# Numerical Flow Simulation on the Virus Spread of SARS-CoV-2 Due to Airborne Transmission in a Classroom

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Approach

#### 2.1. Modelling of the Viral Load

_{2}in Hartmann and Kriegel [21], results in deviations between 2–3%.

#### 2.2. Assessment of Infection Risk

## 3. Application Model

#### 3.1. General Description

#### 3.2. Numerical Treatment of Air Flow

## 4. Results

## 5. Discussion

#### 5.1. Effect of Supply Air Volume on the Risk of Infection

#### 5.2. ‘Green Classroom’ Situation

#### 5.3. Comparison with Available Models

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

RANS | Reynolds-Averaged Navier–Stokes |

RNG | Renormalisation Group |

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**Figure 1.**Illustration of the classroom in modelling high cross ventilation of the occupied area: (

**a**) side view from the simulation model and (

**b**) schematic top view.

**Figure 2.**Resulting virus load at each seat after $t=90\phantom{\rule{0.166667em}{0ex}}\mathrm{min}$ at low and high ventilation while the infectious lecturer is breathing or speaking. (

**a**) Low ventilation, infectious lecturer is breathing (viral load of 10 parts/s). (

**b**) High ventilation, infectious lecturer is breathing (viral load of 10 parts/s). (

**c**) Low ventilation, infectious lecturer is speaking (viral load of 100 parts/s). (

**d**) High ventilation, infectious lecturer is speaking (viral load of 100 parts/s).

**Figure 4.**Total inhaled particle count at each seat over time with low and high ventilation and a breathing infectious lecturer (10 parts/s) for the positions in the room (regions at window side, centre, door side; seat rows of 1 to 5—from front to back).

**Figure 5.**Total inhaled particle count at each seat over time at low and high ventilation and speaking infectious lecturer (100 parts/s) for the positions in the room (regions at window side, center, door side; seat rows of 1 to 5—from front to back).

**Figure 6.**Number of infected persons from available studies compared to the presented study under the conditions from Table 3. (

**a**) Results with low ventilation, infectious person breathing [20]. (

**b**) Results with low ventilation, infectious person speaking. (

**c**) Results with high ventilation, infectious person speaking [20,23,55].

**Figure 7.**Comparison of the presented numerical model and the results of Lam-Hine et al. [55] with regard to the seat-related risk of infection, i.e., which seats are particularly affected. (

**a**) Illustration of infected persons after $t=12\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$ in Lam-Hine et al. [55]. (

**b**) Resulting virus load at each seat after t = 90 min at high ventilation while the infectious lecturer is speaking.

**Table 1.**Investigated ventilation (vent.) conditions (low/high) and viral loads (speaking/breathing) for the analysed 80 square meter (240 ${\mathrm{m}}^{3}$) classroom with 16 persons inside.

Parameter | Low Vent./ Speaking | High Vent./ Speaking | Low Vent./ Breathing | High Vent./ Breathing |
---|---|---|---|---|

Supply air volume flow in ${\mathrm{m}}^{3}/\mathrm{h}$ | 35 | 432 | 35 | 432 |

Air change rate in ${\mathrm{h}}^{-1}$ | 0.146 | 1.800 | 0.146 | 1.800 |

Viral load of lecturer in virus particles/s | 100 | 100 | 10 | 10 |

Parameter | Description | Value |
---|---|---|

Model approach | ||

Virus variant | Virus variant considered in terms of spread and infection | Wildtype |

Viral load | Viral emissions of an infectious person during breathing (a) and speaking (b) | (a) 10 SARS-CoV-2 virus particles/s and (b) 100 SARS-CoV-2 virus particles/s |

Respiration rate | Respiration rate of all persons | $0.36\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}/\mathrm{h}$ |

Calculation of viral distribution | Multispecies model approach for calculating the airborne viral load fraction | Species Transport model |

Threshold for high risk | The risk of infection is assessed as high as soon as the cumulative number of virus particles in a person’s inhalation volume reaches the threshold value | 500 particles |

Application model | ||

Room size | Dimensions of the investigated room | $80\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ ($10\times 8\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$) and $3\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ high |

Window area | Supply airflow into the room through the surface | $1\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ |

Supply air | Volume flow of supply air for (a) low ventilation and (b) high ventilation | (a) $35\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}/\mathrm{h}$ and (b) $432\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}/\mathrm{h}$ |

Open door area | Indoor air escapes through the surface | $0.15\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ |

