# X-ray Particle Tracking Velocimetry in an Overflowing Foam
## 4 Discussion
### X-ray Beam Characteristics  
X-ray radiography was used for visualising the metal beads as radiopaque features of the tracer particles carried by the flowing foam, thus determining the translational velocity and intrinsic rotation of each tracer particle along its motion path. In this regard, measurement uncertainties mainly arise from the X-ray beam characteristics and the principle of transmission imaging in combination with the three-dimensional measurement volume. The tracer particles shown in Figure 4c-e were recorded in the same measurement run, of which the X-ray images in Figures 1, 6 and 7 show a snapshot. As these tracer particles were imaged at different positions (\(x^{i},y^{i}\)) in the field on view, the image background stemming from the X-ray attenuation by the liquid foam is slightly different in each case. Moreover, the tracer particles were added into the flowing foam at different positions \(z_{0}\), i.e., along the X-ray beam direction, by means of small tubes in the bottom left part of the foam channel, which are also visible in Figures 1, 6 and 7. The projections of the metal beads in the X-ray images are affected by geometrical unsharpness: compared to the metal beads of tracer particles that were as close as possible to the scintillation screen (Figure 4d: \(z_{0}=-45\) mm), the blurring is more pronounced the further the tracer particles are away from the scintillation screen (Figure 4c: \(z_{0}=0\) mm, Figure 4e: \(z_{0}=+45\) mm). Geometrical unsharpness is caused by the two-dimensional extension of the X-ray tube focal spot, which cannot be considered as a point-like source. The \(z\)-position of the tracer particle in the measurement volume mainly determines the geometrical unsharpness; it is reduced if the distance between the tracer particle and the scintillation screen becomes smaller. This basically means that a certain geometrical unsharpness is unavoidable when X-PTV measurements are performed in three-dimensional configurations. However, considering the X-ray tube and the radiography arrangement, the geometrical unsharpness can be reduced by a small focal spot and a large distance to the scintillation screen. Here, using the measurement arrangement as described in Section 2.4, the estimated geometrical unsharpness ranges from approximately 0.05 mm (at \(z=-50\) mm, i.e., closest to the scintillation screen) to almost 0.7 mm (at \(z=+50\) mm, i.e., furthest from the scintillation screen), which is comparable to the image pixel size and the metal bead diameter, respectively. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Combination No.** & **1** & **2** & **3** & **4** \\ \hline polymer tetrahedron: edge length & 4 mm & 2 mm & 4 mm & 2 mm \\ metal beads: diameter & 0.5 mm & 0.5 mm & 0.3 mm & 0.3 mm \\ tracer particle: weight-to-area ratio & 14 N m\({}^{-2}\) & 21 N m\({}^{-2}\) & 11 N m\({}^{-2}\) & 9 N m\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of the tracer particles used for the X-PTV measurement (No. 1) and further feasible combinations of the edge length of the polymer tetrahedron and the diameter of the metal beads, as well as the resulting weight-to-area ratio of one tracer particle. As a further consequence of the typical geometrical unsharpness in the X-ray images, the coordinates \(\mathbf{X_{b}^{i}}\) of the metal beads and \(\mathbf{X^{i}}\) of the corresponding tracer particle along its motion path cannot be effectively determined with sub-pixel resolution. Achieving a sub-pixel resolution would be the key to the precision of the angle \(\omega^{i}\), which describes the intrinsic rotation of the tracer particles and could be used as a direct measure of the vorticity of the foam flow. Another option would be to adapt Equation (8) by weighting \(\omega^{i}_{g}\) with the distance between the respective metal beads. As a result, the cases where the projections of the metal beads are very close to each other and the angular rotation \(\omega^{i}_{g}\) is determined to be rather imprecise would have less significance.