Optimizing Energy Efficiency in Deep-Sea Mining: A Study on Swirling Flow Transportation of Double-Size Mineral Particles

Abstract

Deep-sea minerals are regarded as the most economically viable and promising mineral resource. Vertical hydraulic lifting represents one of the most promising methods for deep-sea mining lifting systems. To mitigate the potential for clogging due to the aggregation of particles in vertical pipe transport during deep-sea mining operations, this paper employs numerical simulations utilizing the computational fluid dynamics and discrete element method (CFD-DEM) model to investigate the swirling flow transportation of mineral particles. The characteristics of the swirling flow field and the motion law of double-size particles at different swirling ratios are investigated. The findings demonstrate that, in comparison to axial transport within the pipeline, the particle movement observed in swirling flow transport exhibits an upward spiral trajectory. This phenomenon facilitates the orderly movement of particles, thereby enhancing the fluidization of particles within the pipeline. An increase in the swirling ratio (SR) has a considerable impact on the velocity within the pipe. The tangential velocity distribution undergoes a gradual transition from centrosymmetric to non-centrosymmetric as the distance from the inlet increases. An increase in the SR results in an enhanced aggregation of particles at the wall, accompanied by a notable rise in the local particle concentration. The value of SR = 0.3 represents a critical threshold. When SR exceeds this value, the distribution of particles in the cross-section reaches a relatively stable state, rendering it challenging to further alter the distribution and concentration of particles, even if the SR is augmented. Furthermore, the maximum local particle concentration in the vicinity of the wall tends to be stable. These results provide valuable insights into vertical pipe swirling flow transport for deep-sea mining.

1. Introduction

Pipeline hydraulic transport is the process of transporting water or other liquids from one location to another through a pipeline system [1,2]. Pipeline hydraulic transport technology offers a number of advantages, including high efficiency, controllability, low cost, and environmental friendliness [3,4,5]. As a result, it has gradually become a feasible technical solution to the problem of deep-sea mineral transport. The considerable range of mineral particle gradation and high concentration, in conjunction with external environmental factors, render the internal flow of the pipeline highly complex during the pipeline transport process. Such circumstances may result in an inadequate stability of the hydraulic lifting of the pipeline, with the potential for pipeline clogging. Furthermore, the phenomenon of water hammer in pipeline transport has the potential to inflict damage upon the pipeline system [6]. In order to address the aforementioned issues, swirling flow is employed in hydrodynamic conveying to facilitate the ordered movement of particles, thereby enhancing the fluidization of particles within the pipeline.

In recent years, swirling flow has been employed in a number of industrial applications, including enhanced convective heat transfer [7], production of microbubble generators [8], cyclonic particle separators [9], and swirling flow of particle-bearing fluids [10,11,12]. Pipeline swirling flow transport is a technology that employs the kinematic properties of rotating fluids to facilitate the transportation of materials. By employing tangential velocity to influence the flow field, the particles are transported in a systematic manner along a spiral trajectory, thereby optimizing the flow pattern of the particles.

Experimental studies are frequently employed as a dependable methodology for elucidating the intrinsic mechanisms underlying swirling flows. Zhou et al. [10] conducted an experimental investigation to ascertain the impact of swirling intensity on the pickup velocity of lump coal particles within a pneumatic conveying system equipped with a swirling. It was observed that the pickup velocity of lump coal particles exhibited an initial increase, followed by a subsequent decline, with the intensification of swirling intensity. Li et al. [13] conducted an experimental study of particle characteristics in horizontal dilute swirling flow pneumatic conveying, employing the photographic image technique. The experimental results demonstrate that the particle concentration distribution in swirling flow pneumatic conveying is symmetrically distributed with respect to the axis of the pipe. Additionally, the particle concentration at the bottom of the pipe in swirling flow pneumatic conveying is observed to be lower than that in axial pneumatic conveying. Zhou et al. [14] conducted an investigation into the flow characteristics of a strong swirling gas-particle flow in both a sudden expansion and cyclonic chambers. This was achieved through the utilization of 2-D and 3-D phase Doppler particle anemometers (PDPA). Zhou et al. [15] employed a multifactorial experimental approach to investigate the erosion-corrosion behavior of liquid–solid swirling flow in the pipeline. The findings indicated that the oxygen concentration on the electrode surface exhibited a considerable increase in swirling flow conditions relative to non-swirling flow conditions. Despite the fact that a series of experimental studies of ducted swirling flows are currently being conducted, the experiments in question are constrained in their ability to capture information pertaining to localized flow fields and particle details. This is primarily attributable to the opacity of multiphase flows and the demanding operational conditions.

