## 4 Results and Discussion
### Boundary Conditions and Simulation Settings
#### 4.2.2 Solids Volume Fraction Distribution
As mentioned earlier, when \(\rm Q_{s}=0.8\) m\({}^{3}\)/h, the solids volume fraction in the deposited region is higher (as shown in Figure 10f), and the impact of frictional stress is significant. The simulation results demonstrate that the Dartevelle frictional stress model exhibits the strongest effect in the deposited region, resulting in the highest interparticle energy dissipation, and the lowest solids velocity (as shown in Figure 10b). Consequently, the gas-solid interaction intensifies, leading to a decrease in gas velocity, which also becomes the lowest (as shown in Figure 10a). Figure 10: Distribution of conveying parameters along the height of horizontal pipe predicted by different frictional stress models. (**a**) Gas velocity; (**b**) Solids velocity; (**c**) Gas turbulent kinetic energy; (**d**) Solids turbulent kinetic energy; (**e**) Particle pseudo-temperature; (**f**) Solids volume fraction. Additionally, frictional stress hinders interparticle relative movement, resulting in a decrease in solids' fluctuation velocity. Hence, Dartevelle frictional stress model predicts the lowest solids turbulent kinetic energy (as shown in Figure 10d). Since the solids turbulent kinetic energy has a dominant role, due to the higher solids volume fraction in the deposited region, it also influences the gas turbulent kinetic energy, making it the minimum (as shown in Figure 10c). The particle pseudo-temperature is dependent on the fluctuation intensity of the individual particle. In the high solids volume fraction of the deposited region, the fluctuation of individual particle is limited, resulting in a lower particle pseudo-temperature. Consequently, the Dartevelle frictional stress model, which predicts the lowest solids volume fraction (as shown in Figure 10f), has the highest particle pseudo-temperature (as shown in Figure 10e). The simulation results of the modified Berzi frictional stress model exhibit the opposite trends to the aforementioned results. Figure 10 shows that the three frictional stress models had almost no impact on the simulation results, due to the dominance of KTGF in the suspended region. However, for the transition region, the simulation results of the three frictional stress models were almost contrary to those of the deposited region. This discrepancy arose from the requirement of maintaining consistency in the gas and solids mass flow rate. In short, the influence of frictional stress models on the simulation results of high-pressure dense-phase pneumatic conveying in horizontal pipe was mainly observed in the transition and deposited regions. The frictional stress influences other conveying parameters by affecting the solids volume fraction distribution and the pressure drop in horizontal pipe. Through comparative analysis of the simulation results obtained from the three frictional stress models, it was discovered that the simulation results predicted by the modified Berzi frictional stress model closely aligned with the corresponding experimental data. Additionally, this model better characterized the frictional properties of pulverized coal. These findings of the study led us to the conclusion that the modified Berzi frictional stress model provides a more rational approach to predicting frictional stress in high-pressure dense-phase pneumatic conveying in horizontal pipe.