## ABSTRACT

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.

## HIGHLIGHTS

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.

## INTRODUCTION

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.

## MATHEMATICAL MODELS

### Governing equations

#### Volume fraction

The volume fraction *α _{i}* 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.

#### Conservation equations

*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):where

*ρ*

_{l}and

*ρ*

_{s}are the density of the liquid and solid phases, and and are the velocity of the liquid and solid phases.

*p*is the static pressure,

*p*

_{s}is the solid pressure, is the acceleration of gravity,

*τ*

_{l}and

*τ*

_{s}are the stress–strain tensors of the liquid phase and solid phase,

*λ*

_{l}and

*λ*

_{s}are the bulk viscosity of the liquid and solid phases,

*μ*

_{l}and

*μ*

_{s}are the shear viscosity, is the unit tensor,

*K*

_{sl}and

*K*

_{ls}are the momentum exchange coefficient between two phases,

*K*

_{sl}=

*K*

_{ls}, and are the external volume forces, and are the lift forces, and and are the virtual mass forces.

#### Drag model

*K*

_{sl}is calculated as follows:where

*C*

_{D}is the drag function, Re

_{s}is the particle relative Reynolds number,

*u*

_{r,s}is the end velocity related to particles, and

*d*

_{s}is the particle diameter.

#### Shear viscosity and bulk viscosity of particles

*μ*

_{s}of solid includes collision viscosity

*μ*

_{s,col}, dynamic viscosity

*μ*

_{s,kin}, and friction viscosity

*μ*

_{s,fr}. The friction viscosity

*μ*

_{s,fr}can be ignored for the slurry flow with low concentration. Then, the shear viscosity

*μ*

_{s}can be expressed as (Wang

*et al.*2011):where the collision viscosity

*μ*

_{s,col}is given by:where

*g*

_{0,ss}is the radial distribution function of particles,

*e*

_{ss}is the particle–particle collision recovery coefficient, default 0.9, and Θ

_{s}is the particle temperature.

*λ*

_{s}of the solid phase is based on the theoretical model proposed by Lun

*et al.*(1984):

### Turbulence model

*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):where

*μ*

_{t,m}is the turbulent viscosity coefficient of mixed-phase,

*σ*

_{k,}and

*σ*

_{ε}are the Prandtl numbers corresponding to the turbulent kinetic energy

*k*and dissipation ε, the values are 1.0 and 1.3,

*G*

_{k,m}is the turbulent kinetic energy,

*ρ*

_{m}is the density of mixed-phase, is the velocity of the mixed-phase,

*C*

_{1ε}and

*C*

_{2ε}are empirical constants, and the values are 1.47 and 1.92.

## NUMERICAL METHODOLOGY

### Geometric models and calculation method

*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

*L*

_{01}= 600 mm and

*L*

_{02}= 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

*P*

_{1}and

*P*

_{2}are positioned at a distance of 8 times the diameter (

*L*

_{1}= 520 mm) upstream and 12 times the diameter (

*L*

_{2}= 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/m

^{3}, and a dynamic viscosity of 1.003 × 10

^{−3}Pa·s. The density of sediment particles is 2,650 kg/m

^{3}. 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.

### Mesh generation and independence test

*p*

_{b}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.

### Discrete scheme and boundary conditions

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

*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

*R*

_{c}/

*d*= 1.5, and an equivalent length of

*L*

_{e}= 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/m

^{3}and viscosity of

*υ*= 2.4 × 10

^{−3}kg/(ms). The solid phase was an ice slurry with a density of

*ρ*= 917 kg/m

^{3}and particle size of

*d*

_{s}= 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.

## RESULTS AND DISCUSSIONS

### Pressure loss variation

#### Changes with inlet velocity

*p*

_{b}in this study is defined as the comprehensive pressure loss, which includes local losses, friction losses, and weight effects between sections

*P*

_{1}and

*P*

_{2}(as shown in Figure 1). For the solid–liquid two-phase flow in a combined elbow, the pressure loss Δ

*p*

_{b}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 Δ

*p*

_{b}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).

#### Changes with combined angle

*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

*S*

_{0}, resulting in a reduction in pressure loss.

#### Changes in particle concentration

*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.

#### Changes in particle size

### Velocity distribution at different combined angles

_{1}and section S

_{2}are the same at different combined angles, and the flow velocity distribution in section S

_{1}is not affected by the elbow turn. In section S

_{2}, 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 S

_{3}, 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 S

_{3}. As the combined angle increases, there is a slight decrease in the velocity gradient in section S

_{3}. The velocity distribution in section S

_{4}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 S

_{4}tend to decrease. The flow in section S

_{5}has not fully recovered, and there is minimal fluctuation in flow velocity at different combined angles.

### Particle concentration at different combined angles

_{1}to S

_{3}, the particle distribution is essentially the same regardless of the combined angle. In cross-section S

_{1}, 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 S

_{3}, indicating that a significant number of particles are not transported to the inner pipe wall by the centrifugal force. The particle variation in sections S

_{4}and S

_{5}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.

### Turbulent kinetic energy distribution at different combined angles

_{1}to S

_{3}remains consistent for different combined angles. In the inlet cross-section S

_{1}, the turbulent kinetic energy shows a relatively uniform distribution, indicating that the flow remains unaffected by the turn. Section S

_{2}shows less variability in the gradient of turbulent kinetic energy. A more evident gradient in turbulent kinetic energy is observed in section S

_{3}, indicating the presence of an adjacent influence effect between the two elbows. The distribution of turbulent kinetic energy in sections S

_{4}and S

_{5}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.

## CONCLUSIONS

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.

## ACKNOWLEDGEMENTS

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).

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.