*Article* **Simulation of Two-Phase Flow of Shotcrete in a Bent Pipe Based on a CFD–DEM Coupling Model**

**Guanguo Ma 1,2, Hui Ma <sup>2</sup> and Zhenjiao Sun 1,\***


**Abstract:** To solve the problems in determining the interactions among particles and between particles and pipe walls in pneumatic conveying systems in field tests, this article studied the two-phase flow motion characteristics of shotcrete in pipes based on a CFD–DEM coupling model and field measurement. The movement of the shotcrete, which is affected by the gas phase in the pipe, was simulated for different bend angles, and the velocity of the shotcrete material and pressure distribution within the pipeline were determined. The simulation results show that at the ideal wind pressure, the inelastic collisions among the particles and between the particles and pipe wall cause the accumulation of shotcrete material in the outside area of the bent pipe section, which may block the pipe; nevertheless, the blockage is prevented by the turbulent and secondary flows, which disperse the particles to different degrees. In addition, the wear amounts caused by particles in pipes with different bend angles were quantified. With increasing bend angle, the wear points gradually diffuse radially toward the outside wall of the bent pipe. Additionally, the wear loss decreases and then increases with increasing bend angle. The particle velocity exhibits the minimal loss at a bend angle of 90◦. It was concluded that the energy loss of the aggregate particles in the elbow of the pipe is approximately 30 times that in a horizontal, straight pipe. The results of this study can provide guidance in the construction field and for numerical simulations of the pneumatic conveying process of shotcrete.

**Keywords:** shotcrete materials; pneumatic conveying; CFD–DEM; pipe block; bend angle

#### **1. Introduction**

Significant progress has been made in the theoretical and equipment research fields and the process of improvement and development of shotcrete applications. The wetconcrete spraying technology has been widely promoted and applied to above-ground and underground support structures [1–3]. Pumping is the most important pipeline transportation method for wet shotcrete in coal mines. The movement of the shotcrete material is characterized by the interactions among the particles and between the particles and pipe wall during pneumatic conveying in the rubber hose after pumping. In addition, determining the shotcrete material motion, material blockage, and wear of the bent pipe in the pipeline transportation process in a field test is difficult [4–6]. Thus, the pneumatic conveying process in the section of spraying concrete requires further investigation.

In the wet-shotcrete support project, the concrete material is usually pumped and pneumatically conveyed to the working face by a wet spraying machine (Figure 1). Pneumatic conveying is affected by many factors such as workability of concrete, environmental conditions, pipe assembly, actual pressure, and pump outlet speed; therefore, the field situation is more complex. Identifying the relevant factors, introducing them into numerical models, and evaluating their impacts on conveying performance are crucial and

**Citation:** Ma, G.; Ma, H.; Sun, Z. Simulation of Two-Phase Flow of Shotcrete in a Bent Pipe Based on a CFD–DEM Coupling Model. *Appl. Sci.* **2022**, *12*, 3530. https://doi.org/ 10.3390/app12073530

Academic Editor: N.C. Markatos

Received: 2 March 2022 Accepted: 29 March 2022 Published: 30 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

challenging. Therefore, the numerical simulations do not focus on the entire wet spraying system; instead, they are limited to the local area of the concrete flow, i.e., the piping units. The continuous spraying of underground concrete causes the bending deformation of the rubber hose. The impact of the shotcrete material on the bent pipe, their collision, and the shotcrete accumulation have adverse effects on the long-term use of the rubber hose and its spraying performance. Therefore, the movement characteristics of shotcrete material in the pipeline during pneumatic conveying must be studied.

Owing to the rapid advances in computer technology, numerical simulations have been widely used in the pneumatic conveying field. The discrete element method (DEM) is a numerical method that is used to simulate a great number of mutually interacting particles based on their equation of motion in a system. The DEM is usually coupled with computational fluid dynamics (CFD) in the study of gas–solid two-phase flows [7–12]. For example, Zhao [13] combined the CFD and DEM to simulate numerically the pneumatic conveying process in a horizontal channel and to study the effects of the wall roughness and random effects of turbulence on the particle diffusion. Du [14] used the CFD–DEM to simulate numerically the flow characteristics of gas–solid two-phase flows in a pneumatic conveying pipe with a bent section. Moreover, Oesterle [15] established a transport equation for the interface of gas–liquid coupled multiphase flow and used the numerical simulation software Fluent to simulate the velocity and pressure distributions at the interface between slug and gas–liquid two-phase flows.

