Analysis on Inner Flow Field and Hydrodynamic Force on Flexible Mining Pipeline Under Bending States

Abstract

To investigate the internal flow characteristics of particles during hydraulic lifting in deep-sea mining risers, this study developed a three-dimensional curved riser multiphase flow model based on the Eulerian–Eulerian framework and the RNG k-ε turbulence model. The effects of particle distribution and pressure loss in the curved section, as well as the influence of curvature radius, were analyzed. Results indicate that particle distributions take concave circular or crescent-shaped patterns, becoming more uniform with larger curvature radii. Pressure on the extrados is consistently greater than on the intrados, with pressure loss increasing in the bend and peaking at the midpoint. A larger curvature radius leads to greater total pressure loss but lower frictional loss. Additionally, the bend experiences a restoring force toward the vertical position, which increases as the curvature radius decreases.

1. Introduction

Deep-sea polymetallic nodules (such as manganese nodules and cobalt-rich crusts), as strategic mineral resources, are rich in critical metallic elements, including nickel, copper, cobalt, and lithium. They hold irreplaceable strategic value for new energy industries, high-end equipment manufacturing, and defense technologies. Global estimates indicate over 3 trillion tons of deep-sea polymetallic nodules exist on the ocean floor, with the Pacific Ocean’s Clarion–Clipperton Fracture Zone (CCZ) alone containing approximately 21 billion tons of manganese nodules [1]. As terrestrial mineral resources deplete and ore grades decline, deep-sea mining has become a new focal point in global resource competition. However, the commercial extraction of deep-sea minerals faces numerous technological challenges, among which efficient ore particle transport is a core enabling component for resource development [2,3,4].

In deep-sea mining hydraulic lifting systems, flexible risers demonstrate significant advantages over rigid risers. Flexible risers possess superior corrosion resistance and fatigue performance, coupled with a lightweight structure (weighing over 60% less than rigid pipes of equivalent specifications), substantially enhancing transportation and installation efficiency [5,6,7]. In contrast, rigid risers are susceptible to electrochemical corrosion, exhibit higher risks of fatigue crack initiation under cyclic loading, and suffer from excessive self-weight, leading to sharply increased installation complexity [8]. Based on these characteristics, flexible risers, owing to their curved adaptability and tolerance to dynamic loading, serve as the core transport conduit connecting seabed collectors and surface support platforms. However, over several thousand meters of transport distance, risers must withstand extreme marine environmental loads. Heave motion of the support vessel induced by wave action causes low-frequency, large-amplitude deformation in the pipeline [9]. Deep-ocean current impacts trigger vortex-induced vibration (VIV) [10], compounded by the ore’s self-weight and uneven axial tension. Engineering measurements and the literature indicate that under dynamic loading, the curvature radius of flexible risers can decrease below 15D (where D is the pipe diameter) [11]. Consequently, curved risers represent the normal operational state for risers, inevitably altering the internal flow field. Therefore, research and analysis of the flow characteristics of solid–liquid two-phase flow within curved risers are essential.

