Effect of indoor temperature on the velocity fields and airborne transmission of sneeze droplets: An experimental study and transient CFD modeling

Abstract

The spread of the COVID-19 pandemic through the airborne transmission of coronavirus-containing droplets emitted during coughing, sneezing, and speaking has now been well recognized. This study presented the effect of indoor temperature (T_∞) on the airflow dynamics, velocity fields, size distribution, and airborne transmission of sneeze droplets in a confined space through experimental investigation and computational fluid dynamic (CFD) modeling. The CFD simulations were performed using the renormalization group k-ε turbulence model. The experimental shadowgraph imaging and CFD simulations showed the time evolution of sneeze droplet concentrations into the turbulent expanded puff, droplet cloud, and fully-dispersed droplets. Also, the predicted mean velocity of droplets was compared with the obtained experimental data to assess the accuracy of the results. In addition, the validated computational model was used to study the sneeze complex airflow behavior and airborne transmission of small, medium, and large respiratory droplets in confined spaces at different temperatures. The warm room showed more than ∼14 % increase in airborne aerosols than the room with a mild temperature. The study provides information on the effect of room temperature on the evaporation of respiratory droplets during sneezing. The findings of this fundamental study may be used in developing exposure guidelines by controlling the temperature level in indoor environments to reduce the exposure risk of COVID-19.

Graphical abstract

1. Introduction

Coronavirus (SARS-CoV-2), which was identified as the cause of the global COVID-19 pandemic by the World Health Organization (WHO), is spread through respiratory droplets exhaled during talking, coughing, sneezing, and breathing (Zhou et al., 2020; Ferretti et al., 2020; Wang et al., 2021). The saliva droplets smaller than 50 μm evaporate into aerosolized droplet nuclei <5 μm, which can remain airborne for several hours, while droplets larger than 100 μm settle on the ground within a few seconds (Anfinrud et al., 2020). The droplets with diameters between 5 and 100 μm, referred to as medium droplets, exhibit an intermediate behavior depending on their size and content of nonvolatile matter (Lieber et al., 2021). The physical properties of respiratory droplets, such as the initial size (Han et al., 2013), size distribution (Morawska et al., 2009; Anand and Mayya, 2020), and the number concentration (Bourouiba et al., 2014), as well as environmental parameters such as the indoor temperature (T _∞) and relative humidity (RH _∞), affect the airborne transmission and lifetime of the virus-containing aerosols (Stadnytskyi et al., 2020; Ghoroghi et al., 2022; Ren et al., 2022; Ismail et al., 2022).

Disease transmission is an interdisciplinary process involving biological and environmental sciences (Chen et al., 2020). From a biological viewpoint, the SARS-CoV-2 virus becomes inactive in less than half an hour at 323 K (Dhand and Li, 2020). van Doremalen et al. (2020) reported that the SARS-CoV-2 remained viable in aerosols for a median of about 2.7 h with a reduction in infectious titer from 10^3.5 to 10^2.7 TCID₅₀/L (defined as median tissue culture infectious dose) per liter of air. From the environmental viewpoint, the lifetime of aerosols is considerably higher than the larger droplets that sediment under gravity (Stadnytskyi et al., 2020; Chen, 2020; Dao and Kim, 2022). In addition, environmental conditions affect the evaporation rate of respiratory droplets (Chen, 2020). An initial estimate is that the respiratory droplet shrinks to half its initial diameter, which is known as the equilibrium diameter (Liu et al., 2021a).

The equilibrated diameter of droplets may be less than half of the initial one, based on the amount of nonvolatile compounds in droplets (Lieber et al., 2021). In the case of sneeze droplets, the final droplet size is estimated to be ∼20 % of the initial droplet size due to various environmental conditions (Drossinos et al., 2021). Given the importance of the final droplet size in determining its transport and deposition properties, understanding the effect of T _∞ and RH _∞ on the evaporation of sneeze droplets is of significant interest. Lieber et al. (2021) reported that the lifetime of saliva droplets smaller than 50 μm was mainly determined by the equilibrium droplet size, while the lifetime of droplets with diameters between 50 and 150 μm was affected by environmental conditions. In particular, the droplet lifetime increases at the lower RH _∞ and higher T _∞. After evaporation, all initially present viruses stay in the droplet nuclei (Morawska et al., 2009; Bourouiba et al., 2014; Balachandar et al., 2020).

The large droplets produced by sneezing could break up into smaller droplets with a higher lifetime than the initial droplets (Balachandar et al., 2020). Vuorinen et al. (2020) found that the droplets with diameters smaller than 50 μm evaporate in <3.0 s at RH _∞ < 50 %. The evaporation of droplets at different T _∞ influences the survivability of airborne droplets carrying SARS-CoV-2 viruses (Liu et al., 2021a; Chatterjee et al., 2020; Mikszewski et al., 2022). Bhardwaj and Agrawal (2020) found that the chance for the survival of viruses is enhanced, and the growth rate is augmented when a droplet evaporates slowly at the RH _∞ range of 20–30 %. Higher humidity increases the survivability of viruses when they are inside the droplet (Bhardwaj and Agrawal, 2020).

In order to reduce the risk of COVID-19 spreading through the transmission of respiratory droplets, a “social distancing” guideline of 1.83 m (6 ft) was recommended based on the earlier work of Wells (Wells, 1934). It should be emphasized that the noted social distance was based on the static ambient conditions and neglected the effects of T _∞ and RH _∞, which is unrealistic. It is assumed that most large respiratory droplets with potentially higher levels of viral particles fall on the ground within the noted distance. In contrast, aerosolized droplets with low Stokes numbers are significantly influenced by the surrounding airflow and travel much farther than 1.83 m from the emission source (Finely, 2001; Mousavi Tilehboni et al., 2013; Feng et al., 2020; Ismail et al., 2022). Feng et al. (2020) found that the 2 m social distancing policy was insufficient to prevent airborne transmission of SARS-CoV-2 because of the complexity of environmental conditions. Pendar and Carlos Páscoa (2020) suggested that social distancing must be increased to ∼4 m during sneezing. Bourouiba et al. (2014) reported that the turbulent gas cloud generated by a human sneeze could travel approximately 7–8 m. Mariam et al. (2021) indicated that the “rule of thumb based safe distance approach” cannot be a general mitigation strategy for infection control. They also suggested using “air curtains” to provide safe spaces in indoor environments (Mariam et al., 2021).

It is well-known that fresh air introduced by the ventilation system can reduce the viral load in indoor environments (ASHRAE, 2020). Greenhalgh et al. (2021) presented strong evidence that the SARS-CoV-2 virus spreads by airborne transmission, so its spread is higher indoors than outdoors and is substantially reduced by a ventilation system. However, the ventilation airflow increases the complexity of the social distancing policy and affects the infection risk to people due to airborne droplets containing the SARS-CoV-2 virus (Cortellessa et al., 2021; Ghoroghi et al., 2022).

There is no doubt that the experimental measurements of the behavior of respiratory droplets and their motion under different indoor environmental conditions can provide guidelines for understanding the transmission pathways of virus-containing droplets (Shao et al., 2021). However, due to the limitations in time, costs, and complexity of experiments, the numerical modeling approach provides an additional opportunity to study the transmission pathways of airborne respiratory droplets. The computational fluid dynamic (CFD) approach has been used to study the airborne transmission of respiratory droplets (Gao and Niu, 2006; Wan et al., 2009; Chen et al., 2010; Li et al., 2018; Jacob et al., 2019; Yang et al., 2021). In the earlier Eulerian-Eulerian or Eulerian-Lagrangian numerical studies, the droplets' evaporation was not always considered, which led to limitations on the validity of droplet trajectory predictions. Using an Eulerian-Lagrangian approach, Chen et al. (2010) reported that the T _∞ and RH _∞ play a minor role in the dispersion of human exhaled 0.1–200 μm droplets in ventilated indoor environments. Other numerical studies show that environmental conditions play an important role in the dispersion of respiratory droplets in the presence of ventilation. That is, the presence of a ventilation system could significantly reduce the total aerosol number concentration and the lifetime of droplets (Kumar and King, 2022; Lordly et al., 2022).

For simulating cough and sneeze airflow, a variety of Reynolds-Averaged Navier-Stokes equation (RANS), Reynolds Stress, and Large Eddy Simulation (LES)-based models were used in the literature (Seymour et al., 2000; Wan et al., 2009; Li et al., 2018; Jacob et al., 2019). The turbulent flow regime assumption is appropriate due to the high Reynolds number value of the exhaled airflow during sneezing or even coughing (Bourouiba et al., 2014). Gao and Niu (2006) reported that the renormalization group (RNG) k-ε model better predicts the respiratory flows than the standard and realizable k-ε models. Despite the extensive studies on the application of different models in predicting the dynamic behavior of respiratory droplets, the dynamic change in droplet size and its effect on the airborne transmission of droplets remains a significant challenge in CFD simulations. In addition, the simulations performed with varying ventilation rates can challenge the recommended guidelines of increasing the ventilation rate for better removal of particles (Mariam et al., 2021). The sneeze droplets are affected by the inertia, drag, buoyancy, and gravity forces, where the inertia force gradually decreases due to evaporation, and the gravity controls the movement of larger droplets (Pendar and Carlos Páscoa, 2020). However, the buoyancy effect of the droplets ejected from the mouth is reported to be small in the early ejection stage (Liu et al., 2021b).

Although several studies on the effect of turbulent airflow on the airborne transmission of respiratory droplets have been reported in the literature, the influence of environmental conditions is not fully understood. In addition, the local variation in aerosol concentration due to changes in environmental conditions is less understood. As noted before, the rate of virus transmission in indoor environments is much higher than in outdoor environments. Therefore, understanding the effect of T _∞ on the evaporation of sneeze droplets is helpful in developing exposure guidelines by controlling the temperature level in indoor environments. This study investigated the effect of T _∞ on airflow dynamics, droplet size distribution, and airborne transport in a confined space using CFD modeling. The simulation results using the RNG k-ε turbulence model were compared with the performed experimental data for verification and test of accuracy. The validated model was used to simulate the airborne propagation of sneeze-generated aerosols and medium and large droplets in indoor environments. This study's findings could be used to determine the required social distancing to reduce the exposure risk of COVID-19 for different indoor room temperatures.

