# X-ray Particle Tracking Velocimetry in an Overflowing Foam
## 2 Materials and Methods
### Overflowing Foam  
The flow experiment on the overflowing foam was performed in a laboratory-scale flow channel. Its design was motivated by the froth flow on the top of an industrial-scale froth flotation cell. In the view of the X-ray radiographic measurements, the flow channel had a square cross-sectional area of \(100\,\mathrm{mm}\times 100\,\mathrm{mm}\) and was entirely made of acrylic glass (PMMA) with a wall thickness of \(10\,\mathrm{mm}\). The X-rays had to penetrate the channel walls having a total thickness of \(d_{w}=2\cdot 10\,\mathrm{mm}=20\,\mathrm{mm}\) perpendicular to the X-ray beam direction, thus reducing the X-ray intensity by approximately 30% (Section 2.4). At the bottom of the flow channel, a bubble generator was submerged in a surfactant solution. The essential parts of the bubble generator were four tubes made of high-density polyethylene with a hydrophobic surface. The tubes had an outer diameter of \(19\,\mathrm{mm}\) and a wall thickness of approximately \(3\,\mathrm{mm}\). Produced by sintering, the tube wall was porous, having a pore size of \(60\,\mathrm{\SIUnitSymbolMicro m}\). Four sections of this tube, each \(12.5\,\mathrm{mm}\) long, were aligned vertically and placed equidistantly on a 3D-printed basis, which served for feeding and distributing compressed air into these tube sections. Charging the bubble generator with compressed air at a constant volumetric flow rate \(\dot{V}=0.05\,\mathrm{L}\,\mathrm{s}^{-1}\) by means of a mass flow controller (FMA-2608A, Omega) resulted in \(0.5\,\mathrm{cm}\,\mathrm{s}^{-1}\) superficial gas velocity, which is comparable to industrial-scale froth flotation cells [43]. The bubbles were generated continuously and homogeneously distributed in the  
Figure 1: Overflowing foam is imaged in a laboratory-scale experiment by means of X-ray radiographic measurements employing customised tracer particles. Local flow velocities are determined using X-ray particle tracking velocimetry. Near the weir, the liquid fraction of the foam is monitored by measuring its electric conductivity between three different electrode pairs (EP1, EP2, EP3). The red, green and blue lines highlighted in the X-ray image on the right indicate the motion paths of three different tracer particles, particularly their radiopaque features. surfactant solution. Accordingly, above the liquid surface, the foam flow developed similar to a plug flow, i.e., no major inhomogeneity in the superficial gas velocity was visible. The fresh foam moved upwards in a vertical direction in the flow channel, then flowed off at a weir, i.e., at a one-sided horizontal overflow, and into the open surrounding. After passing the weir, the foam slid down at the outer channel wall, accelerated by gravity. The filling level of the surfactant solution was 50 mm above the bubble generator and 250 mm below the weir. When the foam overflowed, liquid was transported out of the flow channel and, therefore, the filling level above the bubble generator was decreasing. At the given volumetric flow rate (\(\dot{V}=0.05\) L s\({}^{-1}\)) and liquid fraction of the foam (\(\phi_{l}\approx 10\%\), Section 2.2, Figure 3), the estimated decrease of the filling level equalled 15 mm during a 30 s measurement run. To slow down the decrease of the filling level, the flow channel was connected via a tube to a reservoir. Its cross-sectional area was approximately five times larger than the cross section of the flow channel, and, consequently, the filling level decreased by only a few millmeters per measurement run. Furthermore, the reservoir was placed on a laboratory lifting stage in order to re-adjust the filling level before starting each measurement run. To prepare the surfactant solution, sodium dodecyl sulfate (SDS) was dissolved in tap water at room temperature (\(\theta=20\,^{\circ}\)C). The SDS concentration was varied as listed in Table 1: either 1.0 g L\({}^{-1}\) or 0.5 g L\({}^{-1}\). In both cases, the concentration was below the critical micelle concentration \(CMC=2.3\) g L\({}^{-1}\)[44].