## 3 Mathematical Model
### Boundary Conditions and Solution Controls  
The boundary conditions consisted of two inlets for the return airflow from the system, an inlet duct for atmospheric airflow with particle movement, an outlet duct for the particle intake orifice, and walls of the collection head. The average value of the circulating airflow in the system was 15 m/s for inlet ducts D2 and D3 of the pickup head. For the atmospheric airflow and particle movement, the airflow velocity at the inlet was 35 m/s. The expansion surfaces were exposed to the atmosphere; therefore, the value of the inlet pressure was 0. Thus, the inlet return airflow from the system was set to the pressure inlet, and the inlet static pressure was 0 Pa. The pickup head suction port was set to the pressure outlet, and the exhaust pipe outlet and centrifugal fan were connected according to the actual pressure measurement of the dust hopper. The wall surface at the bottom of the pickup head was a nonslip stationary wall. When the \"reflect\" condition was applied to each wall to set the boundary conditions, it was assumed that the particles with the \"escape\"condition would be exported to the remaining inlet and outlet ducts. For the k-\(\varepsilon\) model, a feasible standard wall function was assumed. For the calculation, a steady-state solution method was assumed. The solution methods, a pair of pressure-velocity, Green-Gaussian cells based on gradient discretization, and second-order upwind of momentum were assumed [7, 17, 20]. The boundary conditions in the airflow and the top and bottom of the gridding region were selected as the specific velocity, inlet flow, outflow, and no-slip wall conditions, respectively (Figure 4). The quality of these shapes can be analyzed using indicators such as curvature and aspect ratio. We used the SIMPLEC method for the pressure-velocity coupling because of the complicated airflow. The effects of particle dispersion were simulated using a discrete random walk model. As a result, the under-relaxation factors were set to 0.2-0.5, to allow for easy convergence. The convergence criteria consisted of checking residuals and monitoring the integrals of the inlet and outlet pressure surfaces. The convergence factors of the energy equation were set to \(1\times 10^{-8}\), and those of the other variables were set to \(1\times 10^{-5}\). Jet sources were used to remove particles from the inlet of the particle suction ports, and the boundary positions were set for escape particles. There is a boundary condition inside the assimilation port and a trap on the output boundary. A particle was trapped at the exhaust duct outlet during the simulation, and the particle tracking stopped. The following methods were used to evaluate the effectiveness of particle removal. The particle suction efficiency is calculated by counting the number of particles trapped (trap) at the exit and the number that escape (escapes) [5, 8]. In the first order of determining the efficiency of assimilation of particles, we use the following equation:  
\[\eta_{1}=\frac{trap}{tracked} \tag{22}\]  
where \(\eta_{1}\) is the particle suction efficiency and \(track\) is the total number of particles incident. According to [PERSON] et al. [17], the total removal efficiency represents the ratio of the particles removed to the particles injected into the pickup head, which can be calculated based on the input and output of the total mass flow. The overall removal efficiency was determined using the following equation:  
\[\eta_{2}=\frac{G_{1}}{G_{2}}\cdot 100\% \tag{23}\]  
where \(\eta_{2}\) is the overall removal efficiency, \(G_{1}\) is the inlet mass flow rate (kg/s), and \(G_{2}\) is the mass flow rate (also known as the mass flow rate) (kg/s). To determine the efficiency of particle separation from the surface during an experiment, it was necessary to determine the weight of the particles. In this experiment, a special sieve was used to measure the size and quantity of the particles, allowing us to accurately determine the size of the particles.