Prediction of Erosion of a Hydrocyclone Inner Wall Based on CFD-DPM

Abstract

The erosion mechanism of hydrocyclones under air column conditions is still unclear. In this paper, Computational Fluid Dynamics–Discrete Phase Model (CFD-DPM) technology is adopted to perform transient simulations of the three-phase flow (liquid–gas–solid) within a hydrocyclone. The Reynolds Stress Model (RSM) and Volume of Fluid (VOF) model are adopted to simulate the continuous phase flow field within the hydrocyclone, while the DPM coupled with the Oka erosion model is used to predict the particle flow and erosion mechanisms on each wall within the hydrocyclone. The particle sizes considered are 15 μm, 30 μm, 60 μm, 100 μm, 150 μm, and 200 μm, respectively, with a density of 2600 kg/m³. The particle velocity is consistent with the fluid velocity at 5 m/s, the total mass flow rate is 6 g/s, and the volume fraction is less than 10%. The results indicate that the cone section suffers the severest erosion, followed by the overflow pipe, column section, infeed section, and roof section. The erosion in the cone section reaches its maximum value near the underflow port, with an erosion rate approximately 6.8 times that of the upper cone section. The erosion distribution in the overflow pipe is uneven. The erosion of the column section exhibits a spiral banded distribution with a relatively large pitch. The erosion rate in the infeed section is approximately 1.47 times that of the roof section.

1. Introduction

A hydrocyclone is a device commonly used in mineral sorting and water treatment, which is mainly used in mineral processing, wastewater treatment, the chemical industry, and other industries [1,2,3]. It works by centrifugal force generated in the rotation of the fluid, which separates particles of different sizes to achieve the separation effect [4,5,6]. The hydrocyclone is widely used in coal preparation plants, mainly for coal separation [7,8,9]. In the process of its use, due to the high-speed rotation of the internal fluid and the impact and friction of the solid particles, the erosion of the particles on the wall is unavoidable. The end-of-life cycle of hydrocyclone ceramic liners in the coal separation process is about 6 months to 1 year, which may be shortened to 4 months in some special working conditions [10]. When the hydrocyclone suffers erosion, its structural size changes, which affects the flow field distribution and separation efficiency [11,12]. Therefore, it is necessary to study the erosion characteristics of the hydrocyclone inner wall.

Currently, researchers have studied the problem of inner wall erosion in hydrocyclones from various perspectives. Liu et al. [13] studied the inner wall erosion of a hydrocyclone under liquid–solid two-phase flow by the numerical simulation method, verified it by an experiment, and proposed using the DPM and general erosion model to analyze the particle movement and the erosion degree of each component. Gao et al. [14] conducted flow experiments under four different valve openings to analyze the wall erosion by particles at different inlet velocities of a hydrocyclone, and the results showed that the increase in inlet velocity increased the probability and number of collisions between solid particles and the wall, and its erosion rate increased with the inlet flow rate. Azimian et al. [15] adopted the Euler–Lagrange method to predict the erosion of the inner wall of a hydrocyclone made of stainless steel subjected to sand particles, which treats the solid particles as discrete phase particles to be tracked in a Lagrangian reference frame.

In addition, Oka et al. [16,17] experimentally investigated the wear of different materials under particle erosion and proposed an erosion prediction method based on material hardness, fracture toughness, and impact angle. Thiana et al. [18] adopted the Euler–Lagrange method and RSM to numerically simulate the gas–solid two-phase flow in a cyclone separator. Oka and DNV erosion models were used to study the inner wall erosion, and it was found through experimental validation that the simulation results of the two erosion models were in good agreement with the experimental data. Zhang et al. [19] adopted the Oka erosion equation and CFD technique to study the inner wall erosion of a cyclone separator and analyzed the effect of the inner wall erosion of the cyclone separator on its flow field and performance. Dianyu E et al. [20,21] investigated the effects of different inlet structure parameters on multiphase flow and separation performance within hydrocyclones through numerical simulation.

In summary, numerous scholars have studied the inner wall erosion of hydrocyclones through a combination of numerical simulation and experimental verification. However, in the existing numerical simulation studies, the air column existing in the actual operating conditions of hydrocyclones has not been considered. As an inevitable phenomenon in the internal flow field of hydrocyclones, the air column generally affects its separation efficiency and flow field stability [22,23,24,25,26]. Additionally, the presence of the air column changes the flow field distribution within the hydrocyclone, thereby affecting the distribution and erosion locations of solid particles and consequently influencing the erosion characteristics [27,28]. Therefore, in this paper, numerical simulation is adopted to predict the erosion characteristics of the inner wall of a hydrocyclone under air column conditions.

2. Research Methods

2.1. Geometric Modeling and Meshing

In this study, a numerical simulation is carried out for a Φ150 mm type hydrocyclone in a coal preparation plant, whose structural parameters are shown in

Table 1. Structural parameters of the hydrocyclone.

Structure	Unit	Parameters
Inlet Size	a × b (mm)	40 × 50
Hydrocyclone Diameter	D (mm)	150
Column Height	H (mm)	188
Overflow Diameter	d1 (mm)	44
Overflow Insertion Depth	h (mm)	158
Overflow Wall Thickness	ε (mm)	25
Underflow Diameter	d2 (mm)	12
Angle of Cone	θ (°)	20

and Figure 1.

