Numerical assessment of erosion wear in Pelton turbine injectors

The Pelton turbine is the main type of turbine used to develop hydropower resources in high-head sections and is more sensitive to sediment erosion. To analyse the causes of asymmetric erosion distribution on the injector surface, a solid-liquid-gas three-phase numerical study of Pelton turbine injectors is carried out. The results indicate that as the sediment diameter increases, the differences in particle discharge distribution among the jets of each branch also increase, and the asymmetric erosion distribution on the injector surface becomes stronger. The particle trajectory is influenced by the vortex structure, and fine particles mainly cause banded erosion, while coarse particles cause sheet erosion. At constant inflow conditions, the particle residence time is related to erosion morphology, and the particle velocity and the number of impacts affect the erosion rate on the injector surface. It is recommended to avoid long-term operation of the turbine under conditions of excessively small or large sediment diameters and to pay attention to the sediment erosion degree of the middle branch needle and the end branch nozzle of the water supply mechanism. The research findings of this paper can provide references for further optimization of the design, operation, and maintenance of the Pelton turbine.


Introduction
Hydropower, as the main force of renewable energy generation, possesses significant advantages such as rapid start-up, easy scheduling, and so on.With the implementation of the "dual carbon" target, China is gradually establishing a new energy system dominated by clean energy.There are plans to develop and construct large-scale hydropower bases in high-head and sediment-rich river basins in Southwest China [1,2].When the design head of the hydropower station is 500m or above, the Pelton turbine has more significant advantages than the Francis turbine in terms of service life, operation and maintenance costs, and design parameter matching [3].The jet velocity at the nozzle outlet of a Pelton turbine operating at a high head section usually exceeds 100m/s.Sand-laden jets enter the bucket's working surface and rapidly spread, with relative velocities between sediment particles and a bucket reaching up to 50m/s [4].Therefore, when the unit operates under sediment-laden flow conditions for a long time, solid particles exert a strong cutting effect on the injector and the runner, resulting in pit corrosion or cracks.It causes large-scale material spalling, not only damaging the flow components of the turbine and significantly reducing power output but also shortening the unit maintenance period and increasing the maintenance cost of the hydropower station, thereby posing threats to the safe and stable operation of the unit [5,6].
Research on the characteristics of sediment erosion in Pelton turbine injectors is mainly conducted through experimental testing and numerical simulations [7].In terms of experimental research, Thapa [8] and Boes [9] found that the average particle diameter of sediment carried by inflow from the hydropower station decreases with its increasing concentration and considered that high-concentration sand-laden jets primarily consist of fine particles, which are more likely to cause severe erosion damage to the injector surface.Bajracharya et al. [10], based on the flow analysis through the needle surface in the injector, found an annual erosion rate of 3.4 mm on the main flow components surface in the Chilime Hydropower Plant, resulting in a significant decrease in unit efficiency over the years.Compared to two-phase flow, solid-liquid-gas three-phase flow is more complex, and the randomness of particle spatial distribution is greater.Therefore, monitoring experimental data on sediment erosion is challenging, and it is difficult to elucidate the erosion mechanism of sediment in the Pelton turbine solely through experimental research [11,12].
In terms of numerical research, scholars initially focused on the erosion rule of bucket surfaces.However, in recent years, the wear characteristics of the injector have gradually become a new research focus.The injector is an important flow component for energy conversion in the Pelton turbine.The geometric damage to the nozzle and needle caused by sediment erosion can result in an additional loss of 2%~5% of unit efficiency [4].The distorted jet impinging on the runner intensifies the flow interference between the buckets, further reducing the unit efficiency by about 9% [5].Messa et al. [13] conducted a systematic study on the influence of the needle vertex angle and stroke on component erosion rates and found that as the needle stroke decreases, the particle impact frequency and velocity increase, and the erosion rate of the needle exhibits a magnitude increase.When the needle vertex angle decreases, the surface erosion rate of the injector is primarily determined by the particle impact frequency and impact angle.Din et al. [6] investigated the sediment erosion mechanism on the surface of the injector under different needle stroke conditions, divided the surface of the nozzle outlet seat ring and needle tip into different erosion morphological areas, and considered that particles with small impacting angles tend to create ripple erosion on the wall surface, while particles with large impacting angles tend to cause plowing and craters.Guo et al. [14] suggested that the secondary flow caused by the shedding of attached vortex and extended vortex downstream of needle guides induces particles to impact the needle surface with high frequency and speed, resulting in obvious erosion zones downstream of the needle guide.Tarodiya et al. [15,16] have shown that solid particles with non-uniform diameters can cause an asymmetric erosion distribution on the injector.In summary, the unstable flow in the Pelton turbine injector has a certain influence on the dynamic behavior of sediment particles, which not only alters the distribution of solid particles within the flow passage but also affects the erosion morphology on the injector surface.
To further analyze the relationship between the unstable vortex structure in the water supply mechanism of the Pelton turbine and the movement of sediment particles, as well as the reason for the asymmetric erosion distribution on the injector surface, a numerical simulation of the water supply mechanism of a six-nozzle Pelton turbine model is carried out in this paper.The influence of the particle hydrodynamics parameters on the local erosion rate of the injector is analyze, and the relationship between the overall erosion rate of the nozzle and needle at different positions and the sediment diameter is discussed.The findings of this research contribute to a further understanding of the formation mechanism of the asymmetric erosion on the injector surface, thus providing a theoretical foundation for studying the asymmetric erosion on the rotating bucket surface during the unit operation.Furthermore, it can offer practical references for the optimization design and maintenance of the Pelton turbine in the sediment-laden river.