2. Experimental procedure

2.1. Participants and their conditions

Ethics approval for this study using human volunteers was granted by the guidelines and protocols of the Islamic Republic of Iran, Ministry of Health and Medical Education (IR.IUMS.REC, reference no. 1396/367). The participants in the sneezing analysis experiments consisted of ten healthy 22–23 years-old male volunteer students. The average height of participants was 1.71 m. The characteristics of all participants are given in the Supporting Information (Table S1). The thermographic tests were done at the beginning of experiments on all participants, and they all had negative PCR tests. All participants were healthy and non-smokers and provided written consent before participating in the study. All participants had no history of significant pulmonary diseases or chronic respiratory illness. Each participant was asked to sneeze freely across the mirror to capture their sneeze images. A pepper stimulant was used to help the volunteers start to have a weak sneeze. All tests were performed 2 h after breakfast to minimize the effect of the food on the physical properties of the saliva and respiratory sneezing droplets. The participants were not allowed to drink water before the tests to avoid the possibility of changes in the density and viscosity of sneeze salvia droplets. Experiments were performed individually to prevent the virus transmission risk. The ventilation system was off during the experiments. After each experiment, the ventilation system was turned on for an hour at the air exchange rate of ACH = 8 for safety reasons to remove possible remaining droplets in the indoor air.

2.2. Imaging system

The shadowgraph imaging system was used to determine the axial distance and dispersion of respiratory droplets expelled by the participants during sneezing. The schematic image of the experimental setup is shown in Fig. 1a. The system includes an anemometer, a manual adjustable white light-emitting diode (LED) lamp, a spherical concave mirror with a radius of 40 cm and a focal length of 20 cm, a high-speed digital camera (CCD, Ophir-Spiricon Inc. Model: BA 150, shooting rate: 250–500 fps, resolution: 640 pixels × 480 pixels), and a computer system. A LED light source was supplied by the halolux device (halolux LED-15, STREPPEL). The CCD camera was set at 60^o in the direction of the LED light, while the volunteer subject was positioned at 90^o towards the reflected light. A minimum shooting time of 1000 femtoseconds was selected to generate images with good resolution (Scharfman et al., 2013). A concave mirror was attached to the support base and LED light was reflected from the mirror during experiments. The height of the mirror relative to the ground was adjusted based on the average height of the participants.

Fig. 1.

Schematics image of (a) shadowgraph imaging experimental setup, and (b) PIV measurement set-up.

The airflow velocity was measured by a portable MC 4.0 Multiprobe anemometer. The airflow velocity data are obtained by placing the anemometer tip just in front of the participant's mouth at different longitudinal distances. Anemometer accessories included a temperature and humidity sensor (HygroAir 20). The sensor measured the T _∞ in the range of 283–363 K (±0.3 K) and RH _∞ in a percentage (%) of 10–90 % (±1.8 %). The droplet velocity distribution around the mouth of the sneezing subjects was measured by the non-intrusive Particle Image Velocimetry (PIV) technique (LaVision FlowMaster). A schematic diagram of the PIV experimental set-up is shown in Fig. 1b. The PIV setup consisted of the Nd:YAG solid-state laser (power: 120 mJ per pulse, wavelength: 532 nm, frequency: 15 kHz), a laser sheet to illuminate the droplets in the domain, and a CCD camera. For safety reasons and due to physical constraints of the laser optics, the laser sheet was aligned, pointing away from the volunteer person. The laser beam leaves the laser head with a 1 to 4 mm diameter. A laser sheet <1 mm thick was used to ensure the measured flow field could be kept in a plane. Two cylindrical and spherical lenses with focal lengths of 15 and 500 mm, respectively, were used to reduce the light-sheet thickness. The optimized time separation between pulses was adjusted by 1 ms.

Commercially available software (LaVision Inc.'s DaVis 10) was used to process the raw images. The PIV image pairs were taken at a frequency of 5 Hz. A broad-spanning measurement domain with dots of 20 mm diameter regularly spaced at 60 mm was used to calibrate the CCD camera. A standard set of procedures for PIV data processing, including image background subtraction followed by multi-pass PIV calculations, was used to study the motion of sneeze droplets within the domain of measurement. The relative velocity of droplets at different longitudinal distances is measured by placing the CCD camera at different points of the sneezing zone and averaging the data. The mean droplet velocities were evaluated by repeated sneezing at least 4 times for the fixed position of the PIV setup from the participant's mouth. The cross-correlation analysis was used for interrogation windows of 32 pixels × 32 pixels with a 50 % overlap, yielding a 74 × 99 array of the velocity vector (Recursive Nyquist Grids). Erroneous vectors were removed by the standard deviation filter followed by the local median filter. Thus, this filtering process resulted in <5 % of the vectors being removed.

2.3. Imaging experiments

The imaging experiments were performed in a dark room with a dimension of 3 (length) × 4 (width) × 3 m³ (height), equipped with a mixing air ventilation system with an air change per hour of ACH = 15. The ventilation inlet and outlet register were on the ceiling of the laboratory. The air ventilation system kept the room temperature at the desired value of T _∞. An inlet airflow rate of 160 L/s, recommended by WHO (2020), was used for the safety of the isolation rooms. The imaging experiments were carried out when the ventilation system was off. For all cases, the room temperature was set at four T _∞ levels of 288, 298, 308, and 318 K, while RH _∞ was fixed at 30 %. According to the ASHRAE standard (55-2017) for the thermal environmental conditions for human occupancy (ASHRAE, 2017), the desired range of indoor temperature is 296–300 K during summer and 292–296 K during winter. The suggested temperature ranges for different indoor rooms are presented in the Supporting Information (Table S2). In indoor spaces, temperatures <288 K are not recommended for indoor environments, while the maximum acceptable room temperature is 305 K (ASHRAE, 2017). In hot indoor spaces, such as bakeries, kitchens, laundries, boiler rooms, and manufacturing with hot local heat sources, the maximum T _∞ should not exceed a temperature of 315 K due to the high possibility of risk of human heatstroke (ASHRAE, 2017). Therefore, the T _∞ values between 288 and 318 K were selected for this study to cover all relevant temperature ranges in the indoor environments.

The LED light shines on the surface of the concave mirror, and the reflected light illuminates the droplets' clouds generated by sneezing. The LED light passed through the sharp edge of the metal piece located in the reflected light path to remove marginal images. The participants sneezed at a distance of 15 cm from the concave mirror, and imaging created appropriate high-quality photos (Scharfman et al., 2013). The sneezing images were analyzed using an image processor (ImageJ software, version 1.49). The sequence of different frames was used to explore the droplet cloud boundaries from every single sneeze. The images with good contrast were obtained at T _∞ lower than the individuals' exhalation temperature T _exh (309.5 K), while there were some difficulties in processing and analyzing images at T _∞ ≥ T _exh.

2.4. Determination of the droplet size distribution

The size distribution of droplets generated by sneezing of different subjects was determined by a laser particle sizer (Malvern Instruments Ltd.; UK). The schematic image of the laser system used in this study is shown in the Supporting Information (Fig. S1). The laser beam (beam diameter: 0.012 m) supplied by the helium-neon laser system passed through the measurement zone. Forty-two optical sensors were installed to examine the light diffraction pattern. The sampling frequency was set at 2.5 kHz, corresponding to 0.4 ms, to ensure that the data was obtained in a real-time acquisition process. The t-test was employed to analyze the difference in average sneeze droplet sizes for males and females of different age groups. The experiments were repeated at indoor temperatures of 288, 298, 308, and 318 K, while RH _∞ was fixed at 30 %. The number concentration of sneeze droplets suspended in the air was determined by dividing the total number of respiratory droplets of size, i, by the total number of droplets. Assuming that all the sneezing droplets are spherical, the number fraction of sneeze droplets of size i (P _n,i) is determined by the ratio of the total number of respiratory droplets of size i to the total number of droplets. That is,

$$P_{n,i}=\frac{N_i}{N}=\frac{P_{Vi}{D}_i^{-3}}{{\sum }_i{P}_{Vi}{D}_i^{-3}}$$ (1)

Here N _i is the total number of droplets of size i, where i = 1, 2, …, n; N is the total number of respiratory droplets, $V={\sum }_i{P}_{Vi}$ is the total volume of droplets, and D _i is the aerodynamic diameter of droplets with a size class of i. P _V,i is the ratio of the total volume of droplets with diameters in the size class i, to the total volume of all droplets of any size V.

In addition, the total number of droplets is estimated to be 5535 ± 43 (corresponding to 5.535 ± 0.043 #/cm³). The estimated value is evaluated by averaging the data obtained from the weak sneeze experiments through four repeated sneezes of each of the 10 participants (sneezing 40 times), which is reported in the Supporting Information (Table S3). This result was in agreement with the data reported by Duguid (1946) in the case of a single simulated weak sneeze produced by forming the sound “ttsch” explosively, while the number of droplets in the strong sneezes can reach >39,000.

