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Keywords:

  • Droplets;
  • Droplet nuclei;
  • Dispersion/distribution;
  • Indoor environment;
  • Ventilation;
  • Computational fluid dynamics (CFD)

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Numerical modeling of indoor droplet dispersion
  5. Numerical analysis
  6. Discussion
  7. Conclusion
  8. Acknowledgements
  9. References

Abstract  This study employs a numerical model to investigate the dispersion characteristics of human exhaled droplets in ventilation rooms. The numerical model is validated by two different experiments prior to the application for the studied cases. Some typical questions on studying dispersion of human exhaled droplets indoors are reviewed and numerical study using the normalized evaporation time and normalized gravitational sedimentation time was performed to obtain the answers. It was found that modeling the transient process from a droplet to a droplet nucleus due to evaporation can be neglected when the normalized evaporation time is <0.051. When the normalized gravitational sedimentation time is <0.005, the influence of ventilation rate could be neglected. However, the influence of ventilation pattern and initial exhaled velocity on the exhaled droplets dispersion is dominant as the airflow decides the droplets dispersion significantly. Besides, the influence of temperature and relative humidity on the dispersion of droplets can be neglected for the droplet with initial diameter <200 μm; while droplet nuclei size plays an important role only for the droplets with initial diameter within the range of 10 μm–100 μm.

Practical Implications

Dispersion of human exhaled droplets indoor is a key issue when evaluating human exposure to infectious droplets. Results from detailed numerical studies in this study reveal how the evaporation of droplets, ventilation rate, airflow pattern, initial exhaled velocity, and particle component decide the droplet dispersion indoor. The detailed analysis of these main influencing factors on droplet dispersion in ventilation rooms may help to guide (1) the selection of numerical approach, e.g., if the transient process from a droplet to a droplet nucleus due to evaporation should be incorporated to study droplet dispersion, and (2) the selection of ventilation system to minimize the spread of pathogen-laden droplets in an indoor environment.


Nomenclature
c

the molar diffusion constant of water vapor (m/s)

Cc

Cunningham coefficient caused by slippage

cp

the heat capacity of discrete phase material (J/kg·K)

dp

the particle diameter (m)

dep

the equilibrium diameter of the completely evaporated particle (m)

d0

the initial diameter of droplet (m)

Dm

the diffusion coefficient for species in the mixture (m2/s)

fD

the Stoke’s drag modification function for large aerosol Reynolds number

Fp,i

the gravitational force (m/s2)

g

the acceleration of gravity (m/s2)

H

the heat transfer coefficient between the two phases (W/m2·K)

Hfg

the latent heat of discrete phase material (J/g)

J

the mass diffusion flux in turbulent flow (kg/m2·s)

L

the distance from the human mouth to the floor (m)

mp

the mass of particle (mg)

n

the ventilation rate (h−1)

N

the molar diffusion of water vapor (mol/m2)

NuAW

the Nusselt correlation at the air–water interface

Nu

the Nusselt number

psat(Tp)

the saturated vapor pressure (Pa)

pop

the operating pressure (Pa)

P

the air pressure (Pa)

Pr

the Prandtl number

R

gas constant [m2/(s2·K)]

Rep

the aerosol Reynolds number

Sct

the turbulent Schmidt number

Sc

the Schmidt number

tevaporation

the evaporation time of droplet (s)

tevaporation*

the normalized evaporation time of droplet

tlife

Whole life time of droplet (s)

ts

the gravitational sedimentation time of droplet (s)

ts*

the normalized gravitational sedimentation time of droplet

tn

the room nominal constant time (s)

Tp

the particle droplet temperature (K)

T

the local bulk temperature in the gas (K)

up,i

the averaged particle velocity in i direction (m/s)

vs

the gravitational sedimentation velocity of droplet (m/s)

V

the air velocity vector (m/s)

Xi

the local bulk mole fraction of species

Y

the local mass fraction of species

τp

the aerosol characteristic response time (s)

μt

the turbulent viscosity (kg/m·s)

ρp

the particle density (kg/m3)

ρ

the air density (kg/m3)

κeff

effective conductivity (W/m·K)

Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Numerical modeling of indoor droplet dispersion
  5. Numerical analysis
  6. Discussion
  7. Conclusion
  8. Acknowledgements
  9. References

There has been strong and sufficient evidence to demonstrate the association between ventilation and the control of airflow directions in buildings and the transmission and spread of infectious diseases such as measles, tuberculosis, chickenpox, anthrax, influenza, smallpox, and severe acute respiratory syndrome (SARS) (Li et al., 2007). Breathing, coughing, and sneezing by an infected person can generate pathogen-containing particles of saliva and mucus with diameters <10 μm (Nicas et al., 2005; Morawska, 2006). It has been believed that droplets larger than 20 μm rapidly settle onto surfaces (Gold and Nankervis, 1989), while droplets between 0.5 and 20 μm remain in the air for long periods and are more likely to be captured in the respiratory tract and produce infection (McCluskey et al., 1996). If particles carrying pathogens are inhaled by a susceptible individual and deposited in a suitable location in the respiratory tract, infectious disease may occur. The risk of airborne infection could be estimated by the product of the emission rate, intake fraction, and health risk factor. Human exposure, or intake fraction, is closely related to the infectious droplet concentrations in indoor environment (Nazaroff, 2008). On the other hand, the larger droplets may also transmit infection diseases via indirect contact when they deposit onto indoor surfaces as well as via droplet transmission. Thus the study of larger droplet dispersion is still meaningful for this study. Therefore, it is critical to investigate how the human exhaled droplets from infected human bodies transport or disperse indoors and how to control them to maintain a safe indoor environment.

