Analysis of Sediment Erosion in Pelton Nozzles and Needles Affected by Particle Size

: The sediment erosion of Pelton turbine components is a major challenge in the operation and development of high-head water resources, especially in mountainous areas with high sediment yield. In this paper, a study using numerical simulation was conducted with different sediment particle sizes in the fine sand range. And the erosion mechanism of the Pelton turbine injector was analyzed. The Eulerian Lagrange method was adopted to simulate the gas–liquid–solid flow. The Mansouri’s model was applied to estimate the injector erosion. The predicted erosion results were in accord with field erosion photographs. In particular, the asymmetrical erosion distribution on the needle surface was physically reproduced. With the sediment particle size increasing from 0.05 mm, the needle erosion rate decreased, while the nozzle casing erosion rate increased dramatically. In order to clarify this tendency, the characteristics of the three-phase flow were analyzed. Interestingly, the results show that with the rise in particle size, the separation of particles and water streamlines became more serious in the contraction section of the nozzle mouth. Consequently, it caused the enhancement of erosion of the nozzle surfaces and weakened the erosion of the needle surfaces. Significant engineering insights may be provided for weakening Pelton injector erosion with needle guides in the current study.


Introduction
Hydraulic turbines operating on rivers with heavy sediment, especially Pelton turbines, are subject to sediment erosion and cavitation erosion, which leads to deformation and damage of flow components and induces problems such as efficiency loss and vibration, which will threaten the safe operation of the turbines and cause huge economic losses [1][2][3].Pelton turbines have the advantages of simple structure, high efficiency and a wide range of applicable head, so they are widely applied in the development of high-head and low flow rate hydropower resources.Suitable for super high water head and wide range applications (30-3000 m), Pelton turbines are deemed to play a crucial part in exploiting untapped potentials [4].At the same time, Pelton turbines can be used in the application of ternary units, such as the Kops II HPP in Austria, in which Pelton turbines enable the optimal control range to be between zero and full load without major loss of efficiency.The ternary units can be widely employed in super-high-head pumped storage units [5].One of the major challenges in employing Pelton turbines in heavy sediment regions is the sediment the major challenges in employing Pelton turbines in heavy sediment regions is the sediment erosion of the hydraulic flow components, which often leads to efficiency loss and results in lower electricity generation [6,7].The nozzle and needle are the core components of the injector.Although their structures are simple, they are the main devices used to control the Pelton turbine flow rate [8].When the flow channel contracts, the accelerated highvelocity jet will allow sediment erosion to occur easily, making it one of the most severe erosion parts in the unit [9,10].Due to the complex flow behaviors in a Pelton turbine, when the flow contains sediment particles, it is difficult to carry out experiments to observe the motion of particles with small diameters and low concentrations [11].At present, there are several scholars adopting indirect methods to study particle flow behavior by studying erosion characteristics.The more common methods include suspension testing [12], erosion measurement [13][14][15] and the establishment of erosion prediction models [16].
The early studies on sediment erosion in Pelton turbines were mainly experimental.Bajracharya et al. [17] presented the profile of an eroded injector and analyzed the needle surface erosion distribution in detail.Padhy et al. [7] studied the influence of particlesʹ physical parameters on the Pelton turbine sediment erosion experimentally and found that the increase in particle size and concentration would increase erosion.Rai et al. [3,16] expanded Padhy et al.'s experimental research and studied the influence of particle size and other factors on Pelton turbine components' erosion.It was found that a rise in particle diameter did not always increase the erosion rate, and when the particle size was larger than the critical value, the erosion rate would decrease, as shown in Figure 1, but there was still no reasonable explanation.Moreover, in this sediment particle size range, the erosion law is also the most controversial.With the development of CFD, great progress has been made in the sediment-carrying flow simulation method of Pelton turbines.For the nozzle and needle of Pelton turbines, the VOF and Lagrange methods have been successfully applied to their internal flow with sediment simulation and erosion prediction [22,23].Bajracharya et al. [17] conducted a long-term tracking measurement and analysis of sediment erosion in Chilime HPP and presented the quantitative relationship between the erosion amount of the flow components and the turbine efficiency decline.Ge et al. [24] analyzed sediment erosion characteristics of a Pelton turbine numerically and experimentally.Xiao et al. [25][26][27][28] studied the three-phase flow behaviors in a Pelton turbine and analyzed the motion process of sediment particles in it separately.Leguizamon et al. [29] developed a three-phase flow With the development of CFD, great progress has been made in the sediment-carrying flow simulation method of Pelton turbines.For the nozzle and needle of Pelton turbines, the VOF and Lagrange methods have been successfully applied to their internal flow with sediment simulation and erosion prediction [22,23].Bajracharya et al. [17] conducted a long-term tracking measurement and analysis of sediment erosion in Chilime HPP and presented the quantitative relationship between the erosion amount of the flow components and the turbine efficiency decline.Ge et al. [24] analyzed sediment erosion characteristics of a Pelton turbine numerically and experimentally.Xiao et al. [25][26][27][28] studied the threephase flow behaviors in a Pelton turbine and analyzed the motion process of sediment particles in it separately.Leguizamon et al. [29] developed a three-phase flow simulation for Pelton turbines based on the Lagrange method and indirectly verified the calculated Energies 2024, 17, 1635 3 of 15 results with measured erosion values.Tarodiya et al. [30,31] conducted erosion simulations of a Pelton turbine injector using CFD-DEM.Liu et al. [32] analyzed sediment erosion using numerical simulations.
Sediment erosion in turbines is an instantaneous and multiphase complex phenomenon, which is affected by the interaction of various parameters, such as particle size, density, hardness, shape, fluid viscosity and density [33][34][35][36][37], as well as the particle impact properties on the surface, that is, the impact angle, impact velocity and particle impact number [38][39][40].Due to the difference in inertia force and drag force, particles with different sizes have different performances when exchanging momentum with the carrier fluid, resulting in differences in particle distribution, trajectory, impact angle and impact velocity, which thereby affect the erosion behaviors [41].
In this paper, the Eulerian Lagrange method combined with the VOF model was adopted to simulate the three-phase flow behaviors and erosion patterns for the Pelton nozzle and needle.Three typical medium particle sizes were selected in Figure 1 for numerical calculation and analysis, focusing on the influence of sediment particle size on the nozzle and needle erosion characteristics.Finally, through comprehensive analysis of the change in impact angle, impact number and impact velocity, the sediment erosion patterns of the nozzle and needle were quantitatively predicted and compared with the measured results.Interestingly, the results show that the opposite erosion variation law appears on the surface of the needle and nozzle casing with a rise in particle size.The novelty of this paper is that through the study of the three-phase flow behaviors in a Pelton turbine injector, the effect of sediment particle size on the nozzle and needle is disclosed.With the rise in particle size in the range (0.05, 0.25) mm, the average erosion rate of the needle gradually declines, while the average erosion rate of the nozzle casing gradually increases, which is significantly different from the previous general understanding.It is significant to study the sediment erosion characteristics of Pelton turbines to improve the turbine efficiency and operating stability of the units.