Currently, as a research foundation, the study of solid–liquid two-phase flow in horizontal, vertical, and inclined straight pipes is quite thorough. Relatively mature theoretical frameworks (such as particle force models, interphase drag models, and turbulence modulation mechanisms) and numerical simulation methods (such as the Euler–Euler multifluid model and the Euler–Lagrange discrete particle model) have been established [12,13]. At the application level, CHEN et al. [14] conducted numerical simulations based on the Euler–Euler two-fluid model, focusing on the influence of particle volume fraction and flow velocity changes on the pressure drop characteristics and flow pattern transition of solid–liquid two-phase flow. Their Computational Fluid Dynamics (CFD) numerical simulation results showed high agreement with theoretical calculations and experimental data. Moreover, compared to the experimental data, the simulation errors were smaller and could accurately reflect the slurry flow patterns within the pipeline, providing an important reference for the study of solid–liquid two-phase flow in horizontal pipelines. Similarly, SU et al. [15] employed a coupled approach of liquid-phase CFD and particle-phase Discrete Element Method (CFD-DEM). Using a scaled vertical lifting pipeline model as the research subject, they simulated solid–liquid two-phase flow in vertical pipelines. They found that an increase in flow velocity could significantly improve particle distribution within the pipeline and enhance particle following behavior. Concurrently, disturbances generated by particle collisions and mixing due to changes in particle concentration significantly impacted both the velocity distribution of the two-phase flow and the pipe wall pressure distribution. Furthermore, DAI et al. [16], utilizing the coupling interface between Fluent (ANSYS 2021R2)and EDEM, performed numerical simulations of the internal flow field in pipelines under different flow rates and particle concentration conditions. Laboratory experiments validated that internal flow velocity and particle concentration significantly affect the stress and transport performance of mining pipelines. WAN et al. [17] used machine learning techniques to predict the pressure drop in solid–liquid flow, and the results also demonstrated high accuracy.

When the research focus shifts from straight pipes to the curved pipes commonly found in flexible risers, the situation becomes much more complex. Secondary flows, centrifugal effects, phase separation, and asymmetric flow induced by curvature significantly increase both flow complexity and simulation difficulty. Therefore, although studies on straight pipes provide an important foundation, numerical investigations of solid–liquid two-phase flow in curved pipes (particularly in curved risers) remain relatively limited. Relevant research mainly includes Ling et al. [18], who studied multiphase flow distribution in horizontal bends and found that flow velocity is the dominant influencing factor; Gao et al. [19], who investigated phase separation in solid–liquid turbulent flow within bends, analyzing the effects of particle size, flow rate, and curvature on phase distribution, and elucidating the dynamic contributions of centrifugal force, drag force, and other mechanisms; Xie et al. [20], who, based on Hamilton’s principle, developed a nonlinear dynamic model for flexible pipes conveying variable-density fluids subjected to CF–IL coupled VIV; Pickles et al. [21], who conducted experiments on composite J-shaped risers and found that multiphase flow induces significant in-plane vibrations, with higher liquid flow rates (or lower gas flow rates) leading to greater mean strain in the pipe; Zhang [22], who identified an M-shaped velocity distribution in horizontal U-shaped pipes; and Yang [23], who investigated natural gas hydrates, analyzing the influence of flow velocity and pipe diameter on particle volume fraction. Overall, existing studies on solid–liquid multiphase flow in curved pipes are mainly focused on phase distribution characteristics at the inlet of horizontal pipes and the erosive effects of solid particles on bends. However, studies addressing curved riser systems with vertical inlets in hydraulic lifting processes, particularly concerning key flow parameters such as pressure loss and structural loading in curved sections, are still relatively scarce.

Although the above studies have made considerable progress in straight pipes and partially curved pipes, certain limitations remain. On the one hand, most existing research on curved pipes has focused on horizontal inlet conditions, the results of which cannot fully reflect the flow characteristics in curved risers with vertical inlets. On the other hand, scholars have not reached a consensus on the key influencing parameters. In addition, some studies have paid insufficient attention to the overall force distribution and pressure loss mechanisms in curved sections. These shortcomings indicate that the current theoretical and numerical frameworks are still insufficient to fully reveal the complex flow behavior of solid–liquid two-phase flows in curved risers, leaving room for further investigation.

To address the aforementioned research gap, this paper focuses on the flexible bent riser in a hydraulic lifting system. Section 2 introduces the numerical model for solid–liquid two-phase flow (the Eulerian–Eulerian multiphase model). Section 3 conducts numerical simulations of the solid–liquid two-phase flow transporting ore particles within the bent riser and analyzes the internal flow field of the bent riser, including flow field distribution, pressure loss, and force distribution on the bent section. Section 4 specifically focuses on the influence of the curvature radius of the bent pipe section on the internal flow field.

