loss of kinetic energy across the mask.

The size of droplets released during breathing are smaller than $40 μm$⁠. It may be noted that even the largest of the droplets released during breathing will be invisible to the naked eye and many of the experimental visualization techniques. Therefore, the flow patterns during normal breathing may seem counter intuitive from day-to-day experience. Nevertheless, we do see such flow patterns in cold environments, where the water vapor from breath condenses rapidly and becomes visible.

The person P2 is chosen to study the effects of coughing/sneezing at different instances during and after the event. The process is viewed from behind P1 and the scene is clipped beyond $0.75 m$ of P2 so that the droplets released by P3 and P4 do not clutter the pictures. The coughing process of P2 begins at 20 s in the simulation, initiated by a deep inhalation for 1 s (20 to 21 s) and a rapid exhalation for 0.5 s (21 to 21.5 s). The air flow due to deep inhalation pull the nearby floating small droplets closer to the mouth of P2, which can be seen in the animation. The peak flow rate during the exhalation is at 21.25 s, which is shown in Fig. 10 (Multimedia view). It is evident from the figure that the mass flow and the size of droplets during coughing are significantly larger than the normal breathing. Also, the droplets travel much further during the coughing event compared to normal breathing. The droplets of up to $200 μm$ are released during the exhalation process without the mask. Droplets larger than $125 μm$ are filtered by the mask and are therefore not visible in the right side figures. Another interesting observation is that with a mask, many droplets of the range $40 μm$ fall downward close to the person's body, while without the mask such droplets are pushed away by the exhaled air.

The situation at 6.5 s after the end of cough/sneeze is shown in Fig. 11. In the figure, the distance traveled by the droplets on the table and the spread of the droplets is also marked. The droplets travel slightly more than $0.8 m$ from the mouth without a mask, while they tend to spread more and are confined closer to the body when a mask is used. The advantage of using a table with a central pit (slot) is also obvious here as the droplets are captured in the pit instead of traveling to the person on the other side of the table.

In contrast to coughing, talking is a longer event spanning 10 s, thus releasing more respiratory droplets. Also, due to prolonged release of droplets, they travel further compared to coughing. Here, P5 sitting in the corner is observed from 1 s into the talk until 10 s after the talk, as shown in Figs. 12 (Multimedia view)–14. At 21 s, i.e., 1 s into the talk, the droplets look similar in size and quantity with and without the mask. However, eventually the spread and distribution differ drastically after the talk. Due to the loss of momentum and filtration offered by the mask, droplets tend to fall quickly and do not travel as far as without the mask. This phenomenon can be clearly observed in Fig. 14, where distances traveled by the droplets are marked. The droplets travel about $1 m$ distance on the table and wet the entire area in front of P5 without the mask. Also, without the mask the droplets travel higher into the room compared to little or no vertical flow of droplets with the mask.

The droplet pattern during the talk is similar to the pictures shown in Sec. III A 3. Here, the flow pattern after the talk (and cough) is only displayed; nevertheless, the entire event can be seen in the animations. A talking scenario is observed at two different times after the talk to understand the flow of droplets released by multiple people where P8 is talking while P1, P2, and P9 surrounding P8 are breathing normally, as shown in Fig. 15 (Multimedia view) and Fig. 16. Similar to the previous observations, droplets fall and travel shorter distances due to the mask. It is evident from the observations that the number of droplets reduce and slow down due to the mask. Also, the simulations highlight the usefulness of the mask while interacting face to face since, in the absence of the mask, droplets travel away from the body quickly, eventually increasing the risk of infections. The usefulness of the central pit is obvious in this scenario as it prevents the droplets from crossing over to the other person, which can be seen clearly in the animation of this scenario. The droplets falling inside the pit are manually counted in post-processing to understand its usefulness. It is observed that about 4% of droplets in the range $10$ to $50 μm$ fall into the pit after P8's talk without the mask. The remaining about 96% of droplets in this range fall closer to P8 along with 100% of larger droplets. It is observed that the smaller droplets move upward due to natural convection and evaporate very quickly (Fig. 17, Multimedia View).

As seen in the pictures, during the talking and coughing/sneezing the droplets land on surfaces in front of the person, such as the table top surface, drinking glass, and possibly other objects, such as electrical connectors. Many of such objects in the room are impermeable and hence can stay contaminated for many hours after the meeting compared to porous materials, such as papers or cloth, as reported in literature.¹⁶ 

The air-conditioning system blows fresh air into the room at $11 kg/s$ at a temperature of $25 °C$⁠. This produces four vortex structures in the room as shown in Fig. 18. Since the outlet vent of the room is located at the top of the room, air tends to flow vertically and gives rise to the vortices due to the input from the two air conditioning systems. The flow of the droplets and their evaporation are greatly influenced by the air-conditioning system. Figure 19 (Multimedia view) shows the flow of droplets in the room due to the air-conditioner after P2's cough and during P5's talk. It shows that droplets spread quickly into the room, and their sizes are more significant without the mask. The vertical motion of the droplets can be clearly observed in Fig. 20 (Multimedia view), where P8 talks. Although the droplets are less in number with masks, their presence is seen all over the room. When the air-conditioning system is switched off, the larger droplets tend to stay in the vicinity of the person producing them, as shown in the upper section of Figs. 19 and 20. In contrast, with the air-conditioner working, even the larger droplets (⁠$∼200 μm$⁠) released during coughing and talking get trapped in the vortices and stay suspended in air for more than 10 s. Such droplets evaporate slowly due to their larger size and can potentially contaminate far away surfaces, which is difficult to predict due to the turbulence generated by the air-conditioning system.