3. Computational methodology

3.1. Governing equations

3.1.1. Carrier (air) phase

The balance of momentum, conservation of mass, and energy equations govern the airflow and thermal conditions. The governing equations of airflow are:

Mass conservation

$$\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho {\overline{u}}_i\right)}{\partial x_i}=0$$ (2)

where ρ is the air density, and${\overline{u}}_i$is the airflow mean velocity in the i-direction. The airflow density was determined by defining a second-order polynomial function obtained by fitting the thermodynamic density/temperature data of air:

$$\rho ={10}^{-5}{T}^2-0.0048T+1.2926$$ (3)

Momentum balance

$$\rho \frac{\partial {\overline{u}}_i}{\partial t}+\rho {\overline{u}}_j\frac{\partial {\overline{u}}_i}{\partial x_j}=\frac{\partial }{\partial x_j}\left[{\mu }_{\mathit{eff}}\left(\frac{\partial {\overline{u}}_i}{\partial x_j}+\frac{\partial {\overline{u}}_j}{\partial x_i}\right)-\frac{2}{3}{\mu }_{\mathit{eff}}\frac{\partial {\overline{u}}_k}{\partial x_k}{\mathrm{\delta }}_{\mathit{ij}}\right]-\frac{\partial P}{\partial x_i}+\rho g_i$$ (4)

where P is the mean pressure, μ _eff = μ + μ _T is the effective dynamic viscosity, μ is the airflow viscosity, μ _T is the turbulent (eddy) viscosity, g _i is the acceleration of gravity. The turbulent diffusion of droplets is simulated by the Discrete Random Walk model, in which the instantaneous velocity of the airflow u _i is expressed as:

$$u_i={\overline{u}}_i+u_i^{\prime }$$ (5)

where ${\overline{u}}_i\text{and}{u}_i^{\prime }$ are, respectively, the mean airflow velocity and fluctuating velocity components. ${\overline{u}}_i$ is evaluated using the RANS approach and the RNG k-ε turbulence model.

Energy conservation

The motion of the exhaled airflow into the air is accompanied by energy transfer. Also, the evolution of the droplet diameter is significantly affected by the airflow temperature surrounding the droplets. The initial temperature of the air flow was considered near to the body temperature of the human body.

$$\frac{\partial T_{\infty }}{\partial t}+{\overline{u}}_i\frac{\partial T_{\infty }}{\partial x_i}=\frac{\partial }{\partial x_i}\left[{\lambda }_{\mathit{eff}}\frac{\partial T}{\partial x_i}\right]+S_h$$ (6)

where λ _eff is the effective thermal conductivity, and S _h is the heat source.

Turbulence model

The RNG k-ε turbulence model is also used in the simulations. The transport equations for the RNG k-ε turbulence model are defined as:

$$\rho \left(\frac{\partial k}{\partial t}+{\overline{u}}_i\frac{\partial k}{\partial x_i}\right)=\frac{\partial }{\partial x_i}\left(\left(\mu +\frac{{\mu }_T}{{\sigma }_k}\right)\frac{\partial k}{\partial x_i}\right)+G_k+G_b-\mathit{\rho \epsilon }-Y_M+S_k$$ (7)

$$\rho \left(\frac{\partial \epsilon }{\partial t}+{\overline{u}}_i\frac{\partial \epsilon }{\partial x_i}\right)=\begin{array}{l}\frac{\partial }{\partial x_i}\left(\left(\mu +\frac{{\mu }_T}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_i}\right)+C_{1\epsilon }\frac{\epsilon }{k}\left(G_k+C_{3\epsilon }{G}_b\right)-C_{2\epsilon }\rho \frac{{\epsilon }^2}{k}\\ -R_{\epsilon }+S_{\epsilon }\end{array}$$ (8)

where k is the turbulence kinetic energy, ε is the turbulence dissipation rate, and G _k, and G _b, respectively, represent the generation of turbulence kinetic energy due to the mean velocity gradients and buoyancy. Y _M represents the effect of compressibility on turbulence, and S _k and S _ε are source terms for k and ε equations, respectively. σ _k and σ _ε are the turbulent Prandtl numbers for k and ε, respectively.

3.1.2. Discrete (droplet) phase

The CFD modeling was used to evaluate the influence of T _∞ on the velocity field and the propagation of the droplets expelled during sneezing at a fixed temperature of T _exh = 309.5 K. The main assumptions are:

• (a)The droplets were considered to be spherical. Also, the initial velocity of exhaled droplets is assumed to be the same as the sneeze airflow velocity just at the mouth of the model. This assumption is confirmed by the turbulent regime of the initial jet phase close to the mouth (Bourouiba et al., 2014).
• (b)The effect of the humidity field generated by the dispersed droplets on their evaporation was not considered because the amount of vapor generated is small, and the jet flow exhaled by sneeze disperses quickly in the room air (Han et al., 2013).
• (c)As the momentum of inhaled air by breathing before sneezing is relatively small, the effect of breathing on the velocity fields before sneezing was ignored in this study. For simplicity, only the airflow created by sneezing was assumed to be exhaled from the human mouth.
• (d)The effect of the non-volatile compounds, such as salt and lipids, on the size change during the evaporation of the sneeze droplets, was ignored.
• (e)For simplicity, only one sneeze was modeled.
• (f)A constant injection angle of 25°, relative to the horizontal direction, was assumed in the simulations. This value was selected based on the average of experimental data for male participants, which was in good agreement with the data reported in the literature (Bourouiba et al., 2014). However, in the experiment, the participants may shake or rotate their heads during sneezing, which leads to uncertainties in the simulation results.
• (g)As the droplets are small, their surface tension is sufficiently strong to make them behave as small spherical rigid particles (Balachandar et al., 2020).
• (h)The exhalation airflow temperature, T _exh, was fixed in the simulations and does not change due to the vapor generated by human breathing and water evaporation.

The motion of sneeze droplets was evaluated by solving the Lagrangian force balance equation to track their trajectories. The equation of motion of particles was determined by equating the droplet inertia with the summation of the gravitational force ${\overrightarrow{F}}_g$, Stokes drag force ${\overrightarrow{F}}_D$, buoyancy force ${\overrightarrow{F}}_B$, and frictional force ${\overrightarrow{F}}_f$. That is,

$$m_d\frac{d{\mathit{u}}_d}{\mathit{dt}}={\overrightarrow{F}}_g+{\overrightarrow{F}}_D+{\overrightarrow{F}}_B+{\overrightarrow{F}}_f=\left({\rho }_d-\rho \right)V_d\mathit{g}\mathbf{-}\frac{\mathbf{3}}{4}{C}_D\frac{\rho }{{\rho }_d}\frac{m_d}{2R_d}\left|\mathit{u}-{\mathit{u}}_d\right|\left(\mathit{u}-{\mathit{u}}_d\right)$$ (9)

where u and u _d are the instantaneous airflow and droplet velocity vectors, respectively. Here, ρ and ρ _d are the densities of the air and droplets, respectively. In Eq. (9), m _d, V _d, and R _d, are the mass, volume, and radius of the sneeze droplets, respectively. C _D is the drag coefficient, which can be calculated as a function of the droplet Reynolds number as

$$C_D\mathbf{=}\left\{\begin{array}{c}\frac{\mathrm{24}}{{\mathit{Re}}_d}\left(\mathit{Re}<\mathrm{1}\right)\mathit{Stokes\; regime}\\ \frac{\mathrm{20}}{{{\mathit{Re}}_d}^{0.7}}\left(\mathrm{1}<\mathit{Re}<\mathrm{1000}\right)\mathit{transition\; regim}\mathrm{e}\\ \frac{\mathrm{10}}{\sqrt{{\mathit{Re}}_d}}\left(\mathit{Re}>\mathrm{1000}\right)\mathit{turbulent\; regime}\end{array}\right)$$ (10)

where Re _d = (ρ |  u − u _d|  d/μ).

An enhanced two-layer wall boundary condition is used instead of the standard wall function, consistent with the near-wall mesh refinement strategy proposed by Chao et al. (2009) in modeling aerosol particles settling due to gravitational force. The “Eddy Lifetime” model is used for stochastic tracking in the ANSYS-Fluent code. This model assumes successive encounters of particles with discrete turbulence eddy. Accordingly, u _i′ is given as:

$$u_i^{\prime }=\mathrm{\xi }\sqrt{\frac{2}{3}k}$$ (11)

where ζ is a unit variance Gaussian random number selected after the “eddy lifetime” duration, and k is the turbulence kinetic energy. The Lagrangian particle integral time scale T _L is given as

$$T_L={\int }_0^{\infty }\frac{u_{\mathit{di}}^{\prime }\left(t\right)u_{\mathit{di}}^{\prime }\left(t+s\right)}{{\left(u_{\mathit{di}}^{\prime }\right)}^2}\mathit{ds}$$ (12)

The Lagrangian integral time scale can be estimated by the airflow Eulerian integral time scale. That is,

$$T_L=0.3k/\epsilon$$ (13)

In the absence of airflow, Eq. (9) leads to droplets' gravitational settling with a terminal velocity of

$$u_{\mathit{di}}^T=g_i\tau$$ (14)

where τ is the droplet relaxation time given as:

$$\tau =\frac{C_c{\rho }_d{d}_d^2}{18\mu }$$ (15)

where C _c is the Cunningham slip correction factor, which is given by: (Allen and Raabe, 1985)

$$C_c=1+\mathit{Kn}\left(2.514+0.8exp\left(-0.55/\mathit{Kn}\right)\right)$$ (16)

where Kn (Knudsen number) is the ratio of the mean free path of the air molecules (λ) to the radius of the droplets (d _d/2).

The evaporation of liquid droplets was considered by solving each droplet's mass and energy balance (Feng et al., 2016; Haghnegahdar et al., 2019; Feng et al., 2020). That is,

$$\frac{\partial m_d}{\partial t}=-{\int }_0^{s}n\mathit{dA}\approx -\left(\overline{n}·A\right)$$ (17)

where $\overline{n}$ is the average mass flux of evaporable component w, on the droplet surface, which can be given by (Chen, 2020),

$$\overline{n}=\frac{\rho D_w\mathit{Sh}{C}_m}{d_d}\mathit{ln}\frac{1-Y_{w\infty }}{1-Y_{ws}}$$ (18)

where D _w is the mass diffusivity of evaporable component w, and Y _w∞ and Y _{w s}, respectively, are the mole fractions of evaporable component w, on the droplet surface and in the gas phase far from the droplets (dry air). Y _w∞ was set based on the RH _∞ = 30 %, while Y _{w s} was determined by using the modified Raoult's law, which is defined in the following equations:

The mole fraction of evaporable component w, in the gas phase far from the droplets, is defined as:

$$Y_{\mathrm{w},\mathrm{s}}={\gamma }_w{x}_{ws}{K}_e\frac{P_{\mathit{vap}}\left(T_s\right)}{\mathit{\rho R}{T}_s}$$ (19)

where γ _w is the activity coefficient of component w, x _w,s is the mole fraction of w in the droplet surface, R is the universal gas constant, T _s is the droplet temperature at the surface, P _vap(T _s) is the saturation pressure of component w at the temperature T _s, and K _e is the correction factor for the Kelvin effect, which can be given by:

$$K_e=exp\left(\frac{4\sigma M_w}{{\rho }_{d}R{T}_d{d}_d}\right)$$ (20)

where σ is the surface tension of the droplet, M _w is the molar mass of component w, and ρ _d (m/V _d) is the droplet density. In Eq. (18), Sh is the Sherwood number, which is expressed as (Clift et al., 2005),

$$\mathit{Sh}=2+0.552{\mathit{Re}}_d^{1/2}{\mathit{Sc}}^{1/3}$$ (21)

where Sc = μ/ρD _d is the Schmidt number. For the droplet sizes under consideration, the Stokes number St = (ρ _d/ρ) d _d ² u/9 ν , which is defined as a ratio of the inertia force of droplets to the viscous fluid force. For small Stokes numbers, the droplets follow gas flow streamlines closely.