Shotcrete is a mixture of cement slurry and has aggregates with a certain viscosity. Wu, Knut, and Li [16–18] used the DEM to model concrete materials with good accuracy. Moreover, Zhang [19] and Karakurt [20] established DEM models for self-compacting concrete in the filling state of the rock-fill structure and verified the model with experiments; in addition, they studied the workability and feasibility of the self-compacting concrete. Remond [21] developed a hard-core soft-shell model to simulate the concrete flow and determined the relationships among the main parameters of the mechanical model and the macroscopic rheological properties of the concrete numerical model; however, the concrete material was greatly simplified in the simulations. Hærvig [22] proposed a method for predicting the collisions of viscous particles based on the Hertz method to reduce the particle stiffness in the DEM simulations; this model improves the Johnson–Kendall– Roberts model (JKR model) and reduces the calculation cost; thus, the movement of a great number of viscous particles can be simulated. Ji [23,24] conducted a field test to study the pressure decrease and flow characteristics of solid–gas two-phase flow in a horizontal tube; the pressure decrease was measured with a pressure transmitter. In addition, they verified the test results with the coupled CFD–DEM and analyzed the effect of the pressure decrease on the bent section; however, the model was greatly simplified in the simulation, and the accelerated transportation stage of the conveyed material was not investigated.

Much effort has been devoted to investigating the flow characteristics of shotcrete in experiments [25–29]; nevertheless, the multiphase flow characteristics have not been thoroughly clarified. Because of the limiting experimental conditions, the microscopic mechanism of the particle–fluid interactions has not been fully understood. Although the coupled CFD–DEM has been applied in the modeling of pneumatic conveying processes, the pneumatic conveying process of shotcrete has rarely been studied, and simulations of the entire pneumatic conveying motion of shotcrete materials based on the CFD–DEM coupling model has not been reported.

In this study, the coupled CFD–DEM was used to simulate numerically the pneumatic conveying of shotcrete material in a rubber hose and to determine the characteristics of the gas phase flow field. Furthermore, the motion characteristics of the concrete material were studied, and the mechanical properties of the particles for different rubber hose structures were compared.

#### **2. Governing Equations**

In the pneumatic conveying process of wet-shotcrete materials, the movement state of wet-shotcrete materials and gas in the pipeline is extremely complex. To ensure the feasibility of simulation, shotcrete materials are usually discretized, as studied by Jiang, Chen [30–32], et al. Therefore, the process is simplified as gas–solid two-phase flow in this paper.

#### *2.1. The Continuous Phase Model*

The gas flow is treated as a continuous phase, the equation for conservation of mass, or continuity equation, can be written as follows

$$\nabla \cdot \begin{pmatrix} \stackrel{\rightarrow}{\rightarrow} \\ \stackrel{\rightarrow}{v} \\ \end{pmatrix} = \mathcal{S}\_{\mathfrak{m}} \tag{1}$$

$$\rho \left( \frac{\partial}{\partial t} \left( \stackrel{\rightarrow}{v} \right) + \nabla \cdot \left( \stackrel{\rightarrow}{v} \stackrel{\rightarrow}{v} \right) \right) = -\nabla p + \nabla \cdot \left( \pi \right) + \rho \stackrel{\rightarrow}{\mathbf{g}} + \stackrel{\rightarrow}{\mathbf{F}} \tag{2}$$

where *Sm* is the mass added to the continuous phase from the dispersed second phase, *ρ* →→→

is fluid density, **v** is fluid velocity, *p* is the static pressure, *τ* is the stress tensor (described below), and *ρ* → **<sup>g</sup>** and <sup>→</sup> **F** are the gravitational body force and external body forces, respectively. → **F** also contains other model-dependent source terms such as porous-media and user defined sources.