SolidWorks 2020 software is used to model the hydrocyclone geometric model, which is constructed with the fluid domain considered as a solid for the convenience of subsequent numerical simulation and post-processing. The model is constructed with the center of the overflow pipe as the coordinate origin, which is vertically downward along the negative direction of the Z-axis and has a tangential inlet along the negative direction of the Y-axis. ICEM CFD software was used to structure the meshing and was divided into five sections to observe the wear of each part: overflow pipe, roof section, infeed section, column section, and cone section. The mesh of the cone section and the infeed section is encrypted to improve the accuracy of the calculation. In this study, the standard wall function is adopted to treat near-wall flow. The target y+ value is set to 30, and the mesh is adjusted multiple times to maintain the y+ values of each wall within the range of 30–300. The final mesh height for the first layer is determined as follows: 0.001 m for the column section, cone section, and infeed section; 0.0002 m for the overflow pipe. The specific meshing is shown in Figure 2.

The orthogonality quality is used to evaluate whether the mesh meets the computational conditions, with the specific distribution shown in

Table 2. Orthogonal quality of hydrocyclone meshes.

No.	Orthogonal Quality	Mesh	Percentage, %	No.	Orthogonal Quality	Mesh	Percentage, %
1	0.95–1.0	208,565	86.087	9	0.55–0.6	124	0.051
2	0.9–0.95	19,040	7.859	10	0.5–0.55	174	0.072
3	0.85–0.9	9015	3.721	11	0.45–0.5	116	0.048
4	0.8–0.85	2679	1.106	12	0.4–0.45	72	0.030
5	0.75–0.8	1294	0.534	13	0.35–0.4	100	0.041
6	0.7–0.75	438	0.181	14	0.3–0.35	60	0.025
7	0.65–0.7	330	0.136	15	0.25–0.3	60	0.025
8	0.6–0.65	206	0.085
Total = 233,185, Min = 0.255, Max = 0.1

. As can be seen from the table, the minimum orthogonality quality is 0.255, and most of the mesh qualities are above 0.9, meeting the computational conditions.

2.2. Mesh-Independence Verification

In this paper, a structured meshing method is adopted to generate hexahedral meshes for simulation. Figure 2 shows the structured mesh of the hydrocyclone. To verify the independence of the numerical simulation results from the number of meshes, the mesh is randomly encrypted. Numerical simulations are performed on the hydrocyclone with mesh numbers of 105,552, 233,185, 337,520, and 431,537, with specific boundary conditions detailed in Section 2.4. The tangential velocity and maximum erosion rate are calculated and compared for different mesh numbers. The specific results are shown in Figure 3 and Figure 4. Figure 3 shows the radial distribution curves of tangential velocity at the Z = −300 mm cross-section for different mesh numbers, and Figure 4 shows the maximum erosion rates in the overflow section and column section for different mesh numbers. It can be observed that among the four mesh numbers, the calculation results for the mesh number 105,552 show significant differences from the other three mesh numbers. When the mesh number exceeds 233,185, the increase in mesh number has no significant effect on the flow field and the maximum erosion rate on each wall surface. Therefore, in this paper, a model with a mesh number of 233,185 is used for numerical simulation to improve computational efficiency and accuracy.

2.3. Model Validation

The RSM and VOF models are widely adopted in numerical simulations of hydrocyclones and validated by extensive experimental studies. Therefore, in this study, experimental data on erosion characteristics in the elbow are adopted from Vieira et al. [29] to validate the accuracy of the DPM and Oka erosion model in numerical calculations. Simulation conditions are consistent with experimental conditions. Model reliability is assessed by comparing CFD predictions with experimental data. The comparison results are shown in

Table 3. Comparison between experimental data and simulation results.

	VGAS (m/s)	Sand Size (μm)	Sand Rate (Kg/Day)	Erosion Rate (mm/Year)	Max Erosion Location (°)
Experimental [29]	23	300	227	80.3	47
Simulation	23	300	227	83.6	47

. The simulation results reveal a typical V-shaped erosion morphology (as shown in Figure 5). The maximum erosion rate, converted to mm/year, exhibits a 4.1% error with experimental measurements. Furthermore, the maximum erosion location occurred at 47°, which is consistent with the experimental results. This validates the high accuracy of the adopted model in predicting particle erosion behavior, providing a reliable numerical foundation for subsequent erosion analysis within hydrocyclones.

2.4. Numerical Simulation

Ansys Fluent software is adopted to numerically calculate the internal flow field and the erosion characteristics of the hydrocyclone. Firstly, the VOF model and RSM [30,31,32,33] are adopted to simulate the gas–liquid two-phase flow inside the hydrocyclone, and after the continuous phase is stabilized, the DPM and Oka erosion model [34,35,36] are added for numerical simulation of the particle motion trajectory and wall erosion.

