A Study of Abrasive Solid Particles Erosion for a Centrifugal Pump Operated as a Pump and as a Turbine Using Computational Fluid Dynamics

Abstract

Impeller blades are one of the main parts of a centrifugal pump that affect the performance of the pump. The presence of solid particles in seawater, transported through a centrifugal pump, causes wear in the blade surface that reduces blade lifetime. In the orthogonal direction, this wear is an erosion thickness of the blade. Assuming that these particles have a spherical shape, the erosion rate depends on their velocity, size, impingement angle, and material hardness index. In this work, we investigate the erosion thickness of a low-head centrifugal pump operating in pump and turbine modes, with a particle radius ranging from 4 μm to 50 μm. The numerical simulation used an RNG k–ε turbulence model, assuming a perfect bounce collision between the particle and the rotating solid wall. The study shows that the blade pressure side is impacted by a solid particle concentration higher than the suction side. In pump mode, the erosion thickness on the blade sides increases if the particle radius is above 4 μm and reaches a maximum at 40 μm. In turbine mode, the erosion thickness decreases when the particle radius is greater than 5 μm. The thickness loss is greater in turbine mode than in pump mode. The influence of particle flow rate was investigated. Below a particle radius of 10 μm, particles follow the flow directions and reside for a longer time in the blade channel. Passing from a particle radius of 50 μm to 100 μm, the blade lifetime was decreased by a factor of 11.

1. Introduction

Electricity storage systems (ESSs) help increase the grid’s flexibility and manage the intermittency of renewable generation [1]. However, by using ESS, the excess electricity generated is stored during periods of low demand or low price and released during periods of high demand, enhancing a continuous balance between consumption and production (time shifting) [2].

One of the promising ESS is pumped storage, which is considered a global battery system. Pumped hydropower electricity storage (PHES) involves using a hydraulic turbine between two water reservoirs. Through a penstock, the water is released from the upper reservoir into a lower reservoir to produce electricity. In storage mode, a reversible turbine pumps water from a lower reservoir to an upper reservoir. The capacity of a PHES depends on the water flow (Q_f), available head (H_g), and turbine efficiency (η) [3]. hydraulic turbines are classified into two types: and reaction turbines. The impulse turbine operate with $100 \mathrm{m}<H_g<1770 \mathrm{m}$ and $0.05<n_s (\mathrm{r}\mathrm{a}\mathrm{d})<0.4$, whereas the Francis turbine is used for $20 \mathrm{m}<H_g<900 \mathrm{m}$ and $0.4<n_s (\mathrm{r}\mathrm{a}\mathrm{d})<2.2$ [4]. A radial turbine can operate over a wide range of specific speeds with high efficiency, especially when transitioning from full-load to partial-load operation.

Hydropower systems are classified into three categories: small hydropower, medium hydropower, and large hydropower. Small hydropower has an installed capacity below 10 MW. Another hydropower class is the pico-hydropower with an installed capacity below 10 kW. These systems are used for agriculture, domestic, and industrial applications. The centrifugal pump used in these very small hydropower systems can be operated in turbine mode (Pump as Turbine, PaT) to produce electricity.

The use of a PaT offers a low-cost solution to produce electricity with an existing installed centrifugal pump. In 1931, the first studies were conducted on the feasibility of using a centrifugal pump as a turbine. A pump can be used as a turbine with a head from 7 m to 150 m and fluid flow from 0.003 m³/s to 0.1 m³/s [5]. The study of [6] used the affinity law to predict PaT performance from pump manufacturer data. Gulish et al. [7] studied the effects of fluid characteristics on pump geometry, especially the blade height. He studied geometry design changes from freshwater and wastewater. The research paper [8] studied the effect of blade damage located at 0, 1/3, and 2/3 L on a centrifugal pump performance. The results showed an efficiency of 79.45%, 76.49%, and 76.06%, respectively.

Owing to the presence of sediments in the water flow, the solid particles attack the blade surfaces and cause surface roughness [9]. In the paper of [10], if the surface roughness increases from R_a = 0.12 μm to R_a = 12 μm for a centrifugal pump, the efficiency decreases. Ref. [11] studied the effect of particle size from d_p = 10 μm to d_p = 80 μm on the average erosion rate of the Francis turbine runner. The erosion rate increases linearly with the particle size and concentration for the Francis turbine runner. It was concluded that an optimum particle diameter of d_p = 30 μm minimizes the erosion rate. Ref. [12] studied the effect of stainless steel coating depth on the erosion rate for different water droplet sizes (20 μm, 50 μm, and 100 μm) and compared it to uncoated glass-fiber-reinforced plastic (GFRP) and uncoated carbon-fiber-reinforced plastic (CFRP). It was found that a coating height above 50 μm better prevents material damage from solid particle attack.