Computational modelling techniques are regarded as a valuable and effective means of acquiring data on local flow fields and particle details at a relatively low cost. CFD technology has demonstrated considerable potential for application across a range of fields, offering insights into intricate flow phenomena that are challenging to investigate through conventional experimental techniques. Particularly in vortex studies [16,17], CFD provides intuitive visualization of the flow field, enabling researchers to gain an in-depth understanding of the vortex formation mechanisms and their effects on fluid motion. At this juncture, the numerical methodologies employed to forecast liquid–solid two-phase flow in tubes can be broadly categorized into two principal groups: the Euler-Euler two-fluid model (TFM) and the Eulerian–Lagrangian method. The distinction between these two approaches hinges on the manner in which the granular phase is characterized. The TFM method is a frequently employed technique for modeling dense liquid–solid two-phase flows. In the TFM method, the solid phase is treated as a fluid. This approach allows both the liquid and solid phases to be solved using the Navier–Stokes equations, which greatly reduces the computational effort required by the algorithm [18]. The TFM method has gained widespread acceptance in the study of hydraulic transport in pipelines, largely due to its computational efficiency and convenience [19,20,21]. Liu et al. [22] employed the TFM method to investigate the impact of transverse vibration of the isolation pipeline on the pressure loss of coarse particles hydraulic lifting. Shi et al. [23] investigated the impact of diverse swirling flows on the transport attributes of multi-granular particle flow slurries in a horizontal pipe, employing an Eulerian–Eulerian multiphase methodology. The findings indicated that swirling flow is more conducive to enhancing the multistage solids suspension capacity and transport efficiency in comparison to non-swirling flow. Nevertheless, TFM is unable to characterize certain fundamental properties of the granular phase, including particle size distribution, rotation, dispersion, and collision [24]. Conversely, the Lagrangian framework enables the accurate capture of the discrete features of the solid particle phase through the tracking of solid particle motion. In this framework, the CFD-DEM method is a widely used approach for predicting the characteristics of fluid-particle two-phase mixing flows. Yan et al. [25] conducted a study investigating the use of pneumatic conveying with integrated soft fins. The findings revealed that the incorporation of soft fins markedly diminished the air velocity and energy coefficients in comparison to the conventional pneumatic axial conveying approach. Qi et al. [26] conducted an investigation into the internal liquid–solid flow characteristics in pipeline conveying with a tangential jet inlet, employing the CFD-DEM method. The impact of varying tangential flow ratios on the flow velocity distribution, vortex volume, total pressure, concentration, and resistance of particles of varying shapes was examined.

The occurrence of local particle aggregation and susceptibility to clogging is a common phenomenon during vertical pipe axial transport. The use of pipeline swirling flow transport represents an effective solution to the aforementioned issue. Despite the fact that researchers have conducted some experimental and numerical studies on pipeline swirling flow transport, there is still a paucity of research on the motion characteristics of particles in vertical pipes with varying swirling ratios and the distinctions in the motion of coarse and fine particles within the swirling flow field. Therefore, in this paper, the open-source software OpenFOAM 5.x and LIGGGHTS 3.8.0 are employed to undertake a numerical study of vertical pipe swirling flow transport for deep-sea mining. It is noteworthy that, in order to more accurately simulate the interaction between multi-particle size solid particles and fluids in deep-sea mining swirling flow transport, two distinct particle sizes are employed in this study to investigate the motion characteristics of coarse and fine particles within the swirling flow field. This study investigates the characteristics of the swirling flow field and particle motion of double-size particles under different swirling ratios. The findings of this study may serve as a valuable reference point for the vertical pipe hydraulic transport for deep-sea mining.

