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The transport of solid–liquid two-phase flow is widely used in water conservancy, environmental protection, and municipal engineering. Accurate pressure loss calculation is crucial for hydraulic transport pipelines, particularly in the case of bends, valves, and other deformation parts. These factors directly impact the energy consumption and the investment of the system. This paper employed the Euler–Euler multiphase flow model to investigate the characteristics of solid–liquid two-phase flow in vertically positioned combined elbows. The model was initially validated using data from the literature. Subsequently, based on the validated model, an investigation was conducted to determine the relationship between pressure loss and various factors, including flow velocity, combined angle, particle concentration, and particle size. Finally, the changes in velocity distribution, particle concentration, and turbulent kinetic energy were analyzed. The results indicate that the pressure loss increases with the flow velocity, tends to decrease with the combined angle, and increases with the particle concentration. The relationship between pressure loss and particle size is more complex. The velocity distribution, particle concentration, and turbulent kinetic energy exhibit the variations caused by different factors.

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  • A numerical model for calculating solid–liquid two-phase flow in combined elbows was established.

  • Changes in pressure loss in combined elbows with influencing factors were discussed.

  • The velocity distribution, particle concentration distribution, and turbulent kinetic energy distribution were analyzed.

combined elbow, Euler–Euler model, flow characteristics, pressure loss
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The hydraulic transportation of water containing sand particles has a wide range of applications in the fields of water conservation, environmental protection, municipal engineering, and other related areas. Accurate calculation of pressure loss determines the investment and energy consumption of hydraulic transport systems, especially when considering bends and valves. In transport pipelines, two elbows are often combined to accommodate changes in terrain or fluid orientation. A solid–liquid two-phase flow develops when sand-laden water flows into the combined elbow, resulting in significantly different pressure loss and flow characteristics compared to those observed in single-phase flow. Therefore, it is imperative to investigate the pressure loss and hydrodynamic characteristics of solid–liquid two-phase flow in the combined elbow. The pressure loss and flow characteristics of solid–liquid flow in various pipelines and fittings have been extensively investigated by numerous researchers, with a predominant focus on the hydraulic behavior of horizontal and vertical straight pipes as well as bends with diverse geometries. With the development of computational fluid dynamics (CFD), an increasing number of researchers have adopted numerical simulation to investigate solid–liquid two-phase flow. Compared to experimental approaches, numerical simulation offers distinct advantages such as time and labor efficiency, cost-effectiveness, and ease in replicating working conditions that would be difficult to achieve through experimentation. The accuracy of the numerical simulation has been rigorously validated by means of experimental tests (Kaushal et al. 2005; Hashemi et al. 2014; Zhang et al. 2023). In straight pipes, Chen et al. (2009) applied the Euler–Euler multiphase flow model to simulate the behavior of coal water slurry in a horizontal pipe. They indicated that the flow rate and pressure gradient were the main factors affecting slurry flow properties. Gopaliya & Kaushal (2015) conducted a numerical simulation of sand-laden water flow in a horizontal pipeline with a diameter of 53.2 mm and discussed the influence of particle size on two-phase flow parameters. Singh et al. (2017) applied the Euler–Lagrange multiphase model to investigate the characteristics of slurry pipelines under varying velocities and solid concentrations. Zhang et al. (2022) investigated the coarse particles' transportation behavior in a vertical pipe based on an optimized Euler–Lagrange method. Zhou et al. (2019) and Liu et al. (2023) used the combined CFD and Discrete Element Method (DEM) to study the hydraulic conveying of solid coarse particles. Krishna et al. (2023) utilized the CFD method to investigate the flow behavior in a straight horizontal pipe. It can be found that the study of horizontal branch solid–liquid two-phase flow mainly involves resistance, velocity distribution, and particle concentration distribution, and the main methods are the Euler–Euler method and the Euler–Lagrange method. Recently, research on solid–liquid two-phase flow in elbows has primarily focused on two aspects: the first pertains to pressure drop and flow characteristics, including velocity distribution, and particle concentration distribution, while the second concerns the erosion caused by solid–liquid two-phase flow. In terms of the characteristics of two-phase flow, Ma et al. (2014) employed the Mixture model to simulate the solid–liquid two-phase flow in a horizontal 90° bend and investigated the phenomenon of secondary flow at various cross-sections, as well as its impact on sand concentration distribution. Shi & Zhang (2016) utilized the Euler–Euler and heat transfer models to investigate the solid–liquid two-phase flow of hydrated slurry in a 90° horizontal bend, and they verified the obtained pressure drop and wall temperature. Wang et al. (2018) used the Euler–Euler model to resolve the concentration, velocity, and pressure fields of the ice slurry in the elbow and found that the ice-slurry flow changed in cross-section due to the development of secondary flow. Cai et al. (2019) employed a coupled CFD-Population Balance Model (PBM) to investigate the flow characteristics of ice slurry in a horizontal 90° elbow. Tarodiya et al. (2020) performed the three-dimensional numerical modeling of the conventional 90° bend transporting multi-sized particulate slurry using a granular Eulerian–Eulerian model. The effect of variation in velocity and concentration on pressure drop and flow field of the multi-sized particulate slurry was investigated. Joshi et al. (2024) developed a three-dimensional computational model to explore the transportation characteristics of a bi-modal slurry flowing through a horizontally placed 90° pipe bend. It was observed that the pressure drop increased with both velocity and concentration. In addition, due to the centrifugal force inducing the secondary flow, an accumulation of ice particles was observed in the elbow section. As can be seen, the majority of studies have concentrated on ice-slurry particles within the solid–liquid two-phase flow. Moreover, the effects of solid–liquid flow on elbow erosion have also been studied, including various structural and particle parameters (Sedrez et al. 2019; Xie et al. 2023; Khan et al. 2024). Most of the previous studies have been conducted on solid–liquid two-phase flow in pipes and bends. However, very few studies have focused on the solid–liquid flow in combined elbows. For instance, Nayak et al. (2017) employed the Euler–Euler model to simulate the pressure drop, formation of vortex structures, and heat transfer in a 180° bend for water mortar; however, investigations on other combined angles have not been conducted.