2. Solid–Liquid Two-Phase Flow Numerical Model

2.1. Mathematical Model

The transportation of solid particles within a riser constitutes a multiphase flow. The Euler–Euler approach is currently the prevalent analytical method. This method treats seawater as a continuous medium and solid particles as a pseudo-fluid, assuming that solid particles exhibit continuous pressure and velocity distributions in space. Both phases coexist spatially while interpenetrating, yet each phase maintains its own volume fraction. Slip between phases is permitted, and both phases are described within the Eulerian coordinate system. For analysis, the following simplifications and assumptions can be made regarding solid transportation in the riser: (1) The flow is a steady-state, three-dimensional pipe flow under turbulent conditions. (2) No heat or mass transfer occurs between the solid and liquid phases. (3) Particles are spherical with uniform diameter.

The solid–liquid two-phase flow in the riser should satisfy the following fluid equations:

(1). Governing equations

Assuming constant density during transport, the continuity equation and Navier–Stokes (N-S) equations [24] are expressed as follows:

(1)${\displaystyle \frac{\partial u_i}{\partial x_i}}=0$

(2)${\displaystyle \frac{\partial \rho u_i}{\partial t}}+\rho {\displaystyle \frac{\partial }{\partial x_i}}(u_i{u}_j)=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}(\mu {\displaystyle \frac{\partial u_i}{\partial x_j}}-\rho \overline{u_i^{\prime }{u}_j^{\prime }})$

where x_i and x_j denote coordinate components (i, j = 1, 2, 3), u_i and u_j represent time-averaged velocity components, p is the time-averaged pressure, μ denotes dynamic viscosity, ρ is fluid density, and $\overline{u_i^{\prime }{u}_j^{\prime }}$ is the Reynolds stress component.

(2). Turbulence model

The transport equations for the standard k-ε model [24] are given as follows:

(3)${\displaystyle \frac{\partial }{\partial t}}\left(\rho k\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho k{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]+G_k+G_b-\rho \epsilon -Y_m$

(4)${\displaystyle \frac{\partial }{\partial t}}\left(\rho k\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho k{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\partial \epsilon }{\partial x_j}}\right]+C_{\epsilon 1}{\displaystyle \frac{\epsilon }{k}}\left(G_k+C_{3\epsilon }{G}_b\right)-C_{\epsilon 2}\rho {\displaystyle \frac{{\epsilon }^2}{k}}$

where G_k represents the turbulent kinetic energy generated by mean velocity gradients, G_b denotes the buoyancy-induced turbulence energy, Y_m accounts for the contribution of fluctuating dilatation to the overall dissipation rate in compressible turbulent flows, and σ_k and σ_ε are the turbulent Prandtl numbers for k and ε, respectively, assigned values of 1.0 and 1.3.

2.2. Validation of Multiphase-Flow Numerical Method

The vertical riser model serves as the foundation for calculating curved riser models, equivalent to the limiting case when the curvature radius of the bent section approaches infinity. While extensive experimental data exist for vertical risers, experimental studies on curved risers remain limited. Therefore, validation of computational models for curved risers is performed using vertical riser models.

An ICEM-generated pipe model has a diameter of 10 cm and a length of 30 m. A structured mesh is applied to the model. A grid convergence study was performed for single-phase water flow through the pipe. First, the grid convergence study focused on the pipe length direction, with divisions of 100, 200, 300, and 400 cells along the length. Boundary conditions were set as follows:

(1). Inlet condition: Velocity inlet at 3 m/s, with turbulent intensity and hydraulic diameter calculated and set;

(2). Wall condition: No-slip wall condition applied;

(3). Outlet condition: Pressure outlet with gauge pressure defined as 0, turbulent intensity, and backflow hydraulic diameter set identical to the inlet.

The Reynolds number was approximately 4 × 10⁵. The standard k-ε model was selected. After the calculation, the velocity of single-phase water along the pipe centerline in the length direction was extracted to assess grid convergence along the length.