In this paper, the transportation model selected was the standard *k* − *ε* model. The turbulence kinetic energy, *k* and its rate of dissipation, *ε* are obtained from the following transport equations [33–35]:

$$\frac{\partial}{\partial t}(\rho k) + \frac{\partial}{\partial \mathbf{x}\_i}(\rho k u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_l}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + \mathbf{G}\_k + \mathbf{G}\_b - \rho \varepsilon - \mathbf{Y}\_M + \mathbf{S}\_k \tag{3}$$

$$\frac{\partial}{\partial t}(\rho \varepsilon) + \frac{\partial}{\partial x\_i} \left(\rho \varepsilon u\_i\right) = \frac{\partial}{\partial x\_j} \left[ \left(\mu + \frac{\mu\_l}{\sigma\_\varepsilon}\right) \frac{\partial \varepsilon}{\partial x\_j} \right] + \mathbb{C}\_{1\varepsilon} \frac{\varepsilon}{k} (\mathbb{G}\_k + \mathbb{C}\_{3\varepsilon} \mathbb{G}\_b) - \mathbb{C}\_{2\varepsilon} \rho \frac{\varepsilon^2}{k} + \mathbb{S}\_{\varepsilon} \tag{4}$$

In these equations, *Gk* represents the generation of turbulence kinetic energy due to the mean velocity gradients, *Gb* is the generation of turbulence kinetic energy due to buoyancy, *YM* represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, *C*1*ε*, *C*2*ε*, and *C*3*<sup>ε</sup>* are the turbulent Prandtl numbers for *k* and *ε*. *Sk* and *S<sup>ε</sup>* are user-defined source terms.

#### *2.2. Discrete Phase Model*

The DEM was used to determine the particle phase of the shotcrete material. Based on the force that is acting on the shotcrete material particles, the particle velocity of the shotcrete material can be obtained with Newton's second law. The particle moves in the fluid by translational and rotational motions, and the force acting on the particle comprises several components. The equation of the equilibrium state of the shotcrete particles can be expressed as follows:

$$m\_i \frac{dV\_i}{dt} = F\_{D,i} + F\_{C,i} \tag{5}$$

$$I\_i \frac{d\omega\_i}{dt} = \sum\_{i=1}^n T\_i \tag{6}$$

where *mi* is the mass of particle *i*, *Vi* the velocity of particle *i*, *Ii* the moment of inertia of particle *i*, *ω<sup>i</sup>* the angular velocity of particle *i*, *FC,i* the collision force of particle *i*, *FD,i* the drag force acting on particle *i*, and *Ti* the torque acting on particle *i*.

#### *2.3. CFD–DEM Coupling Model*

In each time step, the DEM is used to calculate the position and velocity of each particle. After the iterative DEM calculation, the porosity and particle–fluid interaction force per unit volume in the fluid grid are calculated. The resulting data are used to calculate the fluid flow field and drag force of the fluid acting on the individual particles. Subsequently, the iterative CFD simulation is run until the value of the result converges; the result is applied to the discrete elements to obtain information about the particle motion in the next time step. The coupled, iterative CFD–DEM calculation process is shown in Figure 2.

**Figure 2.** Flowchart of coupled, iterative CFD–DEM calculation.

The CFD–DEM gas–solid two-phase flow is embodied in the drag force between the gas–solid phases, and the gas–solid phase is coupled by the force acting between the two phases. In this study, the gas drag toward the particle phase and its reaction force were considered. Therefore, the commonly used Wen–Yu drag model was applied:

$$\beta\_{\rm Wen-Yu} = \frac{3}{4} \mathbf{C}\_{\rm D} \frac{\rho\_{\rm g} \varepsilon\_{\rm g} \varepsilon\_{\rm s} \left| \mathbf{U}\_{\rm g} - \mathbf{U}\_{\rm s} \right|}{\mathbf{d}\_{\rm P}} \varepsilon\_{\rm g}^{-2.65} \tag{7}$$