In this study, the Euler–Euler model is employed to investigate the solid–liquid two-phase flow in combined elbows. The effects of velocity, combined angle, particle concentration, and particle size on pressure loss are discussed. The velocity distribution, particle concentration, and turbulent kinetic energy are analyzed at various combined angles. These findings can serve as a valuable reference for the study of hydraulic transportation in two-phase flow.

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Governing equations

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Volume fraction

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The volume fraction αi describes the continuous multiphase flow within the Euler–Euler model. The volume fraction represents the space occupied by each phase, with each term individually satisfying the laws of mass and momentum conservation.

The volume Vi of the phase i is defined as:
(1)
The αi in Equation (1) is established by the following formula:
(2)
The effective density of the phase i is:
(3)
where ρi is the physical density of phase i.

Conservation equations

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The Euler–Euler model is used to simulate the two-phase flow in combined elbows, treating the continuous phase and discrete phase as a continuum and handling each phase using the Euler method (Yeganeh-Bakhtiary et al. 2011). Assuming that each phase shares the same pressure, the continuity and momentum equations are solved for each phase, which can be used to calculate dense particles. The continuity equations for both the liquid phase and solid phase are presented as follows (Jackson 1997):
(4)
(5)
where ρl and ρs are the density of the liquid and solid phases, and and are the velocity of the liquid and solid phases.
The momentum equations of the liquid phase and solid phase are written as (Jackson 1997):
(6)
(7)
(8)
(9)
where p is the static pressure, ps is the solid pressure, is the acceleration of gravity, τl and τs are the stress–strain tensors of the liquid phase and solid phase, λl and λs are the bulk viscosity of the liquid and solid phases, μl and μs are the shear viscosity, is the unit tensor, Ksl and Kls are the momentum exchange coefficient between two phases, Ksl = Kls, and are the external volume forces, and are the lift forces, and and are the virtual mass forces.

