The influence mechanism of asymmetric distribution characteristics of erosion in the injector of Pelton turbines

To analyze the influence mechanism of asymmetric erosion distribution characteristics in the injector of the Pelton turbine, a detailed analysis of erosion amount and distribution characteristics within different regions of the nozzle and needle is conducted, elucidating the motion and distribution characteristics of sediment particles. The Euler-Lagrange method is used to numerically calculate the flow process of sediment particles with a concentration of 2% and diameters of 0.1mm, 0.5mm, 0.75mm, 1mm, and 2mm in the injector, and the erosion amount is estimated. The results show that the sediment particle diameter significantly influences the symmetry of erosion distribution in the injector. When the diameter gradually increases, the symmetrical phenomenon gradually disappears. Furthermore, the influence of sediment particle diameter on erosion degree within different regions of the injector is inconsistent. With the increase of sediment particle diameter, the average erosion rate of the needle increases. While the average erosion rate in the guide plate area of the nozzle decreases at first and then increases, the nozzle throat is reversed. The guide plate affects the distribution and movement of sediment particles at its end, aggravating the asymmetric erosion phenomenon at the nozzle, especially on the small particle size sediment.


Introduction
Hydropower is the main force to promote the transformation of energy structure under the background of the "Carbon Peak and Carbon Neutrality" strategy [1].There is great potential for the development of hydropower resources in the Southwest region of China, mainly high-head resources.The Pelton turbine is widely used in high-head hydropower plants as the main turbine model for the lower reaches of the Yarlung Zangbo River in the southwest region [2].It offers advantages such as a high application head (200-2000m), a wide range of applicable flow rates (1-60m 3 /s), and a broad operating efficiency range [2].However, during the actual operation of Pelton turbines, sediment particles flow through the injector and are accelerated and ejected subsequently, leading to erosion that can affect the normal operation of the turbine.
In 2014, Cao et al. [3] discovered that erosion easily occurs at the nozzle due to a increase in fluid velocity and pressure, with the erosion being more severe on the underside of the nozzle due to the influence of gravity.In 2015, Zeng et al. [4] conducted three-dimensional unsteady numerical simulations of the water-air-sediment flow in the injector, investigating the influence of sediment particle diameter on entrainment characteristics and erosion characteristics.In 2019, Messa et al. [5] studied the erosion conditions of the needle and nozzle at different openings and found more severe erosion on the needle under small opening conditions.In 2020, Guo et al. [6] discovered asymmetric erosion on the surface of the needle while analyzing the relationship between sediment particle impact characteristics and erosion rate.In the same year, Ge et al. [7] conducted numerical and experimental research on the erosion characteristics of the Pelton turbine's injector, exploring the influence of particle size and concentration on erosion properties.In 2021, Xiao et al. [8] conducted numerical research on the water-air-sediment flow inside the Pelton turbine's injector, predicting the sediment erosion conditions on the injector and bucket surface.In 2021, Han et al. [9] used the Euler-Lagrange method to investigate the erosion characteristics of various components of the Pelton turbine.According to the simulation results, they studied the erosion characteristics under different flow conditions and particle properties.In 2022, Deng et al. [10] carried out flow simulations of the Pelton turbine's injector, analyzing the influence of the number of guide plate and the outlet angle of the guide plate on the flow and jet characteristics of the injector.In the same year, Rahul Tarodiya et al. [11,12] also observed the asymmetric erosion phenomenon in the injector.
In summary, scholars from both domestic and international have conducted varying research on the influence of sediment particle diameter, concentration, and nozzle opening on the erosion characteristics of the injector.However, the mechanism behind the asymmetrical distribution of erosion in the nozzle remains unclear.Based on the observed erosion patterns at actual power stations, both the nozzle and needle exhibit asymmetrical erosion.Therefore, this paper studies the influence mechanism of erosion asymmetry of the injector, providing a theoretical reference for the structural design of the multi-nozzle Pelton turbine.