Since the flow becomes fully developed 10 m downstream of the inlet, results are presented along the pipe length from 0 m to 20 m, as shown in Figure 1: it can be observed that when the pipe is divided into 100, 200, 300, and 400 grid cells along the length direction, the velocity at the pipe center remains essentially constant throughout the entire pipe length. The maximum overall error does not exceed 1%. This maximum error occurs in the region approximately 5 m from the bottom end. A closer examination of this region is shown in Figure 2.

As shown in Figure 2, the model with 200 grids along the pipe length already satisfies the convergence requirement in the axial direction. After determining the mesh along the pipe length, it is necessary to verify the mesh convergence at the pipe cross-section. The cross-sectional mesh numbers of the riser were set to 384, 720, 1152, and 1680. The velocity distributions at different x-coordinates along the intersection of the cross-section located 10 m from the pipe inlet and the plane y = 0 were analyzed, as illustrated in Figure 3.

As shown in Figure 3, the cross-sectional mesh resolution has only a minimal influence on the velocity distribution after the flow stabilizes. Even with 384 elements across the section, the requirement for analysis is already satisfied. However, considering future analyses involving larger-diameter pipes and the flow field in curved risers, which demand higher mesh quality, the model with 1152 cross-sectional elements was adopted for subsequent simulations.

The experimental data employed in this study were obtained from the Slurry Pump Hydraulic Lifting Experimental System at the Changsha Research Institute of Mining and Metallurgy [25], as shown in the table below (Table 1 and Table 2).

The frictional pressure gradient reported in the literature refers to the unit-length resistance loss in pipeline transportation, calculated as follows:

(5)$\Delta p={\displaystyle \frac{P_1-P_2-\rho g(h_1-h_2)}{{\rho }_{w}g(h_1-h_2)}}$

where Δp denotes the frictional pressure gradient; P₁ and P₂ represent absolute pressures at elevations h₁ and h₂, respectively; ρ and ρ_w are the mean mixture density in the pipe and density of water; and h is the elevation of the pressure measurement points.

Based on the experimental prototype, a computational model was established with a diameter of 10 cm and a length of 30 m. The domain was discretized with structured grids, as illustrated in Figure 4.

Numerical simulations were conducted using the Eulerian multiphase flow model in FLUENT. Boundary conditions were configured with a weighted average particle diameter of 2 cm: the inlet was specified as a velocity boundary matching experimental flow rates with 10% solid volume fraction in axial orientation, where turbulence intensity and hydraulic diameter were specified. Wall boundaries employed no-slip conditions incorporating surface roughness. The outlet utilized a pressure outlet condition at 0 Pa gauge pressure with turbulence parameters mirroring the inlet configuration. Interphase interactions accounted for the added mass coefficient, drag model, lift model, turbulent dispersion model, and collision coefficients.

During post-processing, simulation data from the stabilized 10~30 m axial region were extracted to eliminate inlet development effects, with subsequent validation against experimental data, as presented in Figure 5.

As shown in Figure 5, the frictional loss between experimental and simulated values—which is a key macroscopic parameter for validating the reliability of the numerical model (including the turbulence model, wall functions, and particle–phase coupling)—has an error within 4%. This indicates that the Eulerian multiphase flow model in FLUENT can accurately simulate the solid–liquid two-phase flow experiment, and can be further used to investigate particle transport behavior within the riser.

3. Analysis of Flow Field in the Curved Riser

3.1. Establishment of Curved Riser Model

The established curved riser model and a detailed view of its grid topology are plotted in Figure 6. To mitigate inlet flow development effects on pressure distribution, the model incorporates a 4 m vertical inflow section (denoted as x₁) preceding the bend and a 4 m inclined outflow section (x₂) downstream. These segments are connected by a curved section with tangential continuity, featuring a curvature radius r = 4 m, arc angle α = 30°, and arc length ≈ 2.09 m (pipe diameter d). Structured meshing was implemented in ICEM CFD, with refined axial discretization within the curved region.