Computational Domain and Mesh
A Pelton injector with an additional blunt body was investigated in this paper, and its parameters are shown in Figure 2 and Table 1.The computational domains including the inlet pipe, injector and air domain are shown in Figure 2a.The diameter of the nozzle outlet is defined as d 0 .The length of the inlet pipe was extended to 15 d 0 to make the water flow rate of the inlet stable.The length of the air domain located downstream of the nozzle outlet is 6 d 0 .The air domain only needs to capture the jet flow pattern, and the diameter value of 1.4 d 0 is sufficient for this purpose.Moreover, 2 profiles of the needle surface and nozzle surface are labeled as "BS" and "CS", respectively, as shown in Figure 2a.Two profiles in the potential erosion regions are applied in the quantitative analysis.To describe the injector opening, the stroke ratio (s n ) defined as S N /d 0 is often used, as presented in Figure 2b, where S N is the needle travel distance.

Numerical Mesh
In order to verify the mesh independence of sediment erosion prediction, this study generated four different sets of mesh.Mansouri's erosion model [42] was adopted to estimate the erosion rate of the injector.As shown in Figure 3a, a mesh independence study is conducted.Its y-ordinate is the relative erosion rate, defined as the erosion rate divided by that of 5.32 million.It shows that the relative erosion rate of the needle does not change significantly beyond 5.32 million cells (circled in red).Consequently, the mesh number of 5.32 million was adopted in Figure 3b.Structured mesh is used in the nozzle regions that are near the outlet.To perform better measurements at the air-water interface, structured mesh is used in the air domain.Local mesh refinement was used in the area close to the needle tip and at the air-water interface, which allowed a more accurate prediction at the water-air interface before entering the rotating bucket domain, as shown in Figure 3c.The y+ of the nozzle and needle walls we are interested in is less than 40.

Numerical Mesh
In order to verify the mesh independence of sediment erosion prediction, this study generated four different sets of mesh.Mansouri's erosion model [42] was adopted to estimate the erosion rate of the injector.As shown in Figure 3a, a mesh independence study is conducted.Its y-ordinate is the relative erosion rate, defined as the erosion rate divided by that of 5.32 million.It shows that the relative erosion rate of the needle does not change significantly beyond 5.32 million cells (circled in red).Consequently, the mesh number of 5.32 million was adopted in Figure 3b.Structured mesh is used in the nozzle regions that are near the outlet.To perform better measurements at the air-water interface, structured mesh is used in the air domain.Local mesh refinement was used in the area close to the needle tip and at the air-water interface, which allowed a more accurate prediction at the water-air interface before entering the rotating bucket domain, as shown in Figure 3c.The y+ of the nozzle and needle walls we are interested in is less than 40.

