# Selecting the Safe Area and Finding Proper Ventilation in the Spread of the COVID-19 Virus

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Test Cases

^{2}ventilation supply and exhaust registers are also shown in the figure. The height of the mouth/nose through which air is inhaled or exhaled is 1.65 m.

#### 2.2. Numerical Model

#### 2.2.1. Boundary Conditions and Simulation

#### 2.2.2. Grid Independence

#### 2.2.3. Results Validation

^{−11}kg/s) expelled by the coughing. The evaporation model is presented in Figure 5. It is assumed that the droplets leave the mouth at a temperature of 37 °C, and the room temperature is 17 °C. Figure 5 shows a good agreement between the present predictions and the results of Li et al. [3].

#### 2.2.4. Step Independence Study

_{Avg}and F

_{Avg}as a function of time, are evaluated in Table 4 and Table 5, respectively. According to Table 4 and Table 5, as the time step gets smaller, the differences between the particle properties becomes negligible. As the values of the properties for the last time steps are very close, the time step of 0.005 is selected for the subsequent simulations.

#### 2.3. Modeling Assumptions

- The droplets stick to surfaces upon contact with no resuspension.
- The evaporation of cough droplets is included in the simulations.
- The mass and the molar fractions of water vapor at the droplet surface assume liquid-vapor equilibrium [18].
- Droplet collisions in dilute concentrations are neglected.
- Cough droplets have a uniform temperature of 37◦C and a spherical shape [19].
- The respiratory droplet with SARS-CoV-2 viruses behaves like water in terms of physical properties [1].
- The number of droplets released when coughing is proportional to the velocity of the airflow [20].
- The heat transfer effects between the surrounding environment and the people’s bodies are ignored.

#### 2.4. Governing Equations

_{C}is the Cunningham coefficient [25,26]. $\dot{\mathrm{m}}$ and ${\mathrm{N}}_{\mathrm{Tn}}$ are the mass flow rate, and the residual cough particles and droplets in space, respectively.

V = 1.93 × 103 t2 + 30.5 × 10 t | 0 < t ≤ 0.077 | (15) |

V = 2.68 × 102 t2 − 1.26 × 10 t + 1.99 × 10 | 0.077 < t ≤ 0.265 | |

V = 4.10 × 102 t2 − 3.14 × 102 t + 5.98 × 10 | 0.265 < t ≤ 0.35 |

^{−5}percent). Furthermore, we assumed cough droplets with a uniform temperature of 37 °C and a spherical shape. The droplets coming out of the mouth during the cough have different diameters, from a few micrometers to hundreds of micrometers. In addition, the velocity of the droplets leaving the mouth is equal to the speed of the cough air flow.

#### Wells–Riley Equation and Risk of Infection

_{0}, $v\left(x,t\right)$ is the average droplet concentration, f (t) is the virus viability (usually taken as 1), and c is the density of the number of quanta in droplets or concentration in the respiratory fluid, and p is the breathing rate.

#### 2.5. Modeling Procedure

## 3. Results

_{Tn}). When the ventilation air flow is along the length of the indoor space (Case 1), it takes 115 s to clear the area of the droplets. However, when there is a ventilation air flow across the width of the indoor space (air curtain), it takes 75 s for the droplets to leave the area (Case 2).

## 4. Conclusions

- In Case 1, it takes 115 s to clear the area of the droplets compared to 75 s in Case 2.
- In Case 1, the droplets spread all over the room after a while, but, in Case 2, they remain only in a part of the space for all the time, so in Case 2, there is always a safe area with a low risk of infection of Covid-19 virus
- When curtain-type air flow patterns are formed in the room, the infection risk of people at the other end of the room decreases significantly. For example, the risk in area A10 decreases from 48% in Case 1 to 0.05% in Case 2, and in area A9 the infection risk is reduced from 30% in Case 1 to 0.05% in Case 2.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | Surface area (m^{2}) |

A_{C} | Average droplet concentration (Kg/m^{3}) |

A_{i} | i-th area (i = 1,2,..,8) |

A_{N} | Number of particles |

C_{Avg} | Volume—average concentration |

C_{C} | Cunningham coefficient (-) |

Cp | Density of the number of quanta in droplets (-) |

D_{d} | Droplet diameter (μm) |

F_{Avg} | Volume—average volume fraction |

F_{B} | Brownian force (-) |

F_{TH} | Thermophoretic force (-) |

F_{L} | Saffman lift force (-) |

f(t) | Virus viability |

G | Gravity (m/s^{2}) |

I | Number of infectors (-) |

I_{I} | Ejection time(s) |

K | Turbulence kinetic energy (J/kg) |

K_{T} | Fluid thermal conductivity (W/m K) |

m_{p} | Mass flow rate (kg/s) |

N | Number of droplets (-) |

N_{T} | Total number of droplets (-) |

N_{t} | Droplets at given times (-) |

N_{Tn} | Droplets remaining in the space (-) |

${N}_{m}$ | Number of mesh elements (-) |

Ns | Total number of quanta (-) |

P | Pressure (Pa) |

P_{Avg} | Average pressure in the domain under consideration (Pa) |

PRT | Particle residence time(s) |

P_{op} | Operating pressure (pa) |

Q | Air supply rate (m^{3}/s) |

Q | Generation rate of quanta (quanta/s) |

RH | Relative humidity (-) |

P_{rt} | Particle removal time(s) |

S | Number of susceptible persons (-) |

Sc | Schmidt number (-) |

S_{T} | Source term (C) |

T | Time (s) |

T | Temperature conditions (C) |

T_{Avg} | Volume average temperature in domain solving (C) |

u, v, w | Velocity components (m/s) |

G_{k}, S_{ε}, S_{k}, α_{k}, α_{ε}, R_{ε}, ${C}_{1\epsilon}$, ${C}_{2\epsilon}$ | Turbulent kinetic energy and modelconstants (-) |

V | Velocity vector (m/s) |

V_{Avg} | Average velocity in the flow domain under consideration (m/s) |

V_{v} | Ventilation velocity (m/s) |

W | Water mass fraction (Kg/mol) |

Greek symbols | |

E | Rate of dissipation of turbulence kinetic energy |

$H$ | Particle removal efficiency (-) |

M | Kinetic viscosity (Pa s) |

P | Breathing rate per person (m^{3}/s) |

ρ_{a} | Density (kg/m^{3}) |

${\rho}_{d}$ | Droplet density |

$v\left(x,t\right)$ | Droplet concentration |

Subscripts | |

Sat | Saturation (-) |

T | Total (-) |

x, y, z | Cartesian directions (-) |

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**Figure 2.**Diagrams of the ventilation air flow patterns in the room with the inlet and outlet registers on the side walls (Case 2); the mannequins are in the same position as in Figure 1.

**Figure 5.**Validation of the present computational results for an evaporating 10 µm droplet by comparison with the data from [3].

**Figure 12.**Particle residence time in the indoor space at different times with two ventilation systems.

**Figure 13.**Time-dependent changes in the fraction of droplets remaining in the air (N

_{Tn}) in the area for Case 1 and Case 2.

**Figure 14.**Infection risk of individuals in areas A1 to A15 for Case 1 and Case 2 ventilation systems in the time interval from 0 to 115s when an infected person is coughing in area C.