Geometric model
The flow components of the Pelton turbine consist of the distributor, nozzle, and runner.The feedwater mechanism, which comprises the distributor and nozzles, allows the flow of water to enter the nozzles at a specific angle, generating a certain flow rate in the main section of the distributor.The flow rate entering the turbine can be adjusted by the movement of the needle, while energy exchange occurs in the contraction section of the nozzle.This process produces a high-speed free jet that impacts the buckets.In this model turbine, the design head is 100m, the unit flow rate is 240L/s, and the unit speed is 39.6r/min.The nozzle has an inlet diameter of 102mm, and the diameter of the runner's base circle is 312.5mm.There are six nozzles and 21 buckets in total.For calculations, an opening of 13.5mm is chosen, and a cylindrical domain is used to replace the jet at the nozzle outlet.The geometric model of the distributor is shown in figure 1.

Computation mesh
Considering the extreme angle of the nozzle and the needle of the Pelton turbine, this study uses the polyhedral mesh in the unstructured grid to mesh the model.The mesh is refined for the bifurcation of the distributor, the contraction section of the nozzle, and the jet region to ensure the accuracy of the calculation.The localized mesh refinement amplification and profile diagram are as follows.When verifying the independence of the feedwater mechanism mesh, steady calculations of the water-air two-phase flow were initially conducted.The average pressure drop from the inlet of the distributor to the outlet of the nozzle is taken as the criterion.When the number of mesh reaches 11.51 million, the average pressure drop from the inlet of the distributor to the outlet of the nozzle only decreased by 0.64% compared to the mush number of 10.05 million.Therefore, a mesh number of 10.05 million was chosen for the numerical calculation of the feedwater mechanism.Figure 3 presents the results of the irrelevance verification of the feedwater mechanism mesh.When defining the Lagrangian phase injector, it is considered that the number of particle beams has a significant effect on the erosion rate and erosion distribution, thus affecting the calculation accuracy.According to the setting requirements of the software, the number of particle beams refers to the number of particle beams at each injection point.This parameter does not affect the mass or volume flow rate of the injector but does impact the calculation accuracy.Therefore, the particle beam is also verified in this paper.As shown in figure 4, when the particle beam reaches 7, the average erosion rate of the nozzle has not increased significantly.Therefore, the particle beam number of 7 is selected for numerical calculation.

Numerical method
In the actual working of the Pelton turbine, the flow in the distributor is a two-phase flow of solid and liquid, and the flow of the free jet entrained with sediment particles impacting the bucket is a more complex three-phase flow of water, air, and sediment [13].Therefore, this paper starts by constructing a three-dimensional model of the feedwater mechanism.The SST k-ω and VOF models are applied to perform numerical calculations.Then, the Euler-Lagrange method is used to conduct numerical simulations of the water-air-sediment flow, and the erosion rate is estimated using the Oka model.
Based on experimental data, Oka and Yoshida [14] considered the impact angle, impact velocity, particle size, and material type in their study.They defined the erosion rate as follows: Where, ( ) E  is the impact damage at the impact Angle  , 3 / mm kg ; ( )  g  is the impact angle function; 90 E And ( ) g  is defined as: Where, K , 1 K and 3 K are constants determined by particle characteristics respectively, and 2 K depends on material hardness and particle characteristics.V , ' V , D , and ' D are the impact velocity used in the experiment 3 ( / ) mm kg , standard impact velocity, particle diameter ( ) m  , and standard particle diameter respectively, Hv is the Vickers hardness ( ) GPa of the material.According to the commonly used base material of hydraulic turbines austenitic stainless steel, Hv is selected as 1.83 [15].

Two-phase calculation setting of feedwater mechanism
In the calculation, water and air are considered as continuous phases for the steady calculation of water-air two-phase flow.The inlet is the total pressure inlet, the design head is 100m, the outlet is an atmospheric pressure outlet, and the reference pressure is a standard atmospheric pressure.At the initial moment, both the distributor and nozzle are filled with water.1.0×10 -8 The analysis of the numerical calculation results of the fourth set of mesh shows that the maximum pressure drop of the 4# nozzle is 19.6 %, as shown in figure 5.

Single nozzle calculation settings
Once the two-phase calculation is completely converged and the flow field becomes stable, sediment particles are added and introduced with a maximum residence time of 3 seconds.
Properties of the sediment particles: Sediment particles are spherical, with a density of 2650 kg/m 3 and a volume fraction of 2%.This corresponds to an inlet mass flow rate of sediment particles is 0.1316 kg/s.The diameters of the sediment particles are 0.1 mm, 0.5 mm, 0.75 mm, 1 mm and 2 mm.The injector type is a component injector, and the component is the inlet of the 4#nozzle.
Boundary conditions: The sediment inlet is treated as a velocity inlet, with the inlet velocity corresponding to the velocity data of the inlet section of the 4# nozzle, and the wall boundary is modeled as a rebound mechanism.