To understand the dispersion characteristics of human exhaled droplets, we should understand the size and the component of these droplets first. There have been a number of studies and excellent reviews (Nicas et al., 2005; Morawska, 2006) on the size distribution of the human exhaled droplets. Some studies indicate that the vast majority of droplets generated through expiratory human activities are in the super-micrometer size (Wells, 1934; Jennison, 1942; Duguid, 1945; Loudon and Roberts, 1967). A recent study by Chao et al. (2009) reported that the expiratory droplets in close proximity to the mouths are in the super-micrometer size. Another recent study by Yang et al. (2007) reported that droplet size spans from 0.6 μm to 16 μm, with the average at 8.35 μm during coughing. Meanwhile, some other studies indicated that the majority of human exhaled droplets are within the sub-micrometer size range (Papineni and Rosenthal, 1997; Morawska et al., 2009). The discrepancy of previous study results on the size distribution of droplets is more because of the instrument and measurement methodology. To address this, a relatively wide range of droplets size is studied in this paper. In addition, there have been different opinions on defining the size of droplet nuclei, which relates to the component of the droplets. Some researchers adopted the solid matter content of 1.8% to define the size of droplet nuclei (Duguid, 1945; Chao and Wan, 2006; Wan and Chao, 2007). However, some other researchers considered that the solution of the droplets should be assumed to be 0.9% NaCl as the physiological solution (Wang et al., 2005; Xie et al., 2007). Nicas et al. (2005) summarized that respiratory droplets are composed of an aqueous solution containing inorganic and organic ions, glycoprotein and protein, and the equilibrium diameter of the completely evaporated particle (dep) is related to the initial diameter (d0) by: dep = 0.44 × d0. Wan et al. (2007) and Chao et al. (2008) adopted the droplet component summarized by Nicas et al.(2005) in their studies.

Some recent work has been conducted to assess the transport of exhaled droplet/aerosol in indoor environment. Li et al. (2004) simulated the bioaerosol dispersion in the hospital ward for analyzing the role of air distribution in SARS transmission during the largest nosocomial outbreak in Hong Kong. Zhao et al. (2005) simulated the transport of non-evaporation particles by normal respiration and coughing/sneezing in an empty mixing ventilation room with an Eulerian method. Qian et al. (2006) studied droplet nuclei’s fate under three different ventilation conditions in a two-bed hospital ward. Zhu et al. (2006) simulated the transport characteristics of saliva droplets produced by coughing with a Lagrangian model in a still indoor space with mixing ventilation. Lai and Cheng (2007) used an Eulerian model to simulate evaporated droplets (droplet nuclei) transport and settling in a similar empty mixing ventilation room. Nielsen et al. (2008) studied the contaminant flow between people under different ventilation conditions. Qian et al. (2008) performed numerical and experimental study on dispersion of exhalation pollutants in a two-bed hospital ward with a downward ventilation system. Gao et al. (2008) investigated the spatial concentration distribution and temporal evolution of exhaled and sneezed/coughed droplets within the range of 1.0–10.0 μm in an office room. Richmond-Bryant (2009) studied the characteristics of the spatial velocity and particle concentration profiles which might result in health care workers’ exposure to a pathogenic agent in an airborne infection isolation room (AIIR). All these studies mentioned neglected the droplet evaporation when doing simulation, or treated the contaminants from persons as gas species. Their results could be better convinced if they considered the droplet evaporation when carrying out the simulations.

Besides, a few studies that consider the evaporation are based on overly simplified indoor environments and influencing factors, such as ventilation rate, indoor air relative humidity, indoor air temperature, ventilation pattern, droplet nuclei size, and exhaled initial velocity. For example, Wang et al. (2005) and Xie et al. (2007) analyzed the droplets evaporation and transport via the exhaled air jets by persons in still indoor space. The effect of airflow caused by ventilation system is not incorporated in the two studies. Chao and Wan (2006), Wan and Chao (2007) studied droplets movement in different kinds of ventilation room with a Lagrangian model while considering the droplets evaporation. Wan et al. (2007) studied the dispersion of expiratory droplets in a general hospital ward with ceiling mixing type mechanical ventilation system under an ‘empty room’ configuration, which also considered the droplet evaporation. Sun et al. (2007) studied the dispersion and settling characteristics of evaporating droplets in simple ventilated room. The studied cases by Chao and Wan (2006), Wan and Chao (2007), Wan et al. (2007) and Sun et al. (2007) are relatively simple: the indoor air is isothermal without any heat sources such as human bodies, which may cause thermal plume and affect the contaminant dispersion indoor. Chao et al. (2008) studied the transport and removal characteristics of expiratory droplets in hospital ward environment, which not only considered the droplets evaporation or ventilation but also heat souses indoor. A series of work by the researchers above have inaugurated a great mode on studying the dispersion characteristics of human exhaled droplets.

