Erosive Wear Caused by Large Solid Particles Carried by a Flowing Liquid: A Comprehensive Review

Abstract

The erosive wear encountered in some industrial processes results in economic loss and even disastrous consequences. Hitherto, the mechanism of the erosive wear is not clear, especially when the erosive wear is caused by large particles (>3.0 mm) carried by a flowing liquid. Current approaches of predicting erosive wear need improvement, and the optimization of relevant equipment and systems lacks a sound guidance. It is of significance to further explore such a subject based on the relevant literature. The present review commences with a theoretical analysis of the dynamics of large particles and the fundamental mechanism of erosion. Then the characteristics of the erosion of various equipment are explicated. Effects of influential factors such as particle size and properties of the target material are analyzed. Subsequently, commonly used erosion models, measurement techniques, and numerical methods are described and discussed. Based on established knowledge and the studies reported, some expectations for future work are proposed.

1. Introduction

Erosive wear is the loss of material from a solid surface due to relative motion in contact with solid particles which are entrained in a fluid or due to the action of streaming liquid, gas, or gas containing liquid droplets. In applications such as oil and gas production, mining, dredge cleaning, and circulation of water in power plants, erosive wear has been recognized as a common phenomenon. Erosive wear may incur performance degradation of the equipment, declined transport efficiency, increased maintenance cost, and even fracture of the impacted components. The replacement of damaged components not only necessitates additional maintenance costs and downtime, but also poses a threat to human safety and the environment. Moreover, erosion implies that a portion of energy has been consumed to generate such a negative effect. This is in contradiction to the currently advocated concept of green and energy-saving societies. In this situation, the mechanism of erosive wear has attracted attentions from both academia and industry.

Erosive wear is influenced by multiple factors, such as the characteristics of particles (size, shape, and hardness), flow patterns, impingement parameters like the impact velocity and impact angle, and properties of the target material. These factors are often inter-related, which increases the complexity of establishing a comprehensive erosion model. The erosion model proposed in Ref. [1] involves deformation and cutting wear, and the result obtained is consistent with the experimental result. The application of scanning electron microscopy (SEM) has enabled the identification of the micromechanism of wear and thereby offers an insight into the physics of wear [2]. However, the erosive wear model generally incorporates several unknown parameters, which presents a challenge for its practical implementation. Moreover, most erosion models are established based on the experimental data obtained in rather simple flows, which raises concerns about their applicability to relatively complex flow fields. For instance, when solid particles are carried by a flowing liquid, the generation of bubbles due to phase change or external gas injection will alter particle trajectories [3]. Moreover, in gas-dominated multiphase flow, liquid can form a film over the solid surface, and the liquid film assumes the function of buffering the impact of particles on the surface. Furthermore, the erosive wear induced by rebounding particles exhibits apparently different characteristics with respect to that associated with the direct impact on the solid surface [4].

In previous studies on erosive wear, the particle sizes are generally less than 3.0 mm, and the solid volume fractions lower than 5%. However, some applications like deep-sea mining require the transport of solid particles larger than 30 mm, and corresponding solid volume fractions are even higher than 10% [5]. For these applications, the erosive wear can aggravate the vibration of the lifting system. Meanwhile, the intensity of the liquid-borne noise is enhanced, and those marine organisms are consequently harmed. Since small particles have been focused on previously, available numerical and experimental results are overwhelmingly related to small particles. In this context, the applicability of existing erosion models for predicting the erosive wear associated with large particles remains uncertain.

Erosive wear is, in essence, a mechanical effect. Particles repeatedly impact the target material, resulting in various patterns of damage to the surface. Explaining erosive wear necessitates the knowledge of multiple disciplines such as mechanics, fluid dynamics, and materials science. When solid particles are carried by a flowing fluid, liquid, or gas, or by the mixture of liquid and gas, the interaction between solid particles and the fluid is inevitable. The phases are considerably different in physical properties, and the kinetic parameters such as velocity are different as well. Thus far, both experimental and numerical methods have been attempted, but the disclosed mechanisms of the inter-phase interaction are rather limited. The numerical methods treating multiphase flow include Euler–Euler and Euler–Lagrange methods. For the Euler–Euler method, all phases are solved within the Eulerian framework [6]. Regarding the Euler–Lagrange approach, the continuous phase is solved in the Eulerian framework, while the particulate phase is tracked in the Lagrangian framework through solving Newton’s motion equations. As a Euler–Lagrange method, the CFD–DEM (computational fluid dynamics combined with the discrete element method) can be categorized as unresolved, fully resolved, and semi-resolved methods based on fluid grid resolution [7]. In the fully resolved CFD–DEM, the solid–liquid interaction force is calculated through directly integrating the fluid stress over the particle surface. Consequently, the fluid grids must be finer than the particle diameter to accurately capture the particle boundary [8]. In the unresolved CFD–DEM approach, the solid–liquid interaction force is calculated through an empirical model with the void fraction [9]. For the conventional unresolved CFD–DEM, the particle-centered method is used to calculate the void fraction, which requires that the grid size be larger than the particle diameter [10]. To address this limitation, various strategies such as the weight function method [11] and the big particle method [12] have been introduced to calculate the void fraction and enhance the unresolved CFD–DEM approach. The unresolved CFD–DEM is recognized for its computational efficiency, particularly when solving solid–liquid flow involving a large number of particles. In recent years, the semi-resolved CFD–DEM has been developed. For such a method, the interaction forces for coarse particles are fully resolved while those for fine particles are calculated through an empirical model [13]. The semi-resolved CFD–DEM provides more accurate results compared to the unresolved CFD–DEM, while being significantly more computationally efficient than the fully resolved CFD–DEM.

To improve the resistance to erosive wear, effort has been devoted to the optimization of the geometry of hydraulic components as well as the operating condition. However, some measures prove effective only in laboratories. When it comes to engineering applications, the results are still ambiguous. This may result in serious consequences. For instance, for deep-sea mining, when all equipment is ready on site, unstable operation or the replacement of damaged components will incur unpredictable losses of time and expenditure.

The present study provides a comprehensive review of the studies on the erosive wear related to large solid particles (>3.0 mm) carried by a flowing liquid. In Section 2, dynamic characteristics of solid particles are discussed. In Section 3, the erosive wear encountered in typical fluid-handling equipment is analyzed. In Section 4, a thorough examination of main factors influencing erosive wear, such as material properties and liquid temperature, is implemented. Generally employed erosive wear models and representative erosive wear patterns are enumerated in Section 5. In Section 6, an introduction of commonly used erosion rigs and measurement techniques is presented. Subsequently, numerical models for solving the solid–liquid two-phase flow and predicting the erosive wear are explained and compared. In Section 7, limitations of current studies are analyzed, and some pieces of advice for future work are proposed.

2. Fundamental Aspect of the Flow Carrying Large Particles

The kinetic characteristics of particles serve as an influential factor for the erosive wear of the target wall. For small particles, they can closely follow the motion of the carrier liquid. Then, the resultant wear can be explained based on the distribution of liquid flow velocity. Even when the solid volume concentration is high, the group characteristics of the particles are still predictable. Regarding large particles, their motion deviates considerably from the liquid flow. In this situation, the behaviors of individual particles depend on multiple factors such as particle size, particle shape, and the liquid flow pattern. The interaction between particles may alter their trajectories and then the position of wear on the impacted solid wall. To gain a comprehensive understanding of erosive wear, it is of significance to explain the hydraulic forces acting on particles and the equations governing the motion of particles.

According to Newton’s second law, the equation of motion for particles in an inertial reference frame can be expressed as [14]:

$$m_s\frac{\mathrm{d}{\mathit{v}}_s}{\mathrm{d}t} = {\mathit{F}}_l+{\mathit{F}}_g+{\mathit{F}}_C+{\mathit{F}}_{other}$$

(1)

where m_s denotes the mass of the particle; v_s is the velocity of the particle; F_l is the hydraulic forces acting on the particle; F_g is the gravitational force acting on the particle; F_C is the force acting on the particle by other particles; and F_other denotes the other forces such as the user-defined forces.

The force acting on the particle by the liquid phase can be decomposed as:

$${\mathit{F}}_l={\mathit{F}}_D + {\mathit{F}}_P +{\mathit{F}}_{LS}+{\mathit{F}}_{LR}$$

(2)

where F_D denotes the drag force; F_P is the pressure gradient force; and F_LS is the shear lift force; F_LR is the rotational lift force.

The drag force is related to the viscosity of the liquid, and its direction is the same as that of the slip velocity of the particle. The drag force is given by [15]:

$${\mathit{F}}_D=\frac{1}{2}{C}_d{\rho }_l{A}_s\left|{\mathit{v}}_{slip}\right|{\mathit{v}}_{slip}$$

(3)

where C_d is the drag coefficient; ρ_l is the density of the liquid; A_s is the projected area of the solid particle; and v_slip is the slip velocity of the solid particle.

The pressure gradient force refers to the force generated in a fluid due to pressure difference, and it points from the high-pressure side towards the low-pressure side. The pressure gradient force is equal to the buoyancy force exerted on the object, and is expressed as [16]:

$${\mathit{F}}_P=-B_s\nabla P$$

(4)

where B_s denotes the volume of the particle, and P is the static pressure of the liquid.

The shear lift force is given by [16]:

$${\mathit{F}}_{LS}=C_{LS}\frac{{\rho }_l\pi }{8}{d}_s^3({\mathit{v}}_{slip}\times {\mathsf{\omega }}_l)$$

(5)

where C_LS is the shear lift coefficient; ω_l is the curl of the liquid velocity; and d_s is the particle diameter.

The rotational lift force is given by [16]:

$${\mathit{F}}_{LR}=C_{LR}\frac{{\rho }_l\pi }{8}{d}_s^2\mid {\mathit{v}}_{slip}\mid \frac{\mathit{\Omega }\times {\mathit{v}}_{slip}}{\mid \mathit{\Omega }\mid }$$

(6)

$$\mathit{\Omega }=\frac{1}{2}\nabla \times {\mathit{v}}_l-{\mathit{\omega }}_s$$

(7)

where C_LR is the rotational lift coefficient and ω_s is the angular velocity of the particle.