Drag model

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In this study, the Syamlal–O'Brien model is utilized for the drag model (Syamlal & O'Brien 1988). The Ksl is calculated as follows:
(10)
where CD is the drag function, Res is the particle relative Reynolds number, ur,s is the end velocity related to particles, and ds is the particle diameter.
The drag function CD is given by the form of Dalla Valle:
(11)
The particle relative Reynolds number Res is calculated as:
(12)
The end velocity ur,s is given as:
(13)
where ⁠, for αl ≤ 0.85 and for αl > 0.85.

Shear viscosity and bulk viscosity of particles

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The shear viscosity μs of solid includes collision viscosity μs,col, dynamic viscosity μs,kin, and friction viscosity μs,fr. The friction viscosity μs,fr can be ignored for the slurry flow with low concentration. Then, the shear viscosity μs can be expressed as (Wang et al. 2011):
(14)
where the collision viscosity μs,col is given by:
(15)
where g0,ss is the radial distribution function of particles, ess is the particle–particle collision recovery coefficient, default 0.9, and Θs is the particle temperature.
The dynamic viscosity is given by the Syamlal model:
(16)
The bulk viscosity λs of the solid phase is based on the theoretical model proposed by Lun et al. (1984):
(17)

Turbulence model

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In this study, the mixture turbulence model is used for calculating the turbulent flow. An extension of the single-phase k − ε turbulence model is employed to investigate the phase separation and stratification (or near stratification) of multiphase flow. The k equations and ε equations are presented as follows (Wang et al. 2011):
(18)
(19)
where μt,m is the turbulent viscosity coefficient of mixed-phase, σk, and σε are the Prandtl numbers corresponding to the turbulent kinetic energy k and dissipation ε, the values are 1.0 and 1.3, Gk,m is the turbulent kinetic energy, ρm is the density of mixed-phase, is the velocity of the mixed-phase, C1ε and C2ε are empirical constants, and the values are 1.47 and 1.92.
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Geometric models and calculation method

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Figure 1 shows the geometric model of the combined elbow with an inner diameter of 65 mm and a curve-to-diameter ratio of R/D = 1.5. The material used is seamless steel with a specified roughness height of 0.03 mm. The combined elbow is placed vertically, and the geometric areas of the calculated model are taken to be L01 = 600 mm and L02 = 850 mm, respectively, to ensure full development of turbulence. The length of the intermediate connecting section is set to 100 mm. To minimize the impact of the combined elbow on both upstream and downstream ranges, the pressure tapping points for calculating the pressure loss P1 and P2 are positioned at a distance of 8 times the diameter (L1 = 520 mm) upstream and 12 times the diameter (L2 = 780 mm) downstream, respectively. The continuous phase of the numerical calculation is liquid water with a temperature of 20 °C, a density of 998.2 kg/m3, and a dynamic viscosity of 1.003 × 10−3 Pa·s. The density of sediment particles is 2,650 kg/m3. The particle size ranges from 0.05 to 0.18 mm, and the particle concentration ranges from 5 to 18%. The velocities of solid and liquid phases are equal. Gravity is considered to act in the negative direction along the z-axis. The geometric models with combined angles of 0, 90, and 120° are shown in Figure 2.
Figure 1
Sketch diagram of combined elbows.
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Sketch diagram of combined elbows.

Figure 1
Sketch diagram of combined elbows.
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Sketch diagram of combined elbows.

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Figure 2
Geometric models of the combined elbow with combined angles of (a) 0°, (b) 90°, and (c) 120°.
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Geometric models of the combined elbow with combined angles of (a) 0°, (b) 90°, and (c) 120°.

Figure 2
Geometric models of the combined elbow with combined angles of (a) 0°, (b) 90°, and (c) 120°.
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Geometric models of the combined elbow with combined angles of (a) 0°, (b) 90°, and (c) 120°.

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Mesh generation and independence test

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The grids are generated using a combination of structured and unstructured meshes. Unstructured meshes with strong adaptivity are used to divide the turning part of the combined elbow, while structured meshes are utilized for dividing the remaining section. Figure 3 displays the grid division diagrams at combined angles of 0 and 90°.
Figure 3
Mesh generation with combined angles of (a) 0° and (b) 90°.
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Mesh generation with combined angles of (a) 0° and (b) 90°.