Some questions may arise when reviewing above researches: Can we ignore the transient process from a droplet to a droplet nucleus due to evaporation when studying the droplets dispersion indoor? This question is particularly critical for numerical simulation of exhaled droplets dispersion, since numerical simulation is regarded as a cheap and effective approach to study indoor droplets dispersion and this method is widely used. And how do those influencing factors, such as ventilation rate, indoor air humidity, indoor air temperature, ventilation pattern, droplet nuclei size, and exhaled initial velocity, decide the dispersion of droplets in indoor environment? We believe the systemic and general analysis about the influencing factors on the droplets dispersion in ventilation room may be helpful to understand the dispersion characteristic of exhaled droplets in indoor environment. Furthermore, if the results indicate that some of these influencing factors could be neglected, it would be helpful to simplify the analyzing method on the dispersion of exhaled droplets in indoor environment. Therefore, the goal of this study is to study the dispersion characteristics of exhaled droplets in ventilated room, tending to answer the questions above. In this study, a Lagrangian model is employed to study the dispersion of droplets after validation by two different experiments (Hamey, 1982; Chao and Wan, 2006). The droplet is assumed to contain 1.8% solid matter (The equilibrium diameter of the completely evaporated particle, dep, satisfies: dep = 0.26 × d0, d0 is the initial diameter of the exhaled droplets), 0.9% NaCl (dep = 0.21 × d0) and complicated components summarized by Nicas et al. (2005) (dep = 0.44 × d0) to compare the difference caused by droplet nuclei size/composition. The main factors influencing on the droplets dispersion, including ventilation rate, ventilation pattern, relative humidity, temperature, and exhaled initial velocity, are numerically analyzed. Two normalized parameters are proposed to characterize the influence of evaporation, relative humidity, temperature, and ventilation rate on droplets dispersion.

Numerical modeling of indoor droplet dispersion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Numerical modeling of indoor droplet dispersion
  5. Numerical analysis
  6. Discussion
  7. Conclusion
  8. Acknowledgements
  9. References

Numerical model

The simulation model has two phases: the carrier phase (air) and the discrete phase (droplets and droplet nuclei). The carrier phase is modeled by Eulerian model whereas the discrete phase is modeled by Lagrangian model.

Carrier phase modeling.  The governing equations of the carrier phase (room air) are based on the Eulerian framework. The renormalization group (RNG) kɛ model (Choudhury, 1993) is used to simulate the turbulence indoor. Standard logarithmic law wall functions (Launder and Spalding, 1974) are adopted to show the connection of the solution variables at the near-wall cells and the corresponding quantities on the wall.

We conducted a grid independence test by calculating the same mode with finer grids until calculated results yielded only small changes during simulations. The tested grid densities and their relative errors for the four studied scenarios are listed in Table 1. Grid convergence index (GCI), which is based on Richardson extrapolation method (Richardson, 1910) and has been suggested by Roache (1994), is calculated to show the relative error of grid independent test.

  • image(1)

where Fs = 3, p = 2, r is the ratio of amounts of fine gird to that of coarse grid. ɛrms is defined as:

  • image(2)

where ɛi,u is defined as:

  • image(3)

u is velocity magnitude. The solutions of u at 64 points equably distributed in the room are selected both in coarse and fine grid cases. The values of GCI(u) are all <5%, which shows that the grids are fine enough.

Table 1.   Grid independent test for studied cases
Case No.Amounts of original gridsAmounts of finer grids GCI(u) [%]Case No.Amounts of original gridsAmounts of finer grids GCI(u) [%]
  1. GCI, grid convergence index.

Case1-1 (standard)90,902201,0403.0Case4-3 (standard)90,902201,0403.0
Case1-290,902201,0403.0Case5-1 (standard)90,902201,0403.0
Case2-1 (standard)90,902201,0403.0Case5-290,902201,0402.9
Case2-290,902201,0403.4Case5-390,902201,0403.1
Case2-390,902201,0404.1Case6-1 (standard)90,902201,0403.0
Case3-1 (standard)90,902201,0403.0Case6-290,902201,0403.0
Case3-2155,340321,0203.6Case6-390,902201,0403.0
Case3-3129,280263,4083.8Case7-1 (standard)90,902201,0403.0
Case3-4147,096302,9103.2Case7-290,902201,0403.0
Case4-190,902201,0403.0Case7-390,902201,0403.0
Case4-290,902201,0403.0    

Mass species (vapor) transport in the carrier phase and the mass diffusion flux in turbulent flow is respectively formulated by the following equation:

  • image(4)
  • image(5)

Where Y is the local mass fraction of species and it stands for water vapor in the air in this study. J is the mass diffusion flux in turbulent flow. SY is the source term of the species conservation equation. Dm is the diffusion coefficient for species in the mixture. μt is the turbulent viscosity. Sct is the turbulent Schmidt number. Here the value is set as 0.7.

