## 3 Results
### The Comparison of the \(\Delta p\) Model Developed by RSM and Ergun Model
#### 3.4.2 The Single Factor Analysis of the \(\Delta p\)  
In order to validate fitting performance of the \(\Delta p_{\text{RSM}}\) and \(\Delta p_{\text{Erg}}\). The second-order polynomial RSM model was transmitted into three first-order quadratic models by substituting the independent variables shown in Table 3 into Equation (8), the single factor pressure drop models based on the \(\Delta p_{\text{RSM}}\) are as follows:  
\[\Delta p_{Mc}=-72.8038915+19.7482445 Mc-0.17375 Mc^{2} \tag{9}\]  
\[\Delta p_{v_{air}}=14.4119253+175.678453v_{air}+645.768v_{air}^{2} \tag{10}\]  
\[\Delta p_{L}=-12.72658565+346.593494L-22.25679L^{2} \tag{11}\]  
Based on the above Equations (6), (9)-(11), the variations of \(\Delta p\) with \(Mc\), \(v_{air}\), and \(L\) can be obtained and, respectively, depicted in the following Figures 7-9. As can be seen from Figure 7 that the \(\Delta p\) is proportional to the \(Mc\), and the variation range of the \(\Delta p\) was ascertained to be 86.38-197.57 Pa. It can be summarized from Figure 2 that the equivalent diameter \(d_{g}\) increases with the \(Mc\) while porosity of the bed layer \(\varepsilon\) decreases with the \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Statistical Indexes**} & **RSM Model** & **Ergun Model** \\ \cline{2-3}  & \(\Delta\,p_{\text{RSM}}\) & \(\Delta\,p_{\text{Eur}}\) \\ \hline Determination coefficient (\(R^{2}\)) & 0.9948 & 0.9868 \\ Mean-square error (\(MSE\)) & 10.9021 & 15.7041 \\ Mean absolute error (\(MAE\)) & 2868.9372 & 7292.4115 \\ Root-mean-square error(\(RMSE\)) & 12.9908 & 20.7115 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The statistical results of the RSM experiment for comparing \(\Delta p_{\text{RSM}}\) and \(\Delta p_{\text{Eur}}\). increase of the \(Mc\), indicating that the volume of the air pathways decreased with the increase of \(Mc\)[40], and further lead to the flow resistance of the air passing through the bed layer will be increased with the increase of the \(Mc\). On the other hand, it can be seen from Figure 7 that the \(\Delta p_{RSM}\) are more closed to the experimental \(\Delta p\) than the \(\Delta p_{Eng}\). Figure 8: The variations of \(\Delta p\) with \(v_{air}\) under the constant \(Mc\) of 18.9 %w.b. and \(L\) of 0.5 m.  
Figure 8 depicts the variations of \(\Delta p\) with \(v_{air}\) under the constant \(Mc\) of 18.9 %w.b. and \(L\) of 0.5 m. Similar with the variation trend of the \(\Delta p\) shown in the above Figure 7, \(\Delta p\) also increases with increase of \(v_{air}\), and the variation range of the \(\Delta p\) was ascertained to be 38.44-352.92 Pa. Moreover, it can be seen from Figure 8 that the \(\Delta p_{Eg}\) with the \(v_{air}\) of 0.1 m/s and 0.15 m/s are more closed to the corresponding experimental \(\Delta p\) than the \(\Delta p_{RSM}\), indicating that the Ergun model has a better prediction performance than the developed RSM model when the \(v_{air}\) is under 0.2 m/s. The same conclusions have been made by [PERSON] et al. [41] and [PERSON] et al. [10]. However, in this case, the developed \(\Delta p_{RSM}\) model has a better performance than the Ergun model when the \(v_{air}\) is above 0.2 m/s, which may be more suitable for guiding the actual process. The variations of \(\Delta p\) with \(L\) under the constant \(v_{air}\) of 0.35 m/s and \(Mc\) of 18.9 %w.b. are depicted in Figure 9. It can be seen from the figure that the \(\Delta p\) linearly increases with the increase of \(L\), and the \(\Delta p\) gets the minimum (27.68 Pa) and maximum value (276.83 Pa), respectively, at the \(L\) of 0.1 and 1m. On the other hand, it can be also seen from Figure 9 that, for the all levels of \(L\), the fitting performance of the \(\Delta p_{RSM}\) is better than that of the \(\Delta p_{Eg}\).