## 5 Conclusions  
Based on the two-fluid model, this study developed a three-zone drag model, modified the classical kinetic theory of granular flows and the Schneider-bauer solids wall boundary condition, and introduced the realizable k-\(\epsilon\)-k\({}_{\text{P}}\)-\(\epsilon_{\text{p}}\) turbulent model to establish a three-dimensional unsteady mathematical model for high-pressure dense-phase pneumatic conveying in horizontal pipe. Using this mathematical model, this study proposed to consider the influence of [PERSON] frictional stress model, [PERSON] and Sundaresan frictional stress model, and the modified Berzi frictional stress model on the simulation results of high-pressure dense phase pneumatic conveying in horizontal pipe. The main conclusions obtained are as follows:  
* The predicted pressure drop in horizontal pipe and its variation with supplementary gas are, using the three frictional stress models, seen to be in good agreement with the corresponding experimental data, with relative errors ranging from \(-4.91\%\) to \(+7.60\%\). In addition, the predicted solids volume fraction distribution contours in the three frictional stress models generally agree with the ECT images in the cross-section of the horizontal pipe. In particular, the predicted variations of the deposited region with supplementary gas are also consistent with those in the ECT images.
* The effect of frictional stress models on the simulation results of high-pressure dense-phase pneumatic conveying in horizontal pipe only presents in the transition region and deposited region. However, the two regions exhibit opposite changes. Frictional stress only exists in the bottom deposited region and diminishes gradually with the rise in supplementary gas.
* The three frictional stress models predict a similar variation range of frictional stress, but their distributions differ, which is a fundamental reason for the variations of the solids volume fraction distribution. This also explains the variation of pressure drop in horizontal pipe. The larger the frictional pressure, the lower the solids volume fraction; and the stronger the frictional viscosity or shear stress, the higher the pressure drop in horizontal pipe.
* After comparing the simulation results of the three frictional stress models, it was observed that Dartevelle frictional stress demonstrates the strongest effect and highest energy consumption in the deposited region. Consequently, this leads to the lowest gas and solids velocity, turbulent kinetic energy, and solids volume fraction, and the highest particle pseudo-temperature. However, the simulation results predicted by the modified Berzi frictional stress model exhibit an opposite trend to the above results.
* Among the three frictional stress models, the simulation results of the modified Berzi frictional stress model are more consistent with the experimental data. Additionally, this model better reflects the frictional properties of pulverized coal. In conclusion, the modified Berzi frictional stress model can provide a more accurate prediction of frictional stress in high-pressure dense-phase pneumatic conveying.  
S.D.: conceptualization, investigation, methodology, validation, post-processing, writing the first draft. H.Z.: paper writing, corrections. W.T.: software, validation, formal analysis and improving the manuscript. R.X.: formal analysis, validation. J.Z.: investigation, review and editing. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare