## 3 Results and Discussion
### Comparing Simulations to Experiments  
The simulation approach to sampling was slightly different to the experiments because the meshing of the domain in the simulation did not allow for the sampling of particles in a vertical plan and thus, particles were sampled horizontally and averaged over space time to get the particle concentration in a line. The simulation ran for 30 s, after which the relevant data was extracted, including the particle concentrations and positions at a defined export plane. In this instance, a reference plane of 220 mm from the silo base, as shown in Figure 16, has been selected. The basis of this selection is from analysing the peak concentration areas in the silo from experimental testing. The exported data was analysed using post processing tools to obtain the CRF value.  
As discussed earlier, the main particle stream was modelled using the 'coarse' particle set. As this is not of interest in the analysis, this region (diameter of 27.5 mm) was excluded from post processing as shown in Figure 16b.  
The CRF was now used to compare data from experiments and simulations. The comparison was based on the following assumptions:  
* The 220 mm plane contains both the C\({}_{\mathrm{W}}\) and C\({}_{\mathrm{H}}\) values, as this is where the laser sheet intensity is greatest. The simulation results compare the nodes at 219.06 mm and 224.04 mm (as shown in Figure 17); this is due to the previously mentioned effects of node averaging.
* The X and Z coordinates for these nodes can be compressed into a 2D line by considering their polar coordinates.  
Figure 15: CRF value for alumina at flow rates of 1.9 \(\times\) 10\({}^{-2}\) kg/s.  
The concentration profile after 30 s of silo filling in the simulation is shown in Figure 16a. At this time step the simulation had reached steady state. Figure 16b provides concentration values obtained from the nodes above and below the sampling plane. The \(X\) and \(Z\) coordinates are converted using Equation (2) so that concentrations of equal distance from the silo centre can be averaged.  
\[\sqrt{X^{2}+Z^{2}}=R \tag{2}\]  
The \(C_{W}\) and \(C_{H}\) data for calculating the CRF values were extracted for each time step of the simulation for the averaged nodes on the plane as shown in Figure 17. These values were then averaged for several time steps after the simulation reached steady-state (assumed due to steady flow rate of material and recirculation) and compared to the CRF values obtained in the experiments. This analysis resulted in an average value of 0.001847 kg/m\({}^{3}\) for the highest concentration and a value of 0.000677 kg/m\({}^{3}\) for the concentration near the wall, see example of alumina data in Figure 18 leading to a CRF value of 0.37.  
Figure 16: (**a**) Isometric view of the 3D silo highlighting the 2D sample plane location. (**b**) Top view of the 220 mm sample plane depicting the particle mass concentrations used in the simulation analysis.  
Figure 17: Node locations considered for the 220 mm plane; side elevation view.  
The values between simulations and experiments for the alumina material are compared in Figure 19. Even so the simulation slightly over predicted the k value obtained in the alumina experiments the values were somehow in a similar region which shows the applicability of the CRF value for comparing CFD simulations with experiments. The difference between the simulation results and the experiments may be due to a number of reasons, the first being the smaller time frame considered. In this work only 16 data points were used; the experiments used the full duration of video footage (2 min at 60 frames per second). Additionally, the simulation has considered the centre of the laser sheet intensity only. When comparing this to the experimental results, the image processing in the experiments considered a wider region (112.5 mm wide as shown in Figure 7). Therefore, a more advanced search algorithm for the CFD data would be required to analyse a wider section of the silo.