Parametric optimisation of two Pelton turbine runner designs using CFD*

ŽIDONIS Audrius, PANAGIOTOPOULOS Alexandros,2, AGGIDIS George A.,

ANAGNOSTOPOULOS John S.2, PAPANTONIS Dimitris E.2

1. Lancaster University Renewable Energy Group and Fluid Machinery Group, Engineering Department, Lancaster University, Lancaster, UK

2. School of Mechanical Engineering, National Technical University of Athens, Athens, Greece

Parametric optimisation of two Pelton turbine runner designs using CFD*

ŽIDONIS Audrius1, PANAGIOTOPOULOS Alexandros1,2, AGGIDIS George A.1,

ANAGNOSTOPOULOS John S.2, PAPANTONIS Dimitris E.2

1. Lancaster University Renewable Energy Group and Fluid Machinery Group, Engineering Department, Lancaster University, Lancaster, UK

2. School of Mechanical Engineering, National Technical University of Athens, Athens, Greece

(Received September 3, 2013, Revised April 8, 2014)

This paper aims to develop a generic optimisation method for Pelton turbine runners using computational fluid dynamics (CFD). Two different initial runners are optimised to achieve more generic results. A simple bucket geometry based on existing bibliography is parameterised and initially optimised using fast Lagrangian solver (FLS). It is then further optimised with a more accurate method using ANSYS Fluent. The second geometry is a current commercial geometry with good initial performance and is optimised using ANSYS CFX. The analytical results provided by CFX and Fluent simulations are used to analyse the characteristics of the flow for different runner geometries.

hydropower, Pelton impulse turbines, numerical modelling, bucket geometry parameterisation, optimisation

Introduction

The persistent increase in fossil fuel energy prices and the global concern about its environmental impact is boosting the development of clean renewable energy technologies. Hydropower is one of the most efficient among them. The UK has an estimated untapped green-field small scale (capped to 10 MW) hydropower capacity of 1.5 GWs[1]. The Pelton turbine (or Pelton wheel) invented by Lester Pelton (US Patent 223,692, October 26, 1880) is an impulse turbine suitable for high head applications. The turbine produces power by utilising water jet momentum impinging on buckets mounted on the periphery of the runner.

Design and development of Pelton turbines was mostly based on trial and error approach which if performed experimentally is very time consuming and expensive, hence limited[2]. There are publications describing numerical optimisation of the turbine performance by modelling and optimising the manifold or the nozzles[3,4]. However, there is a lack of information available in the public domain in terms of bucket design parameters and their influence to the performance of the turbine[5-7]despite the wider use of computational fluid dynamics (CFD) today.

This paper presents methods of applying CFD to develop and optimise Pelton bucket designs. Two parametric optimisation studies of independent bucket designs with different specific speeds are described. Case 1 describes a study performed on design A (having for one jet a specific speed of 12 rpm) which is based on the available literature using fast Lagrangian solver (FLS) coupled with optimisation software for the initial design and then ANSYS Fluent, while Case 2 describes a study performed on design B (having for one jet a specific speed of 26.8 rpm), which is a developed design, using ANSYS CFX as the modelling and performance evaluation tool. Both Cases 1 and 2 investigate the impact of 11 design parameters described in more detail in the following section.

The design optimisation of a Pelton runner is not an easy task, mainly due to the high complexity and unsteadiness of the flow during jet-runner interaction, but also due to the involvement of several design parameters. For this reason, it was decided to perform elaborate modelling work and validation of the resultsbased on numerical tools, prior to constructing and testing a prototype model.

Fig.1 Design parameters

1. Design analysis and modelling

A number of various aspects can cause changes in the performance of the Pelton turbine by reducing or enhancing different phenomena. Therefore the first step required to perform detailed analysis is to describe the modifications in terms of parameters that can be quantified and therefore correlation between the change made and then the performance can be analysed. The authors have identified 11 design parameters of interest that are believed to have influence on performance of the turbine, i.e., efficiency. Some of the proposed parameters control the shape of the bucket whereas others describe the position of the bucket. Even though all of the parameters are related between themselves to a higher or lower extent, due to the current time cost limitations of CFD the authors tried to divide the 11 parameters into smaller groups or investigate them individually as appropriate. Each parameter is presented in Fig.1 and described below.

(a) Bucket length to width ratio: The width is kept constant as it is related to the jet diameter which is acquired from the operational conditions. Therefore the ratio is changed by varying the bucket length.

(b) Bucket depth to width ratio: Again, the width is kept constant. Therefore the ratio is changed by varying the bucket depth.

(c) Bucket exit angle: The bucket exit angle measured in the plane that is perpendicular to the edge of the lips controls at what angle the flow leaves the bucket. It would be ideal to have the water leaving vertically meaning that the flow is diverted by 180° and the most of the power available is given to the bucket. However, this angle is limited by the next bucket as the exiting flow needs to clear away and do not impinge at the back of the next bucket.

(d) Splitter inlet angle: It is anticipated that the angle at which the jet is being split and diverted might have an influence on pressure distribution inside the bucket hence cause variation in the overall performance.

(e) Splitter level: A noticeable variation in the splitter level was observed between different bucket designs. Therefore it was decided to investigate its hydraulic impact on the overall efficiency. The splitter level is defined as the depth of the bucket measured from the splitter tip to the deepest point divided by the depth of the bucket (parameter 2).

(f) Splitter tip angle: The observed variation of the splitter tip angle between various designs has called for a study to identify the importance of this parameter.

(g) Splitter tip geometry: Splitter tip is usually the first part of the bucket that meets the jet. Therefore it is important that this contact is smooth and does not cause any negative torque.

