# Analysis of Erosion Minimization for a Slurry Pump Using Discrete Phase Model Simulations
## 4 Erosion Optimization
### Results and Analysis  
We derived two regression equations. Equations (5) and (6) were for predicting the amount of erosion at the two points with the most wear on the wear map, namely, 45\({}^{\circ}\) and 315\({}^{\circ}\), respectively. Analysis of variance (ANOVA) confirmed that all parameters significantly influenced the erosion rate density. A higher value of the regression coefficient \(R^{2}\) indicates that the model equation is well fitted to the data. The values of the regression coefficients were as follows: Equation (5)--\(R^{2}\) = 0.9963 and adjusted \(R^{2}\) = 0.9926, and Equation (6)--\(R^{2}\) = 0.599 and adjusted \(R^{2}\) = 0.9375, where these equations show the best fits of the model with experimental data along with the \(p\)-values of the linear, quadratic, and interaction coefficients.  
Figure 9: Erosion rate density against different values of slurry particle.  
The regression equations for the erosion rate density were  
\[\text{erosion rate density}(45^{\circ})=-3877+1898w+38.0d+6.8v-743.8w^{2}-0.39d^{2}-0.004v ^{2}-0.75 wv \tag{5}\]  
\[\text{erosion rate density}(315^{\circ})=-2695+2675w+36.1d+1.0v-874w^{2}-22.34 wd \tag{6}\]  
where \(w\), distance (mm); \(d\), diameter (\(\mu\)m); and \(v\), impeller rotation speed (rpm).  
Figure 10a,b shows the response surface plots for the erosion rate density of the throat bush in the slurry pump. The curvature effect was confirmed to occur according to these parameters.  
The response optimizer, which is a reaction optimization tool, was used to derive the optimal values for the gap between the impeller and the throat bush, impeller speed, and slurry particle diameter that affect the erosion rate density of the throat bush. Figure 11 shows the optimal condition results based on the parameters. The target function was set to minimize the erosion rate density in the \(\alpha=45^{\circ}\) and \(315^{\circ}\) sections. The optimal process parameters and dimensions were derived by setting the constraint function to a flow rate of 7000 m\({}^{3}\)/h. The optimal conditions derived using the RSM were applied to the slurry pump for modeling, and a DPM simulation was performed. The results derived from the basic model and the DOE were compared with the optimized results. Table 5 presents a comparison of the results. The optimization results confirmed that the erosion rate density in the \(\alpha=45^{\circ}\) and \(315^{\circ}\) sections of the throat bush was lower than that observed in the earlier model.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Run**} & **Distance** & **Diameter** & **Impeller Rotation** & \multicolumn{2}{c}{**Erosion Rate Density**} & \multirow{2}{*}{**Slurry Flow**} \\  & **(w)** & **(d)** & **Speed (v) (rpm)** & & **(10\({}^{6}\) kg/m\({}^{2}\) s)** & \\  & **(mm)** & **(\(\mu\)m)** & & & **(\(\alpha=45^{\circ}\))** & **(\(\alpha=315^{\circ}\))** \\ \hline
1 & 1 & 45 & 650 & 382 & 363 & 8120 \\
2 & 1.4 & 45 & 650 & 235 & 207 & 6841 \\
3 & 1.4 & 37 & 600 & 161 & 130 & 6250 \\
4 & 1.2 & 45 & 600 & 303 & 280 & 7100 \\
5 & 1 & 37 & 600 & 295 & 267 & 7554 \\
6 & 1.2 & 37 & 650 & 298 & 276 & 7776 \\
7 & 1.2 & 53 & 550 & 242 & 313 & 6400 \\
8 & 1 & 53 & 600 & 347 & 480 & 7450 \\
9 & 1 & 45 & 550 & 277 & 241 & 6815 \\
10 & 1.2 & 45 & 600 & 303 & 280 & 7100 \\
11 & 1.4 & 53 & 600 & 187 & 207 & 6123 \\
12 & 1.2 & 37 & 550 & 190 & 159 & 6521 \\
13 & 1.4 & 45 & 550 & 155 & 120 & 6080 \\
14 & 1.2 & 53 & 650 & 330 & 412 & 7625 \\
15 & 1.2 & 45 & 600 & 303 & 280 & 7100 \\ \hline \end{tabular}
\end{table}
Table 4: Results of experimental and predicted values.  
Figure 10: The response surface plots for the erosion rate density: (**a**) erosion rate density at 45\({}^{\circ}\) locations of the wear map; (**b**) erosion rate density at 315\({}^{\circ}\).