When particles travel in a vertically mounted pipe, they tend to occupy the central region of the pipe. Hence, the erosive wear of the side wall is insignificant. However, in a vertically placed elbow, particles can induce severe wear of the outer wall of the elbow. This results from sharp variation in flow direction. A schematic view of the forces, velocity components, and trajectories associated with a particle traveling in an elbow is shown in Figure 1, where the lift and inter-particle forces are neglected. The equations of motion for the particle in x and y directions can be expressed as:

$$F_{P,x}+F_{D,x}=m_s\frac{d{v}_{s,x}}{dt}$$

(8)

$$F_{P,y}+F_{D,y}-F_g=m_s\frac{d{v}_{s,y}}{dt}$$

(9)

Relative to the liquid, particles have greater mass and inertia, which contributes to a slow decrease in v_s,y, the velocity component in y direction, and a slow increase in v_s,x, the velocity component in x direction. This leads to a non-collinear relationship between particle velocity and liquid velocity, denoted as v_s and v_l, respectively, as shown in Figure 1. Such a relationship differs from that associated with the straight pipe. Furthermore, the directions of the slip velocity, v_slip, and the drag force, F_D, no longer coincide with particle velocity and liquid velocity. The substantial inertia of the particle results in a rotational angle smaller than 90° for the particle trajectory S₁. The change in particle mass stimulates an alteration of particle trajectory. With increasing particle mass, the particle trajectory undergoes a leftward shift, as evidenced by trajectory S₂. Conversely, a decrease in particle mass results in a rightward shift of the trajectory, as indicated by trajectory S₃. Therefore, the maximum erosion is generally produced at the outer surface near the elbow outlet, which has been validated through experimental results [17].

For the impeller machinery, the rotating impeller is usually employed as a reference frame for investigating the motion of particles with respect to the impeller. In Figure 2, the forces, velocity, and trajectory associated with a particle in the flow passage of a centrifugal impeller are presented. Since the rotating impeller can be taken as a non-inertial reference frame, it is necessary to account for inertial forces, such as the centrifugal and Coriolis forces. In the impeller passage, the Coriolis force, F_Cori, is directed towards the pressure surface of the blade, perpendicular to the relative velocity of the particle. Two kinds of centrifugal forces are defined: one is denoted as F_Cen,1, which is generated owing to the rotation of the impeller, while the other centrifugal force, denoted as F_Cen,2, is generated due to the curved geometry of the flow channel. The first centrifugal force is perpendicular to the circumferential velocity of the particle and directs towards the impeller outlet, while the second centrifugal force is perpendicular to the relative velocity of the particle and is directed towards the suction surface of the blade. The expressions for the three types of inertial forces are as follows [18,19]:

$${\mathit{F}}_{Cori}=-2m_s\left({\mathit{\omega }}_{imp}\times {\mathit{v}}_{rel}\right)$$

(10)

$${\mathit{F}}_{Cen}=m_s{\omega }_{imp}^2{\mathit{r}}_1$$

(11)

$${\mathit{F}}_{Cen}=m_s\frac{v_{rel}^2}{{\mathit{r}}_2}$$

(12)

where ω_imp denotes the angular velocity of the impeller, and r₁ and r₂ are radii of curvature.

The motion of a particle in the flow passage of the impeller is influenced by various forces, and its trajectory is denoted by S₃, as shown in Figure 2. When the Coriolis force dominates over the centrifugal force, the particle deviates from its initial trajectory and shifts towards the pressure surface, as denoted by S₂, leading to wear of local surface. Conversely, when the centrifugal force is superior to the Coriolis force, the particle trajectory S₄ is deflected towards the suction surface, as results in the erosive wear of the suction surface. The Coriolis force is prominent when small particles are delivered, while the centrifugal force prevails in channels with large curvature and for the transport of large particles. For the pressure and the suction sides, which one undergoes more severe wear depends on the comparison between the centrifugal force and the Coriolis force.

In engineering applications, erosive wear has been encountered in different pipe segments (elbow, tee, sudden expansion, and contraction), pumps, and valves. From the perspective of engineering design and operation, it is imperative to establish a systematic interpretation of erosive wear. For this branch of fluid-handling equipment, the flow passages exhibit diverse features such as the gap, curved channels between rotating blades, and the narrow flow space between rotating and stationary components. These constitute complex boundary conditions for the liquid flow and the motion of the particles involved.

Although some basic understanding of particle motion and erosive wear has been accomplished, practical design has benefited limitedly from available knowledge. Empirical formulas are overwhelmingly applied in the practice of design, and launching a new design to adapt to a new set of performance specifications necessitates a long tentative route. In this situation, auxiliary tools such as numerical simulation have been effectively used, which to some extent relieves the burden on the task of design.

3. Engineering Correspondence

3.1. Pipe Joints

Pipeline transport serves as a cost-effective but reliable method in various industries such as petroleum, chemical, food, and pharmaceuticals. For the lifting of petroleum and oil from a well, the participation of sand is inevitable, which may cause surface damage, especially at locations where the flow direction changes sharply, such as elbows and tees. The eroded pipe joints are illustrated in Figure 3. The erosive wear of pipes may incur operation accidents [20]. Various measures have been employed to mitigate the erosive wear, such as applying erosion-resistant materials and protective coatings and optimizing the structure of the pipeline.

Diverse wear patterns have been evidenced in engineering applications. For instance, for the transport of crude oil, it is necessary to consider the corrosive nature of the fluid. Theoretically, this can be accomplished through introducing the factors of corrosion and the erosion–corrosion interaction into the wear model. In shale gas production, high pressure liquid–solid mixtures are pumped into the well, and the effect of the internal stress of the equipment on erosion is significant [21,22]. In a black water treatment system, low pressure created by the regulating valve leads to entrainment of a large amount of gas into the fluid [3]. Therefore, the multiphase flow field should be considered. In contrast to the transport of particles ranging from about 50 to 250 μm in size, the deep-sea mining is featured by the particles with mean size of 30 mm or even larger. Snapshots of the transport of large particles in a vertical pipe are shown in Figure 4.

The complexity of the erosion mechanism is associated with the diversity of flow regimes [23]. Apart from the liquid flow, the gas–liquid flow can carry solid particles as well. For the latter, at low gas volume fractions, liquid is continuous in the central region of the pipe where scattered solid particles are involved. When the gas volume fraction increases further, the slug flow prevails. In this case, solid particles are transported with intermittent patterns. The churn flow is featured by streamwise oscillation of gas cavities, and such a flow pattern is inherently unstable. In annular flow, gas occupies the central region of the flow passage. Meanwhile, a thin liquid film is formed along the pipe wall, and most solid particles are distributed near the wall as well. The gas–solid–liquid flow maps obtained thus far are applicable only for certain pipe dimensions [24]. The lift of solid particles with the assistance of injected gas remains a new subject. The conclusions have been obtained overwhelmingly in laboratories. Main influential parameters such as the gas flow rate and initial gas pressure have been considered, but the obtained results deviate explicitly from the real situation.

The geometry of the elbow influences the extent of the erosive wear. Bilal et al. [25] have studied the liquid–solid and gas–liquid–solid flows in the 45° and 90° elbows. The paint removal test shows that the time for a standard elbow (90° elbow) to attain the same extent of erosion is 1.5 to 3 times shorter than that for a 45° elbow. This indicates that the standard elbow is more likely to be eroded. A small radius of curvature is responsible for twisted flow passage, which results in intensified flow disturbance and turbulence. This in turn leads to more stochastic particle impingement on the surface, and the erosive wear is more severe. The study on the gas–solid flow has revealed that the maximum erosive wear of the plugged tee is nearly three times that of the elbow under the same operating conditions [26].

The erosion characteristics vary considerably with flow pattern. The injection of gas into a liquid–solid flow can enhance the erosive wear of the elbow, which demonstrates the contribution of the bubbly flow. The presence of bubbles induces the generation and expansion of the secondary flow, causing an increase in both liquid and particle velocities and an alteration of particle trajectories [17]. As a result, the erosive wear associated with the gas–liquid–solid flow is more severe than that arising in the solid–liquid flow, as shown in Figure 5. Cui et al. [27] have investigated the effect of the bubbly flow encountered in the black water treatment system on the erosion of the elbow. It is found that the erosion rate increases sharply when the gas volume fraction exceeds 10%.

In the annular flow, particles are entrained in the gas core. A properly selected gas velocity enables the formation of a film over the inner wall of the pipe, thereby mitigating the erosive wear. That explains why the annular flow is responsible for less severe wear relative to other flow patterns [28]. The thickness of the film can be estimated through empirical formulas [29]. Peng et al. [4] have performed a comprehensive analysis of the effect of the film on erosive wear. As shown in Figure 6, when a solid particle enters the elbow and impacts the surface, the film has a cushioning effect and can absorb the impact energy of the particle, preventing erosive wear. After the first collision, the particle rebounds and impacts the surface at the opposite side. Furthermore, a large proportion of energy is consumed at the first collision so that the second collision of the particle will not induce severe erosion wear [30,31].

Agrawal et al. [20] conducted a numerical study on the effect of erosion-induced cavities on the flow field using the technique of moving mesh. According to their results, the flow and particles recirculate within these cavities, leading to an increased erosion rate. Nevertheless, in other cases such as sudden contraction, erosion can eliminate protrusions on the target surface; therefore, the surface roughness is reduced. Thus far, the patterns of local flow adjacent to the erosion craters have not been disclosed. This will arouse uncertainties in predicting the erosion rate.

Previous research on the erosive wear in various elbows is summarized in

Table 1. Research on the erosive wear of elbows.

References	Geometry	Method	Operating Condition	ds (μm)	fs (%)	r/D
[25]	45° elbow 90° elbow	Experiment Euler–Lagrange	VL = 5.49~7.26 m/s VG = 5.37~6.21 m/s Liquid–solid flow Gas–liquid–solid flow	300	0.35~0.98	1.5~5
[3]	90° elbow	VOF–DPM	Bubbly flow	30~120	3.5~12	6
[27]	45° elbow 90° elbow	VOF–DPM	Bubby flow	75	3.5	3
[17]	90° elbow	Experiment Euler–Lagrange	VL = 5.5~6.31 m/s VG = 2.04~5.46 m/s Liquid–solid flow Gas–liquid–solid flow	300	0.31~0.37	1.5
[28]	90° elbow	Experiment VOF–DPM	VL = 0.04~0.1 m/s VG = 23~31 m/s Gas–solid flow Gas–liquid–solid flow	25~300	≈0.4	1.5
[4]	90° elbow	Experiment VOF–DPM	VL = 0.33 m/s VG = 32.66 m/s Gas–liquid–solid flow	280~315	≈0.8	1.5
[30]	90° elbow	DPM	VL = 5~10 m/s Liquid–solid flow	300~1500	20	1.5~3.5
[35]	90° elbow	Euler–Lagrange	VL = 5~10 m/s Liquid–solid flow	50~200	0.1~3.2	1.5~8
[20]	Choke Blind tee 90° elbow	Euler–Lagrange	Liquid–solid flow Gas–solid flow	25~250	-	-

, where the information about elbow geometry, particle parameters, research methods, and operating conditions is listed. The particle size adopted in these studies is generally less than 300 μm, and the solid volume concentration is lower than 5%. Dynamic behaviors of large particles have been investigated in Refs. [32,33]. With high momentum and large contact area, large particles induce more severe erosion compared to small particles. Meanwhile, for the interaction between large particles and the target surface, a buffer layer is formed over the target surface, and it protects the surface from the erosive wear [34]. The patterns of erosive wear caused by particles of different sizes are different. In an elbow, as shown in Figure 7, the area with high wear rate shifts towards the upstream flow since the ability of large particles to follow the carrier fluid is weak. Furthermore, the nonuniformity of the distribution of the wear rate over the curved surface is enhanced with increasing particle size, as indicated in Figure 7.