3.2. Distribution of Particle Volume Fraction

Boundary conditions for the curved riser were configured as follows: The inlet adopted a velocity inlet boundary with water and particle velocities set at 3 m/s, particle volume fraction at 15%, and flow direction normal to the inlet plane, where turbulence intensity and hydraulic diameter were explicitly defined. Wall boundaries employed no-slip conditions with zero surface roughness. The outlet utilized a pressure outlet condition, with turbulence parameters identical to the inlet. Particles were characterized by a uniform diameter of 1 cm and a density of 2000 kg/m³.

Following computation, to analyze flow field distributions within the curved riser, the inflow section, curved section, and outflow section were each divided into four equal segments as illustrated in Figure 6a. Contour plots of particle volume fraction on cross-sections perpendicular to the riser axis are presented in Figure 7. Spatial orientation conventions are defined as follows: in the inflow section, the right side corresponds to the riser’s right flank; in the curved section, the right side aligns with the riser’s inner curvature (intrados). In the outflow section, the right side denotes the lower flank. Throughout subsequent analysis, “inner side” refers to the right side as defined per section, “outer side” to the left, while “upper” and “lower” denote the vertical flanks.

Comparative analysis of Figure 7b,c reveals distinct differences in particle volume fraction distribution between curved and vertical risers. In Figure 7c, particle concentration stabilizes at the mid-section (2/4 position, the central area is approximately 0.21–0.24, while the peripheral area is approximately 0–0.12.) of the inflow segment. Upon entering the curved section (1/4 position), centrifugal forces drive particles toward the extrados (outer curvature). This migration intensifies through the bend, reaching peak accumulation at the curved-outflow transition interface. Notably, a particle-depleted zone persists within the near-wall region of the extrados.

Downstream of the bend, gravitational settling in the inclined outflow section promotes particle sedimentation toward the lower quadrant. However, steep velocity gradients near the bottom wall generate significant lift forces, maintaining maximum concentration along the lower axis rather than wall deposition. This creates a stable off-axis concentration core suspended above the wall layer.

Figure 7a,b demonstrate crescent-shaped particle concentration profiles resulting from incompressible fluid dynamics in curved sections. Upon entering the bend, centrifugal forces induce a transverse pressure gradient that drives fluid from the intrados (inner curvature) toward the extrados (outer curvature). After impacting the extrados wall, fluid recirculates along the sidewalls back to the intrados, forming counter-rotating Dean vortices (Figure 7b, Dean vortices: a pair of counter-rotating, symmetrical secondary-flow vortices induced in curved pipes by the combined action of centrifugal force and radial pressure gradient).

As slurry exits the bend into the inclined outflow section, persistent secondary flows transport particles accumulated at the extrados toward the intrados along the sidewalls. These particles become trapped within residual Dean vortices, manifesting as high-concentration zones near the vertical flanks at the 2/4 position (Figure 7a, approximately between 0.3 and 0.3).

With progressive attenuation of secondary flows downstream, the vortex pair converges toward the central axis and dissipates. Particles subsequently migrate radially inward under gravitational settling, reforming a stable concentration maximum near the intrados wall. Unlike vertical risers, this sustained particle accumulation adjacent to the intrados wall (Figure 7a, 4/4 position) elevates erosion risks due to continuous particle-wall collisions.

3.3. Pressure Loss

Fluid–structure interactions within the riser are primarily gravity-dominated in vertical sections. However, in curved riser segments, centrifugal forces introduce additional loading that significantly alters wall pressure distributions. Figure 8 illustrates the pressure distribution along the wall of the riser under baseline operating conditions.

Figure 8 reveals isobars parallel to horizontal planes in both inflow and outflow sections, indicating gravity-dominated pressure distributions. In the curved section (Figure 8b, the red-circled part in Figure 8a), slurry pressure at the extrados (outer bend) consistently exceeds that at the intrados (inner bend) for equivalent elevations, demonstrating centrifugal redistribution of internal forces. To further quantify this effect, axial pressure profiles along the riser centerline and corresponding frictional pressure gradients per unit length are extracted from inlet to outlet, as presented in Figure 9.