2. Mathematical Model

2.1. Governing Equations for Fluid Phase

The fluid phase is described by Euler’s method, and the velocity and pressure distribution fields are obtained by solving the mass and momentum conservation equations for the fluid phase. The governing equations are as follows:

$${\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_f{\rho }_f\right)+\nabla ·\left({\alpha }_f{\rho }_f{\mathit{u}}_f\right)=0$$

(1)

$${\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_f{\rho }_f{\mathit{u}}_f\right)+\nabla ·\left({\alpha }_f{\rho }_f{\mathit{u}}_f{\mathit{u}}_f\right)=-{\alpha }_f\nabla p+\nabla ·\left({\alpha }_f{\mathit{\tau }}_f\right)+{\alpha }_f{\rho }_f\mathbf{g}-{\mathit{F}}_p$$

(2)

where ${\rho }_f$ is the fluid density (kg/m³); $\nabla p$ is the fluid pressure gradient (Pa); ${\mathit{u}}_f$ is the fluid velocity (m/s); $\mathbf{g}$ is the gravitational acceleration (m/s²); ${\alpha }_f$ is the void fraction; and ${\mathit{\tau }}_f$ is the fluid stress tensor (N). ${\mathit{F}}_p$ is the is the momentum exchange between the fluid and particle phases. The formula for calculating momentum exchange is as follows:

$${\mathit{F}}_p={\displaystyle \frac{1}{∆V}}\sum _{i=1}^{i=N}{\mathit{f}}_{p-f,i}$$

(3)

where ∆V represents the volume of a computational cell (m³), and N denotes the number of particles present within the control body. ${\mathit{f}}_{p-f,i}$ is the interaction force between the fluid and particle (N), given by:

$${\mathit{f}}_{p-f,i}={\mathit{f}}_{drag,i}+{\mathit{f}}_{lift,i}+{\mathit{f}}_{p,i}+{\mathit{f}}_{vm,i}$$

(4)

where, ${\mathit{f}}_{drag,i}$ is the drag force (N), ${\mathit{f}}_{lift,i}$ is the lift force (N), ${\mathit{f}}_{p,i}$ is the pressure gradient force (N), ${\mathit{f}}_{vm,i}$ is the virtual mass force (N).

In the present study, the standard k-ε model is employed to elucidate the fluid turbulence dynamics within the pipe, while the standard wall function is utilized to capture the fluid motion in proximity to the wall. The turbulent kinetic energy (k) and its dissipation rate (ε) are defined by the following equation:

$${\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_f{\rho }_{f}k\right)+\nabla ·\left({\alpha }_f{\rho }_f{\mathit{u}}_{f}k\right)=\nabla ·\left({\alpha }_f{\Gamma }_k\nabla k\right)+{\alpha }_{f}G-{\alpha }_f{\rho }_f\epsilon +S_k$$

(5)

$${\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_f{\rho }_f\epsilon \right)+\nabla ·\left({\alpha }_f{\rho }_f{\mathit{u}}_f\epsilon \right)=\nabla ·\left({\alpha }_f{\Gamma }_{\epsilon }\nabla \epsilon \right)+{\alpha }_f{\displaystyle \frac{\epsilon }{k}}\left(c_{1}G-c_2{\rho }_f\epsilon \right)+S_{\epsilon }$$

(6)

where ${\Gamma }_k=\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}$; ${\Gamma }_k=\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}$; $G={\mu }_t\nabla {\mathit{u}}_f·\left[\nabla {\mathit{u}}_f+{(\nabla {\mathit{u}}_f)}^T\right]$; $c_1$ and $c_2$ are constants; and $S_k$ and $S_{\epsilon }$ are the source terms of the fluid turbulence kinetic energy and its dissipation rate induced by particles, respectively.

2.2. Governing Equations for Particle Phase

The particle phase is described by the Lagrangian method, which provides a framework for characterizing the motion of a single particle in a fluid. This motion is predominantly characterized by translation and rotation. The governing equations are as follows:

$$m_i{\displaystyle \frac{d{\mathit{v}}_i}{dt}}=m_i\mathbf{g}+{\mathit{f}}_{p-f,i}+\sum _{j=1}^{k_i}{\mathit{f}}_{c,ij}$$

(7)

$${\mathit{I}}_i{\displaystyle \frac{d{\mathit{\omega }}_i}{dt}}=\sum _{j=1}^{k_i}{\mathit{T}}_{ij}$$

(8)

where $m_i$ is the particle’s mass (kg); ${\mathit{v}}_i$ is the particle’s velocity (m/s); ${\mathit{\omega }}_i$ is the particle’s angular velocity (rad/s); ${\mathit{I}}_i$ represents the particle’s moment of inertia (kg·m²); ${\mathit{T}}_{ij}$ is the total torque (kg·m²·s⁻²); and ${\mathit{f}}_{c,ij}$ is the contact force between particles (N).