The solid particles are mainly affected by inertia force, fluid drag force, centrifugal force, gravity, pressure gradient force, Saffman lift force, and virtual mass force during the fluid movement inside the hydrocyclone [37,38,39]. The particle trajectory is changed by the centrifugal force and collides with the wall and rebounds. Before and after the rebound, the momentum and the equation of the particle’s motion are changed, and the change in momentum before and after the collision needs to be expressed by the elastic recovery coefficient [40,41]. In this paper, Equations (1) and (2) are used to define the rebound recovery coefficient between the particle and the wall [13].

$$e_N=0.993-0.0307\theta +4.75\times {10}^{-4}{\theta }^2-2.61\times {10}^{-6}{\theta }^3$$

(1)

$$e_T=0.998-0.029\theta +6.43\times {10}^{-4}{\theta }^2-3.56\times {10}^{-6}{\theta }^3$$

(2)

where $e_{N }$ represents the normal restitution coefficient, $e_T$ represents the tangential restitution coefficient, and $\theta$ represents the collision angle between the particle and the wall.

Since the particle size considered in this study is relatively small and the volume fraction at the inlet is far less than 10%, the effect of inter-particle interaction forces is relatively small. Therefore, inter-particle interaction forces are not considered in this study.

The inlet boundary condition is set as a velocity inlet, the velocity of both fluid and particles is 5 m/s, the turbulence intensity at the inlet is 5%, and the hydraulic diameter is 44.44 mm. The overflow port and underflow port are set as pressure outlets, and the air return coefficient of the overflow port and the underflow port is 1. The overflow port is set as escape, and the underflow port is set as trap.

The wall material used in this study is carbon steel. The wall boundary conditions are set by the wall standard function method, the wall roughness is set to 0.5 [31], the continuous phase flow field is set to have no slip at the wall boundary, the boundary conditions of the discrete phase particles and the wall are set as reflect, and the interaction with the wall is described by the rebound recovery coefficient of the particles and the wall, which is realized by Equations (1) and (2).

The discrete phase particles are selected quartz sand with a density of 2600 kg/m³, and the particles are vertically incident into the hydrocyclone by a face source from the inlet. In this study, the particle size is adopted as the uniform distribution, and the injection source with different particle sizes is established. The total mass flow rate is 6 g/s, and its specific distribution is shown in

Table 4. Particle size distribution.

Particle size (μm)	15	30	60	100	150	200
Volume fraction (%)	10	20	30	25	10	5

.

In this study, a transient simulation is used, with a total simulation time of 10 s. From 0 to 5 s, a continuous phase simulation is performed, and after 5 s, discrete phase particles are added. The SIMPLEC algorithm is adopted for pressure–velocity coupling, with Least Squares Cell-Based selected for the spatial discretization gradient, PRESTO! for the pressure discretization format, Compressive for the volume fraction, and Second Order Upwind for the momentum, as well as turbulent kinetic energy, turbulent dissipation rate, and Reynolds stress to improve the accuracy and stability of the calculation. The time step is 0.001 s, and the maximum number of iterations is 30. After 0.8 s of simulation time, the residual value stays stable below 10⁻³ and meets this convergence criterion at every time step, ensuring the accuracy and reliability of the numerical solution.

• VOF model

The gas–liquid two-phase flow is present inside the hydrocyclone, and the negative pressure generated by the liquid phase cyclone leads to the formation of the air column in the interior of the gas phase. The VOF model [42,43,44,45,46] is suitable for describing the interfacial flow between two or more immiscible phases by solving a set of momentum equations and keeping track of the volume fractions of each phase in the computational region, which accurately captures the changes at the air–water interface [30,31,32,33], especially the dynamic evolution process of the internal air column. The tracking equation for the phase interface is shown in Equation (3).

$${\displaystyle \frac{1}{{\rho }_q}}\left[{\displaystyle \frac{\partial }{\partial t}}\left({\alpha }_q{\rho }_q\right)+\nabla ·\left({\alpha }_q{\rho }_q{\overrightarrow{v}}_q\right)\right]=S_{\alpha q}+\sum _{p=1}^n\left({\dot{m}}_{pq}-{\dot{m}}_{qp}\right)$$

(3)

where ${\dot{m}}_{qp}$ represents the mass transfer from phase q to phase p, and ${\dot{m}}_{pq}$ represents the mass transfer from phase p to phase q. By default, the source term $S_{\alpha q}$ on the right-hand side of the equation is set to zero. However, users can specify a constant or define a custom mass source for each phase. For the primary phase, the volume fraction equation is not solved. The volume fraction of the primary phase is calculated using Equation (4) as a constraint.

$${\sum }_{q=1}^n{\alpha }_q=1$$

(4)

The volume fraction equation can be solved in an implicit or explicit time scheme.

2.

RSM

The internal flow field of the hydrocyclone has strong vortex and strong anisotropic turbulence characteristics; the flow is highly complex, and the components of turbulent stress in each direction are not equal. Compared to the k-ε and k-ω models, the RSM directly solves the Reynolds stress transport equation, making it more suitable for anisotropic turbulent flow [42,43]. It can accurately capture the characteristics of hydrocyclones and enhance the predictive capability for particle separation and wall erosion.