Geometric model and mesh
In this study, the water supply mechanism of a six-nozzle Pelton turbine model is taken as the research object, and numerical simulations are conducted to investigate the sediment erosion characteristics of the injector under different particle diameters.Figure 1 presents a schematic diagram of the threedimensional geometric structure and grid division of the model water supply mechanism, with the six sets of injectors named No. 1 to No. 6 in turn along the flow direction.The model nozzle outlet diameter is D = 35.2mm,and the cylindrical air region diameter outside the nozzle outlet is 3D, with a length of 10D.The computational domain consists of a static domain and an air domain.The mesh for the computational domain is an unstructured hexahedral cell type.Local grid refinement is applied at the bifurcated pipe of the water supply mechanism, the nozzle outlet, and the downstream jet core region of the needle.

Erosion model and particle rebounding model
Erosion rate is a physical quantity used to quantitatively describe the degree of wear on the hydraulic machinery surface, which can be predicted by numerical simulation through erosion models.However, due to the increased complexity of the solid-liquid-gas three-phase flow patterns and the greater randomness in the spatial distribution of particles, it is crucial to reasonably select erosion and particle rebounding models to improve the accuracy of sediment wear prediction for Pelton turbines [17,18].
There are many factors that influence the degree of wear, so there is no unified form for the erosion prediction formula, but they generally include parameters such as particle impact velocity, particle impact angle, and wall material hardness.E/CRC, Generic, and Oka models are widely used at present, and these erosion models are empirical or semi-empirical models based on specific experimental conditions or erosion mechanisms [19].Among them, the Oka model is established based on experimental data of gas-solid two-phase flow impact wear within a wide impact angle range from 5° to 90° and high impact velocities from 50 to 167 m/s.Its formula considers multiple parameters such as particle impact velocity, impact angle, material hardness, and particle size [17].This erosion model is not only the closest to the elbow experiment in terms of the magnitude of the numerical calculation results but also shows significant sensitivity to changes in particle impact angles [20].Moreover, it is suitable for capturing particle motion in high-velocity flows, which satisfies the flow conditions in the injector of the Pelton turbine.Therefore, in this paper, the Oka model is used to predict the erosion rate on the injector, and the formula is as follows: where Hv is the material hardness, V * is the reference velocity, D * is the reference particle diameter, and n1, n2, k1, k2, and k3 are model constants.
The particle rebounding model is a function established based on particle impact experiments to describe particle impact behavior.Because sediment particles will lose some momentum after impacting on a solid wall, this model can describe the trajectory of particles before and after impacting on the wall, thereby affecting the numerical values of the attack angle function in the erosion model.Currently, there are primarily two commonly used particle rebounding models.Comparative analysis of numerical simulations indicates that, under the conditions of using the same erosion model, changing the particle rebounding model has little impact on the final erosion rate of the component surfaces [21,22].Therefore, referring to Reference [18], the Grant particle rebounding model is adopted in this paper.This model [23] is established based on experimental data from 2024 aluminum alloy substrates, and the formula is as follows: where ET is the tangential rebound coefficient, EN is the normal rebound coefficient, and α is the impact angle.