Discrete phase modeling.  Each droplet is tracked individually in a Lagrangian frame. In this study, the only forces needed to consider are the Stokes drag and gravitational force, because the density ratio between the discrete phase and the carrier phase is in the order of O(103). The Lagrangian equations of the motion of the droplets are described as follow:

  • image(6)
  • image(7)

Fp,i is gravitational force in this study. fD is the Stoke’s drag modification function for large aerosol Reynolds number, Rep, which is defined as (Clift et al., 1978):

  • image(8)

τp is the aerosol characteristic response time, which is defined as:

  • image(9)

Cc is the Cunningham slip correction factor for small size, which is defined as (Hinds, 1999):

  • image(10)

The rate of vaporization is governed by gradient diffusion, with the flux of droplet vapor into the gas phase related to the gradient of the vapor concentration between the droplet surface and the bulk gas:

  • image(11)

Where

  • image(12)
  • image(13)
  • image(14)

The mass transfer coefficient in Equation (11) is calculated from a Nusselt correlation (Ranz and Marshall, 1952a,b). Sc is the Schmidt number. R is the universal gas constant. psat(Tp) is the saturated vapor pressure. Tp is the particle droplet temperature. Xi is the local bulk mole fraction of species, pop is the operating pressure, and T is the local bulk temperature in the gas.

The droplet temperature is modeled by the heat balance equation that relates to the sensible heat change in the droplet to the latent heat of evaporation and the convective heat transfer between the droplet and the carrier phase:

  • image(15)

Where

  • image(16)

When simulating droplet nuclei dispersion, equations (6)–(10), (15), (16) is employed. However, if droplet dispersion is simulated, we should take the evaporation effect into account. Thus equations (6)–(16) should be employed for droplet dispersion simulation.

The boundary effects on droplets at solid walls and exhaust outlet are respectively set as ‘trap’ and ‘escape’, which means the trajectory calculations are terminated at the walls and exhaust outlet. The user-defined function (UDF) is implemented into the model to quantify different droplet nuclei sizes due to different droplet components in FLUENT 6.2 (FLUENT, 2005).

Two different Lagrangian discrete phase models are used to simulate the dispersion of droplets in this study. One is Discrete Random Walk (DRW) model used to macroscopically observe the general fates of droplets. Different numbers of particle trajectories were tested in the numerical simulation to obtain reliable results. For each simulation, we increased the number of emitted particle trajectories from 5000 to 10,000. The relative errors of the tests are all <2.2%. Hence, 5000 injections were selected for all studied cases. Table 2 shows the general fates of droplets in each case. The DRW model tracks the particles with instantaneous air velocity as described by Chao and Wan (2006), Zhao et al. (2008). The other is positive track model used to microcosmically observe the trajectory of droplets. The positive track model tracks the particles with averaged air velocity as described by Zhao et al. (2004), which is convenient to compare the particle trajectories under different conditions.

Table 2.   General fates of droplets in each case
Case No.200 μm100 μm10 μm0.1 μm
Floor deposition (%)Ceiling/walls deposition (%)Escape (%)Floor deposition (%)Ceiling/walls deposition (%)Escape (%)Floor deposition (%)Ceiling/walls deposition (%)Escape (%)Floor deposition (%)Ceiling/walls deposition (%)Escape (%)
Case1-1100.00.00.068.331.10.611.375.213.511.175.213.7
Case1-2100.00.00.066.732.70.611.075.014.010.675.713.7
Case2-1100.00.00.068.331.10.611.375.213.511.175.213.7
Case2-2100.00.00.068.528.82.79.574.516.011.972.815.3
Case2-3100.00.00.068.731.30.03.480.416.22.380.717.0
Case3-1100.00.00.068.331.10.611.375.213.511.175.213.7
Case3-293.76.00.353.046.70.318.379.72.018.280.21.6
Case3-3100.00.00.042.78.748.611.153.835.17.856.435.8
Case3-4100.00.00.078.521.50.00.651.048.42.551.546.0
Case4-1100.00.00.068.830.60.611.275.514.311.475.013.6
Case4-2100.00.00.069.229.90.910.575.214.310.874.714.5
Case4-3100.00.00.068.331.10.611.375.213.511.175.213.7
Case5-1100.00.00.068.331.10.611.375.213.511.175.213.7
Case5-2100.00.00.068.231.40.411.974.913.210.875.214.0
Case5-3100.00.00.068.431.00.611.375.513.210.475.514.1
Case6-1100.00.00.068.331.10.611.375.213.511.175.213.7
Case6-2100.00.00.075.423.90.711.886.02.28.089.03.0
Case6-3100.00.00.073.425.80.811.685.62.87.888.24.0
Case7-1100.00.00.068.331.10.611.375.213.511.175.213.7
Case7-2100.00.00.063.232.84.010.877.212.011.974.913.2
Case7-3100.00.00.063.632.63.810.577.512.011.575.013.5

Validation of the numerical model

The dispersion characteristics of droplets depend not only on the effect of evaporation, but also on the effect of airflow induced by ventilation system. Thus the key points of the model validation are the droplets evaporation in indoor air and the droplets dispersion in indoor airflow. Two previous experiments are adopted to validate the numerical model. The first one was by Hamey (1982) who focused the droplets evaporation in indoor still air. The second one was by Chao and Wan (2006). They measured the droplets dispersion in ventilation room, accounting both droplets evaporation and their dispersion in indoor airflow.