The design of the air-conditioning system, such as air flow speed and placement of the units, plays a very vital role in determining whether the room is safe from spread of air transmitted infections during prolonged meetings. It can be observed in Fig. 19 that the droplets produced by P2 during a cough and P8 during the talk are safely and quickly directed away from other people toward the exhaust vent near the room ceiling. This is a favorable scenario for the air-conditioning system. On the other hand, as seen in Fig. 20, the droplets produced during P8's talk are directed toward the standing person, P9, which is an unfavorable situation for an air-conditioned room. These two examples show that the room design and placement of air-conditioning outlets are important considerations which require detailed room-wide simulations as conducted in this work. Even though the situations without air-conditioning were somewhat independent of room design, if there is forced convection each room design (and sitting arrangement) is different and requires thorough investigation. It is clear, after looking at the results, that the air-conditioning systems can be cleverly designed to quickly advect the droplets away from the people, unlike during natural ventilation where the droplets settle on the table top and other nearby surfaces. Such easily accessible surfaces can stay contaminated for long periods of time.³⁹ To approximately quantify the spread of viruses in air-conditioned rooms, we have also manually counted (in post-processing) the larger droplets settling on other people's faces. It is observed that approximately 1% of the droplets, larger than $40 μm$⁠, released during P2's cough reach near other people's face, before being trapped by other surfaces or escape the room through outlet vent. Also, with the mask less number of larger droplets are released, none of the larger droplets released during P2's cough settle on other people's face before getting trapped. However, it is extremely difficult to accurately count and locate smaller droplets in the room, as there are thousands of such droplets floating for a long duration, before they evaporate or get trapped by surfaces, as seen in the animations and figures below.

A study is performed to investigate the influence of air conditioning and face masks on respiratory droplets released during talking and coughing in a conference room. This study considers a realistic scenario where multiple people with masks and without masks talk and cough during a meeting. A complete room geometry with multiple people and multiple events increases the usefulness of this study to understand the spread of respiratory droplets. Moreover, this also eliminates the errors due to application of artificial boundary conditions which are necessary in other single-person localized studies conducted in literature. Although numerical simulations are quicker to setup and modify, compared to instrumentation for visualization in an experimental study, providing suitable simulation parameters is still a challenge. Many of these parameters needed for a numerical study are not directly available in the literature; such parameters and properties for a real life simulation are derived and reported here.

The modeled conference room consists of ten people coughing and talking during the meeting. These events produce respiratory droplets, which get advected due to the natural ventilation or the circulation due to air conditioning. This advection is investigated along with the effect of using a mask on the spread of the respiratory droplets. The mask is modeled using a porous media formulation where the fabric structure of the cloth is also taken into account. The inertial and viscous loss across the mask are computed using a model developed for woven multilayered masks. The discrete phase model is used to study the transport of respiratory droplets, which requires the droplet size distribution function as an input. This probability distribution is obtained by fitting an appropriate curve through the experimental data reported in the literature. The compositions of ambient and exhaled air reported in the literature are used to calculate the air properties with the help of fundamental gas laws to model a multiphase flow. New mathematical expressions are developed for modeling mass flow rates during breathing, coughing, and talking to mimic the experimental measurements. These functions are superimposed to construct a typical meeting scenario in a conference room.

The results show that the small droplets released during normal breathing tend to flow upward without a mask, while with a mask they fall downwards, which eventually fall on the persons' body or the floor. The upward moving droplets during normal breathing move very slowly and evaporate within a distance of $0.5 m$ from the person's mouth. The mass flow rate during normal breathing produces droplets of at most $40μm$ diameter. On the other hand, coughing and talking produce droplets of diversified sizes with higher total mass of the bio-fluid. The talking event produces more droplets as it is a longer event compared to coughing or sneezing. The droplets also travel longer distances during talking, spanning almost $1 m$ on the table, even though the average exhaled air flow rate is lesser compared to coughing. The spread of droplets in the air is very well captured in these simulations due to the use of multiphase mixture model for the ambient and exhaled air. Many of the simulations in literature neglect the fact that the composition of the exhaled air is different from the ambient air and therefore do not capture the spread of droplets very well. The spread of smaller respiratory droplets in air seems to be heavily influenced by natural convection which cannot be captured by a single phase flow model. In this work, the distances traveled by the droplets on the table top surface are also measured and marked in the figures for coughing and talking events. It can be observed that the distances are reduced by using a face mask due to filtration and pressure drop across the mask. An interesting observation is that with the mask the droplets spread more in lateral directions closer to the person. The masks play a vital role in deciding the fate of respiratory droplets as seen in all the results; however, in the real world, a proper fit of the mask and consistent wearing of masks by people cannot be relied upon. Finally, the results with air conditioning show how the droplets spread across the room and how the mask effectively minimizes the room air contamination. It exemplifies the importance of proper design of the air-conditioning system, so as to quickly move the contaminated air away from room occupants. Animations are provided for each of the figures (in Sec. III), which show many of the additional features discussed in the results.

In the future, studies in different environments are essential for better designing of rooms, sitting arrangements, and air-conditioning systems. Moreover, the active removal of droplets from the room using suction devices will be interesting to consider in the future. In this study, it is observed that the central table slot (also called pit) plays a vital role to prevent the droplets from crossing over to the other side of the table, especially during longer talks. In the future, it will be interesting to conduct a detailed study to understand the pit's role in prevention of spread of viruses during meetings. It is to be noted that the parameters computed and models developed in this study are not limited to any specific simulation tool, room setup, and mask materials. Although a typical geometry is considered in this study, the same simulation setup can be used to study and design other human occupied spaces to minimize spread of air-transmitted infectious diseases.

This work is supported by the Indian Institute of Technology Kanpur by providing the necessary computational facilities for the simulations.

The authors have no conflicts to disclose.

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