Although centrifugal pump technology is mature, few scholars studied erosion, pressure oscillations, and cavitation in turbine mode. However, the pump lifespan depended on the number of cycles in turbine and pump modes. In this study, we focus on predicting the erosion thickness for a radial pump in pumping and turbine modes for a small particle size range from r_p = 4 μm to r_p = 50 μm. The thermophysical properties of seawater are taken from standard thermodynamic conditions, T = 25 °C, salinity concentration of 30 g per liter, and P = 1.013 bar. The pump geometry was designed via CFTurbo (v2025 R2.2) software. The Discrete Phase Model (DPM) is used in Simerics MP+ software (v6.0.17) to track the particle trajectory by setting the number of released particles.

The study focuses on analyzing particle trajectories with different sizes in pump and turbine modes. The second goal is the investigation of erosion thickness variation with particle size and particle mass flow rate. From these results, manufacturers can have a better idea about coating thickness, and engineers can decide on the thickness-loss distribution (variable or fixed thickness) alongside the blade channel.

2. Materials and Methodology

The complexity of studying the erosion rate in a rotating machine depends on the flow regime, secondary flows, particle parameters, and material properties. While material properties are given by the pump manufacturers, the two parameters under control are those related to the flow and particles. However, for a given case, we should know the particle shape, particle size, and particle hardness.

To demonstrate that seawater exhibits a specific turbidity level, depending on the presence of solid particles, we employed remote sensing at a randomly selected site in Essaouira, Morocco. Seawater contains solid particles due to the dissolution of minerals with high concentrations of salts. To prove this assumption, we used Sentinel 2 images available from the Copernicus Open Access Hub project [13] for a site near Essaouira (Morocco) [Figure 1]. The presence of gases in the atmosphere reduces the signal energy received by the satellite. Thus, we used bottom-to-bottom images that do not need atmospheric correction.

The reflectance rate in seawater is important in the visible light band (BAND 3), whereas for land, it is high for near-infrared bands. This means that, for seawater, the wavelength l in BAND3 is high.

The NDWI (Normalized Difference Water Index) indicates water reservoirs in a specific area. A negative value of the NDWI means that the area is land, whereas a positive value indicates water. NDWI uses green (B3) and near-infrared bands (B8) to highlight water bodies [14]. Another option in SNAP software (v12.0.0) is to use a mask (sea/land) to select water areas or calculate the NDWI. For the Sentinel 2 image, the NDWI is defined as follows [15]:

$$NDWI={\displaystyle \frac{B3-B8}{B3+B8}}={\displaystyle \frac{Green band-NIR}{Green band+NIR}}$$

(1)

[

Table 1. NDWI values for seawater pins are positive.

Pins	Lon	Lat	NDWI
Pins1: seawater	−9.85	31.38	0.18
Pins1: seawater	−9.84	31.39	0.14
Pins3: land	−9.82	31.40	−0.19

] gives the NDWI for the three selected pins from [Figure 1]. NDWI is positive for seawater.

The turbidity of water measures the clarity of water, which depends on the particle concentration. High turbidity indicates high solid-particle concentrations. The turbidity (T) is estimated from [16]. The concentration of total suspended particles (TSM) is estimated by using the C2RCC (Case 2 Regional Coast Color) processor from SNAP software. Both results are illustrated in [Figure 2].

For this work, we selected a TSM equal to 30 g/m³ of seawater. When the seawater volumetric flow rate is $Q_f=0.0033 {\mathrm{m}}^3/\mathrm{s}$, the particle mass flow rate is $\dot{m_p}=9.9 \mathrm{m}\mathrm{g}/\mathrm{s}$. If we consider that ${\rho }_f=1024 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$, the mass percentage of the particle concentration in seawater flow is then 0.3 ppm.