In Figure 9, the 0~4 m segment corresponds to the riser inflow section, 4~6.09 m to the curved riser section, and 6.09~10.09 m to the outflow section. The figure reveals distinct unit pressure losses across these segments. The pressure distribution stabilizes approximately 0.1 m from the inlet, after which the unit pressure loss per length remains constant. However, at 3.5~4 m from the inlet, due to the influence of the curved section, the unit pressure loss begins to decrease slightly. Upon entering the curved section (4 m), it increases, peaking at the bend midpoint. Approaching the bend exit (5.75~6.09 m), the unit pressure loss drops to its minimum and then increases and gradually stabilizes within the inclined outflow section. The sudden pressure loss at elbows is primarily caused by increased local resistance, which is a normal hydraulic phenomenon. Although there is additional energy loss in these regions, the overall along-the-pipe pressure distribution remains smooth and uniform. Therefore, the impact on process operation is minimal and does not compromise system stability. However, elbow locations are sensitive to flow impact and wear, and long-term operation may lead to localized erosion or fatigue, which could affect the pipeline’s service life.

This study conducted a numerical analysis of the flow characteristics in a curved riser. Results indicate that within the bend, particles are driven by centrifugal forces toward the extrados, reaching peak accumulation near the bend outlet. Subsequently, under the combined influence of residual secondary flows and gravitational settling, particles gradually migrate toward the intrados and lower section, ultimately forming a high-concentration region adjacent to the intrados wall, which increases local erosion risks. Flow field analysis reveals the presence of typical Dean vortex structures in the curved section, leading to crescent-shaped particle distributions across the cross-section and promoting lateral particle migration in the downstream segment. Regarding pressure distribution, gravity dominates in the straight sections, where isobars remain nearly horizontal, whereas in the bend, pressure at the extrados consistently exceeds that at the intrados, highlighting the role of centrifugal effects. Axial pressure drop exhibits a segmented pattern: stabilization in the straight inlet, a maximum loss at the bend midpoint, and gradual recovery downstream. Overall, the curved geometry significantly alters particle distribution and pressure loss characteristics during slurry transport.

4. Influence of Curvature Radius in Curved Section

4.1. Impact on Particle Volume Fraction Distribution

In seawater environments, variations in external hydrodynamic loads or internal solid–liquid two-phase flow parameters can induce changes in the curvature of the riser’s bent section. Accordingly, it becomes essential to evaluate how different curvature radii affect the internal flow characteristics. To address this, a computational model was developed with a riser diameter of 0.2 m, comprising 4 m long inlet and outlet sections and a fixed curved section length of 2.09 m. The curvature radius of the bend was systematically varied (2 m, 4 m, and 6 m) and compared with a reference case featuring an infinite-radius vertical riser. All boundary conditions remained consistent with those described in Section 2.2.

Figure 10 illustrates the particle volume fraction distributions and velocity fields at both the exit of the curved segment and the downstream end of the outlet section for each curvature radius case.

As shown in Figure 10, when the curvature radius of the riser bend is relatively small, the centrifugal effects become pronounced, leading to the formation of strong secondary flows. Under the combined influence of centrifugal force and secondary flow, particles initially concentrated near the outer wall are entrained by sidewall fluid motion and are redirected toward the inner wall through near-wall regions. This results in a concave particle concentration profile across the cross-section. Furthermore, the flow velocity and velocity gradient near the outer wall are relatively large, producing a considerable lift force. Consequently, even with a curvature radius of 2 m, particle–wall collisions on the outer side of the bend are effectively avoided.

As the curvature radius increases, the centrifugal force weakens, and the secondary flow intensity diminishes. Interestingly, the degree of particle accumulation on the outer wall initially increases and then decreases, suggesting a non-monotonic relationship. A larger curvature radius reduces the radial acceleration acting on the particles, thereby expanding the distribution area. When the curvature radius approaches infinity, the particle distribution becomes most uniform, exhibiting an almost perfect circular shape.