Sediment Particle Diameter and Concentration
According to the relation between the relative erosion rate and the particle diameter in Figure 1, the particle sizes D p of 0.05 mm, 0.15 mm and 0.25 mm were chosen for the CFD simulations.Based on general observation data, this paper selected the average sediment concentration of a hydropower station in Southwest China for numerical simulation.The value of sediment concentration was 0.2 kg/m 3 , which was the arithmetical mean value during the three-month monitoring period.The sediment particle material is quartz with a density of 2650 kg/m 3 and a Mohs hardness of 7.

Sediment Particle Diameter and Concentration
According to the relation between the relative erosion rate and the particle diameter in Figure 1, the particle sizes Dp of 0.05 mm, 0.15 mm and 0.25 mm were chosen for the CFD simulations.Based on general observation data, this paper selected the average sediment concentration of a hydropower station in Southwest China for numerical simulation.The value of sediment concentration was 0.2 kg/m 3 , which was the arithmetical mean value during the three-month monitoring period.The sediment particle material is quartz with a density of 2650 kg/m 3 and a Mohs hardness of 7.

Boundary Conditions and Erosion Model
The solver we used was the commercial code Fluent.The SIMPLE algorithm was adopted to couple the velocity and pressure fields.According to the reference [43,44], the turbulence model SST k-ω can be used to simulate the water and air flow in the Pelton turbine more accurately.Therefore, the SST k-ω turbulence model was adopted.Adopting the Eulerian Lagrange method, three-phase gas-liquid-solid flow and erosion simulations for a Pelton turbine injector were carried out.The VOF model was used to simulate water and air two-phase flow.As for boundary conditions, the outlet condition was set as a " pressure outlet " on the surface of the air domain.The inlet condition was set as "total pressure", which had a value equal to the 1000 m operating water head.The volume fraction of air was 0 near the inlet.As for the walls of the injector, the non-slip condition was applied.Steady-state simulations were conducted.
The discrete phase model (DPM) was adopted to track particles.Particles entered the computational domain from the inlet, and the velocity of particles in the inlet was set to be consistent with the velocity of water flow.The sediment mass flow rate of particles was obtained from the sediment concentration mentioned in Section 2.3.
Based on Mansouri's erosion model, the erosion rates on the needle and nozzle were obtained.The formulas for calculating the erosion rate are as follows [40].And the impact

Boundary Conditions and Erosion Model
The solver we used was the commercial code Fluent.The SIMPLE algorithm was adopted to couple the velocity and pressure fields.According to the reference [43,44], the turbulence model SST k-ω can be used to simulate the water and air flow in the Pelton turbine more accurately.Therefore, the SST k-ω turbulence model was adopted.Adopting the Eulerian Lagrange method, three-phase gas-liquid-solid flow and erosion simulations for a Pelton turbine injector were carried out.The VOF model was used to simulate water and air two-phase flow.As for boundary conditions, the outlet condition was set as a "pressure outlet" on the surface of the air domain.The inlet condition was set as "total pressure", which had a value equal to the 1000 m operating water head.The volume fraction of air was 0 near the inlet.As for the walls of the injector, the non-slip condition was applied.Steady-state simulations were conducted.
The discrete phase model (DPM) was adopted to track particles.Particles entered the computational domain from the inlet, and the velocity of particles in the inlet was set to be consistent with the velocity of water flow.The sediment mass flow rate of particles was obtained from the sediment concentration mentioned in Section 2.3.
Based on Mansouri's erosion model, the erosion rates on the needle and nozzle were obtained.The formulas for calculating the erosion rate are as follows [40].And the impact angle is calculated by Formula (4).The numerical methods are consistent with those in our previous literature [25]. (2) Energies 2024, 17, 1635

Three-Phase Flow Behaviors
The analysis of flow patterns was first carried out for different sediment particle sizes with the same needle stroke.Two axial planes at the position of 0 • and 45 • from the needle body were selected as the analysis sections, as shown in Figure 4a.Under the design operating head, the flow behaviors from the tube to the jet are shown in Figure 4b-g, which are the velocity, pressure and total pressure contours, respectively.The water flows from the inlet into the injector and then flows along the flow channel, passing by the needle guides.Near the injector outlet, the flow channel contracts; therefore, the water accelerates here and then becomes a free jet.After leaving the injector, the water flow begins to expand.