2.1. Materials

2.1.1. Centrifugal Pump Design

(1)

Centrifugal pump

A centrifugal pump is composed of an impeller (rotating part), a diffuser, and a volute. A typical impeller has 6 blades. At the inlet, the fluid is absorbed with high absolute velocity and low static pressure. Through the blade length, the velocity decreases, and the total pressure increases due to the diverging blade channel [7]. The centrifugal pump moves fluid from one point to another by increasing its pressure [17].

Using a pump in inverse mode (turbine) has been the subject of different scientific studies. By inverting the rotational direction of the pump, fluid can pass through and decrease its pressure [18]. For this study, we selected the centrifugal pump design parameters shown in [

Table 2. Centrifugal pump design point.

Parameter	Value
Q	0.0033 m3/s
H	3.91 m
N (Rotational Speed)	2400 RPM

].

A 3D view of the pump geometry is shown in [Figure 3]. The mesh characteristics are listed in

Table 3. Mesh characteristics of the selected centrifugal pump.

Grid Geometry Information
Mesh Type	3D
Number of Cells	1,871,944
Number of Faces	7,004,182
Number of Nodes	2,812,228
Surface Info
Number of Faces	290,580

and

Table 4. Time step calculations.

Time Step Definition	Rotating System
Simulation time (duration)	0.125
Number of time steps	1800
Number of iterations	25
Time step method	revolutions
Number of revolutions	5
Time step per revolution	360
Rotational speed	2400

. The binary meshing method is used to create the 3D geometry mesh [19,20]. The convergence test of pump efficiency and head with the number of cells and time step is presented in Appendix A.

The blade’s passage channel increases the fluid pressure from the pump’s leading edge (LE) to the trailing edge (TE). Owing to the flow rotation, it is subject to both peripheral and relative velocity. In fixed coordinates, the velocity is in an absolute frame. However, in the rotational device, the absolute velocity (C) is the sum of the peripheral velocity (U) and the relative velocity (W).

$$\overrightarrow{C}=\overrightarrow{U}+\overrightarrow{W}$$

(2)

$$\overrightarrow{U}=r\ast \overrightarrow{\omega }$$

(3)

The relative velocity has two components: the tangential component C_u and the meridional component C_m. The tangential velocity is the velocity component created by the flow torque. The meridional velocity is the flow velocity through a given blade channel area.

$$C_m={\displaystyle \frac{\dot{m}}{{\rho }_{f}2\pi rb}}$$

(4)

After leaving the blade entrance, C_m is affected by the blade blockage. However, the flow passage area depends on blade thickness and the number of blades. At blade edges, the thickness is 0 due to the rounding of the blade edge. Immediately after the blade leading edge (or before the blade trailing edge), the blade blocks the flow in a certain manner. This blockage is dependent on the blade thickness, the blade angle, and the blade angle distribution, which is hence rather complex with respect to the blade geometry. The [Equation (5)] becomes

$$C_m=C_m\ast e where e={\displaystyle \frac{s}{s-\frac{2\pi r_1}{z}}}$$

(5)

where (e) is the blade blockage coefficient defined as.

(2)

Pump operated as Turbine mode (PaT)

In this section, we introduce the theory to predict turbine performance from pump characteristics. The power consumed by the pump is as follows:

$$P_{pump}={\displaystyle \frac{{\rho }_{f}g{Q}_p{H}_p}{{\eta }_p}}$$

(6)

where the power produced by the turbine is as follows:

$$P_T=\rho g{H}_T{Q}_T{\eta }_T$$

(7)

Theoretically, the power produced in turbine mode equals the energy consumed by the pump. By setting [Equation (7)] equal to [Equation (8)], it follows that, during the inverse mode, we operate at a higher head than in pump mode.

$${\displaystyle \frac{H_T}{H_p}}={\displaystyle \frac{1}{{\eta }_p{\eta }_T}}$$

(8)

When Q_T is different from Q_p and η_T = η_P = η, Equation (15) becomes

$${\displaystyle \frac{H_T{Q}_T}{H_p{Q}_p}}={\displaystyle \frac{1}{{\eta }^2}}$$

(9)

We define the head and flow ratios as follows:

$$h={\displaystyle \frac{H_T}{H_p}}$$

(10)

$$q={\displaystyle \frac{Q_T}{Q_p}}$$

(11)

The variation in h and q with pump-specific speed (n_sp) was studied by [21]. If n_sp increases, h and q decrease. When n_sp is lower than 50, we can achieve good PAT efficiency. From [22,23,24], h and q are defined as

$$h={\displaystyle \frac{1.2}{{{\eta }_p}^{1.1}}}$$

(12)

$$q={\displaystyle \frac{1.2}{{{\eta }_p}^{0.55}}}$$

(13)

$$n_{sp}=1.17n_{st}+1.73$$

(14)

[

Table 5. Models to estimate h and q for a PaT.