4.2. Influence on Pressure Loss

Figure 11 demonstrates that solid–liquid two-phase flow through a curved riser section exhibits significantly higher pressure loss per unit length than in vertical risers. Centrifugal forces induce complex flow restructuring, generating additional pressure loss and frictional dissipation. Thus, the curved section profoundly impacts pressure loss characteristics. Computed pressure loss variations with curvature radius reveal a monotonic increase as radius increases, asymptotically approaching the stable value equivalent to vertical riser pressure loss.

The frictional pressure gradient diminishes with larger curvature radii. This reduction stems from weakened centrifugal forces and attenuated secondary flows with larger radii, which promote more uniform particle concentration distributions and mitigate flow field distortions. Consequently, viscous dissipation at pipe walls decreases substantially. These findings indicate that increasing the riser’s curvature radius during operational deployment can effectively reduce frictional losses.

4.3. Effect on Mechanical Force in Curved Section

When solid–liquid two-phase flow traverses a curved riser section, the pipe must exert centripetal forces to maintain curved trajectory flow, while simultaneously providing the forces required to redistribute particles, imposing additional mechanical demands in vertical risers. Consequently, curved sections experience not only static fluid pressure but also dynamic reaction forces from flow state alterations. To evaluate curvature radius effects on structural loading, forces exerted by the multiphase flow on bends of varying curvature were analyzed. Figure 12 presents x-direction (horizontal) and z-direction (vertical) reaction forces on the curved section, with force vectors oriented radially outward from the bend centerline.

This outward-directed force promotes realignment toward a vertical configuration. Figure 12 demonstrates that smaller curvature radii generate greater restoring forces. Both x- and z-direction forces diminish as the curvature radius increases. At infinite curvature radius (vertical riser), the x-direction force approaches zero while the z-direction force equilibrates to the viscous force inherent to straight-pipe flow. With increasing bend radius, the forces acting on the pipeline (in the x and z directions) gradually decrease. These forces mainly arise from the fluid’s inertial deflection and local pressure differences at the bend. For the process operation, such force variations do not significantly affect fluid transport performance, as the along-the-pipe pressure distribution and overall energy loss remain smooth and controllable. However, for the pipeline itself, these forces indicate that the elbow regions will experience additional loads, which are particularly pronounced at bends with smaller radii.

5. Conclusions

This paper investigates the spatial evolution of particle distribution and the associated pressure/pressure-drop losses within a curved riser transporting solid–liquid two-phase flow, and examines the mechanical response of the curved section. The main conclusions are as follows:

(1). In vertical risers, particles distribute in a circular pattern. Upon entering curved sections, particles migrate toward the outer bend under centrifugal force. Secondary flows reshape the distribution into a concave circular or crescent profile. At the bend exit, particle concentration peaks at the outer bend. In the outflow section, gravitationally settling particles concentrate toward the bottom, migrating inward from both flanks while forming particle-depleted zones (0% concentration) near the outer flank. Near-wall regions maintain low concentrations due to lift forces throughout curved and outflow sections. Larger curvature radii yield increasingly uniform distributions.

(2). In the inflow and outflow sections of the curved riser, the pressure distribution is mainly influenced by gravity, and the isobars are parallel to the horizontal plane. In the curved sections, combined gravity and centrifugal forces elevate pressure at the outer bend versus the inner bend. The pressure loss per unit length along the pipe direction decreases slightly before flowing into the curved section, increases when entering the curved section, and reaches a maximum at the middle position of the curved section. Then, the pressure loss starts to decrease, drops to the minimum when exiting the curved section, and gradually increases and stabilizes in the inclined riser. Total pressure loss decreases with increasing curvature radius, asymptotically approaching a constant value. Frictional loss reduces with larger radii.

(3). Multiphase flow exerts forces on curved sections that promote realignment to vertical configuration. Smaller curvature radii generate greater restoration forces.