Three-Phase Flow Behaviors
The analysis of flow patterns was first carried out for different sediment particle sizes with the same needle stroke.Two axial planes at the position of 0° and 45° from the needle body were selected as the analysis sections, as shown in Figure 4a.Under the design operating head, the flow behaviors from the tube to the jet are shown in Figure 4b-g, which are the velocity, pressure and total pressure contours, respectively.The water flows from the inlet into the injector and then flows along the flow channel, passing by the needle guides.Near the injector outlet, the flow channel contracts; therefore, the water accelerates here and then becomes a free jet.After leaving the injector, the water flow begins to expand.The velocity distribution contours of the 0 • and 45 • sections are given in Figure 4b,c.Affected by the needle guides, the water flow velocity distributions of the two sections in the injector are quite different.With the accelerated flow of water from the nozzle, an obvious low-velocity zone appears in the needle tip downstream because the boundary layer continuously develops on the needle surface.According to the static pressure results distributed in the two sections, there is a high-pressure region in the needle tip downstream due to the low-velocity region, as shown in Figure 4d,e.The low-velocity region of the tip also has an effect on the velocity of the jet domain, that is, the velocity distributed near the jet center is also lower.However, along the flow direction, this influence is gradually weakened, and the velocity of the jet center far away from the needle tip becomes more uniform.The low-velocity region in the needle tip downstream also significantly reduces the total pressure in the central region of the jet, as shown in Figure 4f,g.
In order to analyze the motion law of sediment particles, four sections were selected at four positions along the axis direction, as can be seen in Figure 5a.The coordinate point 0 of this figure is the exit position of the injector.Figure 5b-d shows the velocity and particle distribution contours of four sections with different particle sizes, respectively.The left side of each figure is the projected velocity in this section, and the right side is the particle distribution in this section.
The projected velocity shows that, in the −2d 0 section, the secondary flow occurs positively downstream of each needle guide due to the presence of the needle guides.And there are two opposite vortexes at corresponding positions downstream of the needle guides.In the −d 0 section, the flow velocity component perpendicular to the axis direction is the largest, which is mainly because the section is in the shrinking channel.The contraction of the flow channel will not only accelerate the main stream but also increase the perpendicular direction flow velocity significantly.And the vortex flow at the corresponding position downstream of the needle guides is no longer obvious.As can be seen from the sections of d 0 and 2d 0 , the effect of needle guides increases on the free jet flow, and the secondary flows caused by vortexes around the four needle guides develop significantly in the downstream jet flow, causing multiple large-scale vortexes to appear in the corresponding region of each jet section.The vortexes near the outer edge of the jet interact with the air, thus affecting the shape of the water-air interface.Further downstream of the 2d 0 section, the maximum velocity of the secondary flow gradually decreases, but the shape change of the water-air interface increases gradually.
The right sides of Figure 5b-d show the particle distribution with different particle sizes in four sections.Along the flow direction, with the change in particle size, the particle distribution in each section will also change.In the −2d 0 section, the particle distribution is relatively uniform, and it is mainly different in the corresponding positions downstream of the needle guides.There is a narrow particle-sparse zone close to the blunt body, and with the particle size increasing, the needle guides will have a greater effect on the particle distributions.In the −d 0 section, the particle distribution difference is significant, and the particle-sparse zone near the blunt body also increases significantly.With the rise in particle size, the particle-sparse zone is larger.In the d 0 section, there is a particle-dense zone in the corresponding position downstream of the needle guides, while the particle-sparse zone near the blunt body is isolated into two sparse zones.With the rise in particle size, the areas of these two sparse zones gradually decrease.In the 2d 0 section, the particle concentration area in the corresponding position downstream of the needle guides becomes larger, and the two sparse zones with sparse particle density are larger than in the previous section.With the particle size increasing, the particles tend to converge to the section center gradually, and the number of particles close to the jet surface decreases significantly.Compared with the projected velocity distribution, it is observed that the change in particle distribution is mainly influenced by the secondary flow.