$$\mathbf{C\_D} = \begin{cases} \begin{array}{c} \frac{24}{\mathrm{Re}\_{\mathrm{P}}} \Big( 1 + 0.15 \mathrm{Re}\_{\mathrm{P}}^{0.687} \right) \prime & \mathrm{Re}\_{\mathrm{P}} < 1000\\ & 0.44 \,, \mathrm{Re}\_{\mathrm{P}} \ge 1000 \end{array} \tag{8}$$

where Rep is the Reynolds number of the particle (Rep= ρgεgε<sup>s</sup> Ug−Us dp/μg), dp the particle diameter, CD the drag coefficient, β the phase momentum exchange coefficient, ρ<sup>g</sup> the fluid density, ε<sup>g</sup> the fluid volume fraction, ε<sup>s</sup> the solid volume fraction, Ug the fluid velocity, Us the minimal spouting velocity, F the drag force acting on the particle, and μ<sup>g</sup> the dynamic viscosity of the fluid.

#### *2.4. Particle Contact Model*

The JKR contact theory is the extension of the Hertz contact theory. It assumes that the adhesion effect only occurs on the contact surface. When two particles do not adhere to each other, their contact radius *a*<sup>0</sup> is expressed with the Hertz contact theory; otherwise, the external load is *N*, and the contact surface radius isa>a0.

To determine the effect of the adhesion force on the contact properties, Johnson [36] applied the Griffith energy method to determine the surface energy *Us* = −*πa*2Δ*γ*, where the total energy *UT* of the system is a function of the contact area *A*. When *dUT dA* = 0, *UT* is in the equilibrium state. The equivalent load force *N1* of the two particles affected by the external load *N* and surface adhesion can be determined as follows:

$$N\_1 = N + 3\pi R^\* \Delta \gamma + \sqrt{\left(3\pi R^\* \Delta \gamma\right)^2 + 6\pi R^\* \Delta \gamma N} \tag{9}$$

The corresponding contact surface radius is as follows:

$$a^3 = \frac{3R^\*}{4E^\*} \left[ N + 3\pi R^\* \Delta \gamma + \sqrt{\left( 3\pi R^\* \Delta \gamma \right)^2 + 6\pi R^\* \Delta \gamma N} \right] \tag{10}$$

The amount of overlap *α* of the two particles can be calculated with Equation (11):

$$\alpha = \frac{a^2}{R^\*} - \left(\frac{2\pi a \Delta \gamma}{E^\*}\right)^{1/2} \tag{11}$$

The increase in the normal force Δ*N* with respect to the increase in the amount of overlap Δα can be written as follows:

$$
\Delta N = 2aE^\* \Delta a \left(\frac{3\sqrt{N} - 3\sqrt{N\_c}}{3\sqrt{N} - \sqrt{N\_c}}\right) \tag{12}
$$

Based on Equation (12), there are two conditions:

For <sup>Δ</sup><sup>γ</sup> = 0, Equation (12) can be simplified to the Hertz contact force *<sup>a</sup>*<sup>3</sup> <sup>=</sup> <sup>3</sup>*R*∗*<sup>N</sup>* <sup>4</sup>*E*<sup>∗</sup> without considering the adhesion of the particles.

For *N* = 0, the two particles adhere to each other, and the radius of the contact surface can be expressed as follows:

$$a^3 = \frac{9\pi\Delta\gamma\left(R^\*\right)^2}{2E^\*}\tag{13}$$

When (3*πR*∗Δ*γ*) <sup>2</sup> <sup>+</sup>6*πR*∗Δ*γ<sup>N</sup>* <sup>≥</sup> 0, Equation (1) can be solved. The result is presented in Equation (14):

$$N \ge -\frac{3\pi R^\* \Delta \gamma}{2} \tag{14}$$

Thus, when the external load *N* is negative (the two particles attract each other), the radius of the contact surface decreases; for *<sup>N</sup>* ≥ −3*πR*∗Δ*<sup>γ</sup>* <sup>2</sup> , the particle adhesion is in a critical state, and when the pull force increases again, the two particles separate. The maximal pull force *Nc* required for separating the two particles is expressed as follows:

$$N\_c = \frac{3\pi R^\* \Delta \gamma}{2} \tag{15}$$

#### **3. Geometric Model and Meshes**

Rubber hoses provide flexibility to the pneumatic conveying process of concrete materials. However, bent pipe structures with different angles suffer from internal wear and unstable particle movement. The geometric model in Figure 3 consists of the horizontal pipes L1 and L2 and the bent-pipe structure *R*. The definite mass flow rate can be obtained based on the horizontal pipe L1. The bend angles were set to 30◦, 45◦, 60◦, 90◦, 120◦, and 150◦. For the particle phase, the self-developed DEM code is merged into CFD, a coupling scheme is used between the CFD and DEM, and the explicit time integration method is used to calculate the motions of particles in the DEM. A compromise between efficiency and accuracy is made: spherical particles with diameters of 5 and 8 mm were used to represent the coarse aggregate covered with cement slurry. Based on field tests and related reports in the literature [37,38], the gas flow velocity was set to 64 m/s, and the initial particle velocity was 1.5m/s; subsequently, the particle was accelerated to a stable velocity for relatively stable flow. The applied particle parameters and operating conditions are listed in Table 1 [39–42].

**Figure 3.** Geometric model and grids of a longitudinal section of bent pipe (bend angle of 90◦).


**Table 1.** Basic parameters.

The number of meshes has a great influence on the accuracy of the transient simulation results of the airflow field. Therefore, mesh independence studies are necessary to ensure that both discretization and rounding errors are within acceptable limits and that the mesh used does not significantly affect the simulation results. Three discrete grids with different densities are generated by ICEM, which are denoted as fine meshes, better meshes, and medium meshes, respectively. In all three cases, the mesh quality is above 0.5.

Using these three meshing cases as shown in Figure 3 (A—A section), the air velocities were compared and analyzed. Figure 4 shows the comparison results of velocity at crosssections of the roadway. It can be observed that the simulation results of airflow velocity using three different kinds of grids show similar variation trends. The velocities using better meshes were very close to the values using fine meshes, but the results using medium meshes were quite different from the results using the other two different meshes. Therefore, it can be considered that better meshes have already met the requirement of meshing independence.

**Figure 4.** The comparison of the velocity among three meshing cases (bend angle of 90◦).

#### **4. Calculation Results and Analysis**

#### *4.1. Verification of Simulation*

The gas power source adopts the air compressor with a rated power of 5.5 kW, maximum conveying air volume of 700 m3/h, and rated pressure of 0.5 MPa to convey air. The inlet apparent gas velocity is controlled by the vortex flowmeter, which changes from 48 to 64 m/s. The field experiment is carried out with an SPB-7 wet concrete sprayer, and the pressure drop in the process of pneumatic conveying is measured by the differential pressure method. The layout diagram of the field test pressure transmitter is shown in Figure 5.

**Figure 5.** Layout diagram of the field test pressure transmitter.

The CFD–DEM coupling method can effectively observe the microscopic features that cannot be observed in the experiment, but the results must be consistent with the experimental results. This paper measured the pressure drop per unit length of the shotcrete in the 90◦ bent pipe at different air velocities when the concrete delivery volume was 4 m3/h through field experiments, and simulated experiments using the same parameters, as shown in Figure 6.

**Figure 6.** Variation in pressure drop with air velocity.

The simulation results were compared with the experimental results to verify the accuracy of the numerical method. The calculation results show that in the stable stage of pneumatic transportation of shotcrete, with the increase in airflow velocity, the pressure drop increases. The simulation results are in good agreement with the experimental results, the numerical values are consistent with the experimental values, and the errors are less than 10%. It shows that the model can be used for experimental calculation.