Figure 3
Mesh generation with combined angles of (a) 0° and (b) 90°.
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Mesh generation with combined angles of (a) 0° and (b) 90°.

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The calculation results may be affected by the number of grids. To ensure that the grid number has a slight effect on the calculation results, the grid independence is assessed by evaluating the pressure loss Δpb with an inlet velocity of 2 m/s, a combined angle of 0°, and a particle size of 0.05 mm. The results are shown in Figure 4. The pressure loss slightly changes when the grid number exceeds 600,000. Finally, the chosen grid number for the calculation is 643,350. In this paper, the range of grid division for different combined angles is approximately from 642,946 to 643,350.
Figure 4
Grid independence test.
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Grid independence test.

Figure 4
Grid independence test.
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Grid independence test.

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Discrete scheme and boundary conditions

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The governing equations are solved by the finite volume method based on Ansys Fluent software, and the iterative solution is performed using the Phase Combined Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm. The standard wall function approach is used to deal with the flow near the wall. To ensure calculation accuracy, the momentum equation, turbulent kinetic energy, and dissipation rate are calculated using the second-order upwind scheme, and the volume fraction is calculated using the first-order upwind scheme. The inlet boundary conditions are set as velocity conditions, with particle velocities defined in the same way as for the liquid phase. In addition, the particles are uniformly distributed across the inlet section. Free-flow boundary conditions are applied at the outlet of the combined elbow.

Model verification

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In this study, the experimental results of ice-slurry-liquid flow in a 90° horizontal elbow from Lee et al. (2002) are used to validate the computational results. Considering the similarity in motion between ice-slurry particles without phase change and sand particles, as well as the same pressure loss mechanism observed in two-phase flow through both single and combined elbows, the validated numerical model can be used to calculate the solid–liquid two-phase flow within combined elbows. The experimental elbow performed is shown in Figure 5, with a diameter of d = 0.024 m, a curvature-to-diameter ratio of Rc/d = 1.5, and an equivalent length of Le = 1.52 m. The liquid phase consisted of a mixture of 6.5% ethylene glycol and water, having a density of ρ = 1,008 kg/m3 and viscosity of υ = 2.4 × 10−3 kg/(ms). The solid phase was an ice slurry with a density of ρ = 917 kg/m3 and particle size of ds = 270 μm. The particle concentration was selected at 10 and 20%, and the velocity ranged from 0.75 to 2.25 m/s. The same velocity was applied to both the solid and liquid phases. In addition, the effect of the phase transition was not taken into account. The results obtained from the Euler–Euler model are compared with experimental tests, as shown in Figure 6. It is observed that the simulation results demonstrate a good agreement with the experimental findings, with a maximum relative error of 17.9%. It is evident that the Euler–Euler model is capable of simulating the solid–liquid two-phase flow in elbows. Therefore, the Euler–Euler model is also suitable for calculating combined elbows.
Figure 5
Experimental elbow from Lee et al. (2002).
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Experimental elbow from Lee et al. (2002).

Figure 6
Comparison of numerical simulation and experimental tests for (a) particle concentration of 10% and (b) particle concentration of 20%.
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Comparison of numerical simulation and experimental tests for (a) particle concentration of 10% and (b) particle concentration of 20%.

Figure 6
Comparison of numerical simulation and experimental tests for (a) particle concentration of 10% and (b) particle concentration of 20%.
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Comparison of numerical simulation and experimental tests for (a) particle concentration of 10% and (b) particle concentration of 20%.