C_m is the Fuchs-Knudsen number correction, which is given as (Clift et al., 2005):

$$C_m=\frac{1+\mathit{Kn}}{1+\left(\frac{4}{3{\alpha }_m}+0.377\right)\mathit{Kn}+\frac{4}{3{\alpha }_m}{\mathit{Kn}}^2}$$ (22)

where Kn is the Knudsen number, and α _m is the mass accommodation coefficient.

3.2. Numerical procedure

A three-dimensional model of an isolation room with the dimensions of 4 (length) × 4 (width) × 3 m³ (height) was considered as the computational domain. The ANSYS FLUENT 2020R1 commercial CFD code was used to simulate the airflow generated by a person's mouth during sneezing. As shown in Fig. 2 , a receptor person is standing at a distance of 1.83 m in front of the emitter (in the x direction). Here the y and z axes represent the sideway and the vertical directions, respectively. The selected distance is based on the recommended guidelines (WHO, 2020). This figure shows two standing human models in the computational domain. Due to the complex geometry of the human face, an unstructured tetrahedral mesh was adopted for mesh generation. In addition, the near-wall mesh refinement strategy was used to better represent the complex human body geometry at the extremities of the person's head.

Fig. 2.

(a) Dimensions of isolation room and human models. (b) Mesh generation with a refinement near the heads.

As was noted before, the RNG k-ε turbulent model was used in the simulations. The transient simulations were carried out for both airflow and droplet motion. The unsteady-state airflow solution during sneezing was used as the input for the transient simulation of droplet trajectories using the discrete phase model. The energy equation is activated because of the temperature dependence of air density and droplet evaporation. The stochastic collisions of droplets were omitted due to the dilute concentration of sneezing droplets. As the number concentration of sneeze droplets is low, the one-way coupling assumption, where the airflow is not affected by the motion of the droplets, was considered in the simulation of sneeze airflow from the mouth of the emitter model. In particle-laden turbulent flows, the one-way coupling is generally valid when the volume fraction is <10⁻⁶ (Elghobashi, 1994). As a result, the airflow can be evaluated first and then used to determine the droplet trajectories by solving the Lagrangian equation of motion. The governing equations of the evaporation of droplets in the isolation room were described in Section 3.1.2 by Eqs. (17), (18), (19).

The partial differential equations of airflow and thermal field are discretized using the finite volume approach and solved until the residuals are <10⁻⁶ for all variables. The SIMPLE algorithm was used for pressure and velocity coupling in the carrier phase. The second-order upwind scheme was used to discretize the convective term under the transient state. Since the injected droplets occupy an extremely small fraction volume, the influence of the droplets' finite size and their effects on the surrounding airflow were neglected. As noted before, the governing equations for the discrete phase were solved by the Lagrangian particle tracking method. A time step dependence study showed very little difference in the results at different T _∞ in the time step range of 1 × 10⁻⁵ to 1 × 10⁻⁴ s. Therefore, a time step of 1 × 10⁻⁴ s was used in all simulations. The simulations were performed on eight PCs (Intel Xeon CPU E5–2630 v4, Core 10, 20, Ram: 32 G, Hard STAT: 4TB) using parallel processing. The computational time for simulating the 0.70 s of real-time was approximately 72 h for each simulation run.

3.3. Boundary conditions (BCs) and simulation inputs

The inlet BCs in the computational isolation room with the exact dimensions of the experimental darkroom are specified as consistent with the experimental study. Different ventilation inlet temperatures of T _∞ = 288 K (case 1), 298 K (case 2), 308 K (case 3), and 318 K (case 4) were applied in different simulation cases, while RH _∞ was fixed at 30 %. The input data was selected based on the physical properties of the exhaled droplets in experiments. The main parameters used in the simulations for the air and droplet phases are listed in the Supporting Information (Tables S4 and S5). Based on the experimental data, the initial velocities of simulated droplets for T _∞ of 288, 298, 308, and 318 K are set at 27.56, 27.33, 27.13, and 26.94 m/s, respectively. The initial airflow rate changed in the range of 6.6 × 10⁻³–7.1 × 10⁻³ m³/s depending on the initial velocity of simulated droplets. The total number of sneeze droplets is 5535 ± 43 (corresponding to 5.535 ± 0.043 #/cm³), which was released from the cells at the emission point at the mouth. Three size classes of water droplets with the mean diameters of 5, 100, and 600 μm that are representatives of the exhaled droplets during a sneeze were used in the simulations (Supporting Information, Table S3). The mean droplet diameters of 5, 100, and 600 μm were obtained by multiplying the quantity in each measurement size range of 3–6, 12–275, and 300–1000 μm by the average number of droplets in each of the size classes. Based on our experimental results, the total percentage of droplets with diameters in the size class ranges of 3–6, 12–275, and 300–1000 μm were 1.8, 39.4, and 58.8 %, respectively (Supporting Information, Table S5).

The mouth opening area in the simulations was estimated to be 3.4 cm² during the sneezing, which is consistent with values reported in the literature (Chao et al., 2009). As expected, the time duration of sneeze airflow varies significantly. Duguid (1946), Dbouk and Drikakis (2020), and Zeng et al. (2021) reported a range of 0.01 to 0.25 s for the sneeze airflow. However, Bourouiba et al. (2014) and Bourouiba (2020) estimated duration of 0.5 s for sneezing. Based on the present experimental data, 0.05 s was used for the sneeze airflow and droplet injection duration, while the total simulation time was 0.7 s. The time step for the injection of droplets was 0.0001 s in the simulations. No-slip and no-temperature jump BCs were applied on the wall of the isolation room. The boundary condition for the contact of expiratory aerosols with solid surfaces was set as a “trap,” which indicates that re-suspension was not considered in the simulations. Thus, the droplets were trapped on the walls, as well as the floor or ceiling. As shown in Fig. 2, the human models are away from the walls. The pressure outlet BC was applied at the ventilation outlet, and adiabatic wall BC was used on the room wall.

3.4. Grid independency analysis

The grid-independency analysis was performed by evaluating the velocity of droplets to verify the accuracy of the CFD results. The simulation results of the grid independence analysis in the sneezing zone are presented in the Supporting Information (Table S6). The computational mesh was refined until an acceptable error between two consecutive meshes was seen in predicting the droplet velocity. Then, an analysis of variance was carried out to determine whether the predictions were independent of the cell number. It is seen that the relative error in the predicted droplet velocity by increasing the mesh number to 1,263,854 is below 1.0 %. Thus, considering the computational cost and accuracy, the mesh with 1,263,854 cells was selected for the rest of the simulation cases.

3.5. Regression analysis models

The root-mean-square error (RMSE) and linear regression analysis (R ²-value) were used to examine the performance of regression analysis results through the estimation of the discrepancy between the experimental data and prediction results:

$$\mathit{RMSE}=\sqrt{\frac{1}{n}{\sum }_{i=1}^n{\left(u_{\mathit{exp},i}-u_{\mathit{pred},i}\right)}^2}$$ (23)

$$R^2=1-\frac{\sum _{i=1}^n{\left(u_{\mathit{exp},i}-u_{\mathit{pred},i}\right)}^2}{\sum _{i=1}^n{\left(u_{\mathit{exp},i}-u_{\mathit{avg},i}\right)}^2}$$ (24)

where u _avg,i is the mean measures value, while u _exp,i and u _pred,i are measured and predicted values at location i, respectively.

4. Results and discussion

4.1. Experimental results

4.1.1. Time-evolution of suspended droplets

Fig. 3 shows the images of the time evolution of airborne respiratory droplets after sneezing of a male participant at times of 0.02, 0.09, 0.20, 0.28, 0.45, 0.52, 0.64, and 0.70 s. Here T _∞ = 298 K, and the room's relative humidity was kept fixed at RH _∞ = 30 %. At 0.02 s after sneezing, the concentrated sneezing jet, defined as a “turbulent puff,” is fully ejected from the emitter's mouth. The “turbulent puff” is roughly spheroidal in shape, moves forward by the sneezing jet pulse, and is affected by inertial and gravitational forces (Bourouiba et al., 2014). In the x-direction, the inertial force is dominant, while in the z-direction, the gravitational force plays an important role in the puff motions. The main portion of the turbulent puff remains coherent, except for a minor portion that separates as a vortex ring and travels faster towards the lower right. This observation is explained by the drag force experienced by the droplets, together with the evaporation and reduction in the droplets' diameters along their path. The droplet size reduction leads to a smaller Stokes number, resulting in the entrainment of sneeze droplets in the turbulent puff (Bourouiba et al., 2014). As was noted before, the chance for collision and coalescence of droplets is slight because of the dilute nature of sneeze droplets, while evaporation is significant (Han et al., 2013).

Fig. 3.

The time evolution of airborne droplets after sneezing of a 23 years-old male student at different times for T_∞ = 298 K and RH_∞ = 30 %. (The listed lengths are the maximum spreading distances traveled by the droplets measured by the CCD camera.)