(h) Backside of the splitter: Backside of the splitter can cause negative torque when entering the jet therefore it needs to be designed carefully to remove this unwanted effect.

(i) Inclination angle: This parameter controls the position of the bucket. Usually the buckets are mounted to the hub inclined at some angle. Therefore it was decided to investigate this parameter and quantify its effect.

(j) Radial position of the bucket: This parame- ter controls the position of the bucket. Usually the jet axis and the deepest point of the bucket are aligned. However, authors felt that investigation can be done to identify what the best efficient radial position is, keeping the pitch diameter fixed.

(k) Number of Buckets: This parameter controls the position of the bucket relative to the neighbouring buckets as the spacing angle decreases when the number of buckets is increased.

2. FLS numerical modelling

The FLS is an alternative meshless methodology customised to simulate the flow on impulse water turbines[8,9]. The method is developed at the laboratory of National Technical University of Athens (NTUA) and it is based on the Lagrangian approach. The method is not as accurate as other confirmed methods but its high speed allows to be used in combination with optimisation software and calculate the best geometry of the runner according to the simulation. So the method can be used for early design stage solutions and assist the more accurate and computationally expensive me-thods during final numerical design optimisation steps.

The FLS method is based on the track of the trajectories of an adequate number of representative fluid particles. The method integrates numerically the particle motion equations using additional adjustable terms in those particle motion equations to account for the hydraulic losses and pressure effects in order to simulate the water flow on the internal surface of the bucket. The jet is considered ideal with uniform velocity and inviscid flow. The main drawback of the FLS simulation seems to be the inability to calculate the impact of the sharp edge of the cut at the jet when the water is entering the bucket. The negative torque values caused due to the interaction of the jet with the backside surface close to the cut cannot be modelled. Moreover, the impact of the water exiting from one bucket, on the backside of the next bucket cannot be simulated, making the method unable to optimise the exit angle of the buckets.

A modern parameterisation technique based on NURBS regression polynomials is developed and used for the representation of the 3-D curved inner surface of the bucket. A number of design control points are properly introduced for the definition of the surface. Overall, the construction or the modification of the entire inner surface of the bucket is controlled by a number of 13 free geometric variables, while 2 more are added for the position definition of the bucket related to the rotating point. A review on the available literature was done to confirm that by using these variables the formation of almost every acceptable Pelton bucket geometry is possible[10]. Finally, a structured mesh can be constructed over the entire surface to facilitate the surface flow analysis, as shown in Fig.2.

Fig.2 Parameterisation of the inner surface

The maximisation of the runner hydraulic efficiency is the principal target of an improved design investigation. The software includes the FLS and the aforementioned parameterisation tool. So, initially the inner surface of the bucket is being designed from the 15 free design parameters and then the FLS simulates the case. The evolutionary algorithms system (EASY) optimisation software selects values of the free design parameters within prescribed ranges and looks automatically for the set that maximises the cost function (here the efficiency), using populations of candidate solutions instead of a single solution. The passage from a population set to the next one, which has more possibilities to contain better solutions, mimics the biological evolution of species generations[11]. The convergence rate of the optimisation algorithm is being decreased as the number of the free design variables increases. For the present case, where more than 10 geometric parameters are used to describe the bucket shape, an adequate convergence may require a few thousands of evaluations.

To achieve the best possible result the free design parameters were grouped and constraints were imposed on the exit angle and the basic dimensions. The minimum value on the exit angle was estimated based on the simulation of the initial bucket using Fluent software so as the exiting water from the first bucket slightly hits the next bucket as it is suggested from the literature[10]. The basic dimensions were also required to be within the literature suggested limits[12]. The free design parameters were separated into three groups:

(1) the 3 basic dimensions,

(2) the rest 10 parameters related with the shape of the inner surface and last,

(3) the 2 parameters that define the exact position of the bucket.

All three groups were investigated separately. For the (1) and (3) cases it was confirmed that the evaluator could not find any better solution after some hundreds of evaluations (even after thousands of evaluations) a strong indicator that the optimum solution had been reached. The adequate convergence for the (2) case required some thousand evaluations because of the amount of the free parameters. The idea of optimising the runner by simulating thousands of different cases is applicable because of the speed of the FLS which can simulate a case within 10 s on a personal computer.In conclusion, by testing thousands of different geometries it was possible to arrive at the optimum one according to the FLS simulation with the target being the maximisation of the efficiency. Three different geometries of the inner surface and the corresponding relative efficiencies according to Eq.(1)

Fig.3 Resultant inner surface geometries during optimisation (normalised to initial efficiency)

are shown in Fig.3. The backside surface of the optimum bucket was designed according to the literature and the case was simulated with a confirmed method using Fluent. The initial and final geometry as well as the relative calculated efficiencies using Fluent are shown in Fig.4. The similarity of the efficiency estimation of the concluded geometry using Fluent and FLS as well as the huge increment of the runner efficiency which almost reaches the highest accomplished values for Pelton turbines according to the literature[12]confirms the success of the presented optimisation methodology.

Fig.4 3-D design of initial and final Pelton bucket geometry (efficiency calculated by Fluent)

3. Computational modelling

The FLS method is a very cost effective tool in order to produce a preliminary good design of the runner in a short computing time, and hence it is used in the first of the two-level optimisation, where the performance of thousands of flow evaluations is not affordable by the other software. The FLS does not solve the Navier-Stokes flow equations, as the CFX and Fluent codes do. Hence, its accuracy is restricted and it cannot be used for the final optimisation stage. But starting from an already “good” design, the required computational work with the Fluent and CFX codes to produce the final design and to carry out various parametric studies