## 4 Results and Discussion
### Mathematical Modeling
#### 4.2.1 Ergun Equation Modifying Method
Since the bed expansion ratio is 2%, \(H_{\rm mf}\), \(r_{\rm mf}\), and \(\varepsilon_{\rm mf}\) can be determined by the following equations:  
\[H_{\rm mf}=1.02H_{0} \tag{14}\]  
\[r_{\rm mf}=r_{0}+H_{\rm mf}\rm{tan}(\alpha) \tag{15}\]  
\[\varepsilon_{\rm mf}=1-\frac{W}{\rho_{\rm s}\pi H_{\rm mf}\Big{(}r_{0}^{2}+r_ {0}r_{\rm mf}+r_{\rm mf}^{2}\Big{)}/3} \tag{16}\]  
Under the condition of \(U_{0}=U_{\rm 0 mf}\), the original Ergun equation is as follows:  
\[\frac{dP}{dh}=150\frac{\left(1-\varepsilon_{\rm mf}\right)^{2}}{\varepsilon_{ \rm mf}^{3}}\frac{\mu U_{\rm h}}{\left(\phi d_{\rm p}\right)^{2}}+1.75\frac{ \left(1-\varepsilon_{\rm mf}\right)}{\varepsilon_{\rm mf}^{3}}\frac{\rho_{ \rm g}U_{\rm h}^{2}}{\phi d_{\rm p}} \tag{17}\]  
Figure 6:0 Schematic diagram of the dimensional parameters of the conical cylinder. To increase the prediction accuracy of \(U_{\text{0 mf}}\) and \(\Delta P_{\text{mf}}\), the original coefficients at two sites are changed. The coefficients \(C_{1}\)/\(C_{2}\) in Equation (1) are changed at the first site, whose original values are 150/1.75, respectively. At the second site, considering that the conical wall may support the material, a multiplicative correction coefficient \(C_{3}\) is added to Equation (21) as follows:  
Figure 6:1  
Therefore, Equation (22) is modified as follows:  
Figure 6:2  
where \(A=\)\(C_{1}\frac{\left(1-\varepsilon_{\text{mf}}\right)^{2}}{\varepsilon_{\text{mf}}^{3}} \frac{\mu}{\left(\left\langle\delta\text{d}r\right\rangle\right)^{2}}\) and \(B=\)\(C_{2}\frac{\left(1-\varepsilon_{\text{mf}}\right)}{\varepsilon_{\text{mf}}^{3}} \frac{\rho_{\text{s}}}{\phi\Phi_{\text{p}}}\). Equation (25) is the equation used in the following fitting operations. All the calculations are performed by MATLAB software, a method of exhaustive optimization is used, and the optimization goal is to minimize the sum of the RMSE values of \(U_{\text{0 mf}}\) and \(\Delta P_{\text{mf}}\) (RMSE\({}_{\text{sum}}\)). The specific calculation processes are shown in Figure 6:3 First, a preset value of 0.8 is given for \(C_{3}\), then \(C_{1}\)/\(C_{2}\) are obtained by fitting the experimental data using a linear regression method (\"regress\" function in MATLAB). Then, this \(C_{1}\)/\(C_{2}\)/\(C_{3}\) combination is used to calculate the value of RMSE\({}_{\text{sum}}\), and the \(C_{1}\)/\(C_{2}\)/\(C_{3}\) combination and RMSE\({}_{\text{sum}}\) are recorded. Then, add 0.0001 to the previous \(C_{3}\) preset value, and perform the same calculation and recording for each \(C_{3}\) preset value. When the \(C_{3}\) preset value reaches 1.2, the exhaustive optimization calculation is ended. Finally, the \(C_{1}\)/\(C_{2}\)/\(C_{3}\) combination corresponding to the minimum value of RMSE\({}_{\text{sum}}\) is found. The final results are \(C_{1}=46.61\), \(C_{2}=3.25\), and \(C_{3}=0.8995\). The coefficients and modeling accuracies of the original and modified  
Figure 6:4 Comparison of the calculated and experimental values of (a) \(U_{\text{0 mf}}\) and (b) \(\Delta P_{\text{mf}}\) for the original Ergun equation. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & \multicolumn{2}{c}{\(U_{\text{0 mf}}\)} & \multicolumn{2}{c}{\(\Delta P_{\text{mf}}\)} \\ \cline{4-8}  & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \begin{tabular}{c} RMSE \\ **(m/s)** \\ \end{tabular} & \begin{tabular}{c} RMSE \\ **(x/a)** \\ \end{tabular} & \
\begin{tabular}{c} RMSE \\ **(x/a)** \\ \end{tabular} &
\begin{tabular}{c} RMSE \\ **(x/a)** \\ \end{tabular} \\ \hline Original & 150 & 1.75 & 1 & 0.0872 & 0.4209 & 0.1076 & 0.6908 \\ Modified & 46.61 & 3.25 & 0.8995 & 0.0402 & 0.8771 & 0.0291 & 0.9774 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Coefficients and prediction accuracies of the original and modified Ergun equations.