3.2. Pumps

Pumps are used to impart energy to the transported medium, which thereby gains enough energy to overcome the frictional and local resistance. The pump conveying the mixture of liquid and solid particles is commonly referred to as the slurry pump. An impeller-type slurry pump is shown in Figure 8a. The concentration of the slurry transported can be as high as 30%. Solid particles in the slurries have high erosivity, and the resistance of the pump to erosion can be achieved through optimizing the hydraulic components and properly selecting materials for the components.

Moreover, in recent years, the displacement pump has been applied for transporting slurries. In Figure 8b, a diaphragm pump is schematically shown. Such a pump can handle slurries containing sand particles of 4.0 mm in median diameter. To avoid the erosive wear, the geometry of the gaps between the upper and lower ball valves and the pump casing should be specifically designed. Generally, apart from the erosive wear of the ball valves and the diaphragm, the risk of blockage of the inlet and outlet segments is conceivable.

In the 1970s, Ocean Management Company (OMC) successfully mined about 1000 tons of manganese nodules from a depth of 5000 m in the Pacific Ocean using a submersible impeller pump [36]. Afterwards, studies on the deep-sea mining pump and even the whole system have been conducted by European Union [37], Japan [38], South Korea [39,40], Australia [41], and China [18,42].

Both the impeller pump and the diaphragm pump have been applied in deep-sea mining. The impeller pump has the ability to convey very large particles (>30.0 mm). However, they are susceptible to significant wear and blockage of the flow passage. For the diaphragm pump, it is only capable of transporting the slurry containing fine particles. In this situation, before entering the pump, large particles must be processed into small ones after being collected from the seabed, which incurs additional cost. Currently, the deep-sea mining pump is still being developed in several countries. However, substantial advancement has rarely been reported. One of the main reasons lies in that dynamic characteristics of large particles have not been well interpreted.

The equipment handling the solid–liquid two-phase fluid is exposed to the risk of particle settlement. This is induced when the energy received by the particles cannot sustain their settling velocity. The pump is responsible for imparting energy to particles. Generally, the pump head and efficiency are reduced when water is replaced with slurry [43,44,45,46], as illustrated in Figure 9. Moreover, due to wear of the hydraulic components, the pump may operate under off-design conditions. In this case, excessive hydraulic loss will be induced, and the wear will be aggravated.

The erosive wear for the pump has been studied using different methods. Experimental investigations enable an identification of the location with the most severe erosion. Meanwhile, the pits and indentations left after removal of large pieces of material can be quantified. For the impeller pump, erosion is typically more pronounced at the blade pressure surface, especially near the blade inlet and outlet, as shown in Figure 10a. The volute tongue is worn earlier relative to other parts of the volute (see Figure 10b). In comparison, wear of the impeller is more severe than that of the volute. The erosion morphology of the impeller and the volute is illustrated in Figure 11. Pits and cracks at the leading edge of the impeller are observed in Figure 11a. When particles enter the blade channel, they impact the blade inlet with high velocity, causing the wall to either experience deformation to form small pits or undergo brittle fracture. Scratches on the impeller are produced by the cutting action of high-velocity particles with small impact angles, as shown in Figure 11b. Indentations on the surface near the volute tongue are shown in Figure 11c. These indentations are produced when particles impact local wall with large impact angles. As shown in Figure 11d, the removal of material in this region away from the tongue is attributed to the effects of cutting and ploughing, which are induced by particles with small impact angles.

The erosion of the impeller of a centrifugal pump is generally greater than that of the volute, which is related to high particle velocity in the flow passage of the impeller. Moreover, this is related to the impact angle. When the impact angle varies between 30° and 60°, it is highly possible that the volute undergoes a higher erosion rate compared to that of the impeller. Chen et al. [49] concluded that the blade with the initial position close to the volute tongue experiences greater wear. The area of erosion increases with the flow rate [49]. The faster the impeller rotates, the greater the erosive wear. In this situation, the energy imparted to the medium from the rotating impeller plays an important role [50].

An increase in the solid volume fraction, f_s, may intensify the erosive wear of the pump. Another consequence is the increase in the frequency of the impacts of particles on pump components [51]. However, the phenomenon known as the wear plateau has been observed. Initially, the wear rate of a material increases with f_s, but then it decreases to its minimum at a critical f_s before ascending again [44]. This behavior is related to the formation of a protective layer on the surface of the material, which alleviates the impact of particles on the surface. An increase in particle size results in increased mass loss, and the area of erosion is enlarged as well [50]. For the impeller blade, the erosion of its pressure side becomes prominent with increasing particle size [49,51]. This is attributed to the fact that particles of small size tend to concentrate at the pressure side and suffer from a Coriolis force superior to the centrifugal force [52]. Liu et al. [53] investigated the erosive wear of a deep-sea mining pump. The maximum particle diameter is 40.0 mm. The results show that the wear rate of the impeller increases with the flow rate. Meanwhile, the wear rate of the guide vane varies slightly. As reported in Ref. [54], an increase in the particle diameter from 5.0 to 20.0 mm is responsible for continuously increasing wear rate of the impeller, while the extent of wear of the guide vane remains declining. For the wear rate of the volute, it varies considerably with particle size [50,55].

At the inlet of the centrifugal pump, low pressure may stimulate the occurrence of cavitation. High-intensity pressure waves are produced with the collapse of cavitation bubbles, imposing transient loads onto adjacent wall [56]. In addition, the presence of cavitation bubbles induces secondary flows and alters the trajectories of solid particles. This leads to recirculation of particles in some regions. Alternatively, the number of particles in these regions increases. Consequently, the rate of wear increases. This is consistent with the conclusion obtained in Ref. [57].

A paint removal test has been performed for a centrifugal pump with sand particles of 6.0 mm in median diameter. The resultant patterns of the erosive wear of the impeller are shown in Figure 12. It is observed in Figure 12a that the most severe wear occurs at the leading edge and the impeller hub. Upon entering the impeller, particles undergo a transition of motion from axial to radial direction under the action of the centrifugal force. Subsequently, the particles impact the blade leading edge. With high inertia, the particles are unable to rapidly complete the transition of motion. In this situation, they impact the impeller hub at large impact angles, causing significant erosive wear of the impeller hub. As shown in Figure 12b, the hub and the pressure surface of the impeller have been severely eroded. For the hub, a large piece of paint has been removed, and for the blade pressure surface, apart from the removal of paint in a sheet pattern, erosion pits are identifiable.

In contrast to large-angle impacts experienced by the inlet part of the hub, the outlet part undergoes primarily small-angle impacts, as is beneficial for alleviating the erosive wear. Wear of the blade pressure surface is concentrated in the central zone. This is relevant to the curvature of the flow passage, which weakens the centrifugal force on the particles [58]. In this case, the Coriolis force takes precedence, forcing the particles to migrate towards the central zone of the pressure surface. Consequently, such a local area is severely eroded.

Numerically obtained wear patterns of the impeller of a deep-sea mining pump are illustrated in Figure 13 and Figure 14. When particles of 15.0 mm are transported, the impeller shroud exhibits severe erosive wear, which is concentrated near the blade pressure side. In contrast, erosion of the hub occurs primarily near the suction surface. Furthermore, the leading edge, the middle part of the suction surface, and the outlet of the pressure surface are susceptible to erosive wear. When particle size increases from 15.0 to 30.0 mm, the area of wear on the shroud is reduced. Meanwhile, the erosive wear of the blade suction surface is intensified, which is explicitly related to the increased centrifugal force.

A summary of previous numerical studies on wear of the impeller pump is shown in

Table 2. Research on the erosive wear of the impeller pump.

References	qV (m3/h)	H (m)	Method	Erosion Model	ρp (kg/m3)	dp (μm)	Cv (%)
[55]	-	-	Euler–Euler	Finnie	-	50~1250	5~25
[49]	135	60	DPM	Oka	-	50~450	0.2~0.6
[57]	61,200	240	Euler–Lagrange	Finnie	1550~3550	1530	-
[51]	485	14	DPM	Generic	2740	40~427.5	0.36
[59]	-	-	DPM	Finnie	2300	20~70	≈1.7
[60]	420	100	Euler–Lagrange	Generic	-	6000	7.5
[50]	6800	-	DPM	Finnie	-	37~53	-
[44]	-	-	Euler–Euler	Finnie	-	-	5~20
[61]	9000	34	Euler–Lagrange	Finnie	1950	200	5~10
[62]	26	11	-	-	2900	600	5~25
[63]	485	14	DPM	Generic	-	20	0.4
[64]	165	32	DPM	E/CRC	2650	25	0.43~4.3
[47]	1260	43.5	DPM	E/CRC	2650	4~250	0.5~6.1
[65]	180	26	DEM	Archard	2500	100	1
[66]	11,100	50	Euler–Lagrange	Tabakoff	-	25	0.1~1
[67]	68.4	8.3	DEM	Oka	2600	1	2
[68]	100	40	Euler–Euler	-	2650	1	10
[69]	5000	24	DDPM	Oka	-	300	4~12
[70]	690	≈40	Euler–Euler	Finnie	-	500	5
[52]	15	4	CFD–DPM	E/CRC	2500	500	0.1~2.5
[53]	420	80	Euler–Lagrange	Finnie	-	0~40,000	5.14
[54]	420	80	Euler–Lagrange	Finnie	-	5000~40,000	5.14

. Based on the table, a comprehensive analysis is implemented through considering the operating parameters, multiphase flow models, erosive wear models, and particle properties. Heretofore, few studies have emphasized the conditions with particle sizes larger than 10.0 mm. Nevertheless, in deep-sea mining, the lift pump is required to deliver particles larger than 30.0 mm, and the solid volume concentration is as high as 12%. After passing though the lift pump, the particles continue ascending in a vertical pipe, and both blockage and erosive wear are possible.

