## 4 The Polydisperse Drag Model
### Drag Model Validation  
To verify the validity of the ANN with OCRA drag model in the polydisperse system, we compare the prediction results of ANN that used OCRA with the well-known BVK formula [27], Cello et al. formula [21], and HS formula [30] in polydisperse and monodisperse systems. The empirical formula and prediction results are shown in Table 10 and Figures 8 and 9. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Training Samples** & **Contribution Rate** & **Step** & \(E_{MSE}\) & \(E_{MAE}\) & **CPU Time (s)** \\ \hline
1290 (100\%) & 1 & 0 & 0.2955 & 0.6879 & 278.5 (100\%) \\
1092 (85\%) & 0.9375 & 4 th & 0.4725 & 0.7891 & 223.3 (80\%) \\
1067 (83\%) & 0.921875 & 6  
As shown in Figure 8a,b, when we explore the drag force prediction of \(\varepsilon_{g}=0.35\) in the polydisperse system, the ANN with OCRA model achieves the best predicted accuracy for medium-sized granules (\(d=3d_{0}\)) compared to the Cello et al. and HS formulas. The BVK formula suffers the lowest accuracy of all four drag models. For the larger-sized granules (\(d=4d_{0}\)), [PERSON] et al. 's formula shows the best performance among the 4 drag models; especially, these advantages will become more obvious when \(Re_{p}\in[300,550]\). On the other hand, the ANN with OCRA has the lowest predicted accuracy compared to the HS and BVK drag models. However, when the 4 drag models were considered in a polydisperse diluted granular system (\(\varepsilon_{g}=0.7\)), the accuracy of the ANN with OCRA drag model is better than the drag model by HS, Cello et al., and BVK in general. In addition, when we explore the drag prediction of these models in a monodisperse granular system (see Figure 9 and Table 11), the accuracy of ANN with OCRA is better than the BVK and HS drag models in terms of medium-sized granules (\(d=3d_{0}\)) with void fractions \(\varepsilon_{g}=0.5\) and \(\varepsilon_{g}=0.85\); the Cello et al. drag model has the lowest predicted accuracy. Similar results are  
Figure 8: Predictions of different drag models in polydisperse system [21,27,30]. shown in the granules (\(d=4d_{0}\)) of a monodisperse granular system; ANN with OCRA has the best prediction accuracy, followed by the HS drag model, and the BVK and Cello et al. drag models have the lowest prediction accuracy for granules (\(d=4d_{0}\)) in \(\varepsilon_{g}=0.5\) and \(\varepsilon_{g}=0.85\), respectively. The simulation results demonstrate that ANN with OCRA has both advantages and disadvantages in different physical conditions. For the advantages, to start with, the ANN with OCRA drag model has obvious advantages in predicting the drag force of medium-sized granules in the polydisperse granular system compared to the other three analytic drag formulas. Next, ANN with OCRA performs well in predicting the larger-sized granules in a relatively dilute polydisperse granular system. Finally, the performance of ANN with OCRA is generally satisfied with the predicting results in a monodisperse granular system. For the disadvantages, ANN with OCRA does not achieve the ideal predicted results in the relatively dense granular system for larger granules. Taking into account this, it is advised to adopt [PERSON] et al. 's drag model for prediction. Therefore, it can be concluded that the ANN with OCRA drag model is good at the prediction of  
Figure 9: Predictions of different drag models in monodisperse system [21, 27, 30]. polydisperse granules with a relatively wide concentration and has better compatibility in a monodisperse granular system. There may be three reasons that we have inferred. First, due to the analytic formula, which the other three formulas hold, having a fixed structure, a fixed functional structure has difficulty accurately describing the characteristics of two different flow systems indeed. Second, the ANN has the advantages of self-learning ability and adequate compatibility. The drag force model based on ANN might be easier for switching accurately between monodisperse and polydisperse systems.