Hamey (1982) studied droplets freely falling in humid, still air at the temperature of 293 K and relative humility of 70%. Figure 1 shows the comparison of predicted diameter decrease of droplet with the experimental data. As in Figure 1, the prediction agrees well with the experimental results of the diameter decrease of droplets due to evaporation.

image

Figure 1.  Comparison of predicted diameter decrease of droplets with experimental data

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The tested ventilation space by Chao and Wan (2006) is a room with dimensions of 4.8 × 4.8 × 2.6 m (Width × Length × Height), which is ventilated by a unidirectional downward flow system. The aerosol injection point is located at the center of the room, 0.8 m above the floor. Aerosols are injected vertically upward. Figure 2 shows the comparison of the predicted vertical position trajectory of droplets by DRW model with the experimental data. Overall, the prediction agrees well with the experimental results of the predicted vertical position trajectory of droplets, which depends on the interaction of droplets evaporation and airflow.

image

Figure 2.  Comparison of the predicted vertical position trajectory of droplets with the previous experimental data. The lines are simulation results

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The dispersion characteristics of droplets in ventilation room depend on the interaction of evaporation and airflow. We believe that this is similar for the experimental cases and the studied cases below. Thus, the numerical model could be used to analyze the cases in the following sections.

Numerical analysis

  1. Top of page
  2. Abstract
  3. Introduction
  4. Numerical modeling of indoor droplet dispersion
  5. Numerical analysis
  6. Discussion
  7. Conclusion
  8. Acknowledgements
  9. References

Case description

A ‘standard’ case is modeled first followed by varying the influencing parameters to compare the different conditions on the dispersion of droplets. Figure 3 shows the sketch of the ‘standard’ case. It is a room with dimension of 5 × 4 × 3 m (Lenth × Width ×Height), which is ventilated by a typical ceiling supply and ceiling return ventilation system. One person with the only heat and mass (vapor and droplets) sources is simulated. In the ‘standard’ case, ventilation rate, supply air temperature, supply air relative humidity and exhaled initial velocity are set as 5 ACH (air change per hour, h−1), 288 K, 88% and 1 m/s, respectively. The droplet initial temperature is 307 K (human body temperature). Droplet nuclei size is defined by dep = 0.44 × d0, according to the result of Nicas et al. (2005). The supply air concentration of droplets is set as zero. The human heat and moisture generating rate are set as 75 W and 96 g/h. Table 3 shows the studied cases with varying influencing factors.

image

Figure 3.  The sketch of ‘standard’ case

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Table 3.   Studied cases Thumbnail image of Thumbnail image of

According to the previous results about the exhaled droplet size (Wells, 1934; Jennison, 1942; Duguid, 1945; Loudon and Roberts, 1967; Papineni and Rosenthal, 1997; Nicas et al., 2005; Morawska, 2006; Chao et al., 2009), the initial diameters of droplets are set as 0.1 μm, 1 μm, 10 μm, 50 μm, 100 μm and 200 μm. In this study, the influence of exhaled jet by breathing on the background velocity field is neglected, because the momentum of exhaled air by breathing is relatively small. However, the influence of exhaled jet by coughing and sneezing on the background velocity field could not be ignored due to the large momentum of exhaled air by coughing and sneezing. Therefore, the unsteady flow caused by exhaled air jet is considered when the exhaled modes are cough and sneeze. For simplification, both cough and sneeze flows are assumed to be exhaled from the human mouth.

Results analysis

Can we ignore the transient process from a droplet to a droplet nucleus due to evaporation?  Two different methods to treat the exhaled droplets are compared. One considers the evaporation and the other only considers the droplet nuclei without evaporation. The diameters of corresponding droplet nuclei are 0.044 μm, 0.44 μm, 4.4 μm, 22 μm, 44 μm, and 88 μm, respectively, for the studied initial diameters of droplets. Table 2 indicates that the general fates of droplets calculated by these two methods are similar.

To further investigate the dispersion characteristics of droplets, the specific trajectory of each droplet is observed. Figure 4 shows the comparison of droplet trajectory calculated by considering droplet evaporation with that calculated without evaporation. The droplets with the initial diameter of 0.1 μm, 1 μm, and 10 μm move upward with plume flow at the beginning and finally escape from the exhaust outlet. The droplets with the initial diameter of 50 μm and 100 μm also move upward with plume flow at the beginning but finally settle onto the floor. The 200 μm droplets fall down immediately and finally settle onto the floor. The trajectories are very similar when the initial diameter of droplet is 0.1 μm, 1 μm, 10 μm, and 50 μm. When the initial diameter of droplet is 100 μm, the two trajectories are also similar as a whole except at the very beginning. When the initial diameter of droplet is 200 μm, the trajectories are quite dissimilarity. The droplets move a longer distance than the nuclei do, which results in the difference of the trajectories.