Model Name	h	q
Sharma	${\displaystyle \frac{1}{{{\eta }_{P,BEP}}^2}}$	${\displaystyle \frac{1}{{{\eta }_{P,BEP}}^{0.8}}}$
Stepanoff	${\displaystyle \frac{1}{{\eta }_{P,BEP}{\eta }_{T,BEP}}}$	${\displaystyle \frac{1}{{\eta }_{P,BEP}}}$

] presents the Sharma and Stepanoff models to predict the performance of the pump operated in turbine mode.

Ref. [25] reported that reducing the rotational speed in the case of a pump operated in turbine mode increases hydraulic efficiency. To optimize our design, we computed hydraulic pump efficiency at different specific speeds. The maximum total-to-total efficiency was reached at n_sp = 49 [Figure 4].

2.1.2. Abrasive Solid Particle Erosion

Erosion is the material mass lost due to the surface attack of solid particles. When a fluid passes through a blade channel, it is in contact with the blade surfaces. The hit energy of particles has two components: normal hit energy and tangential hit energy. As erosion in this case is supposed to be abrasive, the tangential hit energy is higher than the normal hit energy. The amount of erosion thickness is linear with the total number of hit particles. The erosion rate (ER) is the amount of material removal divided by particle mass.

The erosion rate (ER) is defined as follows:

$$ER={\displaystyle \frac{material mass loss}{eroded particles mass}}$$

(15)

According to [26,27,28], the ER depends on the particle velocity, the particle attack angle, and material properties. The present study used the Finnie model to predict thickness losses due to particle attack. However, this model overestimates the erosion rate compared to the Grant–Tabakoff model, which brings us to predict the highest material mass loss [29,30]. For a ductile material, the ER is high for attack angles between 30° and 40°.

$$ER=\{\begin{array}{c}{\displaystyle \frac{{{\rho }_{m}Vp}^2}{8H_v}}(\sin (2\theta )-3{\sin (\theta )}^2) if \theta \le 15°\\ {\displaystyle \frac{{\rho }_m{Vp}^2}{24H_v}}\cos 2(\theta ) if \theta >15°\end{array}$$

(16)

In this work, the assumption of an elastic collision is made between the wall and the particle [31,32]; the momentum and kinetic energy are conserved. The erosion model is simulated by using the perfect bounce model. The effect of the particle incidence angle on the wall rotational speed was investigated. When the incidence angle is less than 40 degrees, the reflection angle is 3 degrees below the incidence angle [33]. The research in [34,35] revealed that under elastic theory, the theoretical erosion rate is close to the experimental result. However, when a particle collides with a solid wall, it is an elastic collision. In this work, we do not consider collisions between particles.

According to Newton’s law, three forces act on the particle’s motion: the drag force, the rotational forces (correct and centrifugal), and the gravitational force.

$$\overrightarrow{F_p}=\overrightarrow{F_D}+\overrightarrow{F_G}+\overrightarrow{F_R}$$

(17)

In the case of spherical particles, the particle mass (m_p) can be computed as follows:

$$m_p={\rho }_p{\displaystyle \frac{1}{6}}\pi {d_p}^3$$

(18)

The drag force is computed as follows:

$$F_D=A_p{C}_D{\rho }_p{\displaystyle \frac{{(V-V_p)}^2}{2}}$$

(19)

where A_p is the particle surface, V is the flow velocity (m/s), and V_p is the particle velocity (m/s). C_D is the drag coefficient that depends on the particle’s Reynolds number (R_ep) [Equation (20)].

$$R_{ep}={\displaystyle \frac{d_p\rho (V-V_p)}{\mu }}$$

(20)

For R_ep less than 0.1, the C_D is equal to (64/R_ep); otherwise, it is constantly equal to 0.44. The gravitational force is the net buoyancy (net mass force) between the particles and the surrounding fluid mass.