At 0.09 s, the volume of the exhaled puff grows, and the length of the puff becomes significantly larger compared to its size at 0.02 s. At 0.09 s, the bulk of the “expanded puff” remains coherent, while a small portion of large droplets moves slightly faster than its bulk due to their high initial velocity. Liu et al. (2021b) reported that the large droplets overshoot the puff, reach farther distances, and quickly fall out. As the puff volume expands with time, the probability of droplet collisions is further reduced; instead, the droplets' evaporation continues. Also, the fraction of large droplets separated from the bulk puff continuously varies over time. The airborne motion of puff is also associated with its volume expansion due to turbulent diffusion. The image processing showed that the volume of the expanded puff varies over the range of 0.0002 to 0.0025 m³, while the maximum detectable spreading distances traveled by the airborne droplets are 0.42 ± 0.03 and 0.74 ± 0.05 m in the period of 0.02 to 0.09 s after the sneeze, respectively.

In the period of 0.20 to 0.28 s, the main structure of the expanded puff is distorted, and the resulting “droplet cloud” travels downward at an angle of 25 ± 3°, relative to the horizontal direction, based on the average of the repeated sneezing experiments. The largest droplets follow a ballistic trajectory unaffected by the airflow, while the smallest droplets disperse due to the turbulence fluctuation expanding to varying degrees within the droplet cloud (Tan et al., 2021). The large droplets with potentially higher levels of viral particles fall to the ground at short distances due to gravity (Liu et al., 2021a). However, the smaller droplets travel further distances with a maximum spreading distance of 1.88 ± 0.04 m. The “droplet cloud” travels forward by the action of drag, inertial, and gravitational forces (Bourouiba et al., 2014). In the horizontal direction, the inertial and airflow drag move the droplet cloud forward, whereas the gravity and resistance drag control the motion in the vertical direction. The image analysis showed that the droplet cloud lost over 34 % of its initial ejected liquid mass at 0.20 s after the sneeze due to evaporation and gravitational sedimentation.

In the period of 0.28 to 0.45 s, the droplet cloud expands further in the air to form “fully-dispersed droplets” with a maximum detectable spreading distance of 2.73 ± 0.05 to 2.82 ± 0.02 m, respectively. The distances mentioned are the maximum visible distance traveled by the droplets, which the CCD camera could detect; however, the actual transmission distances were larger than the noted values. The image processing showed that the volume of the fully-dispersed droplets estimate varies over the range from 0.10 to 0.20 m³.

A minor change in the spreading distances traveled by the droplets was seen in the period of 0.52 s (2.87 ± 0.02 m) to 0.72 s (2.95 ± 0.02 m), which represents the “dilute-dispersed droplets.” Zeng et al. (2021) reported that the sneeze droplets could travel up to 2.5 m, consistent with the present study's data. However, Bourouiba et al. (2014) found that sneezing droplets could travel to >7–8 m in the horizontal direction. Aerosolized droplets contain a high fraction of droplets in the period of 0.52–0.70 s, which can remain suspended in the air over prolonged periods (Bourouiba et al., 2014). In the horizontal direction, the airflow drag moves the suspended aerosols forward. The smaller droplets are more influenced by airflow than the gravity force. In contrast, gravity and inertia are more dominant for larger droplets. The Stokes drag force varies linearly with diameter, while the inertial and gravitational forces are proportional to the cube of particle diameter.

A minor change in the maximum spreading distance of the droplets in the time interval of 0.45–0.70 s indicated a significant loss in the sneeze pulse flow velocity. Generally, due to the dilute nature of sneeze droplets, the possibility of coagulation and interaction of droplets is negligible, while the evaporation of droplets is a common phenomenon, which led to the reduction of Stokes number of droplets, resulting in the entrainment of these droplets in the turbulent puff (Bourouiba et al., 2014). The large droplets were mainly formed from the liquid layer coating the respiratory passages by the explosive sneeze flow. Balachandar et al. (2020) indicated that the droplets <30 μm have small Stokes numbers and follow the gas flow streamlines closely. At 0.64 s, the evaporation of droplets continues. At 0.70 s, the concentration of suspended droplets decreases significantly due to the evaporation process. The image analysis showed that the “dilute-dispersed droplets” lost over 69 % of their initial ejected liquid mass at 0.70 s, and the dilute aerosolized droplets remained suspended in the air.

4.1.2. Size distribution of suspended droplets

Fig. 4 shows the size distribution of sneeze droplets averaged over all participants at 0.09 s, 0.20 s, 0.45 s, and 0.70 s for T _∞ = 318 K and RH _∞ = 30 %. The number concentration count per person is measured at 10 mm. As can be seen, the number concentration of large droplets decreases over measuring time, while the number concentration of airborne medium droplets and aerosols increases. At 0.09 s, the droplet puff shows a bimodal distribution, with a dominant peak distribution in the range of 410–750 μm, and a second weaker peak distribution is observed in the range of 9–230 μm. Two peaks of the bimodal distribution observed in this study are similar to that reported by Duguid (1946) and Han et al. (2013). Data analysis showed that the peaks are at ∼90 and 590 μm, with the dominant peak being about 1.5 times that of the smaller peak. This finding is nearly consistent with the earlier data of Han et al. (2013), which reported that the sneeze generates a bimodal distribution curve with a dominant peak at ∼500–700 μm and a secondary peak at 80–100 μm.

Fig. 4.

The size distribution of sneeze droplets averaged over all participants at 0.09 s, 0.20 s, 0.45 s, and 0.70 s after the sneeze. Here T_∞ = 318 K and RH_∞ = 30 % [the number concentration count per person measured at the 10 mm distance].

At 0.20 s, a bimodal distribution curve is seen, while a change in the droplet cloud behavior is observed. Fig. 4 shows that the first strong peak of the first distribution in the range of ∼1–55 μm is ∼20 % higher than the weaker peak of the second distribution in the range of 85–390 μm. This finding indicates that the size of suspended droplets and their peaks tend to decrease with time. Data analysis showed that the change of the expanded puff to the droplet cloud led to a decrease in the number concentration of the large droplets by ∼60 % and the increase of medium droplets by ∼45 %. These changes are due to evaporation that reduces the diameter of larger droplets to the smaller ones.

At 0.45 s, a bimodal distribution curve with a sharp peak in the size range of ∼0.2–22.0 μm and a weaker secondary peak in the range of ∼32–235 μm are observed. This trend indicates that most large droplets fell to the ground due to gravitational sedimentation, and the resulting state represents the fully-dispersed droplets. Data analysis shows that the bulk of the droplets consist of medium droplets (∼62–68 %), while the remaining large droplets are <5 % of total droplets. Furthermore, the number concentration of medium droplets is not significantly changed in the period of 0.45–0.70 s. This finding implies that the evaporation of medium droplets might have been reduced as the droplets reached their equilibrium nuclei sizes at 0.70 s (Fig. 3). Morawska et al. (2009) reported that the equilibrium size distribution of evaporating respiratory droplets with diameters between 0.5 and 20 μm occurred within 0.7 s. At 0.70 s, the dilute aerosolized droplets have a bimodal distribution curve with a sharp peak in the size range of ∼0.2–20.0 μm and a second weaker peak in the range of ∼17–120 μm. It is seen that the size distribution of sneeze droplets at 0.7 s is similar to the one at 0.45 s. However, the secondary distribution has shifted to smaller-sized droplets and a milder peak. Data analysis showed that the percentage of aerosols in the dilute-dispersed droplets had increased significantly (∼45 %) compared to the fully-dispersed droplets.

Fig. 5 shows the mean size distribution of sneeze droplets averaged over all subjects at the room temperature of T _∞ of 288, 298, 308, and 318 K. Since the size distribution of sneeze droplets varies with time, the data were recorded at 0.70 s after the end of the sneeze to investigate the effect of evaporation on the size distribution of droplets. The number concentration count per person is measured at 10 mm distance. Fig. 5 shows that the size distribution of sneeze droplets is in the range of ∼0.1–1000 μm, while the number concentration of the bulk of droplets lies in the range of ∼1.5–90.0 μm. The number fractions of aerosols (0.1–5.0 μm) are ∼28.1–36.3 % of total droplets, while the medium and large droplets are ∼62.1–67.6 % and ∼2.9–4.11 % of the total number concentration of droplets, respectively. That is, the number concentration of large droplets is significantly lower than the aerosols and medium droplets, which is attributed to the gravitational sedimentation and evaporation of large droplets (Morawska et al., 2009; Anand and Mayya, 2020; Maggiore et al., 2021).

Fig. 5.

The mean size distribution of sneeze droplets emitted by all subjects at 0.70 s after the end of the sneeze for room temperatures T_∞ of 288, 298, 308, and 318 K [The number concentration count per person measured at the 10 mm distance from the mouth].

Comparisons of the results for different room temperatures in Fig. 5 show that the large droplets are 4.11 % of the total number concentration of droplets for T _∞ = 288 K. However, the number concentration of large droplets reduces to <2.9 % at 318 K. Fig. 5 also indicates that the effect of increased T _∞ on the reduction of the diameter of medium droplets was approximately twice that of the aerosols. The data analysis reveals that the mean diameter of medium droplets decreases from 39.6 to 13.5 μm by increasing T _∞ from 288 to 318 K. In comparison, the mean diameter of aerosols decreases slightly from 4.3 to 2.6 μm by increasing the room temperature from 288 to 318 K. Also, the distribution peak diameter decreases from 10.4 to 2.0 μm when the room temperature increases from 288 to 318 K, which also indicates the equilibrium diameter of the droplets decreases by increasing the T _∞. Previous results showed that the droplet lifetime (time duration that droplet remains airborne) increases by decreasing the equilibrium droplets size (Balachandar et al., 2020; Stadnytskyi et al., 2020; Lordly et al., 2022). On the other hand, an increase in the number of aerosols is associated with higher viral loads (Parhizkar et al., 2022). The mean residence time of the aerosols carrying viruses increases nonlinearly with the viral load, ranging from 100 to 150 s at viral loads of <10⁴ RNA copies/ml to about 1100–1250 s at viral loads of >10¹¹ RNA copies/ml (Srinivasan et al., 2021). That is, an increase in the aerosolized droplet lifetime leads to an increase in the risk of transmission of respiratory diseases and virus survival in the indoor environment (Liu et al., 2021a; Liu et al., 2021b).