(4). Regarding the influence of curvature radius, we acknowledge that comparing our results with studies using different curvature radii (or conducting an ablation study by varying radius while keeping other parameters constant) would further strengthen the analysis. In the current work, we focused on a representative radius to isolate and clarify the particle distribution and pressure loss mechanisms. As a next step, we plan to extend the model to different curvature radii and include a systematic comparison, which will help generalize the conclusions. As for the study limitations, we recognize that the present simulations adopt simplified particle assumptions (e.g., uniform diameter and density), and the curved riser geometry was considered under idealized boundary conditions. Future work will incorporate polydisperse particle distributions, realistic riser geometries, and varying curvature radii under more complex operational conditions to enhance the applicability of the findings.

(5). The primary objective of this study is to establish an efficient analytical method for solid–liquid two-phase flow in pipes. As such, the model incorporates simplified assumptions such as steady-state flow, uniform particle size, and no heat/mass transfer to highlight the applicability and effectiveness of the proposed method itself. Currently, complex factors in long-distance transportation have not been thoroughly considered, though these factors would indeed influence the results. In subsequent research, we will progressively introduce elements such as unsteady effects, particle size distribution, and energy exchange to further enhance the model’s accuracy and engineering applicability.

Although this study reveals the influence of curvature radius on particle distribution and pressure loss in solid–liquid two-phase flow within curved pipes and proposes an efficient analysis method for in-pipe solid–liquid two-phase flow, the complexity of deep-sea mineral transport processes is still somewhat diminished by the simplified assumptions employed, including steady-state flow, homogeneous spherical particles, and neglecting heat/mass transfer. Future research needs to further consider the transport characteristics of real mineral particles with multi-size and multi-shape properties under deep-sea high-pressure and low-temperature conditions, quantify erosion risks in critical regions, and investigate the effects of unsteady phenomena, energy exchange, and fluid–structure interaction vibrations on the long-term operational reliability of the system. This will provide stronger theoretical support for the optimized design and safe operation of curved risers in deep-sea mining engineering. Furthermore, the Fluent-EDEM numerical method and two-phase transport test in the bend pipe should be performed to verify our current work.

Author Contributions

Conceptualization, L.W.; Methodology, W.-H.W. and L.-J.W.; Software, L.-J.W. and Z.-H.Z.; Validation, C.L. and L.-L.D.; Formal analysis, G.L. and K.L.; Investigation, W.-H.W., Y.-Y.W. and H.-B.S.; Data curation, L.W.; Writing—original draft, W.-H.W.; Writing—review & editing, L.W.; Visualization, C.L.; Supervision, C.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Acknowledgments

The authors are grateful to all organizations that funded the research in this paper.

Conflicts of Interest

Author Kun Li was employed by the company Dalian Shipbuilding Industry Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Footnotes

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Figures and Tables

Figure 1 Variation in riser center velocity with height in different grids.

Figure 2 Variation in velocity of riser center with height of 4–6 m in different grids.

Figure 3 Variation in riser velocity with x-coordinates at different grid heights.

Figure 4 Riser model with numerical meshes.

Figure 5 Comparison of experimental and simulation results of 10% ore volume concentration.

Figure 6 Curved riser model with numerical meshes.

Figure 7 Distribution of particle volume fraction at different cross sections in curved riser.

Figure 8 Pressure distribution on the curved riser.

Figure 9 Pressure distribution and pressure loss per unit length along the riser.

Figure 10 Particle volume fraction distribution at curved section with radius of curvature.

Figure 11 Influence of curvature radius on pressure loss and friction loss in curved section.

Figure 12 Influence of curvature radius on the force in x and z directions.

Table 1

Particle size composition of experimental ore with 10% ore volume concentration.

Particle Size (mm)	7.5	12.5	17.5	22.5	27.5
Solids Content	4.2%	18.8%	16.6%	31.4%	29.3%

Table 2

Velocity and friction of 10% ore concentration in 150 mm static tube.

Velocity (m/s)	Frictional Coefficient
2.660	0.056
3.101	0.072
3.549	0.090
4.290	0.128

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