$$\overrightarrow{F_G}=\overrightarrow{g}(m_p-m)$$

(21)

which can be written as follows:

$${\displaystyle \frac{\overrightarrow{F_G}}{m_p}}=(1-{\displaystyle \frac{\rho }{{\rho }_p}})\overrightarrow{g}$$

(22)

In a stationary frame, the equation of particle motion can be written as follows:

$${\displaystyle \frac{1}{m_p}}{\displaystyle \frac{d{V}_p}{dt}}={\displaystyle \frac{{(\overrightarrow{V}-\overrightarrow{V_p})}^2}{{\tau }_p}}+(1-{\displaystyle \frac{\rho }{{\rho }_p}})\overrightarrow{g}$$

(23)

where τ_p is a variable that depends on the particle diameter, drag coefficient, and particle Reynolds number.

$${\tau }_p={\displaystyle \frac{{\rho }_p{d_p}^2}{18\mu }}{\displaystyle \frac{24}{C_D{R}_{ep}}}$$

(24)

According to Stokes’ law [36,37], particles with a Stokes number less than 1 are considered passive tracers. They follow the flow without any resistance to changes in the flow direction. The Stokes number is the ratio between the particle relaxation ratio (${\tau }_p)$ and fluid time (${\tau }_f)$. It is defined as follows:

$$St={\displaystyle \frac{1}{18}}{\displaystyle \frac{{\rho }_p-{\rho }_f}{{\rho }_f}}{\displaystyle \frac{{d_p}^2}{{(\frac{{V_p}^3}{ϵ})}^{1/4}}}$$

(25)

The study of [38] introduces a variant of [Equation (25)] for a centrifugal pump:

$$St={\displaystyle \frac{{\rho }_p{V}_p{d_p}^2}{18 \mathsf{\mu }\mathrm{L}}}$$

(26)

L is the chord length of the blade. In CFTurbo, the chord length is calculated from blade loading parameters as follows:

$$L={\displaystyle \frac{l}{t}}\ast {\displaystyle \frac{t}{IML}}\ast IML=194 \mathrm{m}\mathrm{m}$$

(27)

In this study, we focus on large particles with radii ranging from 4 to 50 μm. Figure 5 shows the corresponding Stokes number. Particles with a diameter above 220 μm have a Stokes number greater than 1, corresponding to the maximum particle velocity. Figure 6 illustrates the particle density in blade sides. It figure out that blade pressure side is more affected by particle concentration.

The work of [39] studies the effect of Stokes on the exit nozzle and buckets of an impulse turbine. For high Stokes numbers like those with St above 0.1, there is an asymmetry of particle trajectories. It means that particles with a higher Stokes number resist the flow direction more than those with a lower Stokes number. Thus, the erosion damage is correlated with particle size and impact velocity. In a centrifugal pump blade, increasing the particle density increases the Stokes number and reduces the erosion rate on the blade suction side [40].

In a rotational frame, Coriolis and peripheral forces act on the particle surface. From [Equation (27)], we have the following:

$${\displaystyle \frac{d^2{x}_p}{d{t}^2}}={\displaystyle \frac{\overrightarrow{V_f}-\overrightarrow{V_p}}{{\tau }_p}}+\overrightarrow{g}(1-{\displaystyle \frac{{\rho }_f}{{\rho }_p}}){\rho }_p+{\overrightarrow{F}}_R$$

(28)

with

$$\overrightarrow{F_R}=m_p(-2\overrightarrow{w}\times \overrightarrow{V_p}-\overrightarrow{w}\times \overrightarrow{w})-m_f(-2\overrightarrow{w}\times \overrightarrow{V_f}-\overrightarrow{w}\times \overrightarrow{w})$$

(29)

To solve [Equation (29)], the DPM (Discrete Phase Model) is used under the following assumptions:

• No mass or heat transfer occurs between the particles and the flow.
• The particle volume does not affect the fluid direction.
• The diameter and release position (center of the face) remain constant.

The RNG turbulence k-ε model was used with a rotating mesh for CFD analysis. The boundary condition between the particle and the wall is assumed to be a perfect bounce. The particle velocity remains constant before and after the wall attack.

2.2. Methodology

The Euler–Lagrangian method is used to track particle trajectories in the blade channel in turbulent flow. In pump mode, particles are injected at the interface between the inlet stator and blades, whereas they are injected from the volute extension interface in turbine mode.