3.3. Valves

In a hydraulic transport system, valves assume an important function of regulating the pressure and the flow rate, which is pivotal for ensuring reliable and safe operation of the whole system [71,72,73,74,75]. The direct interaction between solid particles and the components of the valve results in wear of the components, which can give rise to safety issues such as sealing failure and decline in the operating efficiency. Especially in chemical and nuclear engineering, the consequence can be disastrous. Generally, valves are categorized into various types, including but not limited to the governing valve, throttle valve, butterfly valve, and ball valve. The patterns of wear vary with the valve structure, medium, and operating parameters.

Xu et al. [76] and Lin et al. [77] investigated the wear of a ball valve using experimental and numerical methods. Effects of particle size and the opening are considered. They evidenced that the flow coefficient of the valve is reduced when a single liquid is replaced with the mixture of liquid and solid particles. Furthermore, when the opening of the valve is higher than 65%, the flow coefficient is even smaller. The wear rate is high at low openings, which are responsible for high particle velocity, as shown in Figure 15. Consequently, severe damage of the internal wall of the valve is demonstrated. When the mass flow rate of particles remains constant, the number of particles increases with decreasing particle size. The interaction between particles and the valve components is strengthened, and the area of erosion is enlarged.

4. Main Parameters Influencing Erosive Wear

4.1. Material Properties

The mechanical properties of materials, as represented by the stress–strain curve plotted in Figure 16, are significant for understanding the wear mechanism. As marked with B, the ductile material like the mild steel is featured by a yield point, and it sequentially experiences the elastic process, yielding process, strain hardening process, necking process, and fatigue process with increasing stress and strain. When stress is applied onto a material, the deformation process is closely related to the magnitude of the stress and mechanical properties of the material. When the stress magnitude is lower than the elastic limit of the material, σ_s, the deformation is elastic, which signifies that the material can recover its original shape when the load is withdrawn. When the stress exceeds the yield stress of the material, σ_y, the deformation is plastic, demonstrating that there is a permanent deformation even after the stress is removed. If the stress is further increased, it can eventually reach the fracture stress of the material, σ_f, causing the material to fracture. Compared to the ductile material, the brittle material, marked with A, may fracture at low strain with little or no plastic deformation. The area under the stress–strain curve, which serves as an indicator of toughness, is typically smaller for the brittle material compared to that of the ductile material. Brittle materials have a limited ability to resist the plastic deformation and absorbing energy before fracturing. The stress–strain curves plotted in Figure 16 are obtained from tensile tests while the wear of the tested object is produced through the method of indentation.

4.2. Impingement Parameters

4.2.1. Particle Impact Angle

In his wear model, Finnie [1] proposed two concepts known as cutting wear and deformation wear to describe how the characteristics of wear vary with the impact angle. However, it is easy to confuse the two concepts with the abrasive (cutting) wear and the deformation wear discussed in Section 5.1. The cutting wear and the deformation wear in the erosion model represent the wear caused by the tangential and normal velocity components of particles at any impact angle, respectively. The concepts of the cutting wear and deformation wear apply to both ductile and brittle materials, as shown in Figure 17. For ductile materials, the deformation wear causes the deformation of the target material, while the cutting wear is responsible for creating the groove and chips. Regarding brittle materials, both the deformation wear and the cutting wear can lead to the generation of cracks and material failure.

To account for the variation of the erosive wear with the particle impact angle, the following relationship has been used:

$$W_T=W_D+W_C$$

(13)

where W_T, W_D, and W_C represent the rate of the total wear, deformation wear, and cutting wear, respectively.

Clark and Hartwich [79] conducted an experimental investigation on the relationship between the total wear, deformation wear, and cutting wear and the impact angle for different materials, and the result is diagrammed in Figure 18. With increasing particle impact angle, the rate of deformation wear increases continuously and reaches a peak for both ductile and brittle materials. The total wear rate increases first and then decreases for ductile materials. For the deformation wear, the trends of variation are similar for the two materials [79], as illustrated in Figure 18a,b. This is attributed to the fact that the cutting wear can be neglected in brittle materials while it is dominant in ductile materials. With the combination of the two mechanisms of the cutting wear and deformation wear, the erosion-effective impact angle is about 30–45°, which has been validated in several studies [80,81,82,83,84].

4.2.2. Particle Impact Velocity

The erosion rate has been argued to be positively related with the particle impact velocity [84,85,86]. The power–law relationship between the erosion rate and the particle impact velocity is expressed as:

$$ER\propto v_s^n$$

(14)

where ER denotes the erosion rate, and n is a velocity exponent.

Finnie [1] proposed n = 2 based on a theoretical derivation. Oka and Yoshida (2005) concluded that n depends on both material hardness and particle properties. For particles of SiC, n = 3.0(H_v)^0.085, where H_v is the Vickers hardness of the target material. According to Ref. [87], for the cutting and deformation wear, n = 2.41 and 2, respectively. Alam concluded that n varies from 0.34 to 4.8, depending on the properties of the carrier liquid [88].

An increase in particle velocity can lead to work hardening and strain localization, which then influence the erosion rate. Under high strains caused by continuous impacts, the work-hardened layers are created. The soft surface layer is heated by the thermal energy transformed from the kinetic energy of incident particles, which prevents it from work hardening. When the distance from the heated surface increases, the hardness remains increasing until attaining its maximum. Then, the hardness decreases progressively. Particles can travel through the soft ductile layer but encounters a resistance from the work-hardened zone [89].

Adler and Doğan [90] observed that strain localization occurs at particle impact velocities higher than 100 m/s. The strain localization leads to several consequences such as an increase in the height at the edge of the crater (see Figure 19), decrease in dimensions of the plastic zone, and a decline in dynamic hardness. Work hardening can enhance the resistance of materials to erosion. Nevertheless, in the presence of strain localization, erosion can be promoted.

4.3. Particle Properties

4.3.1. Particle Size

In general, as the density of solid particles approaches that of the carrier liquid, particles of small size are prone to follow the motion of the liquid, which is responsible for a decrease in the impact velocity. This, in turn, mitigates the erosive wear when particles collide with the wall. However, the effect of particle size on the erosive wear depends on multiple factors, each of which has been substantiated by experimental evidence. Thus, the correlation between particle size and the wear rate is not explicit and cannot be described by a simple monotonical relationship.

Tilly [91] proposed a two-stage mechanism of ductile erosion, which involves two distinct modes of material removal. The first stage, namely, primary erosion, is attributed to the impact of particles that are not fragmented upon collision with the surface. The second stage, namely, secondary erosion, is produced with the impact of particle fragments that are generated as a result of the primary erosion. The erosion equation is expressed as:

$$ER={\epsilon }_1{\left(\frac{v_t}{v_{ref}}\right)}^2{\left[1-{\left(\frac{d_0}{d_{ref}}\right)}^{\frac{3}{2}}\frac{v_0}{v_{\mathrm{t}}}\right]}^2+{\epsilon }_2{\left(\frac{v_t}{v_{ref}}\right)}^2{F}_{d,v}$$

(15)

where ER denotes the erosion rate; ε₁ is the maximum primary erosion rate; v_t is the test velocity of particle; v_r is the reference velocity of particle; d₀ is the threshold diameter of particle; d_ref is the reference diameter of particle; v₀ is the threshold velocity of particle; ε₂ is the maximum secondary erosion rate; and F_d,v is the fragmentation factor. The equation implies that there exists a threshold particle size beyond which the wear rate does not increase any more [91]. A comparison between experimental data and the result predicted through Equation (15) is illustrated in Figure 20.

Through discharging particles through a nozzle to impinge the target surface, Nguyen et al. [92] revealed that when the particle diameter exceeds a certain value, a further increase in particle size leads to a decrease in mass loss. Meanwhile, the erosion scar is narrowed but deepened. The stagnation pressure generated by the impingement of the mixture on the target wall tends to push small particles outwards due to their lower inertia, while large particles can maintain their original trajectories. Therefore, an increase in particle size is associated with an increase in the intensity of the erosive wear. When particle size increases further, those large particles rebounding from the surface tend to gather in the central region, where they will interact with incoming particles, and thereby the frequency of the impacts of solid particles on the surface is reduced. In contrast, small particles rebounding from the surface tend to move outwards, resulting in less frequent particle–particle collisions. This is why there is a threshold value of particle size, beyond which the erosive wear attenuates.

Abouel-Kasem [93] studied the erosive wear of AISI 5117 steel using the whirling-arm tester. The particle sizes ranged between 112.7 μm and 516.4 μm, and the impact angles of 30° and 90° were adopted. As shown in Figure 21, the mass loss increases with particle size, which is related to high stresses induced by large particles. Under identical operating conditions, such as the solid volume fraction and impact velocity, a single large particle has a higher kinetic energy than that of a single small particle. Therefore, large particles are more erosion effective. From another perspective, the target material is more resistant to small particles [91].

At impact angles of 30° and 90°, there is an abrupt variation of the mass loss, above which although the mass loss still increases with particle size, the erosion rate declines. This is ascribed to the mitigation of plastic deformation due to work hardening. With the whirling-arm tester, the inter-particle collision is considerably alleviated, so the phenomenon differs from what Nguyen et al. [92] observed. Additionally, maintaining consistent conditions of particle impact, such as the impact velocity and impact angle, is very challenging when particle size varies.

The erosion rate has been reported to have a power-law dependence on particle size, which can be expressed as [78,94,95]:

$$ER\propto {\left(d_s\right)}^n$$

(16)

The exponent n varies between 0.2 and 4.0, depending on several parameters such as the impact velocity, impact angle, and properties of the target material.

4.3.2. Particle Shape

To quantify the effect of particle shape on erosion, Cox [96] defined the particle circularity through the shape factor, S_F:

$$S_F =\frac{4 \pi A_s}{{P_s}^2}$$

(17)

where A_s is the projected area of the solid particle, and P_s is the overall perimeter of the projection of a solid particle.

The sphericity of a particle is defined in Ref. [97] as the ratio of the surface area of a sphere with the same volume as that of the given particle to the surface area of the given particle:

$$S_F=\frac{S_s}{S_p}$$

(18)

where S_s is the surface area of a sphere, and S_p is the surface area of a given particle.

