## 4 Results and Discussion
### Mathematical Modeling
#### 4.2.2 Dimensional Analysis  
To obtain a dimensionless correlation, we first determine the factors that influence the hydrodynamic behavior of the fluidized bed, which are the particle size (\(d_{\text{p}}\)), initial bed height (\(H_{0}\)), bottom diameter (\(D_{0}\)), initial bed voltage (\(\varepsilon_{0}\)), particle sphericity (\(\varphi\)), particle true density (\(\rho_{\text{s}}\)), gas density (\(\rho_{\text{g}}\)), gas viscosity (\(\mu\)), angle of repose (\(\theta\)), Hausner ratio (\(R\)). These factors are recast into several dimensionless groups containing the Reynolds number (\(\text{Re}_{\text{mf}}=\frac{\rho_{\text{g}}k_{0\text{mf}}d_{\text{p}}}{\mu}\)), Archimedes number (\(\text{Ar}=\frac{gd_{\text{p}}^{3}\rho_{\text{g}}\left(\rho_{\text{g}}-\rho_{ \text{g}}\right)}{\mu^{2}}\)), \(\frac{\Delta P_{\text{mf}}}{\rho_{\text{g}}\delta H_{0}}\), \(\frac{\varepsilon_{0}}{\phi}\), \(\frac{H_{0}}{D_{0}}\), \(\frac{d_{\text{p}}}{D_{0}}\), \(\frac{\theta}{R}\), and the correlations among these above dimensionless groups with undetermined coefficients are as follows:  
\[\text{Re}_{\text{mf}}=a_{1}\cdot\left(\text{Ar}\right)^{b_{1}}\cdot\left( \frac{\varepsilon_{0}}{\phi}\right)^{c_{1}}\cdot\left(\frac{H_{0}}{D_{0}} \right)^{d_{1}}\cdot\left(\frac{d_{\text{p}}}{D_{0}}\right)^{e_{1}}\cdot\left( \frac{\theta}{R}\right)^{f_{1}} \tag{28}\]  
\[\frac{\Delta P_{\text{mf}}}{\rho_{\text{g}}gH_{0}}=a_{2}\cdot\left(\text{Ar} \right)^{b_{2}}\cdot\left(\frac{\varepsilon_{0}}{\phi}\right)^{c_{2}}\cdot \left(\frac{H_{0}}{D_{0}}\right)^{d_{2}}\cdot\left(\frac{d_{\text{p}}}{D_{0}} \right)^{e_{2}}\cdot\left(\frac{\theta}{R}\right)^{f_{2}} \tag{29}\]  
To obtain the undetermined coefficients of Equations (28) and (29), both linear regression and nonlinear regression methods can be used. For the linear regression method, first, take the logarithm of both sides of Equation (28) as follows:  
\[\text{lg}(\text{Re}_{\text{mf}})=\text{lg}(a_{1})+b_{1}\cdot\text{lg}(\text {Ar})+c_{1}\cdot\text{lg}\left(\frac{\varepsilon_{0}}{\phi}\right)+d_{1} \cdot\text{lg}\left(\frac{H_{0}}{D_{0}}\right)+e_{1}\cdot\text{lg}\left( \frac{d_{\text{p}}}{D_{0}}\right)+f_{1}\cdot\text{lg}\left(\frac{\theta}{R}\right) \tag{30}\]  
Then, the \"regress\" function in MATLAB can be used to fit the undetermined coefficients, i.e., \(a_{1}\), \(b_{1}\), \(c_{1}\), \(d_{1}\), \(e_{1}\), and \(f_{1}\)  
For the nonlinear regression method, the \"nlinfit\" function in MATLAB is used to fit the undetermined coefficients of Equations (28) and (29) directly, and the results of the linear regression are set as the initial values of iteration. The results of the linear and nonlinear regression methods are shown in Tables 6 and 7 and Figure 11. Based on the values of R\({}^{2}\) and RMSE, there is little difference between the fitting accuracy of the linear and nonlinear regression methods, especially for \(\Delta P_{\text{mf}}\), where the models are almost the same. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \(a_{1}\) & \(b_{1}\) & \(c_{1}\) & \(d_{1}\) & \(e_{1}\) & \(f_{1}\) & \(\text{RMSE}\) (m/s) & \(\text{R}^{2}\) (-) \\ \hline Linear regression & 1 & 0.504 & \(-\)1.355 & 0.326 & 0.236 & \(-\)0.635 & 0.011 & 0.991 \\ Nonlinear regression & 1.033 & 0.509 & \(-\)1.142 & 0.186 & 0.232 & \(-\)0.645 & 0.009 & 0.994 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results of the linear and nonlinear regression methods of \(U_{0\text{mf}}\) for the dimensionless model. Since the linear regression has better convergence ability than the nonlinear regression and does not need to assign initial values, it is more practical.