Mathematical model and numerical solution strategy
In this study, the Volume of Fluid (VOF)-Lagrangian multiphase flow model is used for numerical simulations.The water and gas phases are regarded as continuous phases, and the VOF model is used to capture the water-air interface.The solid particles are considered discrete phases, and the Lagrangian model is used to track the particle trajectories.
The interior of the Pelton turbine is an open, multiphase, unsteady flow field, exhibiting complex flow characteristics and strong transient features [24].The SST k-ω turbulence model is not only suitable for capturing the rotating shear flow at high Reynolds numbers but also for dealing with the complex boundary layer with an adverse pressure gradient [25].Furthermore, by comparing different turbulence models, it is demonstrated that the SST k-ω turbulence model can obtain satisfactory predictive accuracy when simulating complex secondary flow phenomena within the Pelton turbine [26,27].Therefore, the multiphase turbulent governing equations are closed using the SST k-ω turbulence model [28], with its transport equations given by: Where t  is the turbulent viscosity, 1 F and 2 F are mixing functions, and 1  , 2  , and *  are constants.The computational domain is set at the inlet of the water supply mechanism, and the total pressure inlet condition is applied for the continuous phase according to the model experimental water head.The surface of the air domain is set as a static pressure outlet with a reference pressure of atmospheric pressure.The inlet and outlet positions for the discrete phase are the same as those for the continuous phase, and the velocity of sediment particles entering from the computational domain inlet is the same as the inlet flow velocity of water.The wall adopts a fixed no-slip boundary condition, and the interaction between the discrete phase and the wall is set to rebound mode.The sediment density (ρ) is 2650 kg/m³ , and the sediment concentration (Cv) is 1%, corresponding sediment mass flow rate at the inlet is 2.964 kg/s.Five groups of sediment particle diameters (d) are 0.05 mm, 0.1 mm, 0.5 mm, 1 mm, and 2 mm, respectively.The sediment particles are simplified as spheres in the calculations, taking into account the drag force, virtual mass force, and pressure gradient force acting on the solid particles.The particle-fluid interaction is treated as one-way coupling.