Albertson [98] has defined particle shape through the following equation:

$$S_F=\frac{c_s}{\sqrt{a_s{b}_s}}$$

(19)

where a_s, b_s, and c_s are the axes of the particles, which are perpendicular to each other; a_s is the longest axis, and c_s is the shortest axis. In general, c_s is in the direction parallel with the motion of the particle, while a_s and b_s are in the plane accommodating the maximum projected area.

It has been observed that the extent of the erosive wear increases with the sharpness of particles. Angular particles are responsible for much sharper craters than rounded particles [99]. Levy and Chik [100] have conducted an experimental investigation on the erosive wear of AISI 1020 steel, which is induced by angular steel grits and spherical steel shots, both are 100 μm in size, with an impact velocity of 80 m/s and an impact angle of 30°. They have demonstrated that the erosion rate caused by the angular steel grits is four times greater than that caused by the spherical steel shots. Al-Bukhaiti et al. [101] performed an experiment to elucidate the effect of particle size and shape on the erosion of a target material. At an impact angle of 30°, the erosion caused by angular particles is characterized by microcutting and microploughing. In contrast, particles with rounded shape result in ploughing with plain and smooth wear tracks. For the two groups of particles released at an impact angle of 90°, the dominant mechanisms are indentation and material extrusion. However, since particle size and shape are interdependent, the obtained results are not sufficiently convincing.

4.3.3. Particle Hardness

In general, the total erosion rate increases with the hardness of the particle. After a certain value of the total erosion rate is attained, a further increase in particle hardness has little effect on the erosive wear, as evidenced in Ref. [100]. They examined the erosion rate of AISI 1020 steel with a hardness of 150 kgf/mm² impacted by five groups of particles of different hardness. It is observed in Figure 22 that the erosion rate remains nearly constant when the harness of the particle reaches about 700 kgf/mm². Initially, the increase in particle hardness results in intensified erosion, which is characterized by reinforced plastic deformation and emerging lateral fractures. In the later stage, the erosion rate reaches an upper limit and then remains nearly invariant, demonstrating a significant effect of work hardening.

Compared to the hardness of the particle, the ratio of hardness between the particle and target material serves as a more comprehensive factor influencing the erosive wear. Shipway and Hutchings [102] have examined the erosion of brittle materials produced with different particles. For the cases in which the particle is softer than the target material, very low erosion rates are obtained, as shown in Figure 23. When a soft particle impacts a hard target, the particle may undergo plastic deformation, consuming kinetic energy. Consequently, the load exerted on the target surface is not high enough to trigger damage to the target surface.

4.4. Target Wall Properties

The resistance of a mechanical component to erosive wear depends greatly on its microstructure. Such a performance can be improved through treatments such as carburizing or annealing hardening. The matrix structures of steel mainly include ferrite, austenite, martensite, pearlite, and bainite. The proportion of each composition is important for determining the mechanical property of the steel. For instance, AISI 1080 steel exhibits higher hardness compared to those of AISI 1018, API X42, and API X70 steels, owing to high content of pearlite in the microstructure [88]. With heat treatment, the austenite phase can be transformed into relatively hard martensite. Such a transformation enables high resistance to erosion [90]. Abd-Elrhman et al. [80] carburized AISI 5117 steel specimens through placing them in a carburizing box at a temperature of 950 °C. The carburized specimens were subsequently quenched in salt water and tempered at 200 °C for one hour. As a result, the carburized specimens exhibited superior hardness and erosion resistance compared to the untreated specimens. However, according to Ref. [87], increasing hardness makes the material brittle prone to fracture under the impact of particles.

4.5. Properties of the Mixture

4.5.1. Solid Volume Fraction

High solid volume fractions are responsible for great mass loss of the target material, which is attributed to high frequency of particle–wall collisions [34,103]. Nevertheless, as the solid volume fraction increases to some extent, the interaction between particles is intensified, but the kinetic energy of individual particles is reduced, and their ability of damaging the target is degraded [104]. The erosion rate may initially increase and subsequently decrease [22,35,105]. Therefore, it is imperative to elucidate the particle–particle interactions and supplement current knowledge about the solid–liquid two-phase flows characterized by high solid volume fractions [106]. It has been evidenced that neglecting inter-particle collisions results in an overestimation of the erosive wear [107].

4.5.2. Corrosivity of the Carrier Liquid

The synergistic effect of erosion and corrosion has been recognized in ocean and chemical engineering. When the mechanical component is exposed to the oil sand slurry with high corrosivity, the damage to the component is intensified due to the combined effect of erosion resulting from particle impact and corrosion arising from chemical or electrochemical reactions [108]. Corrosion and erosion may promote each other [109]. When particles impact the corroded surface, the protective film that resists further corrosion of the surface is disintegrated due to material removal. Consequently, corrosion is accelerated [110]. The ratio of erosion-enhanced corrosion to pure corrosion is noticeable, especially at high solid volume fractions and high flow velocity [111]. Yang and Cheng [112] concluded that with increasing solid volume fraction and flow velocity, the effect of erosion–corrosion on the impacted steel specimens is enhanced, and the erosion effect becomes dominant. According to Ref. [113], the introduction of corrosion inhibitors helps to suppress corrosion of the pipe wall and reduce the rate of erosion–corrosion. However, the damage is not always intensified as corrosion and erosion coexist. When cavitation erosion occurs in seawater, the damage to the target specimen may be mitigated since corrosion will yield a protective film over the sample surface, which can alleviate the impact induced by the collapse of cavitation bubbles. The protective film is removed at high flow velocity, resulting in the exposure of a fresh surface to cavitation, thereby contributing to rapid aggravation of the erosive wear [114].

A diagram illustrating the development of erosion–corrosion is presented in Figure 24. Corrosion is initiated at small anodic sites distributed randomly on the surface due to inhomogeneous material defects. When a sand particle impacts an anodic site, a small crater may be created. When the soft edge of the crater is impacted repeatedly by particles, a new crater adjacent to the original one is formed. The microturbulence generated at the periphery of these craters will lead to enlargement of the craters. As the impact proceeds, those craters can be merged into larger ones [115].

4.5.3. Liquid Temperature

When their temperature increases, metals may react with oxygen in surrounding air, resulting in the formation of an oxide layer. Such a chemical process is commonly referred to as oxidation, which is considered as a form of corrosion that prevents further corrosion but affects physical properties of the metal surface [116]. Typically, the oxide layer formed during the oxidation process causes an increase in the mass loss of the target specimen impacted by particles. Galiullin et al. [117] studied the erosion–oxidation process for several commercial alloys employed as containment materials in concentrated solar power plants. They have evidenced that each material experienced significant mass loss with increasing temperature, which is ascribed to the removal of the oxide layer due to the impact of particles. For different materials, the sensitivity to temperature variation differs. Gietzen et al. [118] disclosed that the cumulative mass loss of SS316L increases by 433 times when it is exposed to a liquid of 800 °C compared to that obtained at normal temperature, whereas the cumulative mass loss of IN740H increases by only twice. Such a considerable difference in the erosion rate is related to low hardness of the oxide layer and weak adherence of the oxide layer to the original surface of the SS316L specimen.

4.6. Flow Pattern

For the solid–liquid two-phase flow, the flow pattern varies with solid boundary and operating conditions. The effect of flow patterns such as the secondary flow and rotational flow on the erosive wear is significant. However, identifying flow patterns is challenging when the geometry of the solid boundary is complex. Especially when large particles are involved, a stable flow pattern is difficult to obtain.

Averaged flow patterns have been associated with erosion, but the effect of turbulent fluctuations on the motion of particles and erosion has seldom been investigated [25]. Essentially, driven by turbulent fluctuations, small particles tend to impact the target surface in a random manner since the consistency between the motion of small particles and the carrier liquid is high. The ability of solid particles of following the liquid flow is often quantified by a Stokes number, S_t, which is a dimensionless parameter that reflects the relative magnitude of the inertia force and the drag force exerted by the liquid on the particle. S_t is defined as the ratio of the characteristic time of the motion of a particle to that of the liquid flow:

$$S_t=\frac{{\rho }_s{d}_s^2{u}_l}{18{\mu }_l{l}_s}$$

(20)

where ρ_s is the density of the particle; d_s is the diameter of the particle; μ_l is the dynamic viscosity of the liquid; u_l is the velocity of the liquid; and l_s the characteristic length of the particle. At S_t << 1, the particle well follows the liquid flow, and the particle detaches from the liquid flow completely at S_t >> 1 [92].

Stokes number is directly related to the motion of particles, but it has no significant effect on the maximum erosion rate [30]. Stokes number can be adopted to explain where the erosion pattern changes due to variation of particle trajectories. In the solid–liquid two-phase flow in an elbow, small particles with a diameter of d_s = 30 μm have a low Stokes number, which implies that they tend to follow the liquid flow and impact the elbow at large impact angles [28]. Regarding the centrifugal pump, the particles in the flow passage of the impeller with a large Stokes number cause great erosion, which is related to the independence of particle motion from the carrier liquid, so these particles can frequently impact the blades [47]. With a stochastic particle tracking model, Agrawal et al. [20] investigated the effect of turbulent dispersion of particles on the erosion of pipe segments with sudden contraction, sharp 90° bend and elbow. The model incorporates several user-defined parameters that influence the turbulence dispersion forces, such as the time scale factor and the number of injection points. It is demonstrated that Reynolds number insignificantly influences the relationship between the erosion rate and the number injection points. However, particles with a low Stokes number (S_t < 1) are highly sensitive to these particle-tracking parameters. Furthermore, these particles are more likely to be trapped in vortices, leading to an increase in the severity of the erosive wear.

5. Erosive Wear Models

5.1. Description of Wear Mechanisms

The erosive wear has been evaluated through the mass loss, volume loss, or the reduction in thickness per unit time. According to Ref. [119], typical erosive wear models include the abrasive or cutting wear model, deformation wear model, and fatigue wear model, as illustrated in Figure 25. The abrasive wear model describes the wear mechanism when particles come into contact with a surface at an oblique or small impact angle. In this situation, the removal of material from the surface is caused by cutting, and grooves dominate the erosion scars. For the effect of ploughing, material is not removed from the surface but shifts to the side of the groove, as indicated in Figure 25a. In contrast, with the action of cutting, the chips formed ahead of the erosive particle are directly removed from the surface [78]. The deformation wear is caused by the impacts of particles at large impact angles. When the surface stress induced by particles exceeds the yield stress, the surface will deform and exhibit localized extrusion, which is then eliminated by subsequent impacts of particles, as illustrated in Figure 25b [95].