The exact transport equation for the Reynolds stress $\rho \overline{u_i^{\prime }{u}_j^{\prime }}$ is given by Equation (5) as follows:

$$\begin{array}{c}\begin{array}{cc}\underset{Local Time Derivative}{\underbrace{ {\displaystyle \frac{\partial }{\partial t}}\left(\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right) }}+\underset{C_{ij}\equiv Convection}{\underbrace{{\displaystyle \frac{\partial }{\partial x_k}}\left(\rho u_k\overline{u_i^{\prime }{u}_j^{\prime }}\right)}}& =\underset{D_{T,ij}\equiv Turbulent Diffusion}{\underbrace{{\displaystyle \frac{\partial }{\partial x_k}}\left[\rho \overline{u_i^{\prime }{u_j^{\prime }u}_k^{\prime }}+\overline{p^{\prime }\left({\delta }_{kj}{u}_i^{\prime }+{\delta }_{ik}{u}_j^{\prime }\right)}\right]}}\hfill \\ & +\underset{D_{L,ij}\equiv Molecular Diffusion}{\underbrace{{\displaystyle \frac{\partial }{\partial x_k}}\left[\mu {\displaystyle \frac{\partial }{\partial x_k}}\left(\overline{u_i^{\prime }{u}_j^{\prime }}\right)\right]}}\hfill \\ & \begin{array}{cc}-& \underset{P_{ij}\equiv Stress Production}{\underbrace{\rho \left(\overline{u_i^{\prime }{u}_k^{\prime }}{\displaystyle \frac{\partial u_j}{\partial x_k}}+\overline{u_j^{\prime }{u}_k^{\prime }}{\displaystyle \frac{\partial u_i}{\partial x_k}}\right)}}\end{array}\hfill \\ & -\underset{G_{ij}\equiv Buoyancy Production}{\underbrace{\rho \beta \left(g_i\overline{u_j^{\prime }\theta }+g_j\overline{u_i^{\prime }\theta }\right)}}\hfill \\ & +\underset{{\phi }_{ij}\equiv Pressure Strain}{ \underbrace{ p^{\prime \left({\displaystyle \frac{\partial u_i^{\prime }}{\partial x_j}}+{\displaystyle \frac{\partial u_j^{\prime }}{\partial x_i}}\right)} }}\hfill \\ & -\underset{{\epsilon }_{ij}\equiv Dissipation}{\underbrace{2\mu \left(\overline{{\displaystyle \frac{\partial u_i^{\prime }}{\partial x_k}}+{\displaystyle \frac{\partial u_j^{\prime }}{\partial x_k}}}\right)}}\hfill \\ & -\underset{F_{ij}\equiv Production by System Rotation}{\underbrace{2\rho {\Omega }_k\left(\overline{u_j^{\prime }{u}_m^{\prime }}{\epsilon }_{ikm}+\overline{u_i^{\prime }{u}_m^{\prime }}{\epsilon }_{jkm}\right)}}\hfill \\ & +\underset{User-Defined Source Term}{\underbrace{S_{user}}} \hfill \end{array}\end{array}$$

(5)

3.

DPM

Compared to the Euler–Euler multiphase flow method, the DPM is less computationally intensive and is suitable for engineering applications, as only particle trajectories are computed instead of solving for the entire particle phase. The DPM follows the Euler–Lagrange method, where the fluid phase, as a continuous phase, is handled by solving the Navier–Stokes equations, and the discrete phases are solved by tracking the motions of massive particles in the computational flow field. The particle motion equations are shown in Equation (6).

$$m_p{\displaystyle \frac{d{\overrightarrow{u}}_p}{dt}}=m_p{\displaystyle \frac{\overrightarrow{u}-{\overrightarrow{u}}_p}{{\tau }_r}}+m_p{\displaystyle \frac{\overrightarrow{g}\left({\rho }_p-\rho \right)}{{\rho }_p}}+\overrightarrow{F}$$

(6)

where $m_p$ represents the particle mass, $\overrightarrow{u }$ is the fluid phase velocity, ${\overrightarrow{u}}_{p }$ is the particle velocity, $\rho$ is the fluid density, ${\rho }_p$ is the particle density, $\overrightarrow{F}$ is the additional force, $m_p{\displaystyle \frac{\overrightarrow{u}-{\overrightarrow{u}}_p}{{\tau }_r}}$ is the drag, and ${\tau }_r$ is the particle relaxation time. The calculation formula is as shown in Equation (7) as follows:

$${\tau }_r={\displaystyle \frac{{\rho }_p{d}_p^2}{18\mu }}{\displaystyle \frac{24}{C_{d}Re}}$$

(7)

Here, $\mu$ represents the molecular viscosity of the fluid, $d_p$ is the particle diameter, and $Re$ is the relative Reynolds number, which is defined in Equation (8) as follows:

$$Re\equiv {\displaystyle \frac{\rho d_p\left|{\overrightarrow{u}}_p-\overrightarrow{u}\right|}{\mu }}$$

(8)

4.

Oka erosion model

Compared with the Finne erosion model and the Generic erosion model, the Oka erosion model is based on experimental data and considers factors such as particle impact angle, impact velocity, and material hardness ratio, and it is more accurate in describing the erosion characteristics under different impact angles [41]. In hydrocyclones, since particles rotate at high speeds and impact the wall at different angles, the Oka erosion model is more accurate and applicable, and it can reasonably describe the erosion characteristics of the hydrocyclone inner wall.