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Pressure loss variation

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Changes with inlet velocity

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The pressure loss Δpb in this study is defined as the comprehensive pressure loss, which includes local losses, friction losses, and weight effects between sections P1 and P2 (as shown in Figure 1). For the solid–liquid two-phase flow in a combined elbow, the pressure loss Δpb depends on the losses in the liquid phase and solid phase. Liquid phase losses are caused by the friction and collisions between liquids, and also between the liquids and the walls. Solid phase losses are determined by the friction and collisions between solid particles, particles and walls, and particles and liquids. Figure 7 shows the variation of pressure loss for different inlet velocities in combined elbows. It can be observed that the pressure loss increases with velocity for different combined angles and concentrations. The increasing trend becomes more significant as the velocity increases. At a particle concentration of 8% and a combined angle of 0°, the pressure loss Δpb is 0.99 kPa for the velocity of 1.2 m/s. However, it rises to 5.14 kPa for a velocity of 3.5 m/s. This indicates that the velocity has a significant effect on the pressure loss. When the mixed flow enters the combined elbow, strong mixing flow occurs due to the deflection of the flow caused by the boundary change at the first elbow. The flow exiting from the first elbow does not recover as it enters the second elbow, causing a second deflection. The degree of turbulence in the liquid and solid phases increases with velocity. In addition, the friction and collisions at the two turns are also intense, leading to significant energy consumption. As a result, losses increase in both the liquid and solid phases. Moreover, as the velocity increases, the secondary flow becomes vigorous and has a greater impact on the pressure loss. In the case of combined elbows, the variation of pressure loss with flow velocity aligns with previous studies conducted on horizontal pipelines (Skudarnov et al. 2001).
Figure 7
Effect of inlet velocity on pressure loss for (a) particle concentration of 8% and (b) particle concentration of 15%.
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Effect of inlet velocity on pressure loss for (a) particle concentration of 8% and (b) particle concentration of 15%.

Figure 7
Effect of inlet velocity on pressure loss for (a) particle concentration of 8% and (b) particle concentration of 15%.
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Effect of inlet velocity on pressure loss for (a) particle concentration of 8% and (b) particle concentration of 15%.

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Changes with combined angle

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Figure 8 shows the variation of the pressure loss with different combined angles. As depicted in Figure 8, the pressure loss decreases as the combined angle increases, although the change is not significant. This phenomenon can be explained by analyzing the streamline diagram of the mixed flow. Figure 9 presents the streamline diagrams for a particle concentration of 18%, a particle size of 0.12 mm, and a flow velocity of 2.4 m/s at combined angles of 0, 60, 120, and 150°. It is evident that the flow in the first elbow remains smooth at various combined angles. Notably, a flow gradient occurs at elbow 1 due to the centrifugal force, while a secondary flow appears at elbow 1. It is also shown that the streamline variant at elbow 1 remains consistent at different combined angles. The flow out of elbow 1 does not resume before entering elbow 2 due to their close proximity. In addition, a flow gradient is formed in elbow 2 due to the centrifugal force, which tends to decrease with the increasing combined angle. The pressure loss in combined elbows mainly depends on the variation of vortex zones on both inner and outer walls, as well as the distribution of secondary flow (He et al. 2004). From Figure 9, it can be observed that there are no significant vortex regions in either elbow 1 or elbow 2 at different combined angles. The pressure loss in elbow 2 is primarily influenced by the secondary flow since the streamline variation remains consistent with elbow 1. The secondary flow weakens as the combined angle increases in cross-section S0, resulting in a reduction in pressure loss.
Figure 8
Effect of combined angle on pressure loss for (a) particle velocity of 1.6 m/s and (b) 2.4 m/s.
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Effect of combined angle on pressure loss for (a) particle velocity of 1.6 m/s and (b) 2.4 m/s.

Figure 8
Effect of combined angle on pressure loss for (a) particle velocity of 1.6 m/s and (b) 2.4 m/s.
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Effect of combined angle on pressure loss for (a) particle velocity of 1.6 m/s and (b) 2.4 m/s.

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Figure 9
Streamline diagrams with combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°.
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Streamline diagrams with combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°.

Figure 9
Streamline diagrams with combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°.
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Streamline diagrams with combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°.

