*3.8. Flow Pattern Maps*

The classification results from the most accurate classifier (MLP in this case) were used to develop a flow pattern map for both ethanol and FC-72. Once the classifier predicted the classes for each data point in the testing set (those in the training set were already stored during training), these points were used to estimate the values for the flow pattern maps. As a result, the maps can be used as a graphic tool for visualizing the outcomes from the classification methods. The *x*-axis corresponds to *Bo*0.5 *<sup>l</sup>* , and the *y*-axis corresponds to *Fr*0.5 *<sup>l</sup> We*0.25 *<sup>l</sup>* , in accordance with the correlation between process conditions (velocity and acceleration) and the effect of the different forces acting on the fluid (namely, inertial, external, and related to the surface tension) proposed by Pietrasanta et al. [29]. The resulting flow pattern maps for both fluids developed by the authors are illustrated in Figure 6. These maps can act as a reference for comparison with the flow pattern maps from the MLP classifier.

For both working fluids, a much clearer transition zone was found when comparing the previous flow pattern maps and those based on the MLP classifier. This was due to the inherent improvements brought about by the use of the MLP method, as this algorithm provides a more systematic mean for classification compared to visual categorization or empirical correlations with physical properties.

Figure 7 shows the flow pattern map for ethanol. The map clearly shows a threshold value where the transition from slug/plug to semi-annular flow took place, located approximately where the *x*-axis was equal to 4. Higher values along this axis indicate semiannular flow, where surface tension no longer dominated the fluid flow, and the increased acceleration led to higher bubble lengths. Semi-annular flow can be further identified on the *y*-axis, where for values of *Fr*0.5 *<sup>l</sup> We*0.25 *<sup>l</sup>* lower than 2, a relatively high density of points classified as semi-annular flow was encountered. Lower values on both axes indicated the presence of slug-plug flow, either because the PHP device was not active or because the external forces were not strong enough to prevail over the surface tension of the working fluid while operating.

**Figure 6.** Flow pattern maps for ethanol (**a**) and FC-72 (**b**) proposed by Pietrasanta et al. [29]. Reproduced with permissions.

**Figure 7.** Flow pattern map for ethanol: multilayer perceptron.

In the case of FC-72, its corresponding flow pattern map is shown in Figure 8. Similar to the case of ethanol, a threshold value for the transition zone was found. Here, the threshold was located approximately when *Bo*0.5 *<sup>l</sup>* = 9. This means that slug/plug flow prevailed for higher velocities and bubble lengths when FC-72 was used as working fluid. This can be explained by the differences in the surface tension of both fluids. FC-72 has a lower surface tension than ethanol; hence, for the same dynamic conditions (fluid velocity and acceleration), greater numbers of *Bo*0.5 *<sup>l</sup>* would be reached before the flow regime transition. Abnormal points were found within the slug/plug region for FC-72, which were classified as semi-annular. This phenomenon could have been caused by the propagation of visual errors, as discussed previously. This would also mean that the choice of *Bol* did not properly reflect the surface tension effects, since this number could not capture the regimes for both fluids.

**Figure 8.** Flow pattern map for FC-72: multilayer perceptron.

To overcome the effect of different surface tension in both working fluids, a modified term was used on the *x*-axis of both flow pattern maps. The term *Bo*0.5 *<sup>l</sup>* was scaled using the ratio *<sup>σ</sup><sup>l</sup> <sup>σ</sup>ref* , where *σ<sup>l</sup>* represents the surface tension of the working fluid, and *σref* is a reference surface tension, which in this case was that of ethanol. Surface tension for both fluids was estimated via validated correlations that depend on key operating conditions such as saturation temperature [29]. This correction ratio was calculated for each observation, and updated flow pattern maps for both fluids were developed (using the MLP classifier). Note that, since ethanol was used as reference, no changes were found in its flow pattern map. The updated flow pattern maps for ethanol and FC-72 are depicted in Figures 9 and 10, respectively.

The updated flow pattern maps exhibited more consistent threshold values on both axes, for both working fluids. In the case of ethanol, these values were 6 for the *x*-axis and 2 for the *y*-axis, whereas, for FC-72, these limits were located at 5 for the *x*-axis and 1 for the *y*-axis. These values allowed for more interpretability, as it was now possible to cluster the observations and determine their corresponding flow regime on the basis of their relative location to the threshold values, with a margin of only ±1 unit on each axis.

Overall, the predictions presented good correspondence with experimental results, and the use of modified numbers plus scaling allowed for a clearer differentiation of flow regimes. However, it is worth noting that, although a relatively large number of data were used, this only represents a single PHP design (e.g., the single-loop PHP). Therefore, any attempt to implement these classifiers in a different system would most likely provide less accurate predictions, and a more extensive dataset would be needed.

**Figure 9.** Updated flow pattern map for ethanol: multilayer perceptron.

**Figure 10.** Updated flow pattern map for FC-72: multilayer perceptron.