The Oka erosion equation [16,17] adopted in this study is given in Equation (9) as follows:

$$E=E_{90}{\left({\displaystyle \frac{V}{V_{ref}}}\right)}^{k_2}{\left({\displaystyle \frac{d}{d_{ref}}}\right)}^{k_3}f\left(\gamma \right)$$

(9)

where $E_{90}$ is the reference erosion rate at a 90° impact angle, $E_{90}$ = $6.154\times {10}^{-4}$; $V$ and $V_{ref}$ represent the particle impact velocity and the reference velocity, $V_{ref}$ = 104 m/s; $d$ and $d_{ref}$ represent the particle diameter and the reference diameter, $d_{ref}$ = 326 μm; $k_2$ and $k_3$, respectively, represent the velocity and diameter index, $k_2$ = 2.35 and $k_3$ = 0.19; and $f\left(\gamma \right)$ represents the impact angle function.

The impact angle function is defined in Equation (10) as follows:

$$f\left(\gamma \right)={\left(sin\gamma \right)}^{n1}{\left(1+H_v\left(1-\mathit{sin}\gamma \right)\right)}^{n2}$$

(10)

where $\gamma$ represents the wall impact angle (rad), $H_v$ represents the Vickers hardness of the wall material, $H_v$ = 1.8 GPa, and $n1$ and $n2$ represent angular function constants, $n1$ = 0.77 and $n2$ = 1.36.

3. Simulation Results and Analysis

3.1. Analysis of the Dynamic Evolution Process of Air Columns

In this paper, the VOF model is adopted to simulate the dynamic evolution of the air column in the hydrocyclone, and the specific process is shown in Figure 6.

Figure 6 shows the distribution cloud map of fluid volume fraction at the X = 0 profile of the hydrocyclone during the 0–10 s period. The figure reveals that at t = 0.1 s, the fluid is given in at high velocity along the tangential inlet. Under centrifugal force, the fluid initially moves radially toward the wall. As time progresses, part of the fluid gradually migrates toward the center until the initial formation of an air column. At t = 0.5 s, the air column has taken shape, though a small amount of air remains within the fluid, and the bottom of the air column is unstable. At t = 0.8 s, all air within the fluid is completely expelled, and the air column stabilizes, maintaining its shape and dimensions. Additionally, it can be observed that no fluid infiltrates the central region throughout the process. This is due to the persistent low-pressure zone in the center. Until a negative pressure zone forms, external air infiltrates through the overflow port and underflow port into this negative pressure zone, ultimately forming an air column that spans the entire flow field.

3.2. Velocity Field Analysis

In this paper, the velocity field inside the hydrocyclone is analyzed, respectively, through tangential velocity, axial velocity, and radial velocity. Figure 7 shows the velocity distribution cloud map of the continuous phase flow field at the X = 0 profile. The tangential velocity is symmetrically distributed along the Z-axis, forming a typical double-vortex structure composed of internal and external vortices. The trajectory of the maximum tangential velocity forms the natural interface between the “semi-free vortex” and the “forced vortex” in the hydrocyclone. The axial velocity determines the vertical movement of the fluid and particles and plays a crucial role in separation efficiency. It also exhibits an overall axisymmetric distribution. At the wall, the axial velocity is directed downward, with the fluid entering the underflow port. At the axis, the axial velocity is directed upward, forming a return flow, with fine particles entering the overflow port to be discharged. The radial velocity primarily exhibits an uneven distribution, with the maximum value at the axis. In general, its magnitude is significantly smaller than that of the other velocity components.

To accurately analyze the distribution law of the velocity field in the hydrocyclone, the data at the axial position Z = −300 mm, Z = −400 mm, and Z = −500 mm cross-section were taken to be analyzed, respectively, and the radial distribution curve is shown in Figure 8. From the figure, the change rule of the tangential velocity in the column section and cone section shows a decrease with the radius increase and reaches the minimum value near the wall. In the overflow zone, the tangential velocity reaches the maximum value and decreases with the radius. The axial velocity reaches the maximum value at the axis center and shows a trend of decreasing with the increase in the radius from the axis center outward, and then the axial velocity reaches a negative value outward and exits the negative region, which is due to the flow structure of inside upward and outside downward inside the hydrocyclone. The radial velocity reaches its maximum value near the axis center, and its change fluctuates greatly without an obvious distribution law.

3.3. Pressure Field Analysis

Figure 9 shows the cloud map of pressure distribution in the continuous phase flow field at the X = 0 profile, from which it can be clearly seen that the overall distribution of the static pressure in the hydrocyclone is symmetric. There is a negative pressure zone in the center region throughout the hydrocyclone, and it reaches the maximum value in the wall of the column section and the cone section. The dynamic pressure in the hydrocyclone is related to the flow rate. Usually, the dynamic pressure is larger in the region of higher fluid flow rate, and its overall distribution is also symmetrical, reaching a minimum value in the center of the air column and a maximum value in the connection between the overflow pipe and the inside of the column section. The total pressure inside the hydrocyclone is the sum of static pressure and dynamic pressure, and its distribution law is the same as that of static pressure; the overall distribution is symmetrical.

To study the radial distribution law of static pressure, dynamic pressure, and total pressure in the hydrocyclone, the data are, respectively, taken from the axial position Z = −300 mm, Z = −400 mm, and Z = −500 mm cross-section, and the radial distribution curve is shown in Figure 10. From the figure, the static pressure and the total pressure distribution law are basically the same, symmetrical distribution along the Z-axis. The Z-axis direction is basically unchanged, the overall trend increases with radius, and the pressure is at the lowest value in the center region. The dynamic pressure in the center region shows the change law of increasing with the radius, reaching the maximum value near the wall of the overflow pipe, and then the outward direction shows the trend of decreasing with the radius increase.