2.2.1. Turbulence Model

The turbulence solution adopts the RNG k-ε model to solve the full 3D Navier–Stokes equations. The solution used a high precision difference with a convergence accuracy of 0.01. The turbulent dissipation rate used the upwind second-order scheme (Figure A3 in Appendix A). In the Simerics model, the Reynolds average method is employed to decompose the incompressible flow parameters into their time-averaged and fluctuating components.

$$u=\overline{u}+\tilde{u}$$

(30)

In the k-ε model, the turbulent viscosity is computed as follows:

$${\mu }_T=\rho C_u{\displaystyle \frac{k^2}{\epsilon }}$$

(31)

k is the turbulent kinetic energy, C_u = 0.09, and ε is the dissipation ratio that helps to solve the transport equations developed in [41]. The difference between the standard k-ε and RNG k-ε models lies in the effect of the strain tensor. The RNG model may be more accurate than the standard K-Epsilon model for flows with stagnation and separation (case of turbine mode). The following two additional constants are hard-coded for the RNG model: n0 = 4.38 and β = 0.02. Additional constants are given in

Table 6. RNG constants for CFD simulation in Simerics.

Model	RNG
Cmu	0.085
C1	1.42
C2	1.68
E	9.54
Karman Constant	0.41
Turbulent Kinetic Energy	0.7194
Turbulent Dissipation Rate	0.9174
Turbulent Viscosity Ratio	100,000
Turbulent Viscosity Relaxation	0

.

2.2.2. Erosion Simulation

The slurry erosion mechanism depends on drag forces and particle concentration. Large particles resist the flow direction, whereas high particle concentration increases the number of hit particles. In this study, we simulate the erosion thickness based on two scenarios:

• Scenario 1: The variation in the erosion thickness with particle size.
• Scenario 2: The variation in erosion thickness with particle flow rate.

The parameters for each scenario are summarized in the

Table 7. Erosion thickness simulation scenarios.

First scenario: Changing the individual particle’s mass	The number of released particles is constant: 100 Time definition: Rotational mesh Number of time steps = 1800 Number of revolutions = 5 Release mode: Every time step Release position: Cell face center The particle radius is a vector of [4 5 6 7 8 9 10 20 30 40 50] μm
Second scenario: Changing the number of released particles	The particle radius is constant: rp = 10 μm Time definition: Rotational mesh Number of time steps = 1800 Number of revolutions = 5 Release mode: Each time step Release position: Cell face center The number of particles is variable: Np = [200 300 400]

below.

3. Results

3.1. The Effect of Surface Roughness on Erosion Thickness

As the roughness height (Ra) increases, the friction factor increases with increasing Reynolds number. A previous study [10] concluded that a modified pump with a low roughness (R_a) performs better than a pump with a high R_a [42,43].

For this work, two simulations were conducted: one in which the blade surface was assumed to be smooth (R_a = 0). In the second simulation, we consider R_a = 20 μm. The particle mass is computed from

$$Individual Particle mass={\displaystyle \frac{Total eroded particle mass (\mathrm{k}\mathrm{g})}{number of particles (N_p)}}$$

(32)

For these simulations, N_p = 100. To simplify the calculations, the particle shape is assumed to be spherical. According to [44], the particle size is calculated as follows:

$$d_p={\displaystyle \frac{6}{S_V}}$$

(33)

where

$$S_V={\displaystyle \frac{External surface}{particle volume}}$$

(34)

From Equations (33) and (34), for a spherical particle, the particle size is defined by its diameter.

The erosion thickness is the wear damage or material mass loss in the flow normal direction. It is related to material density by this equation:

$$Erosion thickness ({\displaystyle \frac{\mathsf{\mu }\mathrm{m}}{\mathrm{h}}})={\displaystyle \frac{Erosion density (\frac{\mathrm{k}\mathrm{g}}{{\mathrm{m}}^2\mathrm{s}})\ast 3600\ast {10}^6}{{\rho }_m}}$$

(35)

The maximum erosion rates in both scenarios are summarized in [

Table 8. Effect of roughness height on blade thickness loss. As R_a increases, ER increases.

	Ra = 0 μm	Ra = 20 μm
Maximum erosion thickness (μm) at t = 80 ms	${2\ast 10}^{-12}$	${8\ast 10}^{-110}$

].