Predicted Erosion Distribution and Comparison
According to the simulated results of the sediment impact angle, impact number and impact velocity on the surfaces of the injector, the final erosion patterns on the injector surfaces were estimated.The typical erosion patterns are shown in Figure 6, where D p = 0.05 mm. Figure 7 shows the comparison of the predicted erosion results with the measured results from the literature.The photographs show the erosion patterns of the needle and nozzle of the Khimti Power Station (KHP-1) in the Himalayas area.Figure 7a shows the erosion of the needle, while Figure 7b shows the nozzle casing erosion.The predicted erosion distributions of the needle and nozzle show that the erosion of the needle mainly occurs on the needle tip, and severe erosion areas appear in corresponding positions downstream of the needle guides.As for the nozzle, the erosion of the contracting surface near the outlet is severe, but the erosion of the expanding surface is not severe.In general, the main erosion areas of the needle and nozzle are both near the outlet.It is interesting that the erosion distribution of the needle tip is asymmetrical.Xiao et al. and Thapa et al. [12,23,40] believe that the formation of severe erosion areas on needle tips was because of the shedding of von Kármán vortexes.The number of severe erosion areas on the needle tip is consistent with that of the needle guides, and Xiao et al. [27] explain the mechanism in detail.
centration area in the corresponding position downstream of the needle guides becomes larger, and the two sparse zones with sparse particle density are larger than in the previous section.With the particle size increasing, the particles tend to converge to the section center gradually, and the number of particles close to the jet surface decreases significantly.Compared with the projected velocity distribution, it is observed that the change in particle distribution is mainly influenced by the secondary flow.

Predicted Erosion Distribution and Comparison
According to the simulated results of the sediment impact angle, impact number and impact velocity on the surfaces of the injector, the final erosion patterns on the injector surfaces were estimated.The typical erosion patterns are shown in Figure 6, where Dp = 0.05 mm. Figure 7 shows the comparison of the predicted erosion results with the measured results from the literature.The photographs show the erosion patterns of the needle and nozzle of the Khimti Power Station (KHP-1) in the Himalayas area.Figure 7a shows the erosion of the needle, while Figure 7b shows the nozzle casing erosion.The predicted erosion distributions of the needle and nozzle show that the erosion of the needle mainly occurs on the needle tip, and severe erosion areas appear in corresponding positions downstream of the needle guides.As for the nozzle, the erosion of the contracting surface near the outlet is severe, but the erosion of the expanding surface is not severe.In general, the main erosion areas of the needle and nozzle are both near the outlet.It is interesting that the erosion distribution of the needle tip is asymmetrical.Xiao et al. and Thapa et al. [12,23,40] believe that the formation of severe erosion areas on needle tips was because of the shedding of von Kármán vortexes.The number of severe erosion areas on the needle tip is consistent with that of the needle guides, and Xiao et al. [27] explain the mechanism in detail.There is a good consistency of sediment erosion patterns between the results predicted by simulations and the field observations.The numerical simulation also reproduces the phenomenon of asymmetrical erosion of the needle tip.Severe erosion areas appearing in corresponding positions downstream of the needle guides are consistent, circled in red in Figure 7.The good agreement between the numerical results and observations provides confidence in the numerical methods used in this paper.

The Effect of Particle Size on Injector Erosion
When the particle size is different, the movement of particles in the injector is also affected differently by the water flow, so the motion trajectory of particles is different, which leads to the surface erosion patterns also changing.The erosion distributions of the nozzle and needle surfaces with different particle sizes are shown in Figure 8. Figure 8ac show that the main severe erosion regions are distributed on the contracting surface, needle tip and nozzle mouth ring.More interestingly, with the particle size increasing, the erosion rate of the needle tip decreases rapidly, while the erosion rate of the nozzle casing There is a good consistency of sediment erosion patterns between the results predicted by simulations and the field observations.The numerical simulation also reproduces the phenomenon of asymmetrical erosion of the needle tip.Severe erosion areas appearing in corresponding positions downstream of the needle guides are consistent, circled in red in Figure 7.The good agreement between the numerical results and observations provides confidence in the numerical methods used in this paper.

The Effect of Particle Size on Injector Erosion
When the particle size is different, the movement of particles in the injector is also affected differently by the water flow, so the motion trajectory of particles is different, which Energies 2024, 17, 1635 10 of 15 leads to the surface erosion patterns also changing.The erosion distributions of the nozzle and needle surfaces with different particle sizes are shown in Figure 8. Figure 8a-c show that the main severe erosion regions are distributed on the contracting surface, needle tip and nozzle mouth ring.More interestingly, with the particle size increasing, the erosion rate of the needle tip decreases rapidly, while the erosion rate of the nozzle casing increases rapidly, which is different from the conventional understanding.
circled in red in Figure 7.The good agreement between the numerical results and observations provides confidence in the numerical methods used in this paper.

