## 4 Results and Discussion
### Boundary Conditions and Simulation Settings
#### 4.2.2 Solids Volume Fraction Distribution
Figure 7 illustrates the relationship between the frictional pressure and the solids volume fraction, showing that the frictional pressure increases as the solids volume fraction \(\alpha_{s}\) increases. When the solids volume fraction is below 0.43, Johnson & Jackson frictional pressure is the smallest and exhibits the slowest growth rate, while Berzi frictional pressure is the highest initially before its growth rate is eventually surpassed by Dartevelle frictional pressure. When the solids volume fraction is between 0.43 and 0.47, Dartevelle frictional pressure increases rapidly, far exceeding the other two, and its growth rate also becomes the largest. Johnson and Jackson frictional pressure remains the smallest, but its growth rate eventually surpasses Berzi frictional pressure. When the solids volume fraction is between 0.47 and 0.50, Berzi frictional pressure becomes the smallest and exhibits the slowest growth rate, while Dartevelle frictional pressure remains the largest and has the fastest growth rate. Thus, due to the influence of frictional pressure, there are significant differences in the predicted distribution of the solids volume fraction (particularly in the deposited region) in the three frictional stress models (as shown in Figures 5 and 6). In summary, Dartevelle frictional stress model predicts the lowest solids volume fraction and the steepest solids volume fraction gradient in the deposited region of horizontal pipe, while the modified Berzi frictional stress model predicts the highest solids volume fraction and the gentlest solids volume fraction gradient in the deposited region of horizontal pipe. Comparison with the ECT images in Figure 5a shows that the simulation results obtained using the modified Berzi frictional stress model are closer to the experimental data. Figure 6: Solids volume fraction distribution along the height of horizontal pipe simulated by different frictional stress models: (**a**) Dartevelle frictional stress model; (**b**) Srivastava and Sundaresan frictional stress model; (**c**) Modified Berzi frictional stress model. Figure 7: Relationship between frictional pressure \(p\) and solids volume fraction. Frictional viscosity or shear stress is directly proportional to the frictional pressure and mainly reflects the resistance characteristics or shear resistance characteristics during interparticle sliding motion. As the frictional viscosity or shear stress increases, the resistance to interparticle relative sliding motion increases, resulting in an increase in the work done by frictional stress and an increase in energy dissipation during the conveying process. In other words, the solids frictional pressure loss also increases. Therefore, frictional viscosity or shear stress is one of the most important factors influencing the pressure drop of high-pressure dense-phase pneumatic conveying in horizontal pipe. Figures 8 and 9 respectively show interparticle frictional pressure \(\mathrm{p_{f}}\) and frictional shear stress \(\mathrm{\tau_{f}}\) obtained through simulation using the three different frictional stress models under varying supplementary gases. From the two figures, it can be observed that frictional stress, including \(\mathrm{p_{f}}\) and \(\mathrm{\tau_{f}}\), only exists in the bottom deposited region and decreases with increases of supplementary gas. A comparison of Figures 8 and 9 also reveals that while the range of frictional stress obtained using the three frictional stress models is the same, there are certain differences in their distribution. As previously mentioned, this is because frictional viscosity or shear stress arises in conjunction with frictional pressure and is directly proportional to it, resulting in a similar variation for \(\mathrm{\tau_{f}}\) and \(\mathrm{p_{f}}\). When the supplementary gas \(\mathrm{Q_{bc}}\leq 0.8\) m\({}^{3}\)/h, the gas flow rate in the conveying pipeline is reduced, resulting in a lower superficial gas velocity and more pulverized coal particles being deposited at the bottom of the pipeline.