Fig. 5 also shows that the number concentration of droplets at T _∞ = 288 K was higher than that at 318 K at 0.7 s after the end of the sneeze because of the rapid evaporation of droplets. This observation indicates that the large respiratory droplets evaporate at 0.7 s and become smaller, which remain suspended in the air as stable aerosols for a longer time (Dao and Kim, 2022). Our findings indicated that the indoor air temperature affects the droplet's evaporation and, thus, influences the size distribution of suspended droplets. Therefore, the effect of T _∞ on the size distribution of the dilute-dispersed droplets should be included in reporting the experimental data and the numerical model results. The evaporation of droplets changes with the room temperature affecting the droplet diameters and, thus, the size distribution of sneeze droplets.

4.1.3. Distance-dependent of velocity profiles

Fig. 6 shows the variation of the (a) mean airflow velocity and (b) mean sneeze droplet velocity with longitudinal distance from the mouth at different room temperatures. The experimental data were reported based on the averaged data of all subjects. The symbols represent the experimental data, while the dashed lines show the developed regression model. Error bars in this figure represent the standard deviation for at least three measurements, where the number of droplets expelled varies substantially between different sneezes. The airflow velocity decreases gradually with the longitudinal distance in the range of ∼0.20–0.45 m (Fig. 6a), which corresponds to the turbulent puff. Then the rate of reduction of airflow velocity with the longitudinal distance increases in the range of ∼0.45–0.74 m, which corresponds to the expanded puff. Finally, the airflow velocity decreases significantly in the distance range of ∼0.8–2.0 m and reaches near zero in the range of 2.5–3.0 m, where the effect of the initial momentum of jet flow is negligible. For different room temperatures, the data of airflow velocity, u, at various distances from the emitter's mouth, was fitted to a cubic regression model given as,

$$\mathit{u}=u_0+C_{1}x+C_2{x}^2+C_3{x}^3$$ (25)

where C _i are the fitting coefficients and u ₀ is the sneeze airflow velocity at the subject's mouth. For T _∞ of 288, 298, 308, and 318 K, u ₀ are, respectively, 27.56, 27.33, 27.13, and 26.94 m/s. The predictions of the regression model for sneeze velocity versus distance at different room temperatures are by dashed lines in Fig. 6a, and error bars on the top and bottom curves are one standard deviation of raw data from the regression. This figure shows that the regression model predictions for the sneeze jet airflow velocity agree well with the experimental data. In addition, the sneeze airflow velocity slightly decreases by increasing T _∞.

Fig. 6.

The variation of (a) mean airflow velocity, and (b) mean sneeze droplet velocity with the longitudinal distance from the mouth for different T_∞ and RH_∞ = 30 %. (The symbols represent the experimental data, while the dashed lines show the regression model.)

For different room temperatures, the measured variations of the mean droplet velocity with the longitudinal distance from the subject's mouth are shown in Fig. 6b. The values of the mean droplet velocity at different longitudinal distances were obtained by averaging the data of sneeze velocity for 10 participants (case no. 1-10) at room temperature T _∞ of 318 K and RH _∞ = 30 % (see Supporting Information, Fig. S2). As the velocity of sneeze droplets varies with their location, the anemometer was positioned at different distances from the emitter's mouth. Fig. 6b shows that the mean velocity of sneeze droplets follows a reverse S-shaped trend. This trend is consistent with the previously reported results in which the relative droplet velocities depend on their size, and the droplet velocity decreases by decreasing the droplet diameter due to evaporation (Liu et al., 2021a). At a distance of ∼0.2 m from the mouth, the velocity drops slightly by about 0.9–1.8 m/s from the sneeze puff's initial droplet velocity. Since the droplets are mainly in the turbulent cloud that forms the puff, their velocities are close to the puff velocity; however, some large droplets are separated from the turbulent cloud and travel at a faster speed than the bulk puff due to their inertia. Gravitational sedimentation also plays an important role in removing the large droplets from the puff. That is, the fall velocities of large droplets could be quite large. Note that only the horizontal velocities were measured in the present study.

As the sneeze jet flow dissipates, the droplet velocity slows down at distances of ∼0.5–1.8 m from the mouth. The drag on the droplet cloud is a significant force in reducing the distance traveled by the droplets. These trends agree well with the droplet trajectories obtained from the imaging experiments. At distances longer than ∼1.8 m, where the dilute-dispersed droplets are suspended in the air, the velocity further decreases. In the range of ∼2.0–3.0 m, the velocity of the droplets becomes negligible as they lose their momentum and the sneeze jet flow dissipates (Fig. 6b, inset). The settling velocity for the remaining suspended aerosols is small and decreases further with time due to evaporation (Zeng et al., 2021).

For different room temperatures, the experimental data for droplet velocities u _d, at various distances from the emitter's mouth, was fitted to a cubic regression model given as,

$${\mathit{u}}_{\mathit{d}}=u_{d0}+C_{d1}x+C_{d2}{x}^2+C_{d3}{x}^3$$ (26)

where u _d0 = u _d,(x=0) = u _d,mouth is the droplet velocity at the mouth of the emitter and C _di are the fitting coefficients. For room temperatures of 288, 298, 308, and 318 K, u _d0 are, respectively, 26.98, 26.83, 26.71, and 26.64 m/s, which are 1.2–1.5 % lower than the sneeze airflow velocity at the mouth. Thus, it is concluded that the airflow and droplet velocities are close to each other when exhaled from the subject's mouth. Dash lines in Fig. 6b (inset) show the predictions of the regression model given by Eq. (26). Again, it is seen that the model predictions are in close agreement with the experimental data for different room temperatures.

Fig. 6b shows that the droplet velocities decrease as T_∞ increases from 288 to 318 K. This is due to the increase in the droplet evaporation rate that leads to a decrease in the size of the droplets. The effect of T _∞ on the mean velocity of the droplet cloud was more dominant than the puff and fully-dispersed droplets due to the high diameter reduction rate of medium droplets caused by evaporation in the range of ∼0.35 to 1.7 m (Fig. 6b). A significant reduction in the diameter of the droplets and, consequently, a decrease in mean droplet velocity occurred for medium-sized droplets due to evaporation. For distances >1.8 m, the aerosolized droplets stay suspended in the air because of their very small fall velocities (Balachandar et al., 2020; Mikszewski et al., 2022). A comparison of Fig. 6a and b shows that the effect of room temperature T _∞ on the mean velocity of sneeze droplets is slightly higher than that of the airflow velocity, while both velocities of droplets and airflow decrease slightly by increasing T _∞.

4.2. Numerical model

4.2.1. Time evolution of velocity profiles

Fig. 7 shows the predicted time evolution of (a) the airflow velocities and (b) droplet velocity at a distance of 0.2 m from the mouth for different room temperatures T _∞. Fig. 7a shows the airflow velocity first increases with time in the period of 0.014–0.016 s to reach its maximum value and then decreases with time. In the period of 0.014–0.016 s, which corresponds to the turbulent puff regime, the airflow velocity is about 24–25 m/s. It was previously shown that the high-speed airflow jet was the primary driver of respiratory droplet transport during sneezing (Liu et al., 2021a; Liu et al., 2021b). A sharp drop in the airflow velocity is seen in the period of ∼0.05–0.20 s when the sneeze pulse passes the 0.2 m section. During this period, the volume of the exhaled puff grows, and the expanded puff's length becomes significantly larger than its original size. The turbulent puff is initially pushed forward by the sneezing airflow jet and droplets' inertial as it expands over time. The airflow velocity at the distance of 0.2 m from the mouth gradually decreases in the period of 0.20–0.45 s and reaches near zero at 0.7 s, which indicates the sneeze jet pulse left the area and lost up to 90 % of its initial momentum.

Fig. 7.

(a) The time-evolution of the airflow velocity and (b) droplet velocity at different T_∞ for a longitudinal distance of 0.2 m from the mouth. (The symbols represent the simulation results, while the dashed lines show the regression model.)

The predicted time evolution of the mean droplet velocities at a section 0.2 m away from the mouth at different room temperatures are shown in Fig. 7b. The velocity of suspended respiratory droplets follows the same variation as the airflow in the period of 0.05–0.20 s, as most droplets are contained in the exhaled turbulent airflow (Bourouiba et al., 2014; Olivieri et al., 2022). However, a careful examination shows that the droplet velocities are 1.4–1.8 % lower than the airflow velocity in the puff. Based on earlier studies (Bourouiba et al., 2014; Liu et al., 2021a), the Reynolds number for sneeze jet is estimated to be >20,000, representing the turbulence flow regime; thus, the drag force has a dominant role in maintaining the droplets' velocities.

By comparing Fig. 7a and b, it is seen that the trend of velocity variations over time for droplet velocity is similar to the airflow velocity. Based on the regression analysis, droplet velocity and airflow velocity follow the same model in the period of 0.28 s, corresponding to the change from the expanded puff to the droplet cloud. The maximum deviations between the results are seen at the interval of 0.28–0.45 s, corresponding to the fully-dispersed droplets. The impact of T _∞ on the droplet velocity gradually decreases over time because of the reduction in the evaporation rate of droplets as they reach their equilibrium diameters. At 0.2 m from the mouth, the effect of room temperature is more noticeable in the period of 0.09–0.28 s, which corresponds to the expanded puff and droplet cloud. In contrast, the minor effect of T _∞ on the droplet velocity is seen for t > 0.45 s, corresponding to the dilute-dispersed droplets, where a high concentration of sneeze droplets is aerosolized with equilibrium nuclei size.