Analysis of the asymmetrical erosion distribution on the nozzle surface
The sediment particles carried by the high-speed water flow inside the nozzle will impact the nozzle surface, guide plate and needle surface, leading to material erosion.This erosion will inevitably affect the normal operation of the turbine.Therefore, this study selected five particle sizes at a certain concentration as the research object and divided the nozzle  When the particle diameter is 0.1mm, the erosion in the nozzle throat exhibits a symmetrical phenomenon in the up-and-down direction.However, as the particle diameter increases, this symmetry gradually disappears, and an asymmetric erosion phenomenon emerges to a larger extent.Therefore, variations in sediment particle size lead to an asymmetric distribution of erosion in the nozzle, exacerbating the erosion on the nozzle throat region.From figure 9, it can be observed that in Zone 1 and Zone 2, which are closer to the nozzle inlet, both the average and maximum values of the erosion increase with an increase in sediment particle diameter, ranging from 0.1mm to 2mm.The increase in the average value of erosion is the most significant in Zone 1, with a 2mm erosion surface average value increasing by about 16 times compared to 0.1mm.On the other hand, Zone 2 exhibits the largest increase in the maximum value of erosion, and the maximum erosion value of 2mm increased by about 31 times compared to 0.1mm.For Zone 3 in the middle area of the nozzle, the average and maximum values of the erosion surface continue to increase with an increase in particle diameter when it exceeds 0.5mm.However, for particle diameters smaller than 0.5mm, the erosion in Zone 3 does not increase with an increase in particle diameter.Instead, it is more severe at smaller particle diameters, particularly at 0.1mm.Additionally, the difference in both the average and maximum values of the erosion between a particle diameter of 0.1mm and 2mm in Zone 3 is relatively small.2 shows the average and maximum erosion values of the surface of Zone 4 for the nozzle with different particle diameters.Comparing the data from figure 9 and table 2, it is evident that the nozzle throat, specifically Zone 4, experiences the most severe erosion.For sediment particle diameters smaller than 1mm, the average surface erosion value in Zone 4 increases with the increase in particle diameter.However, as the particle diameter increases from 1mm to 2mm, the average surface erosion value slightly decreases.When the sediment particle diameter is 0.75mm, the maximum average surface erosion value is 1.38×10 -5 kg/(m -2 •s).Furthermore, for the sediment particle diameter of 0.5mm, the maximum erosion value is 7.84×10 -2 kg/(m -2 •s).From figure 10, It can be observed that the erosion is most severe at the end of the guide plate for small diameter particles (0.1mm).As the particle diameter increases (0.5-2mm), the erosion becomes more severe both on the guide plate and the needle body, with the erosion area expanding accordingly.
The reason for this phenomenon is that small particles exhibit good flowability, but tend to accumulate and remain in Zone 3, which is the end of the guide plate.Therefore, small diameter particles result in severe erosion in the gradually contracting section of the nozzle.On the other hand, large particles have poor flowability, but they tend to impact the wall surface.When the guide plate is present, most of the particles are deflected onto the guide plate and rebound to the throat of the nozzle, flowing out with the water stream ultimately.Consequently, severe erosion occurs at the end of the needle and the guide plate, corresponding to Zone 1 and Zone 2 respectively.To investigate the erosion characteristics of sediment particles in different zones of the needle, the average erosion surface values in various zones of the nozzle needle were monitored and analyzed.
Figure 11 shows the average erosion surface values in different zones of the needle for various sediment particle diameters, while table 3 shows the average erosion surface values in different zones of the needle for 0.1mm and 2mm sediment particle diameters.Based on figure 11 and table 3, it can be concluded that in Zone 1 and Zone 2, the average erosion surface values continuously increase with increasing particle diameter.Zone 1 exhibits the largest increase in erosion surface values, with a difference of five orders of magnitude between the erosion surface values for 2mm and 0.1mm particles.On the other hand, in Zone 3, the maximum average erosion surface value is 2.08×10 -9 kg/(m -2 •s) for a particle diameter of 0.1mm, which is five times higher than the average erosion surface value for 2mm particles.According to the analysis of sediment particle trajectories in figure 12 and figure 13, It can be observed that when the sediment particle diameter is 0.1mm, the particles exhibit better flowability and carried by the water flow.The distribution of sediment particles on the inner and outer sides of the nozzle is relatively uniform.The impact intensity at the guide plate is relatively low, and the erosion is primarily caused by particles sliding along the nozzle wall.However, the vortex shedding at the end of the guide plate results in a longer residence time for small particles near the nozzle throat, with a maximum residence time of approximately 0.4s.Consequently, the erosion in Zone 3 of the needle is more severe.On the other hand, when the sediment particle diameter is 2mm, the flowability of the particles is poorer, resulting in the nonuniform distribution of sediment particles on the inner and outer sides of the nozzle due to gravity and inertia.The impact angle and velocity of sediment particles at the guide plate are higher, and the residence time of large-diameter particles at the end of the guide plate is shorter, with a maximum residence time of only 0.2s.Therefore, the erosion on the nozzle wall is mainly caused by the impact of particles from the guide plate.Consequently, Sediment particles with different diameters In conclusion, the diameter of sediment particles significantly affects the distribution and extent of erosion in different zones of the injector.The asymmetrical erosion phenomenon at the nozzle throat becomes more obvious with larger sediment particle diameters.Smaller diameter particles, with better flowability, tend to reside longer at the end of the guide plate, resulting in more severe erosion at both the end of the guide plate and the nozzle throat.On the other hand, larger diameter particles, with poorer flowability, tend to be propelled towards the zones of the guide plate and nozzle throat, resulting in more severe erosion.Overall, as the particle diameter increases, the average erosion rate of both the nozzle and the nozzle needle increases.