3.4. Particle Motion Analysis

The erosion of hydrocyclone inner walls is primarily caused by the continuous impact of particles against the wall surface, with particle motion behavior directly determining the location and morphology of erosion. This study analyzed the spatial distribution characteristics of particles with different sizes (15 μm, 30 μm, 60 μm, 100 μm, 150 μm, 200 μm) within 5–6 s, as shown in Figure 11.

The results indicate that all particles exhibit typical spiral trajectories within the hydrocyclone. Finer particles (15 μm, 30 μm, 60 μm), due to their low mass and weak inertia, are significantly affected by fluid drag force. Part of them may enter the center zone via the inner vortex and be discharged through the overflow port. However, as particle size increases to 60 µm, the influence of fluid drag force gradually diminishes. These particles are increasingly governed by centrifugal force, tending to move downward along the outer vortex zone and discharge through the underflow outlet. Consequently, their distribution range within the cone section expands markedly.

In contrast, coarser particles (100 μm, 150 μm, 200 μm) are primarily spiraled downward along the wall toward the underflow port under the combined effects of inertia and centrifugal force, with almost none entering the inner vortex zone. By t = 5.6 s, the positional distribution of particles across all size ranges stabilizes. Coarser particles predominantly accumulate in the cone section, while finer particles concentrate near the overflow pipe, forming a distinct spiral-banded distribution pattern within the cone section. This outcome aligns with the typical movement patterns of particles within a hydrocyclone.

3.5. Erosion Analysis of Different Particle Sizes

In the numerical study of wall erosion within hydrocyclones, the contribution of particles with different sizes to wall erosion is of significant importance. Figure 12 displays erosion cloud maps for six distinct particle sizes. The figure reveals that regardless of particle size, the cone section consistently exhibits the most severe erosion, with maximum erosion concentrated near the underflow port. Erosion in the overflow pipe is primarily driven by fine particles (15 μm, 30 μm, and 60 μm), which possess low inertia and readily rise with the internal vortex. Coarser particles (100 μm, 150 μm, and 200 μm), possessing greater inertia, primarily move downward along the wall in a spiral pattern. They struggle to enter the internal vortex zone, thus contributing more significantly to erosion in the cone section. Their erosion morphology exhibits distinct spiral-banded characteristics.

To quantitatively evaluate the erosion impact of particles with different sizes, this study calculated the area-weighted average erosion rates of particles across various wall surfaces and determined their contribution to overall erosion, thereby analyzing the erosion contribution of each particle size to different walls. Figure 13 presents a heatmap illustrating the erosion contribution from different particle sizes, clearly showing the varying dominant roles of different particle sizes across the wall. In the roof section, 60 μm particles contributed the most (37.7%), while 200 μm particles contributed the least (5.2%); in the infeed section, 100 μm particles contributed the highest (30.9%), and 15 μm particles contributed the lowest (3.1%). In the column section, 100 μm particles again dominate (37.9%), while 15 μm particles contribute only 1.3%. Overflow pipe erosion is primarily caused by 30 μm particles, accounting for 49.8%, while 150 μm and 200 μm particles contribute nothing. In the cone section, 60 μm and 100 μm particles contributed significantly to erosion at 32% and 34%, respectively. Notably, although 200 μm particles had the largest diameter, their limited volume fraction resulted in a relatively minor contribution to overall erosion.

3.6. Erosion Analysis of Each Wall

In this paper, the Oka erosion model is adopted to numerically simulate the erosion patterns on each wall of the hydrocyclone. Erosion data at t = 10 s are selected, primarily analyzing the erosion distribution patterns along the circumferential and Z-axis directions for each wall. They include erosion of the roof section, infeed section, column section, overflow pipe, and cone section, thereby achieving erosion prediction for each wall of the hydrocyclone.

3.6.1. Roof Section Erosion

Figure 14 displays the erosion cloud map of the roof section of the hydrocyclone. The figure reveals that the erosion rate is uniformly distributed along the annular zone at the top of the hydrocyclone, gradually increasing with radial distance and reaching a maximum value of 2.3 g/(m²·h) in the outer ring zone. This phenomenon is primarily attributed to the higher turbulence intensity in this zone, which significantly increases the friction between particles and the wall surface. Additionally, a top ash ring phenomenon occurs in the roof section. This is caused by a large pressure gradient, where fine particles experience stronger upward forces in this zone, propelling them toward the top and causing suspension. Consequently, particles accumulate in this zone, further intensifying erosion.

The distribution characteristics of this roof erosion may impact the separation efficiency of the hydrocyclone. Specifically, when the roof experiences severe erosion, its surface structure may become damaged, altering the flow field and consequently affecting classification performance.

3.6.2. Infeed Section Erosion

As the zone where the particle material first contacts the inner wall of the hydrocyclone, the infeed section endures high-speed particle impact and is one of the most erosion-prone areas inside the hydrocyclone. Figure 15 displays the erosion cloud map of the infeed section. It reveals that the infeed section primarily exhibits an asymmetrical, semi-circular high-erosion zone concentrated on the wall surface opposite the inlet. This occurs because the infeed section has a significant pressure gradient. At the inlet, the kinetic energy of particles is converted into impact force against the wall surface, leading to a marked increase in the erosion rate.