#### *4.2. Gas Phase Flow Field*

Figure 7 presents the global distribution of the gas phase static pressure during the pneumatic conveying process of the shotcrete material. The evident pressure loss within the rubber hose can be noted by comparing Figure 7, Figure 8a, and Figure 9. The pressure distribution in the horizontal, straight pipe section is relatively uniform compared to that near the bent pipe section. The pressures in the inside and outside areas of the bent pipe section are significantly different. The pressure in the outside area is high, and the pressure in the inside area is low, which is due to the centrifugal inertial force; in addition, the viscosity of the fluid on the pipe wall increases the fluid velocity on the wall surface. Moreover, both fluid velocities in the inside and outside areas are lower than that in the bent-pipe center, and the centrifugal inertial force of the fluid in the pipe center is greater than that in the inside and outside areas. The fluid in the pipe center flowing from inside to outside increases the external pressure, i.e., the pressure on the outside wall is much higher than that on the inside wall of the bent pipe. In addition, the gas velocity near the wall surface is relatively low, and the slow-flowing particles near the left and right boundaries move from the outside to the inside along the pressure decrease direction; the backflow at the central axis of the bent pipe results in a secondary flow. Subsequently, the secondary and main flows merge and move along the flow direction of the pipeline. The secondary flow limits the hitting velocity of the solid phase toward the outside wall of the bent pipe, and the central part of the outer surface of the bent pipe experiences more wear.

**Figure 7.** Global distribution of static pressure in pipelines with different bend angles.

According to Figure 8b, during the flow of the shotcrete material through the pipeline, the pressure in the straight pipe section in front of the bent pipe tends to decrease at a constant speed; behind the bent pipe section, the pressure decreases significantly for various bend angles. The pressure decrease in the 90◦ bent pipe is the lowest, whereas the other bent pipes show no evident pressure difference. During pneumatic conveying, the particle velocity within the pipeline varies significantly (Figure 9). Furthermore, the pressure difference affects the gas flow velocity distribution near the bent pipe, and the higher pressure pushes the gas phase to the side with the lower pressure. There are a higher pressure and a lower flow velocity in the outside area of the bent pipe part, and lower pressure and a higher flow velocity in the inside area of the bent pipe part.

#### *4.3. Flow Field of Particle Phase*

Figure 10 presents the particle velocity and particle distribution within the pipe under the action of a high-pressure gas flow. The particles within the horizontal pipe flow in a suspension state, and the particle packing density is not uniform. Under the action of a high-pressure gas flow, the particles have a higher density when they are just picked up by the gas flow. With the continuous generation of particles in the inlet, the pipe cross-section decreases, and the windward side of the particles increases. In addition, the packing density decreases continuously as the particles move forward. After entering the bent pipe, the particles accumulate near the midline of the outside area of the bent pipe; they form a rope-like structure (particularly in the 30◦ and 90◦ bent pipes), which is usually called "the particle rope" [43]; this typical phenomenon in gas–solid two-phase flows in bent pipes is mainly due to the inertia among particles and the inelastic collisions between the particles and wall surface. Under the actions of the subsequent turbulence and secondary flows, the particle rope starts to disperse gradually. The comparison of the particle flow characteristics in the three bent pipes shows that when passing through the 150◦ bent pipe, the particles disperse under the action of turbulence before forming the particle rope; this indicates the strong effect of the different angles of the bent pipes on the dispersion of the particle rope. Furthermore, after the particles pass through the bent pipe structure, their velocities recover at different times. The particles accumulate significantly later for a bend angle of 30◦ than for the other bend angles. Hence, the greater resistances among the particles and between the particles and pipe wall causes the particle velocities to recover more slowly. In addition, the particle accumulation at a bend angle of 30◦ is more severe and may block the pipe.

**Figure 8.** Distribution of static pressure in bent pipes with different bend angles. (**a**)z=0 distribution cloud map of global static pressure in bent pipes with different bend angles. (**b**) Line chart of the global distribution of static pressure with respect to distance.

**Figure 10.** Distributions of moving concrete particles in 30◦, 90◦, and 150◦ bent pipes.

Figure 11a–c shows the variations in the velocities of individual concrete particles of different sizes in pipes with different bend angles over time. The particles comprise coarse aggregates with diameters of 5 and 8 mm (their surfaces are covered with cement slurry) and agglomerates of particles of different sizes. The values of 10 particles of each type were averaged to eliminate the accidental effect of particle sizes. According to Figure 11a–c, the particle velocity suddenly changes behind the bent pipe, and the 5 mm particles exhibit a greater acceleration than the other two particle types; hence, they enter the bent pipe section first. This is because although the particle with a large particle size has a larger area under the action of air, it can obtain more acceleration, but the resistance of air to particles is relatively high. Obviously, the resistance has a greater impact on the particle at an air velocity of 64 m/s.