Jet flow discharges and sediment distribution
Figure 2 shows, at sediment particles release before and after, the variation of the jet flow discharges in each branch of the water supply mechanism with the sediment particle diameters and the comparison of the sediment mass flow rates in each branch.Figure 3 presents the velocity distribution nephogram of the horizontal section of the water supply mechanism.The results in Figure 2 indicate that before the injection of sediment particles, the jet flow discharges in each branch of the water supply mechanism are not uniform.The jets of No. 2 and No. 3, located in the middle of the water supply mechanism, have higher jet flow discharges, while the jets of No. 4, No. 5, and No. 6, which are farther from the inlet, have lower jet flow discharges.After injecting sediment particles, the flow discharges in all branches decrease to some extent, with the jet of No. 6 experiencing the largest average reduction of 0.115% in flow discharge.
In combination with Figure 3, comparing the sediment mass flow distribution in different jets, it can be observed that under the conditions of fine sediment particles with diameters of 0.05mm and 0.1mm, the particles exhibit better flowability, and the distribution of particle flow in each branch is comparatively uniform.The relatively lower particle discharge in No. 3 and No. 4 jets may be attributed to the obvious low velocity region at the inlet of some branches.Thus, flow separation in a large range is caused, which hinders the entry of sediment particles.Under the conditions of mediumsized sediment particles with diameters of 0.5mm and 1.0mm, significant differences in flow distribution among the branches are shown.No. 2 and No.3 jets have lower sediment particle discharges, while the sediment concentration in No.6 jet is obviously increased.The reason for this phenomenon may be due to the dominance of inertial forces when medium-sized sediment particles enter the distributor, and most of the particles continue to move forward along the main flow when the fluid suddenly changes direction upon entering the branch.As the particles reach the middle of the water supply mechanism, the flow velocity increases, and centrifugal force becomes dominant, resulting in the particles entering the other branches with the fluid.The difference in flow distribution of coarse sediment with d = 2.0mm in each branch is more obvious.No.2 jet has the lowest sediment particle discharge, while No.6 jet has the highest sediment particle discharge, reaching 1.5kg/s, which is four times the sediment particle discharge in No.1 jet under the same sediment conditions and three times the sediment particle discharge in the same position jet with 0.05mm fine sediment particles.It indicates that as the sediment diameter increases, the influence of inertial and centrifugal forces on coarse sediment particles becomes significantly enhanced.

Erosion characteristics on the injector
In this study, the third branch with significant vortex structures is selected as an example to further analyze the influence of unstable flow within the water supply mechanism on the sediment particle erosion distribution on the injector surface.Figures 4 and 5 respectively, show the distribution of sediment erosion on the nozzle and needle surfaces under different sediment particle diameters.Figure 6 displays the sediment particle volume fraction and streamline distribution on the main sections within the injector when the particle diameter is 0.05mm.
The results in Figure 4 show that the high erosion rate region on the nozzle surface is mainly concentrated near the nozzle outlet.With an increase in sediment particle diameter, the asymmetry of erosion distribution on the contraction section of the nozzle surface gradually intensifies, and the right wall surface experiences more severe sediment wear.When the particle diameter is 0.05mm or 0.1mm, the erosion region on the nozzle surface appears primarily banded.When the particle diameter exceeds 0.1mm, the erosion region on the right wall surface of the nozzle exhibits significant sheet-like features with a larger extent, while the erosion region on the left wall surface is mainly concentrated near the outlet with a slightly smaller extent.Figure 5 results demonstrate that as the sediment particle diameter increases, the erosion distribution on the needle surface also presents significant asymmetry.The erosion caused by fine sediment particles evenly distributes on the needle surface, while when the sediment particle diameter exceeds 0.1mm, the erosion region on the needle surface gradually concentrates on the left wall surface of the needle body.According to Figure 6, the vortices at the inlet of the bifurcated pipe are divided by the needle guide during the downstream movement.The vortexe's volume decreases while its quantity increases.The main sediment accumulation region corresponds to the vortex structure position, and there is a higher sediment content in the right flow passage.The particle trajectories are influenced by vortex disturbances.Sediment particles in the right flow passage gradually accumulate towards the nozzle wall, while particles in the left flow passage gradually gather towards the needle surface.Due to the better particle flowability of fine sediment particles, under the influence of secondary flow near the wall in the needle guide, fine sediment particles continuously roll along the wall, and then form longer sliding wear scars extending from the end to the tip of the needle.Wear on the head of the needle mainly occurs when the sediment particle diameter is less than 0.5mm.This is primarily due to the centrifugal motion of the particles relative to the needle tip.When the diameter of the needle neck suddenly shrinks, the annular flow passage area increases, the sectional velocity increases, and the fluid convergence enhances.A part of the fine sediment particles impacts the needle tip in the converging section of the injector, causing erosive wear.However, due to the poor particle flowability of coarse sediment particles, they experience stronger inertia and centrifugal forces.Most coarse sediment particles, upon motion through the low-velocity region near the bifurcated pipe inlet, directly impact the downstream wall, leading to the formation of the high-erosion-rate region at the end of the needle.Rebounding coarse sediment particles do not fully circle near the needle guide and tend to stall along the downstream vortex structures of the needle guide.Therefore, after causing intense impact erosion at the nozzle outlet, coarse sediment particles enter the air with the high-velocity jet, exhibiting a tendency to move away from the tip of the needle under the action of inertia force and centrifugal force.So that the degree of erosion caused by coarse sediment particles at the needle tip can be relatively neglected.In summary, the erosion rate on the nozzle is always higher than on the needle, and fine sediment particles pose the greatest erosion threat to the needle tip.