image

Figure 4.  Comparison of trajectories of droplets calculated by considering droplet evaporation with that calculated without evaporation. (a) initial diameter 0.1 μm, (b) initial diameter 1 μm, (c) initial diameter 10 μm, (d) initial diameter 50 μm, (e) initial diameter 100 μm, (f) initial diameter 200 μm. z = 2 m

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Here we define the evaporation time, tevaporation, as the time period since the droplets are exhaled till completely evaporation, to characterize the influence of evaporation on droplets dispersion. Figure 5 presents the evaporation time of droplets with different initial diameters. When the initial diameter of droplet is <50 μm, the evaporation time is below 1 s, which is too short to influence the movement of these droplets. The influence of evaporation of 200 μm droplet is stronger than 100 μm droplet, but its evaporation time is shorter. The reason is that the evaporation is kept during the whole ‘life’ of 200 μm droplets. Because 200 μm droplets settle very fast, the evaporation time or their whole ‘life’ is very short. Thus the normalized evaporation time is defined as:

  • image(17)

where tlife is the average longevity of the droplets tracked in DRW model calculation (5000 droplets as mentioned above). The longevity is defined as the time since droplet is exhaled till being trapped or escaped.

image

Figure 5.  The evaporation time of droplets with different diameters

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inline image is ≤1. Figure 6 shows the normalized evaporation time of droplets with different diameters. In these cases, the normalized evaporation time increases with the increase of the droplet initial diameter. It could be found from the results of trajectories that the results calculated by considering droplets evaporation and that calculated without evaporation are quite similar when the initial diameter is less than or equal to 100 μm. The normalized evaporation time is 0.051 when the initial diameter is 100 μm. Therefore, the transient process from a droplet to a droplet nucleus due to evaporation can be neglected in the modeling when the normalized evaporation time is <0.051.

image

Figure 6.  The normalized evaporation time of droplets with different diameters

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How the ventilation rate affects the dispersion of droplets?  The ventilation rates are set as 5, 10, and 15 ACH respectively based on the ‘standard’ case. The initial diameters of droplets are respectively 0.1 μm, 10 μm, 100 μm, and 200 μm. Table 2 shows that the general fates of droplets with initial diameter of 200 μm at different ventilation rates are relatively similar. However, the general fates of droplets at different ventilation rates differ obviously when the initial diameter is 0.1 μm, 10 μm, and 100 μm. This implies that the influence of ventilation rate might play an important role for droplets dispersion when the initial diameter is <100 μm.

Figure 7 further shows the comparison of droplet trajectories at different ventilation rates. The trajectories are similar when the initial diameter of droplet is 200 μm as the gravitational sedimentation plays the most important role on this size of droplets dispersion. However, when the initial diameter of droplet is 0.1 μm, 1 μm, and 100 μm, the trajectories at different ventilation rates are quite dissimilarity. It seems that the influence degree of ventilation rate relates to gravitational sedimentation, which is related to the size of droplet. Therefore, we defined the normalized gravitational sedimentation time to characterize the influence of ventilation rate:

  • image(18)

where ts is the gravitational sedimentation time, which is defined as:

  • image(19)

where L is the distance from the human mouth/nose to the floor. vs is gravitational sedimentation velocity, which is calculated by (Hinds, 1999):

  • image(20)

where d0 is the initial diameter of droplet. tn is the room nominal constant time, which characterize the ventilation rate. It is defined as:

  • image(21)

where n is ventilation rate.

image

Figure 7.  Comparison of trajectories of droplets at different ventilation rates. (a) initial diameter 0.1 μm, (b) initial diameter 10 μm, (c) initial diameter 100 μm, (d) initial diameter 200 μm. z = 2 m

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Figure 8 shows the normalized gravitational sedimentation time of droplets with different diameters. The trajectories results show that the dispersion of 200 μm droplet, where the normalized gravitational sedimentation time is 0.005, would not be influenced by ventilation rates. If the initial diameter of droplet is larger than 200 μm which would also not be influenced by ventilation rates, the normalized gravitational sedimentation time would be smaller. Therefore, in these cases, when the normalized gravitational sedimentation time is <0.005, the influence of ventilation rate on the dispersion of droplets could be neglected.

image

Figure 8.  The normalized gravitational sedimentation time of droplets with different diameters

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How ventilation pattern affects the dispersion of droplets?  Four different ventilation patterns representing the most commonly used ventilation modes in practices are selected to study the influence of airflow pattern on droplet dispersion. Figure 9 shows the air velocity vector plot at z = 2 m of the four different ventilation patterns. Table 2 shows that the general fates of droplets at different ventilation patterns are very different for all the studied droplets sizes. Figure 10 further shows the comparison of droplet trajectories at different ventilation patterns. When the initial diameter is in the range of 0.1 μm to 200 μm, the trajectories at different ventilation patterns are quite dissimilarity, because the influence of airflow occupies the leading function.