The influence of guide plate on sediment particle distribution
To investigate the influence of the guide plate on the distribution of sediment particles, the S1, S2, S3, S4, and S5 cross-sections were taken in the 4# nozzle to analyze the distribution of sediment particles.15 shows that the volume fraction of sediment particles is greater on the inner side of the nozzle than on the outer side.When the sediment particle diameter is 0.1mm, the distribution of sediment particles around the needle is relatively uniform.However, when the sediment particle diameter is 2mm, the distribution of sediment particles is uneven in the nozzle, with a higher concentration of particles in the upper region near the inner side of the nozzle.
Based on the volume fraction of particles in the S4 and S5, the results of comparing the number of guide plate n=0 and n=2 shows that the volume fraction of particles at the upper end of the guide plate is higher when n=2, especially for smaller particle diameters.Thus, the presence of the guide plate does indeed affect the distribution of sediment particles in the throat of the nozzle.16 (a) and (b) show the residence time of sediment particles at different cross-sections for particle diameters of 0.1mm and 2mm respectively.From figure 16, it can be observed that the closer the cross-section is to the nozzle throat, the longer the residence time of sediment particles.The inner side of the nozzle exhibits longer residence time compared to the outer side.For the particle diameter of 0.1mm, sediment particles distribute more uniformly along the circumference of the needle, with a longer residence time near the wall of the nozzle.This corresponds to the analysis in section 4.2.1 where sediment particles have a higher volume fraction near the nozzle wall.

Residence time of sediment particles Figures
For the particle diameter of 2mm, the distribution of sediment particles inside the nozzle becomes non-uniform, with longer residence time in the upper region near the inner side of the needle.Based on the residence time of particles at the S4 and S5 cross-sections for structures with the number of guide plate n=0 and n=2 shows that the residence time at the end of the guide plate is longer when n=2, particularly at the tip of the needle.This phenomenon is more pronounced for smaller-diameter particles.Therefore, the guide plate has a significant impact on the residence time of sediment particles at the nozzle throat.According to the analysis of 4.2.1 and 4.2.2, it can be concluded that the guide plate has varying degrees of influence on the distribution and movement of sediment particles of different diameters at the end of the guide plate.Particularly, the smaller diameter sediment particle has a greater effect on the distribution and residence time of sediment near the end of the guide plate.However, the volume fraction of sediment particles near the nozzle wall is large.This is attributed to a large velocity gradient and high viscosity stress of the fluid when flowing near the boundary layer of the wall.The aforementioned analysis also indicates that vortices at the end of the guide plate are stronger than other areas.Consequently, the shedding of flow in the end region of the guide plate increases the separation of particles and prolongs their residence time, thereby intensifying erosion in the shedding region.From this, it can also be reasonably explained that the guide plate will cause asymmetric erosion of the nozzle.