The Effect of Particle Size on Injector Erosion
When the particle size is different, the movement of particles in the injector is also affected differently by the water flow, so the motion trajectory of particles is different, which leads to the surface erosion patterns also changing.The erosion distributions of the nozzle and needle surfaces with different particle sizes are shown in Figure 8. Figure 8ac show that the main severe erosion regions are distributed on the contracting surface, needle tip and nozzle mouth ring.More interestingly, with the particle size increasing, the erosion rate of the needle tip decreases rapidly, while the erosion rate of the nozzle casing increases rapidly, which is different from the conventional understanding.By means of a quantitative method, the analysis of erosion rates on the needle and the nozzle casing surfaces with different particle sizes was conducted.Then, the analysis of impact properties including the impact number nimp, impact angle θ and impact velocity vp was carried out, which was used to show the erosion mechanism of the injector with different particle sizes.Figure 2a shows two profiles of the needle surface and nozzle surface labeled as "BS" and "CS", respectively, which were selected for analysis.For convenience of description, a coordinate system is established with the position of the injector outlet as the origin, and the coordinates are dimensionless by the nozzle diameter d0, as shown in Figure 2b.The erosion was mainly located in the range of x ≈ −2d0 to 0.4d0.In Figures 9a and 10a, erosion rates with different particle sizes along these profiles are shown, respectively.Clearly, the maximal erosion rates on the nozzle casing are higher than those on the needle.To analyze the erosion distributions, the erosion-related variables of vp, θ and nimp were shown and analyzed in Figures 9b-d and 10b-d.
In Figure 9a, the tendency of erosion rate along the BS with different particle sizes is almost the same, firstly increasing and then decreasing.Furthermore, the impact properties in Figure 9b-d suggest that more particles impact with higher impact velocity and impact angle along the BS when Dp = 0.05 mm.Because the erosion rate is in proportion to vp and nimp, a larger impact number and impact velocity result in a higher erosion rate along the BS using the smallest particle size.When the sediment size Dp = 0.05 mm, the erosion rate increases slowly in the range of x ∈ (−2, −1) d0 and then increases sharply, reaching a peak value at the position x ≈ −0.07d0 and then falling down rapidly.For Dp = 0.15 mm and 0.25 By means of a quantitative method, the analysis of erosion rates on the needle and the nozzle casing surfaces with different particle sizes was conducted.Then, the analysis of impact properties including the impact number n imp , impact angle θ and impact velocity v p was carried out, which was used to show the erosion mechanism of the injector with different particle sizes.Figure 2a shows two profiles of the needle surface and nozzle surface labeled as "BS" and "CS", respectively, which were selected for analysis.For convenience of description, a coordinate system is established with the position of the injector outlet as the origin, and the coordinates are dimensionless by the nozzle diameter d 0 , as shown in Figure 2b.The erosion was mainly located in the range of x ≈ −2d 0 to 0.4d 0 .In Figures 9a and 10a, erosion rates with different particle sizes along these profiles are shown, respectively.Clearly, the maximal erosion rates on the nozzle casing are higher than those on the needle.To analyze the erosion distributions, the erosion-related variables of v p , θ and n imp were shown and analyzed in Figures 9b-d and 10b-d.
creases slowly in the range of x ∈ (−2, −1) d0 and then increases sharply, reaching a peak value at the position x ≈ −0.07d0 and then falling down rapidly.For Dp = 0.15 mm and 0.25 mm, the erosion rate increases slowly in the range of x ∈ (−2, −0.3) d0 and then increases sharply, reaching a peak value at x ≈ −0.04d0 and then falling down rapidly.It can be seen by comparing the results of using different particle sizes that with sediment particle size increasing, the overall erosion rate and its maximum value decrease gradually.In Figure 10a, the tendency of erosion rate along the CS is almost the same with different particle sizes; firstly, it is almost zero, then increases rapidly and, finally, decreases sharply to 0 along the flow direction, which is in good agreement with the field photograph in Figure 7b.As for the particle impact number, the trend of different particle sizes is similar, generally increasing first and then decreasing.With sediment particle size increasing, the particle impact number generally increases gradually.As for the impact velocity, the trend of different particle sizes is nearly the same, generally increasing first and then decreasing sharply to 0, and with different sediment particle sizes, the impact velocities are almost the same.As for the impact angle, the trend of different particle sizes is almost the same, generally decreasing to 0. With sediment particle size increasing, particle impact angle generally decreases gradually.Figure 10b-d show that during flow channel expansion, particle impact number, impact velocity and impact angle drop to nearly 0, therefore reducing the erosion significantly.When the sediment size Dp = 0.25 mm, the erosion rate increases slowly in the range of x ∈ (−2, −0.5) d0 and then increases sharply, reaching a peak value at x ≈ −0.08d0 and then falling down rapidly.For Dp = 0.05 mm and 0.15 mm, the erosion rate increases slowly in the range of x ∈ (−2, −0.0.5) d0 and then increases sharply, reaching a peak value at x ≈ −0.04d0 and then falling down rapidly.With sediment particle size increasing, the overall erosion rate and its maximum value increase gradually.In Figure 10a, the tendency of erosion rate along the CS is almost the same with different particle sizes; firstly, it is almost zero, then increases rapidly and, finally, decreases sharply to 0 along the flow direction, which is in good agreement with the field photograph in Figure 7b.As for the particle impact number, the trend of different particle sizes is similar, generally increasing first and then decreasing.With sediment particle size increasing, the particle impact number generally increases gradually.As for the impact velocity, the trend of different particle sizes is nearly the same, generally increasing first and then decreasing sharply to 0, and with different sediment particle sizes, the impact velocities are almost the same.As for the impact angle, the trend of different particle sizes is almost the same, generally decreasing to 0. With sediment particle size increasing, particle impact angle generally decreases gradually.Figure 10b-d show that during flow channel expansion, particle impact number, impact velocity and impact angle drop to nearly 0, therefore reducing the erosion significantly.When the sediment size Dp = 0.25 mm, the erosion rate increases slowly in the range of x ∈ (−2, −0.5) d0 and then increases sharply, reaching a peak value at x ≈ −0.08d0 and then falling down rapidly.For Dp = 0.05 mm and 0.15 mm, the erosion rate increases slowly in the range of x ∈ (−2, −0.0.5) d0 and then increases sharply, reaching a peak value at x ≈ −0.04d0 and then falling down rapidly.With sediment particle size increasing, the overall erosion rate and its maximum value increase gradually.Table 2 shows the average erosion rate, vp, nimp and θ of the needle surface and nozzle casing with different particle sizes, respectively.In Figure 9a, the tendency of erosion rate along the BS with different particle sizes is almost the same, firstly increasing and then decreasing.Furthermore, the impact properties in Figure 9b-d suggest that more particles impact with higher impact velocity and impact angle along the BS when D p = 0.05 mm.Because the erosion rate is in proportion to v p and n imp , a larger impact number and impact velocity result in a higher erosion rate along the BS using the smallest particle size.When the sediment size D p = 0.05 mm, the erosion rate increases slowly in the range of x ∈ (−2, −1) d 0 and then increases sharply, reaching a peak value at the position x ≈ −0.07d 0 and then falling down rapidly.For D p = 0.15 mm and 0.25 mm, the erosion rate increases slowly in the range of x ∈ (−2, −0.3) d 0 and then increases sharply, reaching a peak value at x ≈ −0.04d 0 and then falling down rapidly.It can be seen by comparing the results of using different particle sizes that with sediment particle size increasing, the overall erosion rate and its maximum value decrease gradually.
In Figure 10a, the tendency of erosion rate along the CS is almost the same with different particle sizes; firstly, it is almost zero, then increases rapidly and, finally, decreases sharply to 0 along the flow direction, which is in good agreement with the field photograph in Figure 7b.As for the particle impact number, the trend of different particle sizes is similar, generally increasing first and then decreasing.With sediment particle size increasing, the particle impact number generally increases gradually.As for the impact velocity, the trend of different particle sizes is nearly the same, generally increasing first and then decreasing sharply to 0, and with different sediment particle sizes, the impact velocities are almost the same.As for the impact angle, the trend of different particle sizes is almost the same, generally decreasing to 0. With sediment particle size increasing, particle impact angle generally decreases gradually.Figure 10b-d show that during flow channel expansion, particle impact number, impact velocity and impact angle drop to nearly 0, therefore reducing the erosion significantly.When the sediment size D p = 0.25 mm, the erosion rate increases slowly in the range of x ∈ (−2, −0.5) d 0 and then increases sharply, reaching a peak value at x ≈ −0.08d 0 and then falling down rapidly.For D p = 0.05 mm and 0.15 mm, the erosion rate increases slowly in the range of x ∈ (−2, −0.0.5) d 0 and then increases sharply, reaching a peak value at x ≈ −0.04d 0 and then falling down rapidly.With sediment particle size increasing, the overall erosion rate and its maximum value increase gradually.
Table 2 shows the average erosion rate, v p , n imp and θ of the needle surface and nozzle casing with different particle sizes, respectively.The averaged erosion rates of the nozzle casing are about ten times those of the needle surface, except for the case where D p = 0.05 mm.With the rise in sediment size, the average erosion rate on the needle tip will gradually decrease, while the nozzle casing's rate will gradually increase.It is interesting that, for the Pelton injector, the erosion variation law on the needle surface and nozzle casing is opposite to the rise in particle size, which is different from the previous general understanding, as shown in Figure 1.With the rise in sediment size, the impact number on the needle surface decreases gradually, while on the nozzle casing, it increases significantly.The impact number on the nozzle casing is about five times that of the needle surface at D p = 0.05 mm and increases to more than 20 times at D p = 0.25 mm, while their impact angles are almost the same.Furthermore, Figure 10b-d suggest that more particles impact with a higher impact angle along the CS when the D p = 0.25 mm.Because the erosion rate is in proportion to θ and n imp , when the value of θ is small, a larger impact number and impact angle result in a higher erosion rate along the CS with the largest particle size.As shown in Table 2, average erosion rates along the CS are bigger than those along the BS.In addition, since severe erosion regions on the nozzle casing surface are mainly located in a small area whose range is x ∈ (−0.2, 0) d 0 , it is harder to visualize the asymmetrical erosion of the nozzle casing surface.