In the context of liquid–solid two-phase flow, the drag force is of critical importance as a fundamental interphase force in comparison to other, relatively minor forces. In the present study, the Di Felice drag model [27] is employed, whereby the calculation formula is as follows:

$${\mathit{f}}_{drag,i}=0.5C_d{\rho }_{f}A\left|{\mathit{v}}_{p-f}^{rel}\right|{\mathit{v}}_{p-f}^{rel}{\alpha }_f^{1-x}$$

(9)

$$x=3.7-0.65\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\displaystyle \frac{{\left(1.5-{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left({Re}_{p,i}\right)\right)}^2}{2}}\right]$$

(10)

$$C_d=\left\{\begin{array}{c}{\displaystyle \frac{24}{{Re}_{p,i}}}, {Re}_{p,i}\le 1\\ {\left[0.63+{\displaystyle \frac{4.8}{{{Re}_{p,i}}^{0.5}}}\right]}^2, {Re}_{p,i}>1\end{array}\right.$$

(11)

where $C_d$ is the drag force coefficient, ${Re}_{p,i}$ is the particle Reynolds number, ${\mathit{v}}_{p-f}^{rel}$ is the relative velocity (m/s), and A is the particle’s projected area (m²).

3. Simulation Settings

The actual mining transport process encompasses the use of hundreds or even thousands of meters of vertical pipelines, which renders the simulation of full-scale transport an exorbitantly costly endeavor. Conversely, the mineral particles within the lifting pipe interact with the fluid, resulting in the formation of mixed flow. This process occurs over a certain distance after the transport, resulting in the formation of a more stable solid-liquid two-phase flow pattern. In order to reduce the computational cost, the length of the vertical pipe established in this study was set at 5 m with an internal diameter of 0.22 m. Figure 1 illustrates the three-dimensional model of the riser and its associated computational grid. The computational domain mesh is structured. The generation of swirling flow within a pipe is achieved through the application of tangential and axial velocities at the inlet. In this paper, the degree of swirling fluid flow in a pipe is characterized by the swirling ratio (SR), defined as the ratio of tangential velocity to axial velocity. When SR = 0, the transport process within the pipe is axial transport. This study examines the kinematic properties of particles exhibiting diverse swirling ratios. The swirling types of the fluids are delineated in

Table 1. Swirling flow schemes in the pipeline transport simulation.

Scheme	Tangential Velocity Vt (m/s)	Axial Velocity Va (m/s)	SR (Vt/Va)
1	0	3	0
2	0.3	3	0.1
3	0.6	3	0.2
4	0.9	3	0.3
5	1.2	3	0.4
6	1.5	3	0.5

. In the context of transportation, the fluid phase is regarded as an incompressible fluid, with the fluid properties of seawater being employed in this regard. Despite the considerable range of particle sizes of seabed mineral particles, it is inevitable that particles of a larger size will degrade and fragment, thereby resulting in particles of a smaller size. Accordingly, in order to simulate the swirling flow of multi-size mineral particles, two distinct particle types, with diameters of 6 and 12 mm, respectively, were modelled. The initial concentration of the feedstock is guaranteed to be identical for each pellet. The initial feed concentration is regulated by modifying the rate of granule injection at the inlet. The quantity of particles injected is contingent upon the specified feed concentration (by volume) and liquid flow rate. The mixing outlet is situated at the pinnacle of the computational domain and has been designated as a pressure outlet. The pipe wall is subject to the no-slip wall boundary condition.

Table 2. Fluid and particle parameters used in the current simulation.

Parameters	Values
Liquid property
Liquid density (kg/m3)	1025
Liquid viscosity (Pa‧s)	0.0015
Solid property
Young’s modulus (Pa)	$2\times {10}^8$
Poisson ratio	0.33
Particle density (kg/m3)	2000
Particle diameter (mm)	6, 12
Particle–wall sliding friction coefficient	0.1
Particle–particle sliding friction coefficient	0.28
Coefficient of restitution	0.45

presents a detailed description of the properties of the fluid and particle.