The erosion condition in the infeed section is similar to that in the roof section, with the maximum erosion zone located directly beneath the roof section. The maximum erosion rate reaches 3.4 g/(m²·h), approximately 1.47 times that of the roof section. Prolonged exposure to high erosion rates may cause significant structural damage to the infeed section, thereby affecting the classification performance and overall operational efficiency of the hydrocyclone.

Figure 16 shows the distribution curve of the infeed section erosion rate along the circumference, with data taken from the cross-sections at Z = −130 mm, Z = −140 mm, Z = −150 mm, Z = −160 mm, and Z = −170 mm, respectively, to analyze the erosion characteristics in the circumferential direction. In general, the distribution patterns of erosion rates at each cross-section along the circumferential direction are generally consistent. As the azimuth increases, the erosion rate first increases and then decreases. The maximum erosion zone is located between azimuths 225°and 290°, while the minimum erosion rate occurs at the connection between the inlet wall and the circular wall.

Figure 17 shows the distribution curve of the erosion rate along the Z-axis in the infeed section, with data taken at azimuths of 0°, 90°, 180°, and 270°, respectively, to analyze the erosion characteristics in the Z-axis direction. The erosion rate at the 0° azimuth is significantly lower than at others, decreasing with increasing axial depth. At the 180° azimuth, the erosion rate first increases and then decreases with increasing axial depth, with the maximum erosion zone located between axial depths of 125 mm and 135 mm, reaching a maximum value of 2.6 g/(m²·h) at the 132° azimuth.

3.6.3. Column Section Erosion

Within the internal flow field and particle classification process of the hydrocyclone, the column section plays a crucial role in the formation of the initial vortex and the development of a stable vortex. Figure 18 presents the erosion cloud map of the column section, revealing that high-erosion zones exhibit a spiral-banded distribution closely aligned with particle trajectories. The maximum erosion zone is located at the connection between the column and cone sections, with a peak erosion rate of 19.2 g/(m²·h). This phenomenon is primarily attributed to the abrupt changes in fluid velocity and pressure caused by the geometric transition. In this zone, increased flow velocity intensifies particle–wall impact forces, exacerbating erosion. Additionally, pressure gradient variations amplify turbulence intensity, particularly within the spiral bands. Tangential fluid velocities and turbulent effects generate more severe particle–wall collisions, further accelerating erosion.

Localized erosion within the column section not only leads to non-uniform velocity distribution but also affects particle trajectories and classification performance. Therefore, a thorough analysis of erosion distribution within the column section is crucial for optimizing hydrocyclone design, enhancing separation performance, and extending equipment service life.

Figure 19 shows the distribution curve of the column section erosion rate along the circumference, with data extracted from the cross-sections at Z = −200 mm, Z = −230 mm, Z = −260 mm, and Z = −290 mm, respectively, to study their circumferential erosion characteristics. As shown in the figure, the erosion rates at each cross-section exhibit significant differences only in the high-erosion zone of the spiral, while the erosion rates in other zones are relatively small and show little variation. The maximum erosion rate at the Z = −200 mm section is significantly lower than at other sections. Its maximum erosion zone is located between azimuths 180° and 220°. At the Z = −230 mm section, the maximum erosion region is located between azimuths 280° and 310°. At the Z = −260 mm section, the maximum erosion region is located between azimuths 0° and 50°. The maximum erosion zone at the Z = −290 mm section is located between azimuths 170° and 200°.

Figure 20 shows the erosion rate distribution curve of the column section along the Z-axis direction, with data taken at azimuths of 0°, 90°, 180°, and 270°, respectively, for analyzing the erosion characteristics along the Z-axis. The distribution pattern of erosion rates along the Z-axis generally followed that of the circumferential direction. Except for the high-erosion zones, erosion rates along the Z-axis exhibited minimal fluctuations, indicating that erosion distribution along the Z-axis is relatively uniform outside these high-erosion zones. The distribution patterns of erosion rates at different azimuths are generally consistent. The maximum erosion zones are located within the spiral band erosion zones. At azimuth 0°, the maximum erosion zone is located between axial depths of 250 mm and 260 mm, while at azimuth 90°, it is located between axial depths of 270 mm and 280 mm. The maximum erosion zone at azimuth 180° is located between axial depths of 290 mm and 300 mm. At azimuth 270°, the erosion rate sharply increases between axial depths of 300 mm and 310 mm, reaching its peak at an axial depth of 310 mm. This is where the column section connects to the cone section.

3.6.4. Overflow Pipe Erosion

During the classification process in a hydrocyclone, fine particles spiral upward along the inner wall of the hydrocyclone toward the overflow pipe, causing direct impact or tangential scouring against the pipe wall, thereby inducing erosion of the overflow pipe. Figure 21 displays the erosion cloud map of the overflow pipe, revealing an irregular erosion distribution pattern with a maximum erosion rate of 38.3 g/(m²·h). This phenomenon is primarily attributed to the constantly varying impact angles and velocities between particles flowing through the overflow pipe and the wall surface.