When high-velocity particles repeatedly impact a certain local surface, stress concentration will be enhanced, and cracks are thereby formed, as indicated in Figure 25c. When the cumulative stress exceeds the allowable fracture stress, the material will lose its performance due to crack propagation [2]. Such a fatigue wear is generally observed in materials with high brittleness. In contrast, regarding ductile materials, they are able to absorb the kinetic energy of incident particles through plastic deformation, which is beneficial for relieving the effect of the fatigue wear. However, ductile materials can still experience fatigue wear, which is generated due to strain localization. In this case, cyclic loading and unloading have exceeded some endurance limit. In the previous literature, the models of the abrasive wear, deformation wear, and fatigue wear have been comprehensively employed to establish erosion models for both ductile and brittle materials. It is noteworthy that qualitative descriptions are prevalent relative to quantitative predictions.

5.2. Typical Patterns of Erosive Wear

Microforging has been commonly observed at small impact angles and low velocity of particles. In these cases, the impact of particles on the target surface results in plastic deformation and work hardening. Under the action of microforging, the target surface is squeezed by the particles, resulting in the formation of tiny pits, as sketched in Figure 26a. With these pits, the contact area between the surface and particles is reduced, and the hardness of the material increases. The two variations are beneficial for preventing further erosive wear [120]. At small impact angles, the tangential component of particle velocity is dominant, resulting in the formation of deep grooves in the contact zone and ridges on both sides of the particle trajectory, as shown in Figure 26b,c. Repeated impacts of subsequently incident particles may lead to flattening, fracture, and formation of the ridges. The microploughing pattern of the target material has been recognized as a primary erosion mechanism at small impact angles and low velocity of particles [21,91,121].

Microcutting arises when the impact angles of the particles are small and the impact velocity is high. In this situation, the particles have sufficiently high energy to cut off material chips from the target surface, as shown in Figure 26d. Moreover, at high velocities, particles may deviate from their original trajectories when they rebound from the surface. When the velocity of these deflected particles exceeds some threshold value, secondary microcutting will arise when these rebounding particles impact the surface [122]. At even higher impact velocities, some particles are embedded in the target surface, as illustrated in Figure 26e. It has been observed that the embedded particles may trigger cracks when they are impacted repeatedly by subsequently incident particles [123]. These embedded particles may fracture or be removed from the target surface, leaving behind pitting and ridges, as shown in Figure 26f.

Plastic deformation has been detected at both low and high impact velocities of particles [22]. Repeated impacts of particles on the target surface result in surface deformation and the formation of craters, as illustrated in Figure 27a,b. Exposed to subsequent impacts, the platelets will fracture or even detach from the surface, as shown in Figure 27c. At high particle velocities, material removal is boosted due to the fracture of ridges, as shown in Figure 27d–f.

Initially, the repeated impacts of particles lead to the formation of a work-hardened layer beneath the deformed layer. Hence, when impact proceeds, cracks are formed in the work-hardened layer. Exposed to continuous particle impacts, cracks propagate through the deformed layer, leading to the removal of large material fragments [124].

5.3. Computational Model

The computational fluid dynamics (CFD) method serves as a useful tool for treating multiphase flows. To solve the equations governing the liquid–solid two-phase turbulent flow, assumptions are often necessitated. In some cases, the numerical results deviate considerably from the real situation. Furthermore, when dealing with large particles, some numerical strategies only consider kinetic characteristics of a single particle. This is beneficial for revealing the most fundamental mechanisms, but the discrepancy between the result obtained thereby and engineering correspondence is explicit.

Two approaches have been commonly used in the numerical work coping with the solid–liquid flow field, namely the Euler–Euler approach and the Euler–Lagrange approach. For the former, the solid phase is treated as a continuum. In contrast, for the Euler–Lagrange approach, the solid phase is solved through tracking the trajectory of each particle. For the Euler–Euler approach, no erosion model is available. Instead, researchers generally predict the position of erosion through analyzing the distribution of the solid volume fraction. Here, only the Euler–Lagrange approach is discussed since it necessitates the participation of erosion models. Commonly used erosion models associated with this approach are elaborated.

5.3.1. Euler–Lagrange Approach

In the Euler–Lagrange approach, the liquid phase is treated as a continuous medium, and the liquid flow is dominated by N–S equations, while the dispersed phase is solved through tracking the trajectories of particles within the computational domain. Three models associated with the Euler–Lagrange approach are available in commercial CFD codes: the discrete phase model (DPM), dense discrete phase model (DDPM), and discrete element model (DEM). Each model has its own advantages and limitations, depending on specific requirements. The dispersed phase can exchange momentum, mass, and energy with the liquid phase. In the commercial code ANSYS Fluent, the DPM and DDPM are widely employed to treat two-phase flows with small particles. Compared with the DPM and DDPM, the CFD–DEM model can predict inter-particle interaction with high fidelity. The CFD–DEM model can be implemented through coupling ANSYS Fluent and EDEM.

DPM

For the DPM, the equations used to solve particle trajectories are as follows:

$$m_s\frac{d{\mathit{v}}_s}{dt}={\mathit{F}}_D+{\mathit{F}}_g\frac{\left({\rho }_s-{\rho }_l\right)}{{\rho }_s}+{\mathit{F}}_{other}$$

(21)

$${\mathit{I}}_s\frac{d{\mathit{\omega }}_s}{dt}=\frac{{\rho }_l}{2}{\left(\frac{d_s}{2}\right)}^5{C}_{\omega }\mathsf{\Omega }=\mathit{T}$$

(22)

where m_s is the particle mass; v_s is the particle velocity; F_other represents the other forces such as the pressure gradient force, thermophoretic force, rotating reference frame force, and Saffman lift force; I_s is the moment of inertia; ω_s is the particle angular velocity; ρ_l is the liquid density; ρ_s denotes the particle density; d_s denotes the particle diameter; C_LR denotes the rotational drag coefficient; Ω denotes the relative particle-fluid angular velocity; and T denotes the torque imposed on a particle. For the comparison with the DEM, all forces are characterized by the force per unit particle instead of the force per unit mass.

The equations of motion for the solid–liquid two-phase fluid are as follows (Peng and Cao, 2016 [35]):

$$\frac{\partial {\rho }_l}{\partial t}+\nabla \cdot \left({\rho }_l{\mathit{v}}_l\right)=S_{DPM}+S_{user}$$

(23)

$$\frac{\partial \left({\rho }_l{\mathit{v}}_l\right)}{\partial t}+\nabla \cdot \left({\rho }_l{\mathit{v}}_l{\mathit{v}}_l\right)=-\nabla P+\nabla \cdot \mathit{\tau }+{\rho }_l\mathit{g}+{\mathit{F}}_{DPM}+{\mathit{F}}_{\mathrm{user}}$$

(24)

where v_l is the liquid velocity; S_DPM and S_user are the source terms derived from the DPM or through user definition, respectively; $\mathit{\tau }$ is the viscosity stress; g is the acceleration of gravity; and F_DPM and F_user are the forces acting on the liquid from the DPM or through user definition.

Equations (21) and (22) represent a comprehensive consideration of all forces acting on a particle, with the exception of the inter-particle contact force. This limitation can be addressed through adopting a DEM collision model. In commercial CFD codes, six collision force models, including the spring, spring–dashpot, Hertzian, Hertzian–dashpot, friction, and rolling friction models, are available. The term F_user can be modified through adding a collision force in Equation (21). The DEM collision model introduces the concept of parcels. Particles are divided into parcels, and the position of each parcel is determined via the technique of tracking. This simplification significantly reduces the computational time. Nevertheless, the effect of the solid volume fraction on the carrier liquid is not included in Equations (23) and (24). Therefore, a low solid volume fraction is necessitated when applying the DPM model. Conceivably, the DPM model is not suitable for the solid–liquid two-phase flows involving large solid particles.

DDPM

The DDPM is deemed as an extended version of the DPM, and the most noticeable improvement is the suitability to high concentration of the medium. The volume fraction is introduced into the governing equations. The continuity and the momentum equations are expressed as [106]:

$$\frac{\partial }{\partial t}\left({\epsilon }_l{\rho }_l\right)+\nabla \cdot \left({\epsilon }_l{\rho }_l{\mathit{v}}_l\right)=S_{DPM}+S_{user}$$

(25)

$$\frac{\partial }{\partial t}\left({\epsilon }_l{\rho }_l{\mathit{v}}_l\right)+\nabla \cdot \left({\epsilon }_l{\rho }_l{\mathit{v}}_l{\mathit{v}}_l\right)=-{\epsilon }_l\nabla P+\nabla \cdot \mathit{\tau }+{\epsilon }_l{\rho }_l\mathit{g}+{\mathit{F}}_{DPM}+{\mathit{F}}_{user}$$

(26)

where ε,v, ρ, and P denote the volume fraction, density, velocity, and pressure, respectively. The subscript l denotes the liquid phase.

The introduction of the volume fraction to the continuity equation enables the simulation of dense solid–liquid flows. The particle motion equations of the DDPM are the same as that associated with the DPM. For both the DPM and DDPM, the DEM collision model can be used to describe inter-particle interaction. When dealing with the cases requiring high numerical accuracy, the DEM is a better option since it comprehensively considers dynamic interactions between the two phases.

DEM

For the DEM, the motion of dispersedly distributed particles is modelled based on Newton’s second law of motion. For the solid phase, the governing equations are expressed as [125]:

$$m_i\frac{d{\mathit{v}}_i}{dt}={\sum }_j{\mathit{F}}_{c,ij}+{\sum }_k{\mathit{F}}_{nc,ik}+{\mathit{F}}_{ls,i}+{\mathit{F}}_{g,i}$$

(27)

$${\mathit{I}}_i\frac{d{\mathit{\omega }}_i}{dt}={\sum }_j({\mathit{M}}_{t,ij}+{\mathit{M}}_{r,ij})$$

(28)

where m_i denotes the mass of solid particle i; v_i is the translational velocity of solid particle i; F_c,ij denotes the contact forces between particles i and j; F_nc,ik is the non-contact forces between particles i and k; F_ls,i denotes the force acting on particle i by liquid phase; F_g,i denotes the body force acting on particle i; I_i is the inertia moment of particle i; ω_i is the angular velocity of particle i; M_t,ij is the tangential friction moment between particles i and j; and M_r,ij denotes the normal friction moment between particles i and j.

