## 4 Results and Discussion
### Boundary Conditions and Simulation Settings
#### 4.2.2 Solids Volume Fraction Distribution  
Figure 4 illustrates the distribution of the conveying parameters (gas velocity and solids velocity) along the height of the horizontal pipe in various cross-sections, under a typical experimental condition with \(Q_{s}=0.8\) m\({}^{3}\)/h, using the modified Berzi frictional stress model. It can be observed from the figure that the conveying parameters gradually stabilize along the axial direction of the horizontal pipe. In referring to Figure 4a,b, we see the conveying parameters have already stabilized from the section at \(L=120\)\(D\) (where \(L\) represents the distance between the selected section and the inlet section of the pipe, and \(D\) represents the pipe diameter). This indicates that the gas-solid flow in horizontal pipe has achieved sufficient development and entered the fully developed section. To investigate the conveying characteristics of high-pressure dense-phase pneumatic conveying in horizontal pipe, this study will discuss the conveying characteristic parameters at the section \(L=180\)\(D\). Figure 4: Distribution of the conveying parameters along the height of horizontal pipe at various cross-sections. (**a**) Gas velocity; (**b**) Solids velocity. Figure 3: Comparative relationship between the predicted pressure drop in horizontal pipe with supplementary gas, using different frictional stress models and the corresponding experimental data. Figure 5 illustrates the comparative relationship between the predicted solids volume fraction distribution contours by using the three frictional stress models and the ECT images. By comparing Figure 5a-d, it is evident that the simulation results generally agree with the corresponding experimental data. The solids volume fraction distribution in the entire horizontal pipe can be categorized into three regions: the upper suspended region corresponding to dilute-phase flow, the middle transition region corresponding to intermediate flow, and the bottom deposited region corresponding to dense-phase flow. However, the boundaries between these regions are not clearly defined. Figure 5 shows that the predicted variations of the deposited region with supplementary gas are fundamentally consistent in horizontal pipe when the three frictional stress models are used, which is also consistent with the variations of the deposited region shown in the ECT images, and namely the bottom deposited region decreasing with increasing supplementary gas. This strongly validates the high reliability of the frictional stress model used in this study. Influence of the Frictional Stress Model on the Conveying Characteristics of High-Pressure Dense-Phase Pneumatic Conveying in Horizontal Pipe  
The frictional pressure \(p_{f}\) represents the normal stress originating in interparticle continuous contact or compression, and its main function is to prevent an excessive solids volume fraction, which lacks physical significance. Obviously, it is closely related to the solids volume fraction and directly affects the solids volume fraction distribution. Figure 5: Comparative relationship between the predicted solids volume fraction distribution contours using three different frictional stress models and ECT images. (**a**) ECT, (**b**) Dartevelle frictional stress model; (**c**) Srivastava and Sundaresan frictional stress model; (**d**) Modified Berzi frictional stress model. Measurement results indicate that the bulk packing volume fraction \(\alpha_{b}\) is approximately 0.5 for pulverized coal. In this circumstance, the frictional pressure is generated by the weight of the pulverized coal itself. Given that granular flows need the corresponding relative motion space, the solids volume fraction does not generally exceed the bulk packing volume fraction (as shown in Figure 6). [PERSON] et al. [48] stated that when \(\alpha_{s}\)\(\geq\) 0.4, it falls into dense-phase flow, and the particles gradually experience compression and frictional effects. Therefore, this study sets \(\alpha_{s,\min}\) at 0.4.