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Changes in particle concentration

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Figure 10 shows the effect of particle concentration on pressure loss for combined angles of 0 and 90°. It is clear that the pressure loss increases with the increase in concentration. The reason is that as the concentration increases, the number of particles also increases, leading to an increase in the intensity of collisions between particles, as well as between particles and the wall surface. Therefore, the energy consumption increases, and the solid-phase pressure loss increases. At the same time, the intensification of particle movement also leads to an increase in turbulence intensity within the water flow. This change is consistent with the results of horizontal pipelines conducted by Hu et al. (2002). In the horizontal pipe, particles move at a slow pace and stay in a suspended state. As the concentration increases, more energy is required to keep the particles suspended in the flow. At the same time, the intensity of collisions between particles is also enhanced. Therefore, the pressure loss in the horizontal pipeline is caused by the action of both the solid phase and the liquid phase. In the combined elbow, due to the influence of boundary discontinuities, particle motion is more intense. Thus, the variation in pressure loss is mainly governed by the solid phase.
Figure 10
Effect of particle concentration on pressure loss for (a) particle velocity of 2.0 m/s and (b) 2.8 m/s.
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Effect of particle concentration on pressure loss for (a) particle velocity of 2.0 m/s and (b) 2.8 m/s.

Figure 10
Effect of particle concentration on pressure loss for (a) particle velocity of 2.0 m/s and (b) 2.8 m/s.
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Effect of particle concentration on pressure loss for (a) particle velocity of 2.0 m/s and (b) 2.8 m/s.

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Changes in particle size

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Figure 11 represents the effect of particle size on pressure loss at combined angles of 0 and 90°. It can be observed that there is a slight increase in pressure loss when the combined angle is 90°. While at an angle of 0°, the change in pressure loss with respect to particle size is relatively smaller and tends to stabilize. The relationship between pressure loss and particle size is more complex. Under specific concentrations, the number of particles decreases as the particle size increases, while the number increases when the particle size decreases. As the particle size increases, the intensity of collisions between particles and between particles and the wall surface increases. This can result in higher energy consumption and increased pressure loss. However, at the same time, an increase in particle size leads to a decrease in the number of particles. It is possible that larger particles with fewer numbers may consume less total energy through collisions compared to smaller particles with greater numbers, resulting in reduced pressure loss. Consequently, the pressure loss may either increase or decrease with an increase in particle size. Further investigations are required to determine the relationship between pressure loss and particle size.
Figure 11
Effect of particle size on pressure loss for (a) particle concentration of 5% and (b) 10%.
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Effect of particle size on pressure loss for (a) particle concentration of 5% and (b) 10%.

Figure 11
Effect of particle size on pressure loss for (a) particle concentration of 5% and (b) 10%.
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Effect of particle size on pressure loss for (a) particle concentration of 5% and (b) 10%.

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Velocity distribution at different combined angles

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Figure 12 shows the velocity distribution of the mixed flow at combined angles of 0, 60, 120, and 150° for an inlet flow velocity of 2.4 m/s, a concentration of 18%, and a particle size of 0.12 mm. As shown in Figure 12, the velocity distribution of each cross-section varies for different combined angles; however, the overall change is not significant. The changes in section S1 and section S2 are the same at different combined angles, and the flow velocity distribution in section S1 is not affected by the elbow turn. In section S2, the main flow tends to bias toward the inner wall of elbow 1 due to the centrifugal force, resulting in a higher flow velocity at the inner wall and a lower velocity at the outer wall. In section S3, the main flow is biased toward the outer wall, where the flow velocity is higher, while it is lower at the inner wall. It is found that the variations in the combined angle have a marginal effect on the velocity distribution in section S3. As the combined angle increases, there is a slight decrease in the velocity gradient in section S3. The velocity distribution in section S4 shows significant variations at different combined angles, with the maximum velocity occurring at the outer wall of elbow 2. As the combined angle increases, both the range of maximum flow velocity and the variability in the velocity gradient on section S4 tend to decrease. The flow in section S5 has not fully recovered, and there is minimal fluctuation in flow velocity at different combined angles.
Figure 12
Velocity distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.
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Velocity distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.

Figure 12
Velocity distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.
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Velocity distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.