For liquid phase, the governing equations complying with the law of conservations of mass and momentum in terms of local average variables are given by the following:

$$\frac{\partial \left({\epsilon }_l{\rho }_l\right)}{\partial t}+\nabla \cdot \left({\epsilon }_l{\rho }_l{\mathit{v}}_l\right)=0$$

(29)

$$\frac{\partial \left({\rho }_l{\epsilon }_l{\mathit{v}}_l\right)}{\partial t}+\nabla \cdot \left({\rho }_l{\epsilon }_l{\mathit{v}}_l{\mathit{v}}_l\right)=-{\epsilon }_l\nabla P+\nabla \cdot \left({\epsilon }_l\mathit{\tau }\right)+{\rho }_l{\epsilon }_l\mathit{g}-{\mathit{F}}_{ls,i}$$

(30)

The commercial CFD codes have some limitations in calculating the volume fraction whilst implementing the DEM. For instance, the coupling of CFD and DEM through ANSYS Fluent and EDEM is only appropriate for the cases where the liquid cell size is larger than that of particles. In contrast, in STAR-CCM+, the smooth large particle method is introduced to solve this problem. At the coupling interface between ANSYS Fluent and EDEM, the particle-centered method (PCM) is used for calculating the volume fraction:

$${\epsilon }_l=1-\frac{{\sum }_{i=1}^{N_{\mathit{cell} }}{B}_{s,i}}{B_{\mathit{cell} }}$$

(31)

where ε_l is the volume fraction of liquid; N_cell is the total number of particles in the cell; B_s,i is the volume of particle i; and B_cell is the volume of the grid cell.

A schematic view of PCM is shown in Figure 28a. The figure on the left illustrates the spatial relationship between a grid cell and a particle, while the figure on the right displays the computed solid volume fraction within the corresponding grid cell. Grayscale represents the solid volume fraction, which is ranged from 0 to 1. Regarding the PCM method, a particle may span across two adjacent computational grids. However, only the grid that contains the center of the particle is assigned a value of solid volume fraction, while the grid that intersects with the edge of the particle is exempt from such an assignment. An overestimation of the volume fraction of liquid may be caused numerically in cases with large particles [10].

For the CFD code STAR-CCM+, to treat the cases where the particle size exceeds that of the liquid cell, the smooth large particle method is adopted. With respect to this method, for large particles, the solid volume fraction is determined within a mesh cluster produced from the virtual merger of two adjacent fluid cells, as illustrated in Figure 28b. The liquid volume fraction in the mesh cluster is given by:

$${\epsilon }_l=1-\frac{{\sum }_{i=1}^{N_{\mathit{cluster}}}{B}_{s,i}}{B_{\mathit{cluster} }}$$

(32)

where N_cluster represents the total number of particles in the mesh cluster, and B_cluster is the volume of the mesh cluster.

5.3.2. CFD-Based Wear Models

(1)

Archard wear model

Archard pioneered the modeling of erosive wear. At present, his model is still being used to estimate the erosion rate when particles impact a surface. It is supposed that the real area of contact between two objects is far less than the apparent area and is determined by the extent of the deformation of the contact surface under the applied load. Supposing the wear rate is independent of the apparent area of contact and is directly proportional to the applied load [126,127]:

$$ER=K{F}_{n}s$$

(33)

where ER denotes the erosion rate, defined by the volume of removed material; s is the sliding distance; F_n is the applied normal load; and K is a constant related to the probability of the production of wear.

The Archard wear model is characterized by a single constant and is convenient to be used for wear prediction. However, its applicability is limited since it has been developed based on the results obtained through pin-on-ring experiments. Moreover, particle properties and impact parameters are not involved in this model. In some instances, the results obtained with this model deviate clearly from the real situation.

(2)

Finnie wear model

Finnie developed a wear model based on the cases in which abrasive particles impact a ductile surface. Since then, his model has been continuously improved [1,128,129]. Through assuming an idealized pattern of material removal, Finnie derived the volume of the material removed by a particle, ER. The impact angle, mass and velocity are denoted by θ, m_s, and v_s, respectively. The model incorporates the stress in the material as strain arises, σ, the ratio of the vertical force component to the horizontal force component, κ, and the ratio of the depth of the lip to its length, ψ.

$$ER=\left\{\begin{array}{ll}\frac{m_s{v}_s^2}{\sigma \psi \kappa }\left(\frac{{\kappa \cos }^2\theta }{6}\right),& \tan \theta >\frac{\kappa }{6}\\ \frac{m_s{v}_s^2}{\sigma \psi \kappa }\left(sin2\theta -\frac{\kappa }{6}{\sin }^2\theta \right),& \tan \theta \le \frac{\kappa }{6}\end{array}\right.$$

(34)

In accordance with the experimental data, the ratio, ψ, and the critical angle, θ, are set to 2 and 18.5°, respectively.

$$ER=\left\{\begin{array}{ll}\frac{m_s{v}_s^2}{24\sigma }{\cos }^2\theta ,& \theta >18.5°\\ \frac{m_s{v}_s^2}{8\sigma }\left(sin2\theta -3{\sin }^2\theta \right),& \theta \le 18.5°\end{array}\right.$$

(35)

The maximum erosion rate of a material is related to the impact angle and material properties. Therefore, the threshold angle needs to be regulated accordingly. Taking into account both ductile and brittle materials, the Finnie model cannot be applied to brittle materials due to its restriction on the angle function. Additionally, the Finnie model includes several parameters, which are difficult to be completely covered through measurement.

(3)

Oka wear model

The Oka wear model can be applied to estimate the extent of the erosive wear under various impact conditions for different materials [130]. As indicated in Equations (36)–(38), various impact parameters such as particle size, impact velocity, and impact angle have been involved. Meanwhile, the hardness of the target material is incorporated in the model as well.

$$ER={ER}_{90}g\left(\theta \right)$$

(36)

$${ER}_{90}={K\left(a{H}_v\right)}^{k_{1}b}{\left(\frac{v_s}{v_{ref}}\right)}^{k_2}{\left(\frac{d_s}{d_{ref}}\right)}^{k_3}$$

(37)

$$g\left(\theta \right)={\left(sin\theta \right)}^{n_1}{\left(1+H_v\left(1-sin\theta \right)\right)}^{n_2}$$

(38)

where ER₉₀ denotes the erosion rate at normal angle; g(θ) is a function of the particle impact angle, which is defined by the ratio of mass loss due to erosion at an arbitrary angle to that at the normal angle; H_v denotes the Vicker’s hardness of the target material; v_s is the particle velocity; d_s is the particle diameter; v_ref is the reference particle velocity; d_ref is the reference particle diameter; K, k₁, k₂ and k₃ are the constants determined by physical properties of the particle; and n₁ and n₂ are the exponents related to the material hardness and properties of the particle.

Both the actions of deforming and cutting contribute to the erosive wear, and the two factors are related to the impact angle. Although the Oka model was initially designed for ductile materials, it can also be applied to brittle materials through setting n₂ = 0. This neglects the cutting mechanism. Consequently, the impact angle function is in accordance with the erosion pattern observed in brittle materials.

In CFD codes, the Oka wear model takes the following form:

$$ER={ER}_{90}g\left(\theta \right){\left(\frac{v_s}{v_{ref}}\right)}^{k_2}{\left(\frac{d_s}{d_{ref}}\right)}^{k_3}$$

(39)

$$g\left(\theta \right)={\left(sin\theta \right)}^{n_1}{\left(1+H_v\left(1-sin\theta \right)\right)}^{n_2}$$

(40)

Typically, the constants incorporated in the Oka wear model for the steel eroded by sand particles are presented in

Table 3. Empirical constants for Oka wear model.

Material	ER90	Hv (GPa)	n1	n2	k2	k3	vref (m/s)	dref (μm)
Sand-steel	6.15 × 10−4	1.8	0.8	1.3	2.35	0.19	104	326

.

(4)

McLaury (E/CRC) wear model

The McLaury wear model, which is also referred to as the E/CRC wear model, is established based on a series of impact experiments conducted with different particle geometries, impact velocities, and impact angles. The property of the target material is considered through introducing the term Brinell hardness in the equation. According to Refs. [26,131], the E/CRC wear model is defined as:

$$ER=K{F}_s{\left(H_B\right)}^k{v}_s^{n}f\left(\mathsf{\theta }\right)$$

(41)

$$f\left(\theta \right)=\left\{\begin{array}{ll}a{\theta }^2+b\theta ,& \mathrm{for} \theta \le {\theta }_0\\ x{\cos }^2\theta \sin \left(w\theta \right)+y{\sin }^2\theta +z,& \mathrm{for} \theta >{\theta }_0\end{array}\right.$$

(42)

The impact angle function takes another form:

$$f\left(\theta \right)=\sum _{i=1}^5{A}_i{\theta }^i$$

(43)

where H_B denotes Brinell hardness of the target material; F_s is a particle shape coefficient, F_s = 1.0 is set for sand particles with sharp angles, 0.53 for semi-rounded, and 0.2 for fully rounded sand particles; v_s is the particle impact velocity; θ is the impact angle; and A_i denotes the constants related to the impact angle function.

Experiments have been conducted to validate and improve the E/CRC model. Arabnejad et al. [87] argued that the cutting and deformation mechanisms are critical for improvement in the E/CRC wear model.

$${ER}_C=\left\{\begin{array}{ll}{K}_C\frac{v_s^{2.41}\mathrm{s}\mathrm{i}\mathrm{n}\theta \left(2K\mathrm{c}\mathrm{o}\mathrm{s}\theta -\mathrm{s}\mathrm{i}\mathrm{n}\theta \right)}{2K^2},& \theta <ta{n}^{-1}\left(K\right)\\ K_C\frac{v_s^{2.41}{\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta }{2},& \theta >{\mathrm{t}\mathrm{a}\mathrm{n}}^{-1}\left(K\right)\end{array}\right.$$

(44)

$$E{R}_D=K_D(v_s\mathrm{s}\mathrm{i}\mathrm{n}\theta -v_{tsh}{)}^2$$

(45)

$$ER=F_S(E{R}_C+E{R}_D)$$

(46)

Arabnejad et al. [132] attempted to incorporate an initial penetration of sharp particles, ER_A, into the E/CRC wear model, rendering it appropriate for scenarios characterized by exceedingly small particle impact angles (<5°). Equation (46) is then transformed into

$$ER=F_S(E{R}_C+E{R}_D+E{R}_A)$$

(47)

where K_C, K_D, and K are empirical constants, and v_tsh is a critical velocity. The empirical constants and the critical velocity for seven materials provided in Ref. [132] are listed in

Table 4. Empirical constants for E/CRC wear model.

