## 3 Results and Discussion
### Prediction of Droplet Characteristics  
The primitive parameters could not be used as the input variables to predict the droplet characteristics since they would make the prediction model lose the physical meaning. Therefore, this paper only discussed the different solution sets with dimensionless numbers to predict the droplet characteristics. The data sets of Experiments 1 and 2 had been divided into two parts, Region A and Region B, with the ratio of about 4:1, in which the data in Region A and Region B were used to establish and verify the prediction model. In Region A, the data were further divided into the training data and testing data with the ratio of 4:1. The data partition of slug and dripping patterns in Set1 was separated by the line with \(We_{d}\)=0.02 in Experiment 1 (Table 4a,b). The data in Region B was above this line, inversely the data in Region A was below it. Similarly, the data partition of slug and dripping patterns in Set2 and Set3 were separated by the line with \(We_{d}\) = 0.02 and \(Ca_{d}\) = 0.002, respectively  
Table 5 is the data partition of slug and dripping patterns in three sets in Experiment 2. The partition line adopted the same parameters following it in Experiment 1. (Table 4c-f). Figure 4: The flow patterns diagram of 2 group experiments: (**a**) Experiment 1, (**b**) Experiment 2. The BPNN algorithm was guided to predict the droplet characteristics, which were \(\pi_{13}\) and \(\pi_{14}\). Before the data inputting, the normalized \(x^{\prime}\in\) (0,1) was identified to increase the convergence of the algorithm (Equation (5)). \[x^{\prime}\ =\ \frac{x-x_{min}}{x_{max}-x_{min}} \tag{5}\]  
Where \(x\) is the number value, \(x^{\prime}\) is the normalized number value, and \(x_{max}\) and \(x_{min}\) are maximum and minimum values of \(x\), respectively. After normalization, the data in Region A had been input to the BPNN algorithm establishing the prediction model of droplet characteristics in different sets. Then the data in Region B were input to the prediction model predicting \(\pi_{13}\) and \(\pi_{14}\) and getting the result the parameters (\(L_{d}\) and \(f\)) by inversed operation. Table 6 shows the model errors of slug and dripping patterns predicted by three sets in Experiment 1 and Experiment 2. The errors were obtained by comparing the predicted value with the measured value in all samples from Region B. \begin{table}
\begin{tabular}{c  
According to Experiment 1, the model errors of droplet characteristics in slug and dripping patterns are shown in Table 6a,b. Compared with the measured values, the prediction model of droplet length \(L_{d}\) and droplet frequency \(f\) in slug pattern occupied the error range within \(\pm\) 20% both in Set1 and Set2, and more than \(\pm\) 20% in Set3 (Table 6a). The same phenomenon was found in the dripping pattern (Table 6b). The model errors of droplet characteristics, \(L_{d}\) and \(f\), in slug and dripping patterns in Experiment 2 were shown in Table 6c,d. It was clear that, compared with the measured values, the model errors of length \(L_{d}\) and frequency \(f\) in slug droplet were in the range of \(\pm\) 20% regardless of Set1, Set2, and Set3 (Table 6c). Similarly, the prediction model occupied the error in \(\pm\) 20% comparing with the measured values in dripping pattern in all three sets (Table 6d). It was true that the model error of Set3 was worse than it of Set1 and Set2 in Experiment 1; however, that of Set1, Set2, and Set3 in Experiment 2 was similar. The accuracy difference of the prediction model among the three sets was mainly expressed by the relationship between dimensionless numbers and their physical meanings. In all three sets, the predicted parameters of \(L_{d}\) and \(f\) were mapped by the dimensionless numbers of \(\pi_{13}\) and \(\pi_{14}\). It was obvious that the accuracy difference with slug and dripping patterns in different experiments had been strongly related to the values of \(\pi_{13}\) and \(\pi_{14}\) (Figures 5 and 6).