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Figure 9.  Air velocity vector plot at z = 2 m of the four different ventilation patterns. (a) Case 3-1: ceiling supply ceiling return, (b) Case 3-2: down-side supply ceiling return, (c) Case 3-3: up-side supply down-side return (d) Case 3-4: under floor supply ceiling return

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Figure 10.  Comparison of trajectories of droplets at different ventilation patterns. (a) initial diameter 0.1 μm, (b) initial diameter 10 μm, (c) initial diameter 100 μm, (d) initial diameter 200 μm. z = 2 m

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How the relative humidity affects the dispersion of droplet?  The supply air relative humidity is respectively set as 18, 44 and 88% based on the standard case. The selected values of the relative humidity represent the common cases found in actual ventilation or air conditioning rooms. Relative humidity could influence the evaporation of droplet since relative humidity has an impact on the local bulk mole fraction of species (vapor), Xi, in equation 14. Equations 11–14 illustrate that the relative humidity affects the droplets evaporation directly. Table 2 shows that the general fates of droplets at different supply air relative humidity are relatively similar. Also, the droplet trajectories at different relative humidity are very similar for all the studied particle sizes. Due to the length limitation of the article, the figures of the comparison of droplet trajectories at different supply air relative humidity are not listed. The results imply that the influence of relative humidity could be ignored. Figure 11 presents the normalized evaporation time of droplets with different diameters. The results tell that the influence of evaporation could be neglected when the initial diameter is ≤100 μm, as the evaporation time is short enough (shorter than 0.051). However, the 200 μm droplets fall down immediately, which means that the influence of gravitational sedimentation occupies the leading function. Thus the droplet trajectories are similar as the influence of gravitational sedimentation makes them all fall down. Therefore, the relative humidity did not influence the dispersion of droplets with initial diameter of 200 μm. On the whole, the influence of relative humidity on the dispersion of droplets could be neglected.

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Figure 11.  Normalized evaporation time of droplets with different diameters at different supply air relative humidity

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How the temperature affects the dispersion of droplet?  The supply air temperature is respectively set as 288 K, 293 K and 298 K based on the standard case. Table 2 shows that the general fates of droplets at different supply air temperatures are relatively similar. The droplet trajectories at different temperatures are also highly similar for all the studied droplets sizes. Due to the length limitation of the article, the figures of the comparison of droplet trajectories at different supply air temperature are also not listed. Similar with relative humidity, temperature affects the droplet evaporating directly. Figure 12 presents the normalized evaporation time of droplets with different diameters. The results indicate that the influence of evaporation could be neglected when the initial diameter is ≤100 μm due to the very short evaporation time. Similar with the analysis of humidity effect, temperature did not influence the dispersion of droplets with initial diameter of 200 μm as the gravitational sedimentation makes them all fall down. Thus the droplet trajectories of 200 μm droplets are similar. On the whole, the influence of supply air temperature on the dispersion of droplets could be neglected.

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Figure 12.  Normalized evaporation time of droplets with different diameters at different air temperatures

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How the initial exhaled velocity affects the dispersion of droplet?  The exhaled initial air and droplet velocity is changed on the ‘standard’ case for comparison. Three exhaling modes, breath (1 m/s), cough (10 m/s) and sneeze (35 m/s), are studied. Basing on the calculated airflow field causing by ventilation, the mouth was set as an opening with an initial velocity of 10 m/s (cough) or 35 m/s (sneeze). The duration time of cough and sneeze behavior is assumed as 0.1 s, which is mimicked as cough or sneeze once. Under this boundary condition, unsteady airflow calculation was processed. We modeled the exhaled velocity using a puff of exhaled jet with the designated velocity instead of by setting an initial velocity on the droplets when we for ‘cough/sneeze’ cases. While, when we modeling ‘breath’ case, the impact of exhaled jet by breathing on the background velocity field is neglected, because the momentum of exhaled air by breathing is relatively small. The results in Table 2 tell that the initial exhaled velocity has great influence on the dispersion of droplets. The breath process has little effect on the airflow. However, the cough and sneeze behavior may greatly influence the airflow and the dispersion of droplets. Figure 13 shows the comparison of trajectories at different initial exhaled velocity. The droplets move longer with the increase of exhaled velocity in the duration time of expiration behavior, which makes the droplet trajectories differ with each other. Therefore, on the whole, initial exhaled velocity has great influence on the dispersion of droplets in the range of 0.1 μm to 200 μm, which could not be neglected.

