# A CFD Approach for Risk Assessment Based on Airborne Pathogen Transmission

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Eulerian CFD Model

#### 2.2. Lagrangian Discrete Phase Model

#### 2.3. Particle Mass Balance

#### 2.4. Droplet Evaporation

#### 2.5. Particle Energy Balance

#### 2.6. CFD Model Setup

^{−1}, while droplets with a diameter of 100 μm regimented at much smaller distances of about 1 m from the jet inlet. The size of the final domain together with the velocity contour of the simulated air jet is presented in Figure 2b.

^{−3}and a specific heat transfer of 2404.6 J·Kg

^{−1}·K

^{−1}at the standard state temperature of 298.15 K. On the contrary, the evaporative fraction was assumed to have material properties equal to those of water (density of 997.6 kg·m

^{−3}, specific heat transfer of 4181.7 J·Kg

^{−1}·K

^{−1}at the same standard state temperature). In addition, the saturation pressure of the evaporative fraction was set to 3170.3 Pa. Having assumed the multi-component droplets, the mass-weighted mixture was used for the calculation of the density and specific heat of each droplet. In addition, regarding the boundary condition for each droplet’s outer surface, it was assumed that the droplets would stick to any wall surface of the room as they reached them. The Lagrangian model also included the Schiller–Naumann drag force coefficients and pressure gradient force to accurately simulate the droplets’ trajectories.

^{−5}Pa·s, 1003.6 J·Kg

^{−1}·K

^{−1}, and 28.97 Kg·Kmol

^{−1}, respectively. Like the Lagrangian model, the weighted mixture method for the Eulerian model was employed for the calculation of the air–water mixture. With all the previous settings, the cases were solved on a computer cluster at Sogang University using 24 computational cores with Xeon(R) 2.20 GHz CPUs. The typical simulation time for 60 s was about 15 h.

_{0}> 20 from the mouth) was investigated, and the normalized results were validated against the experimental results [36]. The inlet velocity had spanwise (along with discharge hole radii) as well as streamwise (centerline) velocity profiles with the maximum value of 20 m·s

^{−1}. It should be noted that the validation case of the Eulerian model was in the isothermal condition. Different mesh resolutions with hexahedral cells were investigated, ranging from 189 k cells to 4.5 M cells, as shown in Figure 2a. The optimal mesh, 189k-HYBcase, has minimum and maximum cell sizes of 0.06 and 0.2 m, respectively, while its surface growth rate is 2.0. This results in a dense mesh of between 0.78 mm (minimum) near the mouth and 0.8 m (maximum) at the central part of the domain. To ensure the accuracy of the results near the walls, the “All Yplus” option was activated, enabling an automatic blending function between low and high Reynolds number wall treatment methods for the calculation of turbulence qualities, such as dissipation, production, and stress tensor [37]. The summary of the applied boundary conditions to validate the Eulerian model is presented in Table 1.

_{i}is calculated as follows:

_{i}and P

_{i}are the measured and computed values of a given variable, respectively; n is the number of available data points, and W and D are the relative uncertainty and repeatability of the calculated data and their values for velocity assessments are 0.05 and 0.25, respectively. The angular brackets of the FB and NMSE equations represent the average of all measured points [33,34]. Table 2 shows the results of the validation metrics calculated at the centerline velocity diagrams of Figure 3.

## 3. Risk Assessment Method (RAM)

_{t}), as is shown in Figure 4. The planes are then divided into multiple facial cells. In the next step, the algorithm counts the number of droplets passing through each cell at each time step. According to Figure 4, calculation of the accumulated number of droplets in the field needs a separate 3D meshing with a coarser resolution than that of the CFD model, which is called a “secondary mesh”. Accordingly, the ∆L and ∆Z shown in Figure 4 present the resolution of the secondary mesh.

_{1}, the proposed algorithm calculates the accumulated number of droplets passing through each of the facial cells within the secondary mesh. As shown in Figure 4, the droplets that pass the plane ${Z}_{t}$ at ${t}_{i}$ (while not having reached it at ${t}_{i-1}$) should be identified and counted for each vertical plane of the secondary mesh. Here,${t}_{i}$ and ${t}_{i-1}$ represent two consecutive times at which the CFD transient data are collected. For this purpose, the droplets’ positions at the previous time step are approximated using the following equation:

_{t}at each time step to see if it passes the plane Z

_{t}within dt. If yes, Equation (17) is used to estimate the time required for that (i.e., dt*). Then, the calculated dt* is applied to calculate the in-plane positions of droplets (X

_{t}and Y

_{t}) using Equations (18) and (19).