In the CFD-DEM simulation, the Eulerian–Lagrangian method is employed for the description of the pipeline hydraulic swirling transport. Given the unsteady nature of solid-liquid swirling flow, it is necessary to select an appropriate time step in order to accurately predict the particle trajectory. The particle time step in DEM is distinct from that employed in CFD simulations of fluids, with the time step for the particle being typically smaller than corresponding fluid time step. The time step for the particle is constrained by the Rayleigh time, $∆t_R$, as calculated in the following equation [28,29]:

$$∆t_R={\displaystyle \frac{\pi R}{0.1631\nu +0.8766}}\sqrt{{\displaystyle \frac{2{\rho }_p(1+\nu )}{Y}}}$$

(12)

where $Y$, ${\rho }_p$, $R$, and $\nu$ are the Young’s modulus (Pa), density (kg/m³), radius (m), and Poisson ratio of the particle, respectively. In the current simulation, the time step for the fluid is defined as 1 × 10⁻⁴ s, and the time step for the particle is selected as 1 × 10⁻⁵ s.

4. Results and Discussion

4.1. Model Validation

In this paper, the experimental data from Ref. [30] are compared with the numerical simulations. The principal parameters of the computational model were set to align with the experimental parameters. The particle diameter and density were 15 mm and 2000 kg/m³, respectively, and the fluid medium was water. Additional details can be found in Ref. [30]. Figure 2 presents a comparison of the pipe hydraulic gradient at different velocities at an initial particle feed concentration of 10%. It can be observed that an increase in the initial mixing velocity results in an elevated pipeline hydraulic gradient. In order to gain further insight into the performance of the model prediction and its applicability, corresponding error analyses were conducted. Three indicators were employed for the purpose of evaluating the model prediction effect: namely, mean squared error (MSE), maximum relative deviation, and average relative deviation.

$$MSE={\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-y_i^{\prime })}^2$$

(13)

$${\delta }_{max}=max\left({\displaystyle \frac{\left|y_i-y_i^{\prime }\right|}{y_i^{\prime }}}\right)$$

(14)

$${\delta }_{mean}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\displaystyle \frac{\left|y_i-y_i^{\prime }\right|}{y_i^{\prime }}}$$

(15)

where $MSE$ represents the mean squared error; ${\delta }_{max}$ and ${\delta }_{mean}$ denote the maximum relative deviation and average relative deviation, respectively; $n$ is the number of sample points; and $y_i$ and $y_i^{\prime }$ denote the numerical predicted results and experimental data, respectively.

The MSE for the numerical predicted results and experimental data is 0.0002238. This MSE is relatively low, indicating that the discrepancy between the numerical predicted and experimental values is also relatively minor. This implies that the numerical model has a relatively high predictive accuracy. Furthermore, the maximum and average relative deviations of the results for the pipeline hydraulic gradient are approximately 9% and 6%, respectively. This means that the fluctuation of these error values is within an acceptable range of 15% [31]. Therefore, the effectiveness of the CFD-DEM method employed in this study for simulating the hydraulic transport process in vertical pipelines is verified from the perspective of the pipeline hydraulic gradient.

4.2. Characteristic of Swirling Flow

In the context of vertical pipe swirling flow transport, the tangential velocity of the fluid represents a pivotal factor influencing the characteristics of the swirling flow field. Furthermore, the flow characteristics of the swirling flow field exert an influence on the flow regime of mineral particles in the hydraulic lifting. This paper investigates the influence law of varying swirling ratios on the hydraulic lifting of mineral particles. Figure 3 illustrates the pressure distribution within the pipeline for varying swirling ratios. It can be observed that an increase in the SR results in an elevated pressure drop within the pipeline when compared to the pipeline axial flow transport. This phenomenon can be attributed to the fact that by maintaining a consistent axial velocity, the increase in tangential velocity causes additional energy loss during the flow. Also, an increase in the SR results in an elevated formation of vortices and eddies within the pipeline, which intensifies the turbulence of the fluid. This, in turn, elevates the resistance to flow and precipitates an augmented pressure drop. Furthermore, it has been demonstrated that when the SR is greater than 0.3, the pressure at the center of the pipe is markedly lower than that on either side. This results in the formation of a considerable pressure gradient, whereby the particles within this region are compelled to move as far as possible towards the pipe wall, driven by the influence of lateral forces. This ultimately leads to a reduction in the number of particles present in the center of the pipe.

