## 3 Numerical Model  
### Mathematical Model  
The mathematical model for simulating gas and dust flow (two-phase gas-solid) in ANSYS 2020R1 software includes the Discrete Phase Model (DPM), the Eulerian model. The Discrete Phase Model belongs to the Euler-Lagrange method. It requires the particle volume to be not too large [15]. By computing the velocity and turbulent kinetic energy of the continuous flow field, we can observe the trajectories of particles. Currently, the study of particle trajectories is generated from the blasting process in the heading face in continuous flow using the DPM model of the Euler-Lagrange method [16]. The Discrete Phase Model (DPM)-Discrete Element Model (DEM) was constructed based on the gas-solid two-phase flow theory, and the dust pollution characteristics of the shuttle transfer stage were numerically simulated [17]. Figure 1: Tunnel model, mesh, and air duct suspension positions. In this paper, the Euler-Lagrange model is utilized to calculate the flow field and the trajectory of dust particles within the tunnel. The Euler-Lagrange method considers the fluid as a continuous phase and applies [PERSON]'s second law to track the fluid stream. The variation characteristics of dust concentration and particle size distribution are simulated and analyzed using DPM. A pressure-based solver is utilized, and wind velocity and turbulent kinetic energy of the flow are computed using the SIMPLE algorithm. The fundamental governing equations for the particle trajectory, flow field, and dust movement in the field are as follows [18]. The gas phase continuous equation in the two-phase solid-gas flow is as follows:  
\[\frac{\partial p}{\partial t}+\frac{\partial(pu_{i})}{\partial x_{i}}=0 \tag{1}\]  
The conservation equation of momentum is as follows:  
\[\frac  
where \(F_{D(u-up)}\) is the drag force on the particle per unit mass, N; \(C_{D}\) is the drag force coefficent; \(u\) is the fluid phase velocity, m/s; \(u_{p}\) is the particle velocity, m/s; \(\rho_{p}\) is the particle density, kg/m\({}^{3}\); \(d_{p}\) is the particle diameter m; and \(F\)--the addition forces, Including particle-particle collisions, particle-wall collisions, Saffman force, and Magnus force, the DPM model in Fluent integrates DEM-collision to account for the impact interactions between particles and with walls. The contact forces between particles describe direct interactions among particles, particle-surface interactions, and torque forces. The contact force between particles \(i\) and \(j\), denoted as \(F_{n}\), \(i\bar{j}\), is expressed by the formula [20].  
\[F_{n,ij}=\frac{4}{3}Y^{*}\sqrt{R^{*}}\delta_{n,ij}^{3}-F_{n,ij}^{d} \tag{8}\]  
\[F_{n,ij}^{d}=-\sqrt{\frac{5}{6}}\frac{\ln e}{\sqrt{\ln^{2}e+\pi^{2}}}\sqrt{S_ {n,ij}}m*v_{n} \tag{9}\]  
\[S_{n,ij}=2Y^{*}\sqrt{R^{*}\delta_{n,ij}} \tag{10}\]  
where \(F_{n,ij}\)--normal force between particles; \(Y^{*}\) is the equivalent Young's modulus; \(m^{*}\) is the particle mass; \(R^{*}\) is the equivalent radius; \(\delta_{n,ij}\) is the normal overlap; \(S_{n,ij}\) is the normal stiffness; \(V_{n}\) is the normal component of the relative velocity; and \(e\) is the coefficient of restitution. The tangential force \(F_{t,ij}\) depends on the tangential overlap, \(\delta_{t,ij}\), and the tangential stiffness \(S_{t,ij}\), and is limited by Coulomb friction \(\mu_{s}\)\(F_{n,\bar{j}}\), where \(\mu_{s}\) is the coefficient of static friction and \(G^{*}\) is the shear modulus.