Conclusions
The erosion simulation for a Pelton turbine injector has been carried out, and the influence of sediment size on erosion characteristics of the nozzle and needle was discussed in detail.Meanwhile, the flow process of water and the influence of secondary flow on the distribution of particles were analyzed, respectively.
In the sediment size range (0.05, 0.25) mm, as the particle size increases, the surfaces of the needle and nozzle casing show opposite variation trends in erosion rate.With the rise in particle size, the average erosion rate of the needle gradually declines, while the average erosion rate of the nozzle casing gradually increases.Interesting results show that, as for the Pelton injector, the opposite result appears in the erosion variation law on the surface of the needle and nozzle casing with the rise in particle size, which is significantly different from the previous general understanding.The increase in particle size would strengthen the tendency of particle separation in the internal flow of the injector mechanism, which dramatically enhances the impact frequency on the nozzle casing and reduces the impact frequency on the needle surface.Thus, the erosion rate of the nozzle casing and needle is lifted and lowered, respectively.
The sediment particle size varies among different river basins, and sediment erosion of the injector of the Pelton turbine is commonly observed.Sediment erosion causes damage to the surface structure of flowing components and deteriorates the jet quality, affecting the overall efficiency of the unit.Aiming at the goal of reducing the efficiency loss caused by erosion, the results of this study may provide a reference for further methods to be conducted to alleviate sediment erosion, such as focusing on suppressing the erosion caused by particles with smaller sizes, as well as applying surface coatings to areas with severe sand erosion.