Figure 4 illustrates the streamline pattern within the vertical pipe for varying swirling ratios. The maximum value of the flow velocity increases by approximately 21% as the SR increases from SR = 0 to SR = 0.5, indicating that the swirling intensity has a significant effect on the velocity inside the pipe. And an increase in the SR results in a reduction in the pitch of the spinning trajectory, thereby prolonging the period of the fluid that is carrying the particles in a spiral motion. As the distance from the pipe inlet increases, the swirling intensity will diminish along the flow direction due to the reduction in centrifugal force. This phenomenon can be attributed to the fact that the formation and sustenance of the swirling flow necessitate a specific input of energy. As the fluid–particle mixture traverses the pipe, the resistance generated by the interaction between the fluid, the particles, and the pipe wall, coupled with the turbulence within the fluid, leads to a notable decline in the strength and stability of the swirling flow as it deviates from the inlet.

Figure 5 illustrates the variation in axial and tangential fluid velocities along the axial position for different swirling ratios. From the figure, it can be observed that when h/H is 0.2, which indicates that the observation plane is in closer proximity to the inlet, the axial velocity at the center of the pipe exhibits a slight decrease with an increase in the swirling ratio. Meanwhile, the tangential velocity distribution demonstrates a Centro symmetric structure. As the value of h/H increases, the resulting increase in the SR gives rise to larger axial velocities and smaller tangential velocities in general. This phenomenon may be attributed to the fact that energy dissipation is more pronounced in the region situated away from the inlet, due to the friction between the fluid and the pipe wall, as well as the turbulent movement. An increase in the SR indicates that a greater proportion of energy is expended on increasing the axial velocity, with less energy allocated to maintaining the tangential velocity. Consequently, the tangential velocity declines in comparison. Furthermore, the tangential velocity distribution undergoes a gradual transition towards a non-centrosymmetric distribution as h/H increases. This is due to the fact that at the inlet, the fluid may form stable vortices or eddies, which have a certain degree of stability and are capable of maintaining the symmetry of the tangential velocity distribution. However, as the fluid progresses towards the interior of the pipe, a number of factors, including friction, pressure gradient and flow boundary conditions, contribute to a reduction in stability. This, in turn, gives rise to changes in the shape and location of the vortices, which in turn result in a gradual change in the tangential velocity distribution from centrosymmetric to non-centrosymmetric.

4.3. Characteristic of Particle Movement

In the vertical pipe swirling flow transport, the centrifugal force provided by the tangential velocity of the fluid constitutes the primary driving force propelling the particles in a spiral motion. The transportation of particles at varying swirling ratios gives rise to markedly disparate trajectories and distributions of particles, attributable to the interplay between particles and the fluid, as well as collisions between particles and between particles and pipe walls. In order to examine the kinematic characteristics of vertical tube multi-particle size swirling flow transport, two distinct diameters of particles are employed in this study: 6 mm and 12 mm, which are designated as coarse and fine particles, respectively.

The particle trajectories in a swirling flow field are a complex process influenced by a multitude of factors. An investigation into the trajectory characteristics of particles in a swirling flow field can facilitate a more profound comprehension of the particle motion law. Therefore, we select the coarse and fine particles that are initially located in the center of the pipe and subsequently observe their trajectories within the pipe. Figure 6 illustrates the movement trajectories of coarse and fine particles at varying swirling ratios. From an analysis of the path trajectories, it can be determined that when SR = 0, which represents axial transport, the particles do not move in a completely vertical upward trajectory, but rather fluctuate in the radial direction. In contrast, within the swirling flow field, the particles migrate in a radial direction due to the centrifugal force, and the radial displacement fluctuation of the particles increases with the increase in SR. The particles demonstrate a spiral upward movement in response to the swirling flow. Furthermore, the distinction in particle trajectories produced by coarse and fine particles initially situated in the center of the pipe at varying swirling ratios is not readily discernible.

Figure 7 illustrates the cross-sectional distribution of particles along the axial direction under varying swirling ratios. It can be observed that when SR = 0, the distribution of particles of varying sizes at different axial locations is more uniform. As the SR increases, the particles gradually move towards the wall and aggregate more and more at the wall. This phenomenon can be attributed to the fact that an increase in the SR increases the centrifugal force, which causes the particles to move towards the wall as much as possible. When SR = 0.1, the flow pattern of the particles remains largely unaltered during conveying in comparison to axial conveying, whereby the centrifugal force generated is insufficient to propel the particles along the wall. The aforementioned flow pattern of particle distribution along the wall becomes evident when the SR is greater than 0.2. In other words, an SR between 0.1 and 0.2 generates a centrifugal force that is sufficient to drive particles from the pipe center towards the pipe wall. In closer proximity to the inlet location, the particle distribution is observed to be symmetrical. Conversely, in regions distant from the inlet, an asymmetrical distribution of particles is evident. This may be attributed to the fact that in the vicinity of the inlet, the initial flow state of the fluid is relatively more homogeneous and stable, as evidenced by the axial and tangential distributions of the fluid. Consequently, the particle distribution is more likely to exhibit a symmetrical distribution. As one moves further away from the inlet location, the flow field becomes increasingly complex and inhomogeneous. This may result in the particles being subjected to forces of varying directions and magnitudes, which could lead to a change in the symmetry of their distribution.