4.2.2. Airflow velocity vector fields and droplet concentrations

Fig. 8 shows the droplet number concentration contour plots and velocity vector fields at 0.02 s, 0.12 s, and 0.20 s after sneezing. Here, T _∞ and RH _∞ are fixed at 298 K and 30 %, respectively. At 0.02 s after sneezing, the puff formation comprises a relatively dense concentration of aerosols and droplets. The mean velocity of bulk puff represents the mean velocity of droplets. The initial trajectory of the droplet population shows that the droplets travel forward rapidly with nearly the same airflow velocity. However, the dispersion rate of sneeze droplets significantly differs because of their different sizes, which are captured by solving for airflow in the transient mode (Morawska et al., 2009). At 0.02 s, the aerosols and small droplets expelled by the sneezing person disperse in the room, with a maximum spreading distance of 0.45 m. The transport of aerosols is mainly influenced by airflow with the negligible effect of gravity. Katre et al. (2021) reported that the gravitational settling and inertia significantly affect the transport of medium 10–100 μm droplets as they evaporate. Thus, because of their greater inertia, the medium droplets travel longer distances from the source than the aerosols. Therefore, the maximum spreading distance of medium sneeze droplets is somewhat higher than the aerosols. The large droplets tend to separate from the sneeze jet flow and vortex ring and fall on surfaces because of the gravity force.

Fig. 8.

The droplet number concentration contour plots (left panels) and velocity vectors (right panels) at 0.02 s, 0.12 s, and 0.20 s after sneezing. (T_∞ = 298 K and RH_∞ = 30 %.)

At 0.12 s, the maximum spreading distance of droplets is 1.5 m, with the droplet cloud traveling downward with an angle of 25 ± 2°, relative to the horizontal direction. This finding is in agreement with the imaging experiments shown previously in Fig. 3. At 0.20 s, the aerosols and small droplets expelled by the sneezing person reach the breathing zone of the receptor person at a distance of 1.92 m. Fig. 8 also shows that the droplet number concentration significantly decreases at 0.2 s (as compared with 0.12 s), indicating that the number of dispersed droplets in the indoor air decreases with time. On the other hand, the differences in the droplet velocities are reduced, perhaps due to the decrease in the droplets' size differences at 0.20 s. The droplet number concentration contours show that the droplets are transported >1.83 m from the emitter within the 0.2 s.

Fig. 8 also shows the velocity vector fields at 0.02, 0.12, and 0.2 s representing the turbulent puff, expanded puff, and droplet cloud, respectively. These simulated velocity vector fields are similar to the experimental sneeze flow dynamics visualized by shadowgraph imaging. The image processing analysis showed that the volume of the expanded puff was 0.002 m³, which was compatible with the calculated value of the total volume of air expelled, 0.0002 to 0.0025 m³, during a sneeze. At 0.12 s, a maximum droplet velocity (∼20 m/s) is seen in the central region of the puff. Furthermore, the droplet velocity reduces from the core of the puff to its surroundings, where there is a large area with velocities <5 m/s. At 0.12 s, the velocity difference between the core of the expanded puff and its surroundings decreases by about 30 % compared to the turbulent puff at 0.02 s.

In the case of turbulent puff, the Reynolds number is >20,000, which corresponds to the turbulent flow regime, while Re _d becomes <3000, which corresponds to transitional flow. The velocity vector fields show the propagating vortex structures with nonsymmetrical flows with a higher or lower velocity magnitude than the bulk puff. As previously found by shadowgraph imaging experiments, the vortex structures could spread droplets perpendicularly to the primary puff. The vortex structures, which lead to the spread of droplets perpendicular to the main airflow direction, have small velocity values.

Fig. 9 shows the simulation results of the variation of mean droplet velocity with the longitudinal distance from the sneezing source for different room temperatures T _∞. The mean velocity of droplets follows a reverse S-shaped trend, which was also seen in the experimental data. At first, the mean velocity of droplets decreased slightly with the axial distance from the source, which indicates that the sneeze jet flow maintains its velocity, and fluid and particle inertia have a dominant role in the airborne transmission of droplets in the puff structure. Then, the mean droplet velocity reduces considerably with the longitudinal distance from the sneezer's mouth due to the turbulent dispersion of sneeze jet flow, where the puff-shaped structure changes to a droplet cloud. Finally, the mean droplet velocity becomes very small, representing a significant loss in the sneeze's initial momentum due to the dilute nature of aerosols.

Fig. 9.

Comparison of the simulated mean droplet velocity variations with distance from the sneezing source for different room temperatures T_∞.

The droplet composition affects pathogens' survival in droplets; therefore, droplet evaporation is crucial to the resultant risk of infection (Shao et al., 2021; Ghoroghi et al., 2022). As the composition of salvia sneeze droplets contains 98.2 % water and 1.8 % salt, the effect of nonvolatile content was ignored in the present simulations, which was consistent with the previous studies (Li et al., 2020; Katre et al., 2021). However, the earlier results showed that the assumption of pure water in sneeze droplets might lead to under-predicting the mean droplet velocity compared to including the nonvolatile content of respiratory droplets (Li et al., 2020). Furthermore, for mucus droplets, the equilibrium nuclei diameter after evaporation is 0.44 times the original diameter, which indicates the important effect of nonvolatile compounds (Lieber et al., 2021). Fig. 9 shows that the mean velocity of droplets slightly decreases by increasing T _∞. However, the impact of T _∞ on the mean velocity of droplets in the distance range of ∼0.40–1.55 m is higher than the other distances. At distances less than ∼0.4 m, where the Reynolds number is estimated to be >20,000, the sneeze pulse jet dominates the transport of airborne droplets and maintains the momentum of the puff structure. At the longitudinal distance of 2.0–3.0 m, the suspended aerosols have a low mean droplet velocity associated with the Stokes numbers that are less than one. Balachandar et al. (2020) and Chaudhuri et al. (2020) reported that water droplets smaller than 30–50 μm remain suspended in the indoor air after evaporation.

Fig. 10 compares the predicted and measured values of the droplet velocity for different room temperatures T _∞. The regression analysis shows that the simulated and measured values of droplet velocity agreed very well with the RMSE values of 0.9882, 1.0640, 1.1435, and 1.3111 m/s, respectively, for room temperatures of 288, 298, 308, and 318 K. It is seen that the lowest and highest RMSE values were found at T _∞ of 288 and 318 K, respectively. Thus, it is seen that the largest differences between the experimental data and simulation results occur at the highest room temperature of 318 K. The high evaporation rate and size reduction of droplets might be the reasons for the higher deviation. Using linear regression analysis, the adjusted correlation coefficient, R ²-value, was obtained to test the goodness of the fits. The results indicate a high R ²-value for all room temperatures. However, the R ²-value decreased slightly from 0.9959 to 0.9933 when T _∞ increased from 288 to 318 K, which indicates that the model predictions at T _∞ = 318 K are slightly different from the experimental data. The predicted and measured mean values of the size evolution at a distance of 0.2 m from the emitter under different T _∞ are shown in the Supporting Information (Fig. S3). The highest and lowest deviations (relative error, %) between the results are found for the room temperatures of 318 and 288 K, respectively. By comparing the results, it is found that the change in the droplet size leads to the variation in droplet velocity.

Fig. 10.

Comparisons of the predicted and measured values of the droplet velocity for different room temperatures T_∞.

Table 1 shows the relative error (%) of the mean droplet velocity obtained from the experimental data (Fig. 6a) and simulation results (Fig. 9) at five distinct zones (turbulent puff, expanded puff, droplet cloud, fully-dispersed droplets, and dilute-dispersed droplets) for various T _∞. The error analysis results in Table 1 show that the lowest variations in the mean velocity of droplets are found at spreading distances <0.42 m, corresponding to the turbulent puff. The minor changes in the size and velocity of droplets in the turbulent puff could be the reason for the low deviation of results. On the other hand, the highest deviations between the experimental data and simulation results are seen for the longitudinal distance of 1.89–2.23 m, representing the fully-dispersed droplets. The high evaporation rate of droplets and the high variations of droplets' diameters might be the reasons for the higher deviation of the results in the fully-dispersed droplets. It is also seen that the relative error (%) between the experimental data and simulation results increases as T _∞ increases. The lowest and highest relative errors occur at 288 and 318 K. As the number of small droplets increases by increasing T _∞, the mean droplet velocity decreases, increasing the relative error.

Table 1.

The relative error (%) between the experimental data and simulated mean droplet velocities at different longitudinal distances and room temperatures T_∞.

Longitudinal distance z (m)	T∞
	288 K	298 K	308 K	318 K
Turbulent puff (z ≤ 0.42 m)	2.15	2.74	3.18	3.72
Expanded puff (0.42 < z ≤ 0.74 m)	6.95	7.34	8.73	9.24
Droplet cloud (0.74 < z ≤ 1.88 m)	11.22	12.07	13.11	14.68
Fully-dispersed droplets 1.88 < z ≤ 2.23 m	16.66	18.24	19.41	22.04
Dilute-dispersed droplets 2.23 < z ≤ 3.0 m	9.29	10.05	10.88	11.71

4.3. Application of validated numerical model

Fig. 11 shows the time evolution in the normalized fraction of suspended droplets, defined as the total number of suspended droplets divided by the total number of emitted droplets for different T _∞. The normalized fraction of suspended droplets is expected to become one at 0.01 s, and then gradually decrease in the period of ∼0.05–0.2 s. A significant drop in the fraction number of droplets is seen in the time interval of ∼0.2–0.7 s. The data analysis shows that the number fraction of medium and especially large droplets decreases sharply in the interval of ∼0.28–0.45 s. In contrast, the aerosols and small droplets may remain in the air for several minutes. In addition, the aerosolized droplet nuclei forms that remain airborne for several minutes to hours (Anand and Mayya, 2020; Maggiore et al., 2021; Chen et al., 2020).

Fig. 11.

The time evolution of the number fraction of droplets suspended in the air for different T_∞.

Fig. 11 shows that the droplet number fractions decrease with time due to the gravitational sedimentation of large droplets and the evaporation of the droplets, assuming no nuclei are left after evaporation. The effect of T _∞ on the fraction number of droplets is negligibly small in the period of ∼0–0.09 s. In the time interval of ∼0.09–0.28 s, the effect of T _∞ on the fraction number of droplets becomes more noticeable due to the change of puff structure into the droplet cloud, which accelerates the size reduction rate of droplets by evaporation. Most medium and large droplets evaporate or deposit on the floor or surfaces within 0.5 s. As the droplet number fractions further decrease with time > 0.5 s, the effect of T _∞ on the fraction number of droplets is more clearly seen because of the enhanced evaporation rate of the fully-dispersed droplets due to the higher room temperature.