In the bottom zone of the overflow pipe, despite higher flow velocities, the erosion rate is relatively low. This may be related to the particle impact angle and flow field characteristics in this zone. Although high flow velocities increase particle impact energy, the fluid distribution characteristics and particle trajectories result in relatively minor erosion in the bottom zone. A detailed analysis of the erosion distribution in the overflow pipe is crucial for enhancing the overall performance of the hydrocyclone and extending the service life of the equipment.

Figure 22 shows the distribution curve of the overflow pipe erosion rate along the circumference, with data taken from the sections at Z = −20 mm, Z = −120 mm, and Z = −220 mm, respectively. It can be seen from the figure that the overflow pipe erosion rate fluctuates greatly along the circumference and does not follow a clear distribution pattern. The erosion rates at the Z = −20 mm and Z = −120 mm cross-sections exhibit generally consistent circumferential distribution patterns, with the maximum erosion zones occurring between azimuth angles of 50° and 100°.

Figure 23 shows the distribution curve of the overflow pipe erosion rate along the Z-axis, with data taken at azimuths of 0°, 90°, 180°, and 270°, respectively. Between axial depths of 0 mm and 280 mm, the erosion rate first increases and then decreases with increased axial depth. At axial depths between 260 mm and 280 mm, the erosion rates at azimuths of 180° and 270° decrease sharply with the increased axial depth, and their erosion rates are significantly lower than those at other azimuths.

3.6.5. Cone Section Erosion

In a hydrocyclone, particles entering the outer vortex spiral downward along the cone wall, continuously impacting the surface and causing erosion of the cone section. Figure 24 displays the erosion contour map of the cone section. It is evident that the cone section is the most severely eroded part of the hydrocyclone, particularly near the underflow port, where the erosion rate reaches a maximum value of 2721.6 g/(m²·h). This phenomenon primarily stems from the maximum fluid tangential velocity near the underflow outlet at the wall surface, coupled with a sharp change in pressure gradient. These factors synergistically intensify local turbulence, thereby triggering more severe erosion. Therefore, this paper conducts an intensive analysis of the erosion distribution pattern in the conical section, providing a theoretical basis for optimizing hydrocyclone design.

Figure 25 shows the distribution curve of the cone section erosion rate along the circumference, with data taken from the Z = −350 mm, Z = −450 mm, Z = −550 mm, and Z = −650 mm cross-sections, respectively. The erosion rate exhibits consistent circumferential distribution patterns across all Z-axis cross-sections, with minimal variation between different circumferential directions. This indicates that erosion along the cone section is relatively uniform in the circumferential direction. At the Z = −650 mm cross-section, the erosion rate is significantly higher than at other cross-sections, with a maximum erosion rate reaching 1360.8 g/(m²·h). This is because the diameter at the Z = −650 mm cross-section is much smaller than at other cross-sections, resulting in a higher particle concentration on the wall surface and, consequently, a higher erosion rate.

Figure 26 shows the distribution curve of the cone section erosion rate along the Z-axis, with data taken at azimuths of 0°, 90°, 180°, and 270°, respectively. As shown in the figure, the erosion rate remains basically constant with an increased axial depth in the upper part of the cone section (axial depth of 310 mm to 600 mm). In the lower part of the cone section (axial depth of 600 mm to 701 mm), the erosion rate increases sharply with an increased axial depth. Near the underflow port, the erosion rate reaches its maximum value of 2721.6 g/(m²·h) at approximately an axial depth of 675 mm. The maximum erosion rate at this location is about 6.8 times that of the upper cone section wall. At the end of the cone section, between axial depths of 675 mm and 701 mm, the erosion rate decreases again with an increased axial depth. At azimuth 0°, the maximum erosion rate reached 2656.8 g/(m²·h), approximately 1.2 to 1.6 times higher than at other azimuths.

4. Conclusions

In this paper, numerical simulation is used to predict the erosion mechanisms on each wall of the hydrocyclone under air column conditions. Under the simulated conditions, the following conclusions are drawn:

(1) Among each wall of the hydrocyclone, the cone section suffers the most severe erosion, followed by the overflow pipe, column section, infeed section, and roof section. Erosion near the underflow outlet of the cone section is most pronounced, with a maximum erosion rate approximately 6.8 times that of the upper cone section. The maximum erosion rate in the infeed section is about 1.47 times that of the roof section.

(2) Erosion in the roof section exhibits uniform circumferential distribution and increases gradually with radius. The maximum erosion zone in the infeed section is located directly below the roof section. Erosion distribution in the column section resembles that of the cone section, both showing spiral band patterns. Erosion in the overflow pipe, however, exhibits irregular distribution.

(3) Among the contributions of different particle sizes to hydrocyclone wall erosion, 60 μm and 100 μm particles exerted the most dominant influence on cone section erosion, accounting for 32% and 34%, respectively. Meanwhile, 30 μm particles played a leading role in overflow pipe erosion, contributing 49.8%.

This paper presents a numerical simulation method for investigating erosion conditions under air column conditions in hydrocyclones. Future research may consider broader particle size distributions, varying operating parameters, and different wall materials to explore wall erosion under diverse simulated operating conditions, thereby gaining deeper insights into the erosion mechanisms within hydrocyclones.