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Figure 13.  Comparison of trajectories of droplets at different initial exhaled velocities. (a) initial diameter 0.1 μm, (b) initial diameter 10 μm, (c) initial diameter 100 μm, (d) initial diameter 200 μm. z = 2 m

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How the droplet nuclei size affects the dispersion of droplet?  The nuclei size is decided by droplet composition. Here the droplets are assumed to contain 1.8% solid matter (dep = 0.26 × d0), 0.9% NaCl (dep = 0.21 × d0) and complicated components summarized by Nicas et al. (2005) (dep = 0.44 × d0) to compare the difference caused by droplet nuclei size. Table 2 shows that the general fates of droplets at different droplet nuclei sizes are similar when the initial diameter of droplets is 0.1 μm and 200 μm. However, when the initial diameter is 10 μm and 100 μm, there is a little dissimilarity. Figure 14 shows the comparison of droplet trajectories at different droplet nuclei sizes. The droplet trajectories at different droplet nuclei sizes are highly similar when the droplets initial diameter is 0.1 μm and 200 μm. The study by Zhao et al. (2009) suggests that the dispersion characteristics of aerosol particles are highly similar when the diameter is in the range of 0.01 μm to 0.1 μm. So the influence of droplet nuclei size on the dispersion of droplets could be neglected when the initial diameter is 0.1 μm. When the initial diameter is 200 μm, the influence of gravitational sedimentation occupies the leading effect. So the influence of droplet nuclei size on the dispersion of droplets could also be neglected when the initial diameter is 200 μm. However, when the initial diameter is 10 μm and 100 μm, the influence of droplet nuclei size is obvious. In this range, the droplet size is a key factor on the droplet dispersion, because the influence of airflow highly relates to the droplet size, e.g., the drag force is very different and this force is the dominant force on particles deciding particle dispersion. Thus it is necessary to quantify the droplet component for the droplet dispersion study as the component decides the nuclei size.

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Figure 14.  Comparison of trajectories of droplets at different droplet nuclei sizes. (a) initial diameter 0.1 μm, (b) initial diameter 10 μm, (c) initial diameter 100 μm, (d) initial diameter 200 μm. z = 2 m

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Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Numerical modeling of indoor droplet dispersion
  5. Numerical analysis
  6. Discussion
  7. Conclusion
  8. Acknowledgements
  9. References

The current study only considers one person for the heat and water vapor sources in all simulation cases. To further avoid the complexity of room geometry, no furniture is incorporated in this study cases. These factors would affect the indoor airflow, temperature and humidity distribution. The droplet dispersion is decided by airflow (correlative to ventilation rate, ventilation pattern and initial exhaled velocity), droplet evaporation (correlative to indoor air temperature, humidity, initial diameter and composition of droplets) and droplets gravitational sedimentation (correlative to droplets inertia and evaporating time scale). The number of studied cases is limited; however, the normalized evaporation time and the normalized gravitational sedimentation time are defined trying to reflect the certain physical characteristics of the interaction of these influencing factors on droplet dispersion. The normalized evaporation time scale of 0.051 and the normalized gravitational sedimentation time scale of 0.005 are not exact ‘cut-off’ points. To obtain the exact ‘cut-off’ points need recalculation of all the diameters between 0.1 and 200 μm. It is worth for a further investigation in a future study. Nevertheless, it is much easier to perform more cases in similar way in this study.

The shape of human body in these cases has some unlikeness with the real human body. According to the study by Nielsen (2004), the influence of the shape of human body on indoor airflow could be neglected except those studies focus on local airflow around human bodies. How the shape of human body affects the droplets dispersion near/around the human bodies might be another topic for further study.

It should be noticed that when we consider the influence of initial exhaled velocity, the differences of droplets dispersion could depend on the dimension of the room. It could be expected that, in a larger room, the differences of dispersion and deposition under different initial exhaled velocities would be more significant. This factor may be studied in the similar way presented in this study.

The current study focuses on the dispersion characteristics of droplets. In addition, the control strategy on the human exposure to the droplets is quite important in practical engineering. Further efforts are deserved on this interesting and important topic.

Conclusion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Numerical modeling of indoor droplet dispersion
  5. Numerical analysis
  6. Discussion
  7. Conclusion
  8. Acknowledgements
  9. References

This study employed a validated numerical model to investigate some questions about the dispersion of exhaled droplets in indoor environment. Within the scope of this research, the following conclusions can be drawn:

  • 1
     When the normalized evaporation time is <0.051, modeling the transient process from a droplet to a droplet nucleus due to evaporation could be neglected in the modeling.
  • 2
     When the normalized gravitational sedimentation time is <0.005, the influence of ventilation rate on the dispersion of droplets could be neglected.
  • 3
     For the droplet with initial diameter in the range of 0.1 μm to 200 μm, the influence degree of ventilation pattern on the dispersion of droplets is very high, because the influence of airflow occupies the leading function.
  • 4
     The influence of temperature and relative humidity on the dispersion of droplets could be neglected for the droplet with initial diameter in the range of 0.1 μm to 200 μm.
  • 5
     Initial exhaled velocity has great influence on the dispersion of droplet for the droplet with initial diameter in the range of 0.1 μm to 200 μm.
  • 6
    For the droplet with initial diameter in the range of 10 μm to 100 μm, the influence of droplet nuclei size is highly obvious.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Numerical modeling of indoor droplet dispersion
  5. Numerical analysis
  6. Discussion
  7. Conclusion
  8. Acknowledgements
  9. References

This study is supported by National Key Technology R&D Program of China (No. 2006BAJ02A10).

References

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  2. Abstract
  3. Introduction
  4. Numerical modeling of indoor droplet dispersion
  5. Numerical analysis
  6. Discussion
  7. Conclusion
  8. Acknowledgements
  9. References
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