Conclusion
In this paper, the flow process of five kinds of sediment particles with different diameters in the injector of the Pelton turbine is numerically calculated at a certain concentration.The study investigates the influence of sediment particle diameter on the erosion distribution, erosion rate, and erosion asymmetry of the nozzle, providing a reference and theoretical basis for the safe and stable operation of the Pelton turbine.The main conclusions are as follows: 1) The sediment particle diameter has a significant effect on the symmetry of erosion distribution in the injector.When the sediment particle size is small (less than 0.5mm), the erosion at the nozzle throat exhibits a symmetrical pattern.With the increase of the sediment particle diameter, the symmetry of the erosion at the nozzle throat gradually disappears.The erosion on the upper side of the nozzle throat is more serious than the lower side.This asymmetry phenomenon becomes most obvious when the particle diameter increases to 2mm.
2) The influence of sediment particle diameter on the extent of erosion varies within different regions of the nozzle.With the increase of the sediment particle diameter, the average erosion rate of the overall needle area increases.While, the average erosion rate of the guide plate area decreases at first and then increases, and the average erosion rate of the nozzle throat increases at first and then decreases.The maximum average erosion rate of the nozzle throat is 1.38×10 -5 kg/(m -2 •s) for a particle diameter of 0.75mm, and the maximum average erosion rate of the guide plate area is 1.78×10 -8 kg/(m - 2 •s) for a particle diameter of 2mm.
3) The guide plate affects the distribution and movement of sediment particles at its end, thereby exacerbating the asymmetric erosion phenomenon of sediment particles at the nozzle, particularly for smaller particle diameters.When the sediment particle diameter is 0.1mm, sediment particles are distributed more evenly along the circumference of the needle.However, due to the presence of the guide plate, the volume fraction of sediment particles near the nozzle wall is higher.The inner side has a higher volume fraction of sediment particles compared to the outer side.Additionally, there is more deposition of sediment particles at the end of the guide plate, resulting in longer residence time.

Figure 1 .
Figure 1.Geometric model of the Pelton turbine.

Figure 2 .
Figure 2. Mesh of component of feedwater mechanism.When verifying the independence of the feedwater mechanism mesh, steady calculations of the water-air two-phase flow were initially conducted.The average pressure drop from the inlet of the distributor to the outlet of the nozzle is taken as the criterion.When the number of mesh reaches 11.51 million, the average pressure drop from the inlet of the distributor to the outlet of the nozzle only decreased by 0.64% compared to the mush number of 10.05 million.Therefore, a mesh number of 10.05 million was chosen for the numerical calculation of the feedwater mechanism.Figure3presents the results of the irrelevance verification of the feedwater mechanism mesh.

Figure 3 .
Figure 3. Mesh independence verification of feedwater mechanism.When defining the Lagrangian phase injector, it is considered that the number of particle beams has a significant effect on the erosion rate and erosion distribution, thus affecting the calculation accuracy.According to the setting requirements of the software, the number of particle beams refers to the number of particle beams at each injection point.This parameter does not affect the mass or volume flow rate of the injector but does impact the calculation accuracy.Therefore, the particle beam is also verified in this paper.As shown in figure4, when the particle beam reaches 7, the average erosion rate of the nozzle has not increased significantly.Therefore, the particle beam number of 7 is selected for numerical calculation.

Figure 5 .
Figure 5.The proportion of pressure drop of different nozzles.To save computational resources, the 4# nozzle is selected as the research object by truncating it in the fourth mesh model.The schematic diagram is as follows.
and needle into different zones to analyze the erosion degree in different zones.The zone division of nozzle(a) and needle(b) is shown in the following figure.