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Particle concentration at different combined angles

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The particle concentration distribution of the mixed flow is shown in Figure 13 for combined angles of 0, 60, 120, and 150° with an inlet flow velocity of 2.4 m/s, a concentration of 18%, and a particle size of 0.12 mm. It can be seen that there is minimal variation in particle concentration for different combined angles. In cross-sections S1 to S3, the particle distribution is essentially the same regardless of the combined angle. In cross-section S1, the effect of gravity leads to a higher particle concentration at the bottom of the pipe and a lower concentration at the top. This variation persists until section S3, indicating that a significant number of particles are not transported to the inner pipe wall by the centrifugal force. The particle variation in sections S4 and S5 remains significantly higher at the outer wall and lower at the inner wall, indicating that the inertial force has a pronounced effect on particle distribution compared to gravity.
Figure 13
Particle concentration distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.
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Particle concentration distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.

Figure 13
Particle concentration distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.
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Particle concentration distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.

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Turbulent kinetic energy distribution at different combined angles

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The turbulent kinetic energy is related to the intensity of turbulent motion, and regions exhibiting high levels of turbulent kinetic energy indicate vigorous turbulent motion. Figure 14 shows the turbulent kinetic energy distribution in the mixed flow at combined angles of 0, 60, 120, and 150° under conditions of an inlet flow velocity of 2.4 m/s, a concentration of 18%, and a particle size of 0.12 mm. Based on Figure 14, it can be observed that the variation in turbulent kinetic energy distribution across sections S1 to S3 remains consistent for different combined angles. In the inlet cross-section S1, the turbulent kinetic energy shows a relatively uniform distribution, indicating that the flow remains unaffected by the turn. Section S2 shows less variability in the gradient of turbulent kinetic energy. A more evident gradient in turbulent kinetic energy is observed in section S3, indicating the presence of an adjacent influence effect between the two elbows. The distribution of turbulent kinetic energy in sections S4 and S5 exhibits significant variations for different combined angles. With the increase in combined angle, there is a progressive decrease observed in both the maximum distribution area of turbulent kinetic energy and the gradient of turbulent kinetic energy between the two sections. This indicates that elbow 2 experiences a more intense turbulent motion, leading to higher energy consumption and resulting in increased pressure loss. The observed variation in pressure loss at different combined angles primarily results from the flow variability occurring within elbow 2.
Figure 14
Turbulent kinetic energy distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.
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Turbulent kinetic energy distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.

Figure 14
Turbulent kinetic energy distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.
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Turbulent kinetic energy distribution for combined angles of (a) 0°, (b) 60°, (c) 120°, and (d) 150°. The letter ‘R’ represents the right side of the observed section along the flow direction; ‘L’ designates the left side of the section; ‘O’ refers to the outer wall of this section; and ‘I’ indicates the inner wall. S1 ∼ S5 represents different cross-sections.

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In this paper, the pressure loss and flow fields for the solid–liquid two-phase flow in combined elbows were investigated using numerical simulation based on the Euler–Euler multiphase flow model. The conclusions obtained are as follows:

  • (1) The pressure loss of the combined elbow increases as the inlet velocity increases, slightly decreases as the combined angle increases, and rises as the concentration increases.

  • (2) The relationship between particle size and pressure loss is more complex. As the particle size increases, the pressure loss may either decrease or increase. For a given particle concentration, a decrease in particle size leads to an increase in the number of particles, while an increase in particle size leads to a decrease in their number.

  • (3) The velocity distribution is analyzed, revealing that the velocity distribution remains consistent in elbow 1 at various combined angles, while variations in velocity gradients primarily occur in elbow 2.

  • (4) The analysis of particle concentration and turbulent energy distribution reveals that the variation in elbow 1 remains consistently small for different combined angles, while the variation in elbow 2 exhibits a significant magnitude.

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This work was supported by the National Natural Science Foundation of China (51969011), the Natural Science Foundation of Gansu Province (21JR7RA684), and the Lanzhou Jiaotong University-Tianjin University Joint Innovation Fund (LH2023008).

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Data cannot be made publicly available; readers should contact the corresponding author for details.

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The authors declare there is no conflict.

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