## 2 Methodology  
### Physical Model  
Considering the airflow in a subway carriage is slow (\(<<\) 0.3 Ma), the gas phase can be assumed as a noncompressible state, and the Navier-Stokes equation is chosen as the governing equation for the flow field. The renormalized group (RNG) \(k\)-\(\varepsilon\) model exhibits good reliability and robustness in simulating mixed convection at low turbulence levels for indoor environments [50], and it is widely used in engineering applications. Therefore, the RNG \(k\)-\(\varepsilon\) model is chosen for the flow field calculation in the present work. The governing equations for continuity, momentum, and energy can be found in Xu, Qian, Wu, Li, Zhang and Wang [28]. The Boussinesq assumption is also applied to simulate the buoyancy flow induced by the temperature difference in the flow field [22; 23]. The particles' trajectories are tracked using the Lagrangian method. The position and velocity of each particle's trajectory can be calculated using a differential equation:  
\[m_{p}\frac{du_{p}}{dt}=F_{D}+F_{g}+F_{a} \tag{1}\]  
For large density ratios between the particle and fluid, the drag force \(F_{D}\) dominates the particles' behavior [51], and is given as:  
\[F_{D}=\frac{1}{8}\pi C_{D}d_{p}^{2}\rho\bigg{|}u-u_{p}\bigg{|}(u-u_{p}) \tag{2}\]  
where \(u\) and \(u_{p}\) are, respectively, the fluid phase and particle velocity; \(m_{p}\) is the droplet mass; \(\rho\) is the density of the fluid; and \(d_{p}\) is the droplet diameter. \(C_{D}\) is the non-linear drag coefficient [52], which can be described as:  
\[C_{D}=\left\{ \begin{array}{l}24/Re_{p},Re_{p}<1\\ 30 Re_{p}^{-0.625},1\leq Re_{p}\leq 1000\\ 0.45,1000<Re_{p}\end{array} \right. \tag{3}\]  
where the particle Reynolds number \(Re_{p}\) can be obtained using the following equation [53]:  
\[Re_{p}=\rho d_{p}\big{|}u-u_{p}\big{|}/\mu \tag{4}\]  
where \(\mu\) is the dynamic viscosity of the air. The gravitation and buoyancy are computed as follows one total force \(F_{g}\):  
\[F_{g}=g(\rho_{p}-\rho)/\rho_{p} \tag{5}\]  
where \(g\) is the gravitational acceleration vector and \(\rho_{p}\) is the density of the particle. In addition, this work also considers the influence of the load from the thermophoretic force [54], pressure gradient, Saffman lift force [55], and Brownian force [56] on the particle motion, and these forces are abbreviated as the additional force \(F_{a}\) in Formula (1). The evaporation process of the droplets significantly affects their diffusion in the air. To simulate the droplet evaporation process, this study employs the assumption of spherical liquid droplets made of pure water [28; 57]. Under this assumption, the droplets are considered ideal spherical particles, and their volume continuously decreases due to evaporation until it decreases to a certain threshold. At this point, they are regarded as non-evaporable droplet nuclei. The droplet evaporation [19] is defined as:  
\[\frac{dr_{p}}{dt}=\frac{CM_{V}D_{\infty}pSh}{\rho_{p}r_{p}RT_{\infty}}\ln \frac{p-p_{\infty}}{p-p_{\infty}} \tag{6}\]  
where \(r_{p}\) is the droplet radius; \(M_{V}\) is the molecular weight of the vapor; \(D_{\infty}\) is the binary diffusion coefficient away from the droplet; \(R\) is the universal gas constant; \(T_{\infty}\) is the ambient air temperature; \(p\) is the total pressure; \(p_{\infty}\) is the vapor pressure, where the subscripts a and \(\infty\) represent the physical parameters at the droplet surface and away from the droplet surface, respectively; \(C\) is a dimensionless factor that reflects the dependency of the diffusion coefficient on temperature and is defined as Equation (7); and the dimensionless parameter the Sherwood number (\(Sh\)) represents the ratio of the convective mass transfer to the mass diffusion rate. It reflects the influence of the relative velocity between the gas and the droplet on the mass transfer and is defined as Equation (8).