Running CFD visualization for r_p = 1 μm, one can see that the blade pressure side is more impacted by particle concentration.

3.2. Erosion Thickness on the Blade Sides During Pump Mode

(1)

The effects of the particle inflow rate and size on the erosion thickness

A particle size (diameter) above 25 μm causes more serious erosion–cavitation problems than smaller particles [45]. The impact of particle parameters was studied by [46]. The authors concluded that the shape, velocity, and diameter of particles influence the erosion rate. However, triangular particles are more likely to erode the material than spherical particles. However, the erosion rate changes as an exponential function of the particle velocity [46]. Small particles have a greater impact on walls where there is no potential cavitation. Large particles are impacted by secondary flow and the mainstream [47]. In the work of [48], the fluid viscosity increases with fine (small-sized) particles. An increase in fluid viscosity decreases the attack (impingement) angle. Owing to other factors, such as the particle size and rotational frame, the erosion rate increases for small particles [49]. The abrasive particles have a greater impact on the pressure side [50,51]. In this simulation, we change the particle radius to simulate the erosion thickness on the blade surfaces of the centrifugal pump.

As the particle radius increases, the erosion rate increases. However, owing to the inertia of the particle, the erosion rate decreases when the particle radius exceeds 8 μm. The maximum erosion thickness loss is reached with r_p = 40 μm, as illustrated in [Figure 7].

Small particles have longer residence times. Compared with large particles, they resist flow deviation in the impeller wheel. They cause uniform erosion, whereas large particles cause local erosion. [Figure 8] shows the particle density distributions for r_p = 4 µm and r_p = 10 µm.

The small-radius particles are uniformly distributed in the blade channel, leading to greater contact with the blade material. According to Stokes’ law, as particle radius increases, the particle drag forces allow them to resist the flow directions. This is what we can see in Figure 8: particles with a radius of 10 μm tend to stay parallel with the blades. The pressure side of the blade is impacted more by the presence of solid particles than the suction side. For a fixed particle size, the erosion thickness is linear with the particle flow released. As the particle radius increases, the slope of the linear curve increases [Figure 9].

In turbine mode, the flow characteristics are different. The flow enters with high velocity from the volute section and attacks the blade outlet with an angle different from the blade outlet angle. This causes different flow mechanisms in the pump outlet mode because the flow leaves in parallel to the blade. An improved method is to reduce the rotational speed when working in turbine mode to keep high efficiency.

In turbine mode, the flow impinges on the blade inlet at a different angle. The incidence losses depend on the variation between the flow angle and the blade angle. By reducing the rotational speed, the absolute velocity is reduced, which reduces the flow angle incidence.

$$i={\beta }_{fl}-{\beta }_{bl}$$

(36)

Reducing the incidence angle (i) helps achieve high efficiency. In Figure 10, we reduced the rotational speed from the design point to 2300 RPM. The hydraulic efficiency was improved.

The erosion thickness is presented in [Figure 11]. The maximum erosion thickness is reached with a particle radius of 6 μm; then it decreases. The erosion thickness tends to increase for large particles. On the other hand, small particles reside for a longer time in the impeller [Figure 12]. This means that a small particle can cause more problems for a high-head pump.

The particle density is not uniform along the blade length. As r_p increases, the particles accelerate to a nonuniform distribution. To validate the simulation results, we compared our model with the model developed by [51]. The erosion rate is linear with the particle size for 10 μm < r_p < 40 μm; then it increases for r_p = 50 μm. The variation in erosion thickness with particle flow is linear [Figure 13]. As the particle radius increased and N_p increased, the erosion thickness increased. The particle velocity magnitude [Figure 14] is high on the pressure side.

In [15], the particle velocity is high at the blade pressure side. Furthermore, seawater arrives at the incidence location (blade TE) at a high velocity, increasing the erosion potential. The simulation is repeated for different particle inflow rates, showing that the pressure side in turbine mode is more likely to have a high erosion rate. From [Figure 15], it is clear that even in turbine mode, particles have high velocity in the middle of the blade length.

From the thickness loss, one can compute the expected pump blade lifetime from [Equation (37)]. For particles with a radius of 50 μm or greater, blade lifetime is significantly reduced.

$$ER\ast t=s$$

(37)

[

Table 9. Estimated blade lifetimes with different particle radii.

