# Numerical Simulation of Heat Transfer and Spread of Virus Particles in the Car Interior

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Problem Statement of Simulation Liquid Droplets in a Flow

_{i}are internal sources of heat and diffusion; Y

_{i}is local mass fraction of i substance; and ${\mathit{J}}_{i}$ is diffuse flow of the i substance, which occurs due to the gradient of concentrations, temperature, and pressure, and the turbulent case is written as follows:

#### 2.2. Particles Motion Simulation by the Discrete Phase Model (DPM)

_{p}is the particle diameter; Re is the Reynolds number; and C

_{d}is aerodynamic drag coefficient, and in the case of spherical drops can be represented as:

_{i}are constants [60].

_{p}is a moment of inertia of the particle; ${\mathit{\omega}}_{p}$ is a vector of the angular velocity of the particle;

**Ω**is the relative angular velocity; and ${C}_{\omega}$ is the coefficient of aerodynamic resistance to rotation [58]:

_{p}is the heat transfer coefficient; A

_{p}is the surface area of the particle; and B

_{p}is determined by the formula:

_{out}is the temperature of the medium; and T

_{p}is the temperature of the particle.

#### 2.3. Boundary Conditions

**u**

_{n}and

**u**

_{τ}are the normal and tangential components of the velocity vector

**u**.

**u**, pressure p, temperature T, and concentrations Y

_{i}, and coordinates and masses (diameters) of particles.

#### 2.4. Geometry

## 3. Results

#### 3.1. Numerical Analysis

^{−7}m

^{2}, the maximum is 2.7 × 10

^{−3}m

^{2}.

^{−4}.

#### 3.2. Task Parameters

^{2}·K) and the thickness is 0.0312 m.

_{i}and λ

_{i}are the thickness and thermal conductivity of the i-th wall, α is the thermal conductivity coefficient for the walls, and α

_{out}are the external heat transfer coefficients of the medium. For Formula (27), the applied Yurgens formula [22] for the external velocity is used u’ = 2 m/s.

^{2}·K) for thickness 6 mm.

#### 3.3. Numerical Results

## 4. Discussion

- ➢
- “Large” droplets, with a diameter greater than approximately 400 μm, hit the wall and are removed from the calculation in accordance with the sticking condition on the walls of the cabin.
- ➢
- “Small” droplets with a diameter of more than 400 μm, in the case of taking into account the volatility of droplets, evaporate almost instantly.

## 5. Conclusions

- The mathematical model is implemented on the basis of the Navier–Stokes hydrodynamic equations for an ideal gas together with a discrete-element model for modeling particles with a virus in the air. The Euler–Lagrange approach is used to simulate liquid droplets in a flow. In this case, the liquid phase is considered as a continuous medium using the Navier–Stokes equations, the continuity equation, the energy equation, and the diffusion equation. Accounting for diffusion makes it possible to explicitly model air humidity and is necessary, among other things, to consider the evaporation of droplets (changes in the mass and size of particles containing the virus). The discrete-phase DPM model is used to simulate liquid drops.
- The validation of the proposed model was carried out at the stage of debugging the program in comparison with the known theoretical models of a stable flow. The results were obtained on a grid with an orthogonality of 0.16 and minimal residuals of 1 × 10
^{−4}. In addition, the comparison with previous studies of other researchers was made. - In the work, the simulation of the fields of velocities, pressures, temperatures, and humidity in the time domain for various modes was performed. Field stabilization times were found.
- To simulate droplets containing a virus, a discrete-phase DPM model was used, for which the discrete and continuous phases are interconnected through the initial terms in the equations. The following regularities were found:
- “Large” drops, with a diameter greater than approximately 400 microns, hit the wall and were removed from the calculation in accordance with the sticking condition on the walls of the cabin.
- “Small” droplets, with a diameter of more than 400 microns, in the case of taking into account the evaporation of droplets, evaporate almost instantly.
- “Small” droplets, with a diameter of more than 400 μm, in the case of non-evaporability of droplets, propagate through the cabin along the streamline of air velocities, and the dependences of concentrations and the number of particles in the cabin on time are plotted.
- In this work, only one value of air humidity was considered: “non-evaporable” particles. It is also quite interesting to study the influence of the humidity value on the evaporation of particles, and therefore on their behavior in the volume of the cabin. In addition, the question of the influence of the airflow directions of the air deflectors of the climate system and the internal equipment of the cabin remains unexplored. These studies, field tests and comparison of the results are planned to be carried out in subsequent works.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 12.**Trajectories and diameters of pathogen particles for deflectors, the colors on the value scale correspond to the size of particle diameters (