Figure 7 .
Figure 7. Nozzle and needle zone division.4.1.The influence of sediment particle size on the erosion characteristics of the injector 4.1.1.Nozzle erosion characteristics Figures 8(a)-(e) show the erosion distribution contours on the surface of Zone 4 in the nozzle throat for different sediment particle diameters with 0.1mm, 0.5mm, 0.75mm, 1mm and 2mm.It can be observed from the figures that as the sediment particle diameter increases, the erosion in Zone 4 becomes more severe and the extent of erosion also expands.When the particle diameter is 0.1mm, the erosion in the nozzle throat exhibits a symmetrical phenomenon in the up-and-down direction.However, as the particle diameter increases, this symmetry gradually disappears, and an asymmetric erosion phenomenon emerges to a larger extent.Therefore, variations in sediment particle size lead to an asymmetric distribution of erosion in the nozzle, exacerbating the erosion on the nozzle throat region.

Figure 8 .
Figure 8. Erosion contours of the zone 4.To investigate the erosion characteristics of sediment particles in different zones of the nozzle, the average and maximum values of erosion in different zones of the nozzle (according to figure7) were monitored and analyzed.From figure9, it can be observed that in Zone 1 and Zone 2, which are closer to the nozzle inlet, both the average and maximum values of the erosion increase with an increase in sediment particle diameter, ranging from 0.1mm to 2mm.The increase in the average value of erosion is the most significant in Zone 1, with a 2mm erosion surface average value increasing by about 16 times compared to 0.1mm.On the other hand, Zone 2 exhibits the largest increase in the maximum value of erosion, and the maximum erosion value of 2mm increased by about 31 times compared to 0.1mm.For Zone 3 in the middle area of the nozzle, the average and maximum values of the erosion surface continue to increase with an increase in particle diameter when it exceeds 0.5mm.However, for particle diameters smaller than 0.5mm, the erosion in Zone 3 does not increase with an increase in particle diameter.Instead, it is more severe at smaller particle diameters, particularly at 0.1mm.Additionally, the difference in both the average and maximum values of the erosion between a particle diameter of 0.1mm and 2mm in Zone 3 is relatively small.

Figure 9 .
Figure 9.The average and maximum values of the erosion in different zones of the nozzle for different particle diameters.Table2shows the average and maximum erosion values of the surface of Zone 4 for the nozzle with different particle diameters.Comparing the data from figure9and table 2, it is evident that the nozzle throat, specifically Zone 4, experiences the most severe erosion.For sediment particle diameters smaller than 1mm, the average surface erosion value in Zone 4 increases with the increase in particle diameter.However, as the particle diameter increases from 1mm to 2mm, the average surface erosion value slightly decreases.When the sediment particle diameter is 0.75mm, the maximum average surface erosion value is 1.38×10 -5 kg/(m -2 •s).Furthermore, for the sediment particle diameter of 0.5mm, the maximum erosion value is 7.84×10 -2 kg/(m -2 •s).
) and (b) illustrate the erosion contour of the outer and inner surfaces of the needle for different sediment particle diameters respectively.

Figure 10 .
Figure 10.Distribution of erosion on the outside and inside of needles with different sediment particle diameters.

4. 2 . 1 .
The distribution of sediment particles Figures 15(a) and (b) show the volume fraction of sediment particles at different cross-sections for sediment particle diameters of 0.1mm and 2mm respectively.Overall, figure

Figure 15 .
Figure 15.The volume fraction of sediment particles with different particle sizes.

Figure 16 .
Figure 16.The residence time of sediment particles with different diameters.According to the analysis of 4.2.1 and 4.2.2, it can be concluded that the guide plate has varying degrees of influence on the distribution and movement of sediment particles of different diameters at the end of the guide plate.Particularly, the smaller diameter sediment particle has a greater effect on the distribution and residence time of sediment near the end of the guide plate.However, the volume fraction of sediment particles near the nozzle wall is large.This is attributed to a large velocity gradient and high viscosity stress of the fluid when flowing near the boundary layer of the wall.The aforementioned analysis also indicates that vortices at the end of the guide plate are stronger than other areas.Consequently, the shedding of flow in the end region of the guide plate increases the separation of particles and prolongs their residence time, thereby intensifying erosion in the shedding region.From this, it can also be reasonably explained that the guide plate will cause asymmetric erosion of the nozzle.

Table 1 .
Two-phase calculation setting of feedwater mechanism.

Table 2 .
Average and maximum surface values of zone 4 for nozzle.