Particle Radius (μm)	Blade Lifetime (h)
10	100,000
50	2000
100	134

] summarizes the expected blade lifetime with different particle radii.

3.3. CFD Visualization of Particle Flow in Pump and Turbine Modes

For the pump mode, small particles stick to the suction and pressure sides of the blades, whereas for large particles, they stay on the pressure side. The effect of the Stokes number is clearly seen. For a particle radius of 100 μm, the particles are parallel to the impeller blade, resisting the flow passage directions and causing higher erosion in the blade surface. It can be seen from the temporal evolution of the erosion rate. Initially, blades are attacked by high particle density and stay constant, which means that particles are tangential to the blade surface. Smaller particles follow flow directions. Due to flow turbulence, the energy will increase due to the increase in hit particles in the normal and tangential directions.

Due to the inertia of particles, small particles will stay in contact with the blade surface and attack it continuously, whereas large particles cause high erosion thickness, but they stay parallel to the blade surface, causing less impingement angle. The Figure 16 describes the temporal variation in erosion thickness, showing a logarithmic trend for large particles and an exponential trend for small particles. In Turbine mode, the particle distribution I is different than pump mode. However, at a particle radius of 100 μm, particles still follow the flow pathline. It means that the Stokes number in turbine mode is much higher than in pump mode [Figure 17].

To compare erosion thickness in the turbine and pump modes, we ran a scenario with r_p = 100 μm and N_p = 400. Erosion is greater in turbine mode than in pump mode [

Table 10. Comparison of the erosion thickness between the pump mode and the turbine mode.

	Pump Mode	Turbine Mode
Maximum erosion thickness (μm/h), Np = 400	${4\ast 10}^{-4}$	${7\ast 10}^{-4}$

]. For small particles, the erosion in pump mode is greater than that in turbine mode (we operate at a higher flow rate than in pump mode, q > 1) because of the high vorticity magnitude at the blade inlet and outlet, which causes secondary flows.

From CFD visualization, it is clear that particle radius had a greater impact on their trajectory in the blade channel. Blade manufacturers should have a good idea from these results. If the working fluid has a high sediment concentration with large particles, the blade coating is different from that of usual blade manufacturing. On the impeller side, as it is a rotating machine, we have an additional velocity component, which is the relative velocity. In this case, the particle impact velocity is the particle velocity minus the relative tangential velocity. In turbine mode, the magnitude of the relative tangential velocity is greater than in pump mode. As the erosion rate is proportional to the square of particle impact velocity, the erosion rate in turbine mode should be higher than in pump mode.

4. Discussion

The present concentration of solid particles in the centrifugal impeller working fluid causes blade thickness loss. The thickness loss is linear with particle flow rate for both pump and turbine modes. In the case of particle radius changes, the thickness loss is not correlated with particle size. However, changing the particle size changes the drag forces, which affect the particle trajectory. Large particles have higher kinetic energy than small particles, which may cause them to leave the impeller wheel faster.

The particle kinetic energy and flow pattern are the most important parameters in erosion and particle trajectory. However, in pump mode, particles with a size of 100 μm and above are parallel to the blade channel, whereas in turbine mode, these particles follow the flow streamline. Thus, in a centrifugal pump, the Stokes number in turbine mode is higher compared to pump mode.

This study was limited to spherical particles with the Discrete Phase Model (DPM). Applied work can use a two-phase model to better predict erosion thickness.

5. Conclusions

In this work, we studied the effects of particle size and flow rate on the erosion thickness in a centrifugal pump operated in pump and turbine modes. In both modes, the erosion thickness is linear with particle flow rate, whereas it depends on drag forces with particle size (radius). The temporal evolution of the erosion thickness depends on particle size. Particles with a radius below 50 μm follow an exponential evolution. On the other hand, they follow a logarithmic evolution.

The erosion thickness is higher on the blade pressure side than on the suction side. In pump mode, for a particle with a radius of 100 μm, the particle follows the blade channel and resists the flow direction. In turbine mode, these particles still follow the flow path. One can consider that the Stokes number in turbine mode is higher than in pump mode.

For a particle radius of 50 μm and above, the blade lifespan is significantly reduced. Further work should investigate the effect of seawater viscosity on erosion thickness and define Stokes number limits in pump and turbine modes, including dynamic boundary layer conditions.