## 1 Introduction
They explored the impact of material properties, wall roughness, and bend curvature on the conveying characteristics in vertical bends. However, dense-phase pneumatic conveying is highly susceptible to particle deposition in horizontal pipes and bends. Hence, [PERSON] et al. [23] introduced a frictional stress model to study the conveying characteristics of high-pressure dense-phase pneumatic conveying in horizontal pipe, producing simulation results that showed improved accuracy when frictional stress was considered. However, the underlying mechanisms of the frictional stress model were not revealed in this study. In dense-phase pneumatic conveying, the solids volume fraction is extremely high, approaching the bulk packing volume fraction (about 0.5 for pulverized coal). During this process, particles are mainly in sustained contact, resulting in the solids stress primarily originating from frictional stress [24; 25; 26; 27]. Consequently, the impact of frictional stress on high-pressure dense-phase pneumatic conveying is of utmost importance, although there is limited literature on this topic. [PERSON] and [PERSON] [28; 29] were the pioneers in proposing the frictional pressure model, introducing three dimensionless coefficients (Fr, r, s) to characterize the frictional properties of different materials. Simultaneously, on the basis of the von Mises yield criterion, [PERSON] [30] proposed a solids frictional viscosity model. However, [PERSON] et al. [31; 32] found that the von Mises/Coulomb yield criterion, originating from soil mechanics, cannot fully apply to the simulation for dense-phase gas-solid flow of pulverized coal since it can properly predict dilatancy but not consolidation for the solids phase. To overcome this limitation, [PERSON] and [PERSON] [33] incorporated the strain rate term proposed by [PERSON] [34] and soil mechanics theory [35; 36] into modifying the Schaeffer frictional viscosity model on the basis of the extended von Mises yield criterion. They combined the Johnson and Jackson frictional pressure model to establish a comprehensive frictional stress model. Then this model was applied to simulate the gas-solid flow in the fluidized bed, validating its reliability. [PERSON] [25] set up a new frictional stress model by combining the Pitman-Schaeffer-Gray-Stiles yield criterion, which considers both the dilatancy and consolidation of the solids phase in dense-phasegas-solid flow. And the model was validated through simulations of soil particle flows in geophysics. [PERSON] et al. [37] made modifications to the Johnson and Jackson frictional pressure model [28; 29] and the Syamlal frictional viscosity model [38] to establish the frictional stress model. Coupling this frictional stress model with the classical particle kinetic theory, they conducted a three-dimensional simulation of high-pressure dense-phase pneumatic conveying in horizontal pipe. The study provided insights into the flow patterns and pressure drop of high-pressure dense-phase pneumatic conveying in horizontal pipe, and the simulation reliability was verified by comparisons with the experimental data. [PERSON] et al. [39] coupled the classical kinetic theory of granular flows (KTGF) with Schaeffer frictional stress model to simulate bypass dense-phase pneumatic conveying. The results showed substantial improvement in predicting pressure drop after considering the frictional stress model, although the experimental pressure drop remained underestimated. Therefore, they introduced an offset volume fraction to alleviate excessive over-prediction of granular flows and modify the Johnson and Jackson frictional pressure model. Additionally, they used the modified frictional stress model to simulate bypass dense-phase pneumatic conveying of different materials (aluminum powder, fly ash, and sand grains) and obtained reliable validation [23].