## 4. Case Studies

## 5. Results

#### 5.1. Grid Independence of Risk Assessment Model

#### 5.2. Spreading Pattern of Respiratory Jets

#### 5.3. Safe Distance Calculations

#### 5.4. Instantaneous and Accumulative Concentrations

#### 5.5. Time History of Spreading Patterns

#### 5.6. Spreading Pattern Cross-Sections

#### 5.7. Sensitivity of Spreading to Environmental Conditions

## 6. Concluding Remarks

- The obtained results indicate that the proposed RAM method can successfully capture different respiratory events. Hence, the length and height of spread as well as the overall behavior of different exhalation jets can be studied using the proposed RAM.
- While the instantaneous CFD output data of droplets can barely provide any information regarding risky and safe zones inside a domain, the proposed model can represent the evolution of risky zones in time. Comparison of the instantaneous and accumulated droplets indicated that most of the heavy falling particles do not pass through the Z-constant planes through the field, which is used in the accumulated representation. Since these particles do not contribute to the dispersion of droplets in the domain, this adds no error to the calculated spreading length and height of airborne droplets.
- The proposed RAM can be used to derive safe social distances in terms of exposure time. This is especially important in short-duration respiratory activities such as coughing or sneezing in which considerable viral loads are released by an infected person and the safe distance for susceptible persons depends on their exposure time.
- The calculated results confirm that the sensitivity of the predicted spreading patterns to environmental temperature and relative humidity can be investigated by the proposed RAM. According to selected sample cases, both parameters can influence the propagation length and even the mechanism by which droplets are transmitted inside the environment.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Parameter | Unit | Description |

C_{p} | J·(kg·K)^{−1} | Heat capacity |

F | N | Force |

d | m | Diameter |

g | m·s^{−2} | Gravity |

u | m·s^{−1} | Velocity |

P | Pa | Pressure |

T | K | Temperature |

t | s | Time |

Re | -- | Reynolds number |

Greek | ||

µ | Pa·s | Dynamic viscosity |

ρ | Kg·m^{−3} | Density |

ε | -- | Fractional mass transfer |

subscript | ||

p | Particle | |

D | Drag | |

Abbreviations | ||

NHS | National Health Service | |

COVID-19 | Coronavirus disease 2019 | |

BMA | British Medical Association | |

CFD | Computational fluid dynamics | |

HVAC | Heating, ventilation, and air conditioning | |

ADP | Airborne pathogen droplet |

## Appendix A

**Figure A1.**The impact of ∆L size on the accumulated number of droplets in (

**a**) a cough respiratory event (velocity = 8 m·s

^{−1}, temperature = 15 °C, RH = 50%), and (

**b**) in a sneeze respiratory event (velocity = 50 m·s

^{−1}, temperature = 29 °C, RH = 50%).

**Figure A2.**The impact of ∆Z size on the distribution of the accumulated number of droplets in (

**a**) a cough with velocity = 8 m·s

^{−1}, temperature = 15 °C, and RH = 50%, and (

**b**) a sneeze with velocity = 50 m·s

^{−1}, temperature = 29 °C, and RH = 50%.

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**Figure 1.**The framework of the risk assessment model (RAM) of pathogen airborne droplet transmission.

**Figure 2.**(

**a**) Discretized computational domain with unstructured mesh and (

**b**) velocity contours of the respiratory jet with a schematic of droplets. The color of droplets shows their size group.

**Figure 3.**Comparison between centerline velocity of different mesh resolutions in present work with an experiment by [36].

**Figure 4.**Construction of the secondary mesh to calculate the number of droplets passing the facial cells in the vertical planes.

**Figure 7.**The accumulated number of droplets in case of a cough after 80 s: (

**a**) Iso-surface representation, (

**b**) 2D contour plot.

**Figure 8.**(

**a**) Accumulated droplets, and (

**b**) regions with more than 100 accumulated droplets within 60 s of simulation time of an exhalation activity.

**Figure 9.**(

**top**) The accumulated spread pattern until t = 60 s against (

**bottom**) the time-resolved droplet concentration for a cough case at t = 20, 40, and 60 s (velocity = 8 m·s

^{−1}, temperature = 29 °C, RH = 80%).

**Figure 10.**(

**top**) The accumulated spread pattern until t = 60 s against (

**bottom**) the time-resolved droplet concentration for a sneeze case at t = 20, 40, and 60 s (velocity = 50 m·s

^{−1}, temperature = 15 °C, RH = 50%).

**Figure 11.**Time evolution of safe social distance for exhalation event of Case 4 (velocity = 22 m·s

^{−1}, temperature = 11 °C, RH = 19%).