Factors affecting erosion rate
In Figure 7, under the condition of a sediment particle diameter of 0.05mm, it shows the relationship between the erosion rate on the injector surface and the particle hydrodynamics parameters, namely, the particle slip velocity, particle discharge, and particle residence time.The particle slip velocity represents the velocity difference between solid particles and the fluid, with smaller slip velocities indicating better particle flowability.The particle discharge reflects the average number of particle impacts at a certain position, and higher particle discharges result in more particle impacts on the wall.The particle residence time is the contact time between particles and the wall; longer residence times indicate that the wall is primarily affected by sliding wear, while shorter residence times indicate that the wall is mainly subjected to impact wear from the particles.The results in Figure 7 show that the uneven particle distribution on the nozzle and needle surfaces is one of the reasons for the asymmetric erosion on the injector surface.The high erosion rate regions of the injector are mainly concentrated near the contraction section because the particles impact on the nozzle outlet wall and the needle head with high speed, multiple, and short duration and exert a strong cutting effect on the wall.At the same time, there are also banded erosion regions with lower erosion rates and longer lengths on the surface of the needle body and the straight section of the nozzle, corresponding to the particle sampling positions with longer residence times in Figures 7(d) and (h).The particles with better flowability slide along the wall surface at these positions while being influenced by vortices, resulting in sliding wear scars that are formed on the nozzle and needle surfaces along the flow direction and are not parallel to the needle guide.In addition, the remaining localized low erosion rate regions on the injector surface can be explained by multiple low-speed particle impacts on the wall.In summary, when the inflow conditions are constant, the particle residence time is related to erosion morphology, and the particle velocity and the number of impacts on the wall are two key factors that affect the erosion rate on the injector surface.

Overall erosion rate analysis of the injector
Figure 8 shows the variation of the overall erosion rate of the components of the injector in the Pelton turbine with different particle diameters.The results in Figure 8 show that the sum of the overall erosion rates of the six nozzles gradually increases with the particle diameter.Due to the fact that the difference in sediment particle discharge between each bifurcated pipe of the water supply mechanism increases with particle diameter, there is also a significant difference in the overall erosion rates among nozzles.

Conclusions
(1) The distribution of sediment particle discharge is different to some extent due to the influence of the vortex structure at the inlet of the bifurcated pipe.As particle size increases, inertial and centrifugal forces gradually become dominant, which results in the difference in sediment discharge distribution also expanding among the jets in the water supply mechanism.
(2) The erosion distribution on the injector surfaces is asymmetrical.The particle trajectory is influenced by the vortex structure, and fine particles mainly cause banded erosion, while coarse particles cause sheet erosion.In that design phase of the Pelton turbine, the optimization design of resistance to sediment wear can be carried out in aspects such as reducing the degree of geometric variation at the bifurcated pipe of the water supply mechanism or reasonably setting the number and shape of the needle guide.
(3) At constant inflow conditions, the particle residence time is related to erosion morphology, and the particle velocity and the number of impacts on the wall are two key factors that affect the erosion rate on the injector surface.
(4) It is recommended to avoid long-term operation of the turbine under conditions of excessively small or large sediment sizes so as to prevent serious sediment erosion on the injector.During maintenance, particular attention should be paid to the sediment erosion of the needle in the middle branch and the nozzle in the end branch of the water supply mechanism and the timely repair and replacement of the failed components.

