## 3 Results and Discussion
#### 3.2.1 Stress Analysis of Gangue Particles  
Particle size distribution has a significant effect on the stability of slurry [56; 57]. Therefore, it is crucial to study the stress of gangue in slurry under an appropriate mass gradation coefficient for slurry settlement analysis. Figure 10 describes the medium resistance and stress of the aggregate in the slurry, which can be reduced to spherical particles with uniform textures. When gangue particles are suspended in the slurry, the effective gravity of the solid particles must be less than or equal to the resistance of the slurry to the particles. Gravity \(G\) of spherical particles in slurry:  
\[G\ =\ \frac{\pi}{6}d^{3}(\rho_{2}-\rho_{1})g \tag{4}\]  
In this formula, \(\rho_{1}\) is the slurry density and \(\rho_{2}\) is the apparent density of the aggregate. Resistance \(F\) is the resultant force of shear resistance caused by slurry yield stress. The calculation formula is as follows:  
\[\left\{ \begin{array}{l}\mathrm{d}F\ =\ \int\tau_{0}\cos\theta dA\\ F\ =\ 4\pi d^{2}\tau_{0}\int\limits_{0}^{\pi/2}\cos\theta d\theta\ =\ \frac{\pi^{2}d^{2}}{4}\tau_{0}\end{array} \right. \tag{5}\]  
Figure 10: The sinking force of single-particle gangue in static slurry. Figure 9: Calculation of the flow characteristic evaluation model. In this formula, \(A\) is the cross-sectional area of the particle and \(\tau_{0}\) is the slurry yield stress, Pa. When the gauge particles are suspended in the high-concentration slurry, they are in a state of force balance, and the sum of resistance, gravity, and buoyancy is zero. \[G-F\ =\ \frac{\pi}{6}d^{3}(\rho_{2}-\rho_{1})g-\frac{\pi^{2}d^{2}}{4}\tau_{0}\ =\ 0 \tag{6}\]  
\[\tau_{0}\ =\ \frac{2d(\rho_{2}-\rho_{1})g}{3\pi} \tag{7}\]  
The density, particle size, suspension velocity, and viscosity of gangue particles are the properties of the slurry itself, which are determined by the morphology of slurry particles. It can be seen that adjusting the grading coefficient of gangue is an effective strategy for controlling the plastic viscosity and yield stress of slurry and optimizing the suspension performance. For the sedimentation rate of high-concentration slurry particles, the modified Stokes equation is used [58]:  
\[v_{s}\ =\ \frac{(\rho_{2}\ -\rho_{1}\ )d^{2}g}{18\eta}\cdot f(\varphi) \tag{8}\]  
\[f(\varphi)\ =\ (1-\varphi)^{n} \tag{9}\]  
In the formula, \(\eta\) is the viscosity of the fluid, \(\varphi\) is the volume fraction of the particles in the slurry, and \(\varphi\) is a constant. When the slurry mass concentration is constant, for \(\rho_{1}\), \(\eta\), and \(\varphi\) in Formula (6), there are:  
\[\left\{ \begin{array}{l}\rho_{1}\sim D\\ \eta\ \sim D\\ \varphi\ \sim D\end{array} \right. \tag{10}\]  
When the fractal dimension of particles in the slurry increases, the increase in the proportion of fine particles will lead to changes in slurry density \(\rho_{1}\), viscosity \(\eta\), and particle volume fraction \(\varphi\). Through the settlement experiment, the influence of different gradation ranges on the settlement of gangue particles was evaluated, and the optimal particle gradation range was determined. 2.2 The Effect of the Fractal Dimension on the Solid Mass Fraction and Slurry Particle Distribution During the Settlement Process  
Figure 11 is a schematic diagram of particle distribution during gangue settlement. In order to facilitate the analysis, we divided the settlement instrument into three regions: upper, middle, and lower, and analyzed the solid mass fraction \(W\) and the fractal dimension \(D_{\rm S}\) of the particle size distribution in each region in different time periods. The pair (2) can be obtained by taking logarithms on both sides:  
\[\lg\frac{m}{M}\ =\ (3-D_{\rm S})\lg d_{\rm i}+b \tag{11}\]  
Let 3-\(D_{\rm S}\) be equal to a. According to Formula (2), the least squares method is used to fit the best curve of the screened gangue particles. The parameters \(b\) and \(a\) are obtained, and the grading fractal dimension \(D_{\rm S}\) can be solved.