Fig. 12 shows the percentage of droplets (aerosols, medium droplets, and large droplets) at different spreading distances for T _∞ of (a) 288, (b) 298, (c) 308, and (d) 318 K. The presented data are for t = 0.2 s for droplets that are passing through different sections at distances of 0.25, 0.9, and 1.83 m, respectively. These distances fall within the puff, droplet cloud, and the fully-dispersed droplets region of the sneeze droplets. The data analysis shows that the diameter of droplets is about 0.31 times their initial diameters because of evaporation during 0.2 s. As can be seen, the percentage of droplets with diameters of 10–100 μm and >100 μm significantly decreases with the spreading distances. In contrast, the percentage of aerosols increases markedly with distance from the emitter. For example, at a distance of 0.25 m, the bulk of droplets are in size range of 10–100 μm, while the percentages of aerosols and large droplets are approximately equal. However, at a distance of 1.83 m from the mouth, the highest percentage of droplets is in the range of 5–10 μm, while only a few large droplets (∼0.2–0.7 %) are found. This indicates that the large droplets fell from the droplet cloud due to gravity or evaporated and were reduced to smaller droplets.

Fig. 12.

The percentage of 5–10 μm, 10–100 μm, and >100 μm droplets at different distances for T_∞ of (a) 288 K, (b) 298 K, (c) 308 K, and (d) 318 K and at 0.20 s after sneezing.

Fig. 12 also shows that the percentage of aerosols increases as the room temperature T _∞ increases. For example, at T _∞ = 288 K (Fig. 12a), only 12.4 % of droplets are in the size range of 5–10 μm, while 79.6 % have sizes in the range of 10–100 μm. At T _∞ = 298 K (Fig. 12b), the percentage of aerosols at distances of 0.25 m and 1.83 m are 13.2 % and 75.1 %, respectively. At T _∞ = 308 K (Fig. 12c), the mean size of droplets further decreases as compared with the lower room temperatures and the increase of the distance from the source. Finally, at 318 K (Fig. 12d), only a few droplets (∼0.2 %) with a size of >100 μm are found in the breathing zone of the receptor person, while >81.1 % of aerosols can get there. This finding suggests there are more airborne aerosols in the warmer room than in the colder room. Fig. 12 also shows that the effect of T _∞ on increasing the number of aerosols and decreasing medium droplets increases with the distance from the sneezing source.

Fig. 13 compares the percentage of sneeze droplets in the diameter range of 5–35 μm at a distance of 1.83 m from the emitter at 0.2 s after the sneeze for different T _∞. As the fraction number of droplets with diameters >35 μm was negligible, their data are omitted in this figure. The distance of 1.83 m from the source is important as expelled droplets would reach the breathing zone of the receptor person. This figure shows that the droplets are mainly in size range of 5–10 μm at 1.83 m from the emitter at 0.2 s after the sneeze. In addition, the concentration of 5–10 μm droplets increases as the room temperature increases. However, as shown in Fig. 6, Fig. 7, Fig. 9, a rise in T _∞ leads to a decrease in the velocity of droplets. At 288 K, only a few droplets (∼0.3 %) are found in the breathing zone of the receptor person, while >81.3 % of aerosols can reach the breathing zone of the second person at T _∞ = 318 K. Also, an increase in T _∞ leads to a significant drop in the percentage of medium droplets due to evaporation and gravitational sedimentation (Maggiore et al., 2021; Wang et al., 2021). Thus, the receptor person is exposed to a higher percentage of aerosols and small droplets with increasing T _∞, but the exposure to larger droplets decreases significantly.

Fig. 13.

The comparison of the percentage of droplets of different sizes at a distance of 1.83 m from the emitter for different room temperatures T_∞.

The effect of T _∞ on the percentage of droplets in the 5–10 μm size range increases because of the fully-dispersed nature of droplets. Also, the airborne droplet nuclei could travel further than 1.83 m from the emitter. The present results show that an increase in T _∞ significantly reduces the sedimentation of large and medium droplets. As a result, the warm room with T _∞ = 318 K leads to about ∼14 % increase in the percentage of 5–10 μm droplets compared with the room with T _∞ = 288 K.

Fig. 14 shows the streamlines colored by the velocity magnitude on a vertical plane crossing the emitter and the receptor persons 0.45 s after the sneeze for rooms with T _∞ of (a) 288, (b) 298, (c) 308, and (d) 318 K. In all cases, the flow streamlines generated from the emitter mouth have reached the breathing zone of the receptor person. For room temperatures of 288 and 298 K, the streamlines are closer to each other than the rooms at 308 and 318 K, which indicates an increase in the droplet velocity at low temperatures compared to higher temperatures. Fig. 14 also indicates that the volume of the vortex ring increases with increasing T _∞. At T _∞ = 318 K (Fig. 14d), although the size of the vortex rings reached its maximum, the separation of the streamlines and their color indicate a significant reduction in the airflow velocity. As noted before, the number concentration (%) of aerosols increases with T _∞. Therefore, the concentration of aerosols increases in the room with higher temperatures. Thus, social distancing in an isolation room with a temperature of 288 K is more important than in a warmer room with T _∞ = 318 K. The numerical results show that the number of virus-containing aerosols increases by about 14 % at short intervals for each sneeze when room temperature increases from 288 to 318 K.

Fig. 14.

The flow streamlines on the vertical plane 0.45 s after the sneeze for room temperatures T_∞ of (a) 288, (b) 298, (c) 308, and (d) 318 K.

Based on the presented results, the exposure risk of airborne virus-containing aerosols in the warm room is higher than in the colder room due to the higher number concentration of aerosols. The infection control guidelines describe the safe distance between people to reduce disease transmission. WHO suggests a 1 m distance, while NHS and CDC guidelines recommend 2 m. The present study showed that the aerosols travel much farther than 1.83 m from the emission source. Therefore, the “social distancing” guideline of 1.83 m (6 ft) is insufficient to prevent exposure to the viruses emitted by sneezing of an infected person.

5. Conclusions

In this study, the effect of indoor temperature (T _∞) on airborne transmission, size distribution, and mean velocity of sneeze droplets in a confined space was studied experimentally and numerically. Computational fluid dynamics (CFD) simulation showed the formation and evolution of turbulent puff, expanded puff, droplet cloud, fully-dispersed droplets, and dilute-dispersed droplets phases, as seen in the shadowgraph imaging experiments. The results showed that the number concentration, and size distribution of droplets depend on the time and distance from the source. The maximum spreading distance depends on the droplet's size, initial velocity, and room temperature. A significant loss in the velocity and spreading distance of droplets was found in the transition phase from the expanded puff to the droplet cloud. The mean velocity of droplets decreased slightly with the increasing T _∞. The effect of T _∞ on the mean velocity of the droplet cloud phase was more noticeable than the puff and fully-dispersed droplets phases due to the reduction of medium droplets' diameters caused by evaporation. In addition, the evaporation rate of droplets in the droplet cloud became more noticeable than in the expanded puff.

In contrast, the impact of T _∞ on the number of droplets was negligible in the turbulent puff, while the number of medium and large droplets significantly decreased in the dilute-dispersed droplets. The indoor temperature T _∞ had different effects on the respiratory droplets. A rise in the T _∞ led to a decrease in the size and velocity of droplets. On the other hand, a rise in T _∞ led to an increase in the number of aerosols reaching the breathing zone of the receptor person. The effect of T _∞ on the reduction of large droplets' diameter was lower than those of the medium droplets and aerosols due to the low concentration of large droplets. The results indicated that the warm room showed more than ∼14 % increase in airborne aerosols compared to the cooler room. The study's findings could help with exposure guidelines in controlling temperature levels in indoor environments to reduce the exposure risk of COVID-19. However, providing a better understanding of the effect of residence times of virus-carrying droplets on infecting people >6 f. away, in the absence and presence of the ventilation system, and also the effect of sneeze duration time on the dispersion of respiratory droplets are left for future studies.

Nomenclature

Acronyms

CFD

Computational fluid dynamic

LED

Light-emitting diode

LES

Large eddy simulation

RANS

Reynolds Average Navier-Stokes

RH

Relative humidity

RNA

Ribonucleic acid

WHO

World Health Organization

Symbols

C_c,

Cunningham slip factor (−)

D_i, d

Aerodynamic mean diameter (μm)

F_D

Drag force (N)

F_g

Gravitational force (N)

g

Gravitational acceleration (m/s²)

Kn

Knudsen number (−)

m

Mass (kg)

N_i

Total number of droplets (−)

P

Pressure (Pa)

P_n,i

Number fraction (−)

P_V,i

Ratio of the total volume of droplets (m³)

Q_v

Volumetric airflow rate (m³/s)

R

Droplet radius (μm)

S_h

Heat source (W/m²)

T

Temperature (K)

T′

Integral time scale

T_L

Lagrangian integral time scale (s)

t

Time (s)

$\overrightarrow{u}$,$\overrightarrow{v}$

Velocity (m/s)

V

Total volume of droplets (m³)

z

Spreading distance (m)

Greek letters

ζ

Random variable subject to a normal distribution

λ_eff

Effective thermal conductivity (W/mK)

μ

Dynamic viscosity (Pa.s)

ρ

Density (kg/m³)

Δt

Time interval (s)

τ

Droplet kinematic time scale (s)

τ_eff

Effective stress tensor (N/m²)

Subscripts

a

Air

d

Droplet

i

Inlet

o

Outlet

CRediT authorship contribution statement

Alireza Bahramian: Conceptualization, methodology, Validation, Writing and Original draft.

Maryam Mohammadi: Simulation and software, Data analysis, and Investigation,

Goodarz Ahmadi: Supervision, Writing, Review and editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Editor: Pavlos Kassomenos

Appendix A. Supplementary data

Supplementary material

Data availability

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