## 2 The Experimental Dataset  
The experimental facility is a non-pressurized vertical pipe made of Plexiglas@with d\({}_{\text{c}}\) = 0.24 m and H\({}_{\text{c}}\) = 5.3 m (Figure 1). A pressure reducer controls the pressure upstream from the rotameters (1) and (2), used to measure the gas flow rate (accuracy \(\pm\) 2% full scale., E5-2600/h, manufactured by ASA, Italy). The gas distributor is a spider-sparger distributor with hole diameters of d\({}_{\text{o}}\) = 2-4 mm. The spider sparger has six arms made of 0.12 m diameter stainless steel tubes soldered to the center cylinder of the sparger. The sparger has been installed with the six holes located on the side of each arm facing upward. The reader should refer to our other paper [1] for photography of the experimental setup as well as the sparger design. The diameter of the column, its height, and the sparger openings have been chosen considering the well-known scale-up criteria reported in the literature: generally, a diameter greater than d\({}_{\text{c}}\) > 0.15 m, an aspect ratio (the ratio between the liquid free level and the column diameter) larger than 5, and sparger openings larger than d\({}_{\text{o}}\) > 1 mm guarantee results that could be used for scaling up, accordingly with the widely accepted scale-up criteria of [PERSON] et al. [5]. Following the \"[PERSON] scale-up criteria,\" we suppose that our experimental results can be representative of \"industrial-scale\" bubble column reactors. It is worth noting that the column diameter (d\({}_{\text{c}}\) = 0.24 m) makes this facility a large-diameter pipe, considering the dimensionless diameter (see, for example, reference [6]). The experimental facility was used in our previous studies to pursue three lines of research. First, experimental methods to estimate the bubble size distributions and shapes were proposed and tested. Second, the influence of the counter-current operation mode and the presence of internals on the bubble column fluid dynamics was studied. Third, the influence of the liquid phase properties was investigated and, in particular, we studied the effect of (i) viscous solutions and (ii) active compounds. In particular, two kinds of surface-active compounds were studied: (i) organic and (ii) inorganic substances. The reader should refer to our previous paper for further details [1]. In particular, the dataset used in the present paper has already been described  
Figure 1: The experimental setup. in a previous paper [1], where we applied a variety of experimental techniques to investigate the air-water bubble column hydrodynamics: (i) gas holdup, (ii) gas disengagement, (iii) image analysis, and (iv) optical probe measurements. The gas holdup measurements were used to investigate the flow regime transition from the homogeneous towards the transition flow regimes. The gas disengagement measurements were used to further investigate the flow regime transition and study the structure of the gas holdup curve. The image analysis was used to study the bubble size distributions and shapes near the sparger and in the developed region of the column for different gas and liquid velocities (in the \"pseudo-homogeneous\" flow regime). The optical probe was used to acquire radial profiles of the local properties (i.e., local void fraction and bubble rise velocity) to study the flow properties and further investigate the flow regime transition. The reader should refer to our previous study [1] for all the details concerning the experimental data. In this paper, we have used the previously-published dataset (BSDs obtained at 1.9 m above the gas sparger) to set up and validate a PBM approach; in particular, five cases with different gas superficial velocities have been investigated in the batch mode: (i) U\({}_{\text{G}}\) = 0.0037 m/s, (ii) U\({}_{\text{G}}\) = 0.0074 m/s, (iii) U\({}_{\text{G}}\) = 0.0111 m/s, (iv) U\({}_{\text{G}}\) = 0.0149 m/s, and (v) U\({}_{\text{G}}\) = 0.0188 m/s.