## 1 Introduction
In particular, computational fluid dynamics (CFD) has played an important role in analyzing high-pressure dense-phase pneumatic conveying. Compared to experimental research methods, simulation methods can capture real-time comprehensive flow field information of each cell in the computational domain, including detailed micro-characteristic parameters [11], which facilitates thorough analysis and understanding of the conveying characteristics and flow mechanism. Additionally, simulation methods are more efficient, convenient, and cost-effective, serving as an important supplement to experimental research methods, and provide a reliable theoretical foundation and practical guidance for the rational design and comprehensive optimization of high-pressure dense-phase pneumatic conveying. The simulation approaches can be broadly categorized into the Euler-Lagrangian approach and the Euler-Euler approach. The Euler-Lagrangian approach provides microscopic details, such as on particle trajectories and forces acting on individual particle, which are essential for understanding gas-solid systems [12; 13; 14]. Hence, this approach is considered to be more accurate and efficient for gas-solid systems involving particles of different sizes or densities [15]. Numerous simulation studies of pneumatic conveying have been conducted using the Euler-Lagrangian approach by both domestic and foreign researchers [16]. However, due to the requirement for small time steps to accurately simulate particle-particle/wall collisions with relatively high accuracy, the Euler-Lagrangian approach necessitates high computational resources, particularly for dense-phase pneumatic conveying. To reduce the computational cost of this approach, several new methods have been proposed, including the CFD-discrete element method (CFD-DEM), the coarse-grained discrete element method (CG-DEM), and the multiphase particle-in-cell method (MP-PIC). These methods aim to develop scaling theories and accelerate simulations of gas-solid systems [11; 17]. For instance, MP-PIC considers a specific group of particles with similar properties (e.g., species, size, density, temperature) as a parcel, to decrease the number of particles involved in the computations [18]. Nevertheless, these new methods have excessively high computational requirements for large-scale industrial applications, such as dense-phase pneumatic conveying systems with billions of particles. Furthermore, the scaling theories, derived from force balances between coarse-grained or parcel particles and their corresponding original particles, introduce uncertainties in the modeling process, particularly in determining parcel particle properties and precise particle-particle/wall interactions [12; 19]. The continuum approach, represented by the two-fluid model (TFM), is more suitable for dense-phase pneumatic conveying due to its computational convenience and efficiency. As a result, an increasing number of researchers have used the Euler-Euler approach for simulating high-pressure dense-phase pneumatic conveying. [PERSON] et al. [20] conducted simulation research on horizontal bends using two-fluid modelling, producing simulation results that indicated the two-fluid modelling has great potential in predicting the conveying characteristics of dense-phase pneumatic conveying. Inspired by [PERSON]'s work, many scholars have gradually improved the constitutive relationships of dense-phase pneumatic conveying and conducted relevant simulation research from different perspectives. For instance, [PERSON] et al. [21] introduced the kinetic theory of granular flow to simulate the gas-solid two-phase flow in the spouted bed and restructured a three-dimensional steady-state mathematical model to investigate the effects of non-ideal particle collisions on the restitution coefficient. Subsequently, [PERSON] et al. [22] further considered the unsteady-state conveying characteristics and established a three-dimensional unsteady-state mathematical model for dense-phase pneumatic conveying.