## 4 Results and Discussion
### Mathematical Modeling  
#### 4.2.1 Ergun Equation Modifying Method  
The fluidized state could be observed through the glass window embedded in the sidewall of the conical cylinder, and the expansion of the bed surface could be clearly observed with the help of a thin film ruler that was pasted on the inner wall of the conical cylinder. When \(U_{0}<U_{\rm 0 mf}\), the bed was in a fixed state. When \(U_{0}\) approached \(U_{\rm 0 mf}\), slight expansion could be observed in the bed, and the bed height expansion ratio was approximately 2%. After \(U_{0}\) reached \(U_{\rm 0 mf}\), bubbling was observed in the central part of the bed surface. As \(U_{0}\) continued to increase, significant bubbling occurred throughout the entire cross section, and the fluidization height also increased as more particles were brought into the freeboard. To simplify the model, the following assumptions are made for the point at which \(U_{0}=U_{\rm 0 mf}\): the material expands uniformly, the bed height expansion ratio in the height direction is 2%, and the surface of the bed is flat. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material Number**} & \multicolumn{5}{c}{\(U_{\rm 0 mf}\) (m/s)} \\  & \multicolumn{5}{c}{\(\Delta P_{\rm mf}\) (kPa)} \\ \cline{2-6}  & **Material Mass** & **Material Mass** & **Material Mass** & **Material Mass** & **Material Mass** \\  & **2 kg** & **2.5 kg** & **3 kg** & **3.5 kg** & **4 kg** \\ \hline \multirow{2}{*}{\#1} & 0.403 & 0.408 & 0.413 & 0.419 & 0.433 \\  & 0.650 & 0.810 & 0.964 & 1.064 & 1.213 \\  & 0.302 & 0.328 & 0.344 & 0.361 & 0.375 \\  & 0.640 & 0.793 & 0.933 & 1.093 & 1.213 \\  & 0.232 & 0.245 & 0.249 & 0.255 & 0.258 \\  & 0.615 & 0.756 & 0.903 & 1.043 & 1.170 \\  & 0.106 & 0.125 & 0.136 & 0.149 & 0.157 \\  & 0.643 & 0.771 & 0.899 & 1.020 & 1.116 \\  & 0.113 & 0.116 & 0.125 & 0.134 & 0.143 \\  & 0.665 & 0.829 & 0.966 & 1.117 & 1.189 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: \(U_{\rm 0 mf}\) and \(\Delta P_{\rm mf}\) of each experiment. Figure 6: 3D views of the particle size and material mass versus (a) \(U_{\rm 0 mf}\) and (b) \(\Delta P_{\rm mf}\). From Equations (4) and (5), it can be further deduced that \(U_{\rm h}\) is the function value of \(r_{\rm h}\), and the deduced equation is as follows:  
\[U_{\rm h}=U_{0}\frac{S_{0}}{S_{\rm h}}=U_{0}\frac{r_{0}^{2}}{r_{\rm h}^{2}} \tag{10}\]  
As Figure 7 shows, according to the trigonometric function equation:  
\[r_{\rm h}=r_{0}+h\rm{tan}(\alpha) \tag{11}\]  
\[r_{1}=r_{0}+H_{0}\rm{tan}(\alpha) \tag{12}\]  
To obtain a precise \(H_{0}\) value for each experiment in Table 3, the following equation is drawn based on the relationship of volume times density is equal to mass:  
\[\frac{1}{3}H_{0}\pi(r_{0}^{2}+r_{0}r_{1}+r_{1}^{2})(1-\varepsilon_{0})\rho_{ \rm s}=W \tag{13}\]  
By solving the combined equations of Equations (12) and (13), the \(H_{0}\) value can be obtained. In this study, an exhaustive method is used, and the programming and calculation are conducted in MATLAB. The specific method is to assign all the values of every 0.0001 in 0-0.1 to \(H_{0}\) (unit: m) and substitute \(H_{0}\) into Equations (12) and (13); then, calculate the equation error of Equation (13); finally, the best-matched \(H_{0}\) value can be obtained by finding the minimum absolute equation error. The calculated \(H_{0}\) values of all experiments are listed in Appendix A.