Figure 8 illustrates the localized particle concentrations along the axial direction for varying swirling ratios. As illustrated in the figure, the local concentration at various locations along the axial direction remains largely consistent throughout the process of axial conveying. In comparison to axial conveying, an increase in the SR has been observed to result in a notable elevation in the local particle concentration, with a maximum increase of 25% at SR = 0.5. As the SR increases, the local particle concentration at different locations tends to increase. However, when the SR is greater than 0.3, the increase in the local particle concentration at different locations is not significant. This may be attributed to the fact that the particles are subjected to more pronounced rotational and centrifugal forces as the SR increases. This results in an increased tendency for particles to aggregate at different locations, which gives rise to an increasing trend in the localized particle concentration. When the SR is greater than 0.3, the distribution of particles in the cross-section is relatively stable. Consequently, it is challenging to achieve a significant alteration in the distribution and concentration of particles, even if the SR is increased. Furthermore, as the distance from the inlet increases, the local concentration of particles tends to rise and then decline due to the gradual reduction in swirling intensity.

Figure 9 depicts the local concentration of double-sized particles in the vicinity of the wall. In this instance, the radius, r, of the region in question is situated between 0.09 and 0.11. In axial conveying, the local concentration of particles in the vicinity of the wall is observed to be lower, indicating that particles tend to concentrate more in the center of the pipe. As the SR increases, the local particle concentration in the vicinity of the wall also increases. Furthermore, larger particles exert a more pronounced inertial effect than their smaller counterparts. When subjected to the influence of a swirling flow field, smaller particles are more prone to migrate towards the wall, resulting in a higher concentration of small particles in the vicinity of the wall compared to that of larger particles. Additionally, an SR value of 0.3 represents a critical threshold, with the maximum localized particle concentration near the wall reaching approximately 0.08 when SR is greater than 0.3.

5. Conclusions

In this paper, a numerical study of liquid–solid two-phase swirling flow in a vertical pipe is carried out using the open source codes OpenFOAM and LIGGGHTS. The characteristics of the swirling flow field and the particle motion characteristics are analyzed for different swirling ratios. The main conclusions are as follows:

(1)

In comparison to pipeline axial transport, the particle movement observed in swirling flow transport is characterized by a spiral trajectory. An increase in the SR results in an elevated pressure drop within the pipe.

(2)

The maximum value of the flow velocity is observed to increase by 21% as the SR is increased from SR = 0 to SR = 0.5, thereby indicating that the SR exerts a significant effect on the velocity within the pipe. The tangential velocity distribution undergoes a gradual transition from centrosymmetric to non-centrosymmetric as the distance from the inlet increases.

(3)

An increase in SR results in an augmentation of the centrifugal force exerted on the particles within the pipe, thereby rendering them more inclined to move towards the wall. The weaker inertial effect of small particles means that they are more likely to migrate towards the vicinity of the wall under the action of the swirling flow field. This results in a greater concentration of small particles near the wall than of large particles. Furthermore, the local particle concentration at varying points along the axial direction is observed to increase considerably, with a maximum increase of 25% occurring when SR = 0.5.

(4)

SR = 0.3 represents a critical threshold. When SR is greater than 0.3, the distribution of particles in the cross-section is in a relatively stable state. Even if the SR is increased, it is challenging to achieve a significant change in the distribution and concentration of particles. Furthermore, when SR is greater than 0.3, the maximum local particle concentration in the vicinity of the wall tends to stabilize.

Overall, the limitations of the current work are mainly in the choice of the two particle sizes and the assumption of sphericity of the particles. In actual deep-sea mining, the particles are non-spherical and exhibit a wide particle size distribution, which has a significant impact on the flow behavior of swirling flow transport. In light of the aforementioned considerations, our subsequent objective is to concentrate on the impact of the wide particle size distribution and shape of the particle within the vertical pipe.