## 4 Results and Discussion
### Boundary Conditions and Simulation Settings
#### 4.2.2 Solids Volume Fraction Distribution
As a result, the influence of frictional stress is stronger, and the differences with various frictional stress models are more prominent (as shown in Figures 8 and 9). Consequently, due to different frictional stress models, there is a larger disparity in the predicted pressure drop in horizontal pipe. When \(\rm Q_{bc}\leq 0.8\) m\({}^{3}/h\), the solids volume fraction is higher in the deposited region at the bottom of the horizontal pipe, and it is almost always higher than 0.43 (see Figure 6). At this time, Dartevelle frictional pressure varies most rapidly with the solids volume fraction, and its value is higher than the other two frictional pressure models (refer to Figure 8), as does Dartevelle frictional shear stress (observed in Figure 9). Therefore, Dartevelle frictional stress model predicts the highest pressure drop in horizontal pipe (refer to Figure 4), while the results are reversed for the modified Berzi frictional stress model. When \(\rm Q_{bc}>0.8\) m\({}^{3}/h\), the gas flow rate in the conveying pipeline increases rapidly, leading to an increase in superficial gas velocity and a sharp reduction in pulverized coal deposition at the bottom of the pipeline. The solids volume fraction in the deposited region is almost always lower than 0.47. Especially when predicted by Dartevelle frictional stress model, the solids volume fraction is even lower than 0.43 (refer to Figure 6). Consequently, the influence of frictional stress is significantly weakened, and the difference between various frictional stress models is visibly diminished (as shown in Figures 8 and 9). Therefore, there is a smaller discrepancy in the predicted pressure drop in horizontal pipe, which is due to different frictional stress models. When \(\rm Q_{bc}>0.8\) m\({}^{3}/h\), the modified Berzi frictional pressure is the highest (refer to Figure 8), and the same applies to frictional shear stress (refer to Figure 9). Hence, the modified Berzi frictional stress model predicts the highest pressure drop in horizontal pipe (refer to Figure 3). According to Figure 3, the relative error of pressure drop in horizontal pipe decreases gradually with the increase in supplementary gas. As previously discussed, with the increase of supplementary gas, the deposited region shows a continuous reduction, causing a gradual decrease in interparticle friction and a simultaneous increase in interparticle collision. This discrepancy is attributed to the existing frictional stress model insufficiently considering interparticle frictional properties, such as the oversight of particle shape and particle size distribution, influencing the friction stress model. Therefore, as the supplementary gas decreases, the relative error correspondingly increases. The distribution of various conveying parameters along the height of horizontal pipe is predicted by different frictional stress models. Figure 10 illustrates the typical experimental  
Figure 9: Frictional shear stress \(\tau_{f}\) predicted by different frictional stress models: **(a)** Dartevelle frictional stress model; **(b)** Srivastava and Sundaresan frictional stress model; **(c)** Modified Berzi frictional stress model. conditions with the supplementary gas \(\rm Q_{s}=0.8\) m\({}^{3}\)/h. Among the three frictional stress models, Figure 10f clearly shows the difference in the predicted solids volume fraction distribution. It can be inferred from this figure that the trend of the solids volume fraction is inversely related to the frictional pressure, confirming the above conclusion that the solids volume fraction distribution depends on frictional pressure distribution, which is ultimately determined by the mechanical balance in the conveying pipeline. This is also the fundamental reason for the differences in the predicted pressure drop in horizontal pipe that are observed when the three frictional stress models are used.