Material	KC	KD	K	vtsh (m/s)
Carbon steel 1018	5.90 × 10−8	4.25 × 10−8	0.5	5.5
Carbon steel 4130	4.94 × 10−8	4.94 × 10−8	0.4	3.0
Stainless steel 316	4.58 × 10−8	5.56 × 10−8	0.4	5.8
Stainless steel 2205	3.92 × 10−8	2.30 × 10−8	0.4	2.3
13 chrome steel	4.11 × 10−8	3.09 × 10−8	0.5	5.1
Inconel 625	4.58 × 10−8	4.22 × 10−8	0.4	5.5
Aluminum alloy 6061	3.96 × 10−8	3.38 × 10−8	0.4	7.3

.

6. Wear Equipment and Measurement Method

Regarding the experiment of wear, both the experimental rig and the instrument should adapt to the solid–liquid two-phase flow. An instrument needs to be developed specifically. Furthermore, the measurement uncertainty must be controlled to a reasonably low level. In this aspect, qualitative results prevail in the relevant literature, while quantitative results are difficult to obtain. This is related to both the experimental rig and the measurement method employed.

6.1. Wear Equipment

Various experimental rigs for studying wear have been devised and applied. Two types of experimental rigs have been frequently reported in the relevant literature. One is the rigs testing the resistance of the material to the erosive wear. The target sample is directly impacted by solid particles, and the main operating parameters such as the impact angle and the impact velocity can be adjusted. Various materials with different properties can be tested. The other type is more concerned with practical applications. Models of pipes, tees, valves, and pumps are employed in these types of experimental rigs. The results obtained can be directly used in the optimization design of the equipment handling the solid–liquid two-phase flow.

6.1.1. Jet Erosion Rig

The jet erosion rig enables an immediate examination of the resistance of a material to wear [31,88,105,133,134,135]. Such a rig is mainly composed of a pump, a tank, a sample holder, a nozzle, control valves, and pressure and temperature gauges. The pump delivers the mixture from the tank to the nozzle, and the mixture is discharged from the nozzle towards the target sample. The orientation of the sample holder can be adjusted to vary the impact angle. The impact velocity and flow rate can be regulated through manipulating the opening of the valve and the rotational speed of the pump. A mixer is used to agitate the mixture and ensure homogeneity of the mixture. The mass loss method is commonly used to measure the erosion rate, which is defined as:

$$ER=\frac{\mathrm{mass} \mathrm{loss} \mathrm{of} \mathrm{the} \mathrm{sample}}{\mathrm{mass} \mathrm{of} \mathrm{the} \mathrm{particles} \mathrm{used}}$$

(48)

Measuring particle velocity is of great significance for predicting the extent of erosion. The techniques of laser Doppler velocimetry (LDV) [131] and particle image velocimetry (PIV) [82,118,136] have been applied to measure liquid velocity. A conventional PIV system is composed of a double-pulsed laser, a CCD camera, a synchronizer, and a processor. The double-pulsed laser emits two light sheets with a certain time interval to illuminate the particles in the flow field, and the camera is synchronized with the laser pulse to capture two consecutive images of particles. A correlation algorithm is operated in the processor to calculate the particle velocity. The shortages of PIV in measuring the velocity of solid particles are explicit. Especially at high solid volume fractions or large particle size, spurious results can be generated. This is related to the shield effect of particles on the incident light. Furthermore, three-dimensional particle velocity is difficult to obtain since the particles may escape from the field of view (FOV) plane.

6.1.2. Whirling-Arm Rig

The whirling-arm rig operates differently from the jet erosion rig. With such a rig, the sample is rotated with the aid of a motor, and meanwhile, the sample is impacted by the solid–liquid mixture falling freely from the nozzle [80,93,101,137];. The whirling-arm rig incorporates a vacuum unit, which assumes the function of removing air and other gases from the test chamber, ensuring that the erosion process is free from the influence of environmental factors. The rotational speed of the sample can be regulated through the whirling rotator, which is driven by a variable speed motor. The sample holder is designed with the tilting and locking functions that allow for the adjustment of the angle of inclination for the tested sample. The mass of the particles impacting the sample per revolution can be calculated through a validated formula.

6.1.3. Closed-Loop Rig

The closed-loop experimental rig enables testing the resistance of pipes and pumps to the erosive wear through circulated solid–liquid two-phase medium [17,138,139]. Such a rig has been widely used in fluids engineering [85,140]. A schematic view of a typical closed-loop experimental rig is shown in Figure 29. The attainable superficial liquid velocity exceeds 8.0 m/s, as is dependent on the performance of the delivery pump and the diameter of the pipe. The maximum solid volume fraction is generally lower than 30%. Water is mixed with solid particles in the tank where a mixer is deployed to keep the mixture homogenous and avoid particle settlement at the bottom of the tank. The flow rate of the mixture is measured using an electromagnetic flow meter and can be adjusted through the valves. The erosion rate is obtained through weighing the pipe segment before and after the test [141]. The electrical resistance probe and ultrasonic transducers are installed on the test section to monitor the variations of the wall thickness and the mass loss [131]. A transparent test section is deployed for the visualization of the multiphase flow patterns [142]. When the test is finished, a bypass line equipped with a filter is applied to separating water and solid particles.

Regarding the test involving large particles, several challenges are conceivable. The distribution of large particles in the loop is non-uniform, so the accuracy of the solid volume fraction measured is difficult to evaluate [143]. Meanwhile, a high-power pump is required to ensure that large particles are circulated in the loop without sedimentation. In this case, since the momentum of large particles is relatively high, the pipes and the pump are subject to vibration and erosion. Therefore, the operating stability of the whole experimental system deserves prudent consideration [5]. Heretofore, such an experimental rig has been mainly used to study the operating performance of the pump instead of the wear mechanism.

6.2. Method of Wear Measurement

The weighing method has been widely used in the experimental works of erosive wear. This involves separately recording the mass of the specimen before and after the wear test. Then, the difference between the two measurement results is used to quantify the mass loss. This method is easy to implement, and the accuracy is fairly high. However, such a method is unable to identify the location of the most severe erosion [105]. In this situation, the technique of scanning electron microscopy (SEM) is used as a supplement. Such a combination enables establishing the relationship between the mass loss and the erosion pattern of the target sample. Furthermore, a deep understanding of the wear characteristics such as ploughing, cutting, indentation, and fracture can thereby be accomplished [144].

6.2.1. Electrical Resistance Probe

The electrical resistance probe, as shown in Figure 30, is used to monitor the mass loss caused due to erosion or corrosion [131,145,146]. It is composed of two electrically conductive elements, namely the sample element and reference element. The sample element is directly exposed to the flowing liquid, and the reference element is protected from the liquid through an epoxy coating. As solid particles repeatedly impact the sample element, the element experiences mass loss, and the electrical resistance is therefore changed. Through comparing the resistance between the sample element and the reference element, the mass loss is quantified. The electrical resistance probe may affect the flow pattern since it protrudes in the flow field. In Figure 30, it is seen that two flush head probes are mounted at the curved segment, and an oblique probe is fixed in the straight segment [147].

6.2.2. Profilometer

The profilometer has been applied for measuring the profile of the surface exposed to the erosive wear [26,148]. As the stylus traverses across the surface, its movement is detected and then converted into electrical signals, which are processed using specifically developed algorithms. Through comparing the surface profiles before and after the erosion test, the intensity of the erosive wear can be described in both quantitative and qualitative manners. Furthermore, the data of surface height can be processed using the algorithms embedded in the commercial codes or the self-developed code based on MATLAB R2014a, then the area-averaged parameters such as the surface roughness can be obtained.

6.2.3. Ultrasonic Technique

The ultrasonic technique has been recognized as an effective tool for detecting the thickness of the eroded surface [149]. The ultrasonic probe comprises a sender and a receiver, which assume the functions of generating and receiving ultrasonic waves, respectively. When reaching the target surface, most of the ultrasonic waves emitted by the sender are reflected and subsequently received by the receiver. The thickness of the target can be determined through measuring the time required for the ultrasonic wave to propagate through it. By comparing the thickness before and after the erosion test, the reduction in the thickness due to the erosive wear can be obtained. In terms of the accuracy of the thickness measurement, the ultrasonic testing system is superior to other techniques since it can compensate for the variation of the ultrasonic wave velocity induced by temperature change.

7. Conclusions and Future Work

The present study provides a comprehensive review of the studies on erosive wear when large particles are transported. The analysis covers the mechanisms, engineering correspondence, fluid-handling equipment, numerical methods, and the experimental rig and measurement techniques. The review furnishes an insight into the erosive wear caused by the impact of large particles. Meanwhile, a reference is provided for the design and operation of the equipment and systems handling the solid–liquid medium. Some expectations for future work can be outlined as follows:

(1) Previous research on erosive wear has been primarily focused on small particles (<3.0 mm) and low solid volume fractions (<5%). With the development of some industrial applications such as lifting minerals from deep sea, the transport of large particles (>30 mm) at high solid volume fractions (>12%) poses a new challenge in both academic and technical aspects. The dynamic behaviors of large particles involved in flowing liquid deserve specific theoretical and experimental studies.

(2) When the liquid carrying solid particles exhibits corrosive property, it is essential to investigate the synergistic effect of erosion and corrosion. Experimental works have shown that corrosion significantly affects wear of the equipment or the specimen, yet the mechanism of the synergistic effect has not been well interpreted. In some cases, corrosion promotes wear, while in some other cases, the tendency is overturned. Furthermore, whether numerical methods can be used to describe the synergistic effect is expected to be examined.

(3) Regarding numerical simulation of the erosive wear, few works have considered the effect of the change in local surface pattern due to material removal on the multiphase flow and further erosion. According to previous reports, local structure change can cause an intensification of erosion through promoting particle recirculation in cavities. However, it has also been argued that the erosion is mitigated since the flow field is uniformized. In the future work, the change in geometry of local surface is expected to be incorporated into numerical simulation through applying the mesh motion technique.

(4) Available wear models have been developed based on the experimental data obtained from simple geometries, which may raise concerns about their suitability for predicting the erosive wear of the equipment with complex geometry. As a matter of fact, the flow patterns associated with complex geometries differ significantly from those with simple geometries, leading potentially to imparity between the wear predicted and the real situation. The data of the wear test with curved surfaces or surfaces with irregular shapes should be supplemented, therefore serving as an important reference for the improvement in the wear model.

(5) Experiments on the solid–liquid two-phase flow have been conducted, but substantial help for revealing the wear mechanism pertinent to large particles has not been accomplished. No relationship of similarity has been established between the results obtained at different particle sizes. With emerging industrial applications involving large particles, it is imperative to improve experimental techniques to test both the wear-resisting ability and operating performance of the equipment handling large particles.