Susceptible persons | Number of uninfected persons in the room | 15 |

Distance | Minimum distance between persons | $1.5\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ |

Infectious person | Number and location of the infectious persons | 1 lecturer, standing at the front of the room |

Thermal boundary conditions | Thermal boundary conditions of persons (a), walls (b), fresh air (c), exhaled air (d) | (a) 37 °C, (b) adiabatic, (c) 25 °C, (d) 37 °C |

Viscous model | Model for calculating the flow pattern | k-$\u03f5$ (RNG) model |

Numerical mesh | Number of elements of the mesh | Approx. 2.6 million |

Near-wall treatment | Near-wall treatment of the room walls that are parallel to the flow direction (from the window to the door) and for those of the bodies of people in the room | Enhanced Wall Treatment with the first near-wall node is set to ${\mathrm{y}}^{+}\approx 1$ |

**Table 3.**Important boundary conditions for the risk of infection in the mentioned studies compared to the study presented here with adapted viral load for reasons of comparability.

Parameter | Lelieveld et al. (2020) [20]/MPIC | Kriegel and Hartmann (2021) [23] | Lam-Hine et al. (2021) [55] | Present Study | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Classroom size in ${\mathrm{m}}^{2}$ | 80 | N/A | N/A | 80 | ||||||

Infectious person | 1 ($t=4\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$ before others in room) | 1 | 1 (lecturer) | 1 (lecturer) | ||||||

Others | 15 unmasked | Normal occupancy, unmasked | 24 masked | 15 unmasked | ||||||

Ventilation | $0.146\phantom{\rule{0.166667em}{0ex}}{\mathrm{h}}^{-1}$ | $1.800\phantom{\rule{0.166667em}{0ex}}{\mathrm{h}}^{-1}$ | $25\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}/(\mathrm{h}\xb7\mathrm{person})\left(1.667\phantom{\rule{0.166667em}{0ex}}{\mathrm{h}}^{-1}\right)$ | Windows and door are open; air filter (no ventilation rate given) | $0.146\phantom{\rule{0.166667em}{0ex}}{\mathrm{h}}^{-1}$ | $1.800\phantom{\rule{0.166667em}{0ex}}{\mathrm{h}}^{-1}$ | ||||

Virus variant | Wildtype | Wildtype | Delta | Wildtype | ||||||

Viral load in parts/s | 10 | 100 | 10 | 100 | 100 | N/A | 10 | 100 | 10 | 100 |

Result unit | Infected persons and probability that a least one susceptible person becoming infected | Number of susceptible persons becoming infected | Number of susceptible persons becoming infected | Number of susceptible persons with a high risk of infection | ||||||

Result after 1.5 h | 1.0 ($9.4\%$) | 1.0 ($86\%$) | 1.0 ($3.0\%$) | 1.0 ($46\%$) | No result | No result | 0 | 3 | 0 | 0 |

Result after 6 h | 1.0 ($33\%$) | 6.2 | 1.0 ($11\%$) | 2.2 | 11.5 | No result | 2 | All 15 | 0 | 11 |

Result after 12 h | 1.0 ($55\%$) | 9.8 | 1.0 ($22\%$) | 4.2 | No result | 12 (out of 22 tested) | All 15 | All 15 | 0 | All 15 |

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**MDPI and ACS Style**

Moeller, L.; Wallburg, F.; Kaule, F.; Schoenfelder, S.
Numerical Flow Simulation on the Virus Spread of SARS-CoV-2 Due to Airborne Transmission in a Classroom. *Int. J. Environ. Res. Public Health* **2022**, *19*, 6279.
https://doi.org/10.3390/ijerph19106279

**AMA Style**

Moeller L, Wallburg F, Kaule F, Schoenfelder S.
Numerical Flow Simulation on the Virus Spread of SARS-CoV-2 Due to Airborne Transmission in a Classroom. *International Journal of Environmental Research and Public Health*. 2022; 19(10):6279.
https://doi.org/10.3390/ijerph19106279

**Chicago/Turabian Style**

Moeller, Lara, Florian Wallburg, Felix Kaule, and Stephan Schoenfelder.
2022. "Numerical Flow Simulation on the Virus Spread of SARS-CoV-2 Due to Airborne Transmission in a Classroom" *International Journal of Environmental Research and Public Health* 19, no. 10: 6279.
https://doi.org/10.3390/ijerph19106279