Surface | Variables | Boundary Value | W (Kg/mol) | T(°C) |
---|---|---|---|---|

Ventilation inlet | Velocity inlet | 5 m/s | 0.00726 | 17 °C |

Ventilation outlet | Pressure | 105 KPa | - | 30 °C |

Person’s mouth | Cough velocity | UDF | 0.03534 | 37 °C |

D_{d} (μm) | m_{p} (kg/s) | A_{N} | I_{I} (s) |
---|---|---|---|

0.15 | 4.24 × 10^{−15} | 1800 | 0.35 |

1 | 1.25 × 10^{−12} | 1800 | 0.35 |

10 | 1.25 × 10^{−9} | 1800 | 0.35 |

50 | 1.57 × 10^{−7} | 1800 | 0.35 |

100 | 1.25 × 10^{−6} | 1800 | 0.35 |

150 | 4.24 × 10^{−6} | 1800 | 0.35 |

Grid 1-2 | Grid 2-3 | |||||
---|---|---|---|---|---|---|

1-Fine, 2-Medium | 2-Medium, 3-Coarse | |||||

p | ε_{2,1} (%) | GCI_{2,1}(%) | ε_{3,2} (%) | GCI_{3,2} (%) | ||

Average pressure in the domain | 3 | 3.93 | 0.49 | 0.87 | 1.99 | 7 |

C_{Avg} in Simulation Time (s) | |||||
---|---|---|---|---|---|

0.4 | 0.6 | 0.9 | 1.5 | ||

Time step (s) | 0.009 | 1.312846 × 10^{−7} | 1.743575 × 10^{−7} | 1.7442261 × 10^{−7} | 1.738135 × 10^{−7} |

0.007 | 1.312863 × 10^{−7} | 1.743583 × 10^{−7} | 1.7442299 × 10^{−7} | 1.740361 × 10^{−7} | |

0.005 | 1.312866 × 10^{−7} | 1.743589 × 10^{−7} | 1.7442302 × 10^{−7} | 1.740365 × 10^{−7} | |

0.003 | 1.312867 × 10^{−7} | 1.743590 × 10^{−7} | 1.7442302 × 10^{−7} | 1.740366 × 10^{−7} |

F_{Avg} in Simulation Time (s) | |||||
---|---|---|---|---|---|

0.4 | 0.6 | 0.9 | 1.5 | ||

Time step (s) | 0.009 | 1.364348 × 10^{−10} | 1.765463 × 10^{−10} | 1.765554 × 10^{−10} | 1.759675 × 10^{−10} |

0.007 | 1.364375 × 10^{−10} | 1.765481 × 10^{−10} | 1.765603 × 10^{−10} | 1.760236 × 10^{−10} | |

0.005 | 1.364387 × 10^{−10} | 1.765491 × 10^{−10} | 1.765611 × 10^{−10} | 1.760238 × 10^{−10} | |

0.003 | 1.364389 × 10^{−10} | 1.765492 × 10^{−10} | 1.765614 × 10^{−10} | 1.760239 × 10^{−10} |

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## Share and Cite

**MDPI and ACS Style**

Karami, S.; Lakzian, E.; Shabani, S.; Dykas, S.; Salmani, F.; Lee, B.J.; Warkiani, M.E.; Kim, H.D.; Ahmadi, G.
Selecting the Safe Area and Finding Proper Ventilation in the Spread of the COVID-19 Virus. *Energies* **2023**, *16*, 1672.
https://doi.org/10.3390/en16041672

**AMA Style**

Karami S, Lakzian E, Shabani S, Dykas S, Salmani F, Lee BJ, Warkiani ME, Kim HD, Ahmadi G.
Selecting the Safe Area and Finding Proper Ventilation in the Spread of the COVID-19 Virus. *Energies*. 2023; 16(4):1672.
https://doi.org/10.3390/en16041672

**Chicago/Turabian Style**

Karami, Shahram, Esmail Lakzian, Sima Shabani, Sławomir Dykas, Fahime Salmani, Bok Jik Lee, Majid Ebrahimi Warkiani, Heuy Dong Kim, and Goodarz Ahmadi.
2023. "Selecting the Safe Area and Finding Proper Ventilation in the Spread of the COVID-19 Virus" *Energies* 16, no. 4: 1672.
https://doi.org/10.3390/en16041672