## 3 Results and Discussion
### Single-Factor Experimental Results and Analysis
#### 3.3.3 Effect of Initial Concentration of Fe/Mn on the Iron and Manganese Ion Removal  
As can be seen from Figure 5, with the increase in the initial concentration ratio of Fe/Mn from 2 to 4, the iron ion removal rate in the solution increased from 78.4% to a maximum of 82.9%. The manganese ion removal rate showed a decreasing trend, and the degree of decrease became obvious after the Fe/Mn ion concentration ratio was greater than 4. The above phenomenon indicated that when the concentration of iron in the solution was small, the generated iron hydroxide acted as a coagulant and adsorbed iron, so the removal rate of iron was slightly increased. With the increase in the iron ion concentration, the system had a dynamic adsorption equilibrium. At the same time, there was a competitive adsorption relationship between iron and manganese ions. Excess iron ions dissociated in the solution and affected the removal of manganese ions. Therefore, a higher Fe/Mn concentration ratio was not favorable for the removal of Fe and Mn ions. Based on the above analysis, the reasonable initial Fe/Mn concentration ratio in this experiment was about 4. Based on the single-factor experimental results, 17 groups of comparative experiments were designed by using the response surface method. The experimental results are shown in Table 3.  
Design-Expert 10.0 analysis was performed, and the experimental results are shown in Table 4. The quadratic polynomial regression model for the interaction between pH (A), manganese sand dosage (B), and the Fe/Mn initial concentration ratio (C) was obtained as follows:  
\[\text{Y}_{1}=83.14+1.49\text{A}-0.057\text{B}-0.65\text{C}+0.62\text{AB}-0.41 \text{AC}+0.16\text{BC}-1.42\text{A}^{2}-0.97\text{B}^{2}-1.18\text{C}^{2} \tag{2}\]  
\[\text{Y}_{2}=76.04+1.36\text{A}+1.15\text{B}+0.13\text{C}-0.75\text{AB}-0.70 \text{AC}-2.39\text{BC}-3.16\text{A}^{2}-3.53\text{B}^{2}-2.89\text{C}^{2} \tag{3}\]  
where Y\({}_{1}\) is the predicted Fe removal efficiency (%), Y\({}_{2}\) is the predicted Mn removal efficiency (%), A is the pH value, B is the manganese sand dosage, and C is the initial concentration ratio of Fe/Mn. Figure 5: Effect of the initial concentration ratio of iron and manganese ions on the iron and manganese ions’ removal. R\({}_{1}\) is the iron ion removal rate; R\({}_{2}\) is the manganese ion removal rate. The analysis of variance is an essential statistical method to test the significance and suitability of regression models. The results of the analysis of variance for the regression models of Y1 and Y2 obtained from the response surface are shown in Table 4. The F-value was used for statistical saliency detection, and the \(p\)-value was used for each regression coefficient. If the \(p\)-value in the model was smaller, the experimental results were more significant. In this experiment, the \(p\)-values of the two models were \(<\)0.01, indicating that the model had high significance and was suitable for the optimization study of this parameter. Similarly, the F-value in Table 4 showed that the quadratic model had certain sufficiency and significance. The \(p\)-values of the lack of fit for both models were greater than 0.05, showing that the model was consistent in the regression study. As can be seen from the data in Table 4, the correlation coefficients of the predicted value of the model's removal rate and the experimental value were 0.9972 and 0.9747, respectively. This could better reflect the removal effect of iron and manganese ions. The values of C.V. were 0.16 \(<\) 10% and 1.20 \(<\) 10%, respectively, proving that the experiment had good accuracy and reliability. In the model, the values of Adeq precision referred to the effective signal-to-noise ratio and were considered reasonable if they were greater than 4. The Adeq precision values of the two models, as shown in the table, were 45.006 \(>\) 4 and 15.392 \(>\) 4, which means the model had high accuracy [28].