# Study on the Impact of Ventilation Methods on Droplet Nuclei Transmission in Subway Carriages
## 2 Materials and Methods  
### Numerical Models  
Simulations were conducted to evaluate the performance of different ventilation modes (MV and SFRC) under varying ventilation factors. The Reynolds-averaged Navier-Stokes (RANS) equations were employed to model the airflow field inside the subway carriage. Considering the successful application of the RNG k-\(\epsilon\) model in indoor airflow simulations, this study employed the RNG k-\(\epsilon\) model to improve the accuracy of turbulent vortex simulations [30,31]. The governing equations for continuity, momentum, energy, and turbulence are expressed in the following general form:  
\[\frac{\partial(\rho\varphi)}{\partial t}+div\bigg{(}\rho\overset{\rightarrow} {U}\varphi\bigg{)}=div\big{(}\Gamma_{\varphi}grad\varphi\big{)}+\overset{ \rightarrow}{S}_{\varphi} \tag{1}\]  
In this equation, \(\rho\) represents the density of the air, U represents the velocity vector of the air, \(\varphi\) denotes the transported quantity, \(\Gamma_{\varphi}\) is the effective diffusion coefficient associated with \(\varphi\), t represents the time of particle flow, and \(\overset{\rightarrow}{S}_{\varphi}\) is the source term. After being expelled from the passenger (cougher), the moisture within the droplets rapidly evaporates into the surrounding air. To model the transport and diffusion of the water vapor-air mixture under turbulent conditions, a mass transfer model is employed. This model is represented by the following equation:  
\[\frac{\partial}{\partial t}(\rho Y)+\
abla\cdot(\rho YY)=-\
abla\cdot J+SY \tag{2}\]  
In this equation, Y represents the local mass fraction of water vapor, \(J\) signifies the diffusion flow of water vapor, and SY is the source term for water vapor. Relative humidity is used to calculate the mass fraction of the air and water vapor mixture. Figure 1: General framework of this study. The rate of evaporation is governed by diffusion driven by concentration gradients and is related to the difference in water vapor concentration between the surface of the droplet and the surrounding air [32]. \[\frac{dN}{dt}=c(CS-C\infty) \tag{3}\]  
In this equation, c represents the mass transfer coefficient and CS and \(C\infty\) are the molar concentrations of water vapor at the droplet surface and in the air, respectively. The general Equations (2) and (3) can be used to figure out how the mass fraction of each fluid in the mixture changes as it moves. In the Lagrange framework, the trajectory of each droplet is obtained by solving the momentum equation for the particles. \[\frac{d\overset{\rightarrow}{u}p}{dt}=FD\left(\overset{\rightarrow}{u}- \overset{\rightarrow}{u}p\right)+\frac{\overset{\rightarrow}{S}(\rho p- \rho)}{\rho p}+\overset{\rightarrow}{Fa} \tag{4}\]  
In this equation, \(\overset{\rightarrow}{u}\) and \(\overset{\rightarrow}{u}p\) represent the velocities of the air and the particles, and \(\overset{\rightarrow}{Fa}\) denotes the additional force acting on the particles. Based on the indoor airflow and droplet size, this study considers only the thermophoretic force, Brownian force, and Saffman lift, neglecting the resuspension of particles. The calculation formula for the thermophoretic force is as follows:  
\[F_{th}=-\frac{6\pi d_{p}\mu^{2}C_{s}(K+C_{t}K_{n})}{\rho T(1+3C_{m}K_{n})(1+2K +2C_{t}K_{n})}\
abla T \tag{5}\]  
In this equation, \(K_{n}\) is the Knudsen number = 2 \(\lambda\)/dp, where \(\lambda\) is the mean free path of the fluid, K is the thermal conductivity of the fluid, and \(K_{p}\) is the thermal conductivity of the particles. \(C_{s}\) is 1.17, \(C_{t}\) is 2.18, \(C_{m}\) is 1.14, T is the local fluid temperature, \(\mu\) is the fluid viscosity, and dp is the particle diameter. Forces two orders of magnitude smaller than the pulling force are considered negligible and do not need to be accounted for. It is assumed that the particles are spherical in shape. The Discrete Random Walk (DRW) model is used to predict particle movement under turbulent conditions.