**Figure 12.**Distribution of accumulated droplets until t = 60 s at different cross-sections (Z-planes) of a sneeze after 60 s.

**Figure 13.**Risky regions with more than 100 accumulated droplets from cough cases 7 to 9 after 60 s.

**Figure 14.**Risky regions with more than 100 accumulated droplets of sneeze cases 10 to 12 after 60 s.

Boundary Type | Boundary Condition | Boundary Value | Air Density | Air Dynamic Viscosity |
---|---|---|---|---|

Inlet | Velocity inlet | 20 m·s^{−1} | 1.184 kg·m^{−3} | 1.855 × 10^{−5} Pa·s |

Outlet | Outlet pressure | 1 bar | ||

Walls | No-slip | - | - |

Mesh Configuration | FAC2 | q | FB | NMSE |
---|---|---|---|---|

Ideal Values | 1 | 1 | 0 | 0 |

4.5 M mesh | 1 | 0.875 | −0.032 | 0.012 |

627 k mesh | 1 | 1 | 0.074 | 0.017 |

189 k mesh | 1 | 0.875 | 0.092 | 0.042 |

189 k-HYB mesh | 1 | 0.875 | −0.030 | 0.027 |

3.7 M mesh | 1 | 0.875 | −0.040 | 0.020 |

Case Number | Exhalation Mode | Exhalation Jet Maximum Velocity (m·s^{−1}) | Temperature (°C) | Relative Humidity (%) |
---|---|---|---|---|

1 | Breathing | 3 | 43 | 28 |

2 | Cough | 11 | 2 | 64 |

3 | 14 | 18 | 80 | |

4 | Sneeze | 22 | 11 | 19 |

5 | 38 | 16 | 55 | |

6 | 50 | 29 | 50 |

Case Number | Exhalation Mode | Exhalation Jet Maximum Velocity (m·s^{−1}) | Temperature (°C) | Relative Humidity (%) |
---|---|---|---|---|

7 | Cough | 14 | 15 | 50 |

8 | 14 | 22 | 50 | |

9 | 14 | 29 | 50 | |

10 | Sneeze | 18 | 29 | 20 |

11 | 18 | 29 | 50 | |

12 | 18 | 29 | 80 |

Size Range | Size Class/Mean | DNC of Speaking | DNC of Coughing | DNC of Sneezing |
---|---|---|---|---|

2–4 | 3 | 4.59 | 86 | 0 |

4–8 | 6 | 66.21 | 1187 | 7706.95 |

8–16 | 12 | 22.23 | 444 | 23,491.91 |

16–24 | 20 | 11.33 | 144 | 26,203.62 |

24–32 | 28 | 7.87 | 54 | 25,689.82 |

32–40 | 36 | 4.32 | 50 | 24,933.4 |

40–50 | 45 | 4.47 | 41 | 24,176.97 |

50–75 | 62.5 | 4.57 | 43 | 58,344.43 |

75–100 | 87.5 | 3.44 | 30 | 33,054.23 |

100–125 | 112.5 | 4.52 | 36 | 41,703.14 |

125–150 | 137.5 | 4.31 | 34 | 32,540.44 |

150–200 | 175 | 4.52 | 93 | 41,588.96 |

200–250 | 225 | 3.85 | 53 | 44,129.41 |

250–500 | 375 | 3.45 | 44 | 179,257.9 |

500–1000 | 750 | 1.11 | 30 | 193,444.3 |

Sum | 150.8 | 2368 | 756,265.5 |

**DNC**= Droplet Number Concentration.

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## Share and Cite

**MDPI and ACS Style**

Motamedi Zoka, H.; Moshfeghi, M.; Bordbar, H.; Mirzaei, P.A.; Sheikhnejad, Y.
A CFD Approach for Risk Assessment Based on Airborne Pathogen Transmission. *Atmosphere* **2021**, *12*, 986.
https://doi.org/10.3390/atmos12080986

**AMA Style**

Motamedi Zoka H, Moshfeghi M, Bordbar H, Mirzaei PA, Sheikhnejad Y.
A CFD Approach for Risk Assessment Based on Airborne Pathogen Transmission. *Atmosphere*. 2021; 12(8):986.
https://doi.org/10.3390/atmos12080986

**Chicago/Turabian Style**

Motamedi Zoka, Hamid, Mohammad Moshfeghi, Hadi Bordbar, Parham A. Mirzaei, and Yahya Sheikhnejad.
2021. "A CFD Approach for Risk Assessment Based on Airborne Pathogen Transmission" *Atmosphere* 12, no. 8: 986.
https://doi.org/10.3390/atmos12080986