Figure 2 .
Figure 2. Schematic diagrams of a Pelton turbine injector and the simulation domain.

Figure 2 .
Figure 2. Schematic diagrams of a Pelton turbine injector and the simulation domain.

Figure 3 .
Figure 3. Mesh for the computational domain.

Figure 3 .
Figure 3. Mesh for the computational domain.

Figure 4 .
Figure 4. Distribution of the flow from the tube to the jet.The velocity distribution contours of the 0° and 45° sections are given in Figure4b,c.Affected by the needle guides, the water flow velocity distributions of the two sections in the injector are quite different.With the accelerated flow of water from the nozzle, an obvious low-velocity zone appears in the needle tip downstream because the boundary layer continuously develops on the needle surface.According to the static pressure results distributed in the two sections, there is a high-pressure region in the needle tip downstream

Figure 4 .
Figure 4. Distribution of the flow from the tube to the jet.
(a) Section along the injector flow direction (b) Particle size of 0.05 mm (c) Particle size of 0.15 mm (d) Particle size of 0.25 mm

Figure 5 .
Figure 5. Flow and particle distribution of the injector and jet sections for different particle sizes.Figure 5. Flow and particle distribution of the injector and jet sections for different particle sizes.

Figure 5 .
Figure 5. Flow and particle distribution of the injector and jet sections for different particle sizes.Figure 5. Flow and particle distribution of the injector and jet sections for different particle sizes.

Figure 6 .
Figure 6.Sediment erosion distribution at needle tip and nozzle, where Dp = 0.05 mm.Figure 6. Sediment erosion distribution at needle tip and nozzle, where Dp = 0.05 mm.

Figure 8 .
Figure 8. Erosion distributions of the needle and nozzle casing with different particle sizes.

Figure 8 .
Figure 8. Erosion distributions of the needle and nozzle casing with different particle sizes.

Figure 9 .
Figure 9. Erosion-related variables with different particle sizes on needle surfaces.

Figure 9 .
Figure 9. Erosion-related variables with different particle sizes on needle surfaces.

Figure 9 .
Figure 9. Erosion-related variables with different particle sizes on needle surfaces.

Figure 10 .
Figure 10.Erosion-related variables with different particle sizes on nozzle casing.

Figure 10 .
Figure 10.Erosion-related variables with different particle sizes on nozzle casing.

Table 1 .
Parameters of the injector and Pelton turbine.

Table 1 .
Parameters of the injector and Pelton turbine.

Table 2 .
The average erosion rate, vp, nimp and θ with different particle sizes.

Table 2 .
The average erosion rate, v p , n imp and θ with different particle sizes.