**Figure 11.** Variation in velocity of a single particle concerning time (Particle 1 represents the coarse aggregate with 8 mm diameter, Particle 2 represents coarse aggregate with 5 mm diameter, and Particles 1 and 2 represent agglomerates with two diameters). (**a**) Bend angle of 30◦. (**b**) Bend angle of 90◦. (**c**) Bend angle of 150◦.

The different bent pipe structures have great effects on particle velocity. The particle velocity decreases by approximately 73% behind the 30◦ bent pipe and by approximately 15–35% behind the 90◦ and 150◦ bent pipes. This is because the particle in the 30◦ bent pipe is not affected by the gas phase, and its velocity continues to decrease. When the particle leaves the bent pipe section, it is picked up again by the airflow and accelerated under the actions of turbulence and secondary flows. When the particle passes through the 90◦ bent pipe, its movement is not heavily affected by the turbulence and secondary flows; thus, the particle velocity does not increase further and stabilizes. In addition, the turbulence and secondary flows in the bent pipe act as a resistance to the particles. Because the 90◦ bent pipe provides the lowest resistance, the particle velocity experiences the lowest loss.

The CFD–DEM model provides detailed data about particle movement and collisions. Figure 12 presents the main wear areas and wear effect of the concrete particles on the bent

pipe with different bend angles. The results show that within the same time interval, the 45◦ bent pipe experiences significant wear, and its maximal depth of wear is 0.042 mm. In addition, the main wear position diffuses radially toward the outside wall of the bent pipe with an increasing bend angle. This is because the wall roughness promotes particle–wall collisions. The reflection angle at a rough wall is much greater than that at a smooth wall. The rougher wall causes more particles to diffuse radially along the outside wall of the lifted bent pipe, while the higher axial velocity causes the particles to suspend uniformly. Thus, when the particles hit the bent pipe, they are uniformly distributed across the curved section of the bent pipe, thereby causing elliptical wear; these results are consistent with those presented by Solnordal [44].

**Figure 12.** (**a**–**f**)Amount of wear and main wear position of particles in bent pipes with different angles.

Figure 13 shows the numbers of particle–pipe wall contact times and normal contact forces of the shotcrete particles in the pipes with different bend angles. With increasing bend angle, the number of particles–pipe wall contact times and normal contact force first increase and then decrease. At a bend angle of 45◦, both values exhibit a maximum, which is consistent with the results of Xu [45]. In addition, in pneumatic conveying, the normal contact forces of most particles affect the bent pipe; it is assumed that the 45◦ bent pipe is subjected to stronger impact and wear.

**Figure 13.** Number of particle–pipe wall contact times in the pipe in 0.5 s.

Figure 14 presents the variation in the particle energy loss concerning time. When the particles enter the bent pipe, the particle–pipe wall collisions significantly increase the energy loss. As the particle rope gradually forms, the number of particles–pipe wall contact times decreases, and the energy loss gradually decreases and stabilizes. The particle energy loss is not correlated with the bent pipe structure, and the particle energy loss in the bent pipe is approximately 30 times that in the straight pipe.

**Figure 14.** Variation in particle energy loss concerning time.

#### **5. Conclusions**

In this paper, the flow characteristics of wet-shotcrete materials in the horizontal hose were simulated, and a CFD–DEM model suitable for simulating the flow characteristics of wet-shotcrete materials was developed to study the flow of wet-shotcrete materials in the horizontal hose. In addition, the accuracy of the numerical simulation results was verified through the on-site pressure drop experimental test. The main conclusions are as follows:


**Author Contributions:** Investigation, H.M.; Methodology, G.M.; Resources, Z.S.; Validation, H.M.; Visualization, G.M.; Writing—original draft, G.M. and Z.S.; Writing—review & editing, Z.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was funded by projects such as Major scientific and technological innovation projects of Shandong Province (Grant No. 2019SDZY0203).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare that they have no conflict of interest.

#### **References**