**a**) with volatility and (

**b**) without volatility.

**Figure 13.**Trajectories and particle diameters of pathogens during the driver’s cough, the colors on the value scale correspond to the particle diameter size (

**a**) with volatility and (

**b**) without volatility.

**Figure 14.**Trajectories and diameters of pathogen particles for deflectors versus time, the colors on the scale of values correspond to the size of particle diameters: (

**a**) time 0.4 s; (

**b**) time 1.2 s; (

**c**) time 127 s.

**Figure 17.**Temperature in the central part of the cabin: 1—previous solution [22]; 2—new results.

Material of the Wall | Coefficient of Thermal Conductivity, W/(m·K) | Thickness, m |
---|---|---|

Steel | 58 | 0.002 |

Bituminous mastic layer | 0.27 | 0.0042 |

Cast polyurethane | 0.32 | 0.025 |

Num | Title | Value | |
---|---|---|---|

1 | Incoming flow temperature, °C | 30 | |

2 | Air speed in deflectors, m/s | 1 | |

3 | Cabin air flow temperature, °C | 14 | |

4 | Average temperature on the human surface, °C | 25 | |

5 | Thermal conductivity of water vapor (H_{2}O), W/(m^{2}∙K) | 0.0261 | |

6 | Molar mass of water vapor (H_{2}O), kg/kmol | 18.01 | |

7 | Heat capacity of water vapor (H_{2}O), J/(kg∙K) | 2014 | |

8 | Thermal conductivity of air, W/(m^{2}∙K) | 0.0242 | |

9 | Molar mass of air, kg/kmol | 28.966 | |

10 | Heat capacity of air, J/(kg∙K) | 1006.43 | |

11 | Mass fraction of water vapor (H_{2}O), Y_{H2O} | Cabin inlet | 0.0091 |

12 | Cabin outlet | 0.0091 |

Num | Name of Injection | Deflectors | Driver’s Mouth/Cough |
---|---|---|---|

1 | Minimum diameter value, µm | 2 | 2 |

2 | Maximum value of diameter, microns | 2000 | 2000 |

3 | Average value of diameter, microns | 1000 | 1000 |

4 | Initial speed, m/s | 1 | 5 |

6 | Mass of particles, kg/s | 0.01 | 0.001 |

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**MDPI and ACS Style**

Panfilov, I.; Beskopylny, A.N.; Meskhi, B.
Numerical Simulation of Heat Transfer and Spread of Virus Particles in the Car Interior. *Mathematics* **2023**, *11*, 784.
https://doi.org/10.3390/math11030784

**AMA Style**

Panfilov I, Beskopylny AN, Meskhi B.
Numerical Simulation of Heat Transfer and Spread of Virus Particles in the Car Interior. *Mathematics*. 2023; 11(3):784.
https://doi.org/10.3390/math11030784

**Chicago/Turabian Style**

Panfilov, Ivan, Alexey N. Beskopylny, and Besarion Meskhi.
2023. "Numerical Simulation of Heat Transfer and Spread of Virus Particles in the Car Interior" *Mathematics* 11, no. 3: 784.
https://doi.org/10.3390/math11030784