Figure 1 .
Figure 1.Water supply mechanism model and unstructured grid.

Figure 2 .
Figure 2. The impact of sediment particles on jet flow and its distribution in each branch.

Figure 3 .
Figure 3. Velocity distribution in the horizontal section of the distributor.

Figure 4 .
Figure 4. Erosion contours of the nozzle surface under different particle size.

Figure 5 .
Figure 5. Erosion contours of the needle surface under different particle size.According to Figure6, the vortices at the inlet of the bifurcated pipe are divided by the needle guide during the downstream movement.The vortexe's volume decreases while its quantity increases.The main sediment accumulation region corresponds to the vortex structure position, and there is a higher sediment content in the right flow passage.The particle trajectories are influenced by vortex disturbances.Sediment particles in the right flow passage gradually accumulate towards the nozzle wall, while particles in the left flow passage gradually gather towards the needle surface.Due to the better particle flowability of fine sediment particles, under the influence of secondary flow near the wall in the needle guide, fine sediment particles continuously roll along the wall, and then form longer sliding wear scars extending from the end to the tip of the needle.Wear on the head of the needle mainly occurs when the sediment particle diameter is less than 0.5mm.This is primarily due to the centrifugal motion of the particles relative to the needle tip.When the diameter of the needle neck suddenly shrinks, the annular flow passage area increases, the sectional velocity increases, and the fluid convergence enhances.A part of the fine sediment particles impacts the needle tip in the converging section of the injector, causing erosive wear.However, due to the poor particle flowability of coarse sediment particles, they experience stronger inertia and centrifugal forces.Most coarse sediment particles, upon motion through the low-velocity region near the bifurcated pipe inlet, directly impact the downstream wall, leading to the formation of the high-erosion-rate region at the end of the needle.Rebounding coarse sediment particles do not fully circle near the needle guide and tend to stall along the downstream vortex structures of the needle guide.Therefore, after causing intense impact erosion at the nozzle outlet, coarse sediment particles enter the air with the high-velocity jet, exhibiting a tendency to move away from the tip of the needle under the action of inertia force and

Figure 7 .
Figure 7. Erosion contours of the injector surface (d=0.05mm):Erosion rate (the first column), Particle slip velocity (the second column), Particle discharge (the third column), Particle residence time (the fourth column).
For d = 0.05mm and d = 0.1mm, the overall erosion rates of the six nozzles are almost the same.When d > 0.5mm, the overall erosion rates of No. 2 and No. 3 nozzles decrease slightly, while the overall erosion rate of No.6 nozzle increases significantly.At d = 2.0mm, the overall erosion rate of No. 6 nozzle is 7.58x10 -7 kg/s, which is about 9.8 times the overall erosion rate of the same nozzle at d = 0.05mm.The sum of the overall erosion rates of the six needles decreases first and then increases with the sediment particle diameter, reaching the minimum at d = 0.1mm.For d = 0.05mm, due to the better flowability of particles, although the sediment particle discharges in each jet are basically the same, the particle trajectories are influenced by the vortex structures, inducing an increased frequency of wall impacts.Therefore, the No. 3 needle suffers the most severe sediment erosion with an overall erosion rate of 8.69x10 -9 kg/s, approximately 8.8 times the minimum overall erosion rate under this sediment condition.When d = 2.0mm, the sediment particle discharge is highest in the No. 6 branch, resulting in the highest overall erosion rate for the needle here.During the operation of a Pelton turbine, the nozzle is the most severely worn component of the water supply mechanism by sediment.The turbine is recommended to avoid long-term operation under conditions of excessively small or large sediment diameters so as to prevent serious sediment erosion on the injector.During maintenance, particular attention should be paid to the sediment erosion of the needle in the middle branch and the nozzle in the end branch of the water supply mechanism and the timely repair and replacement of the failed components.

Figure 8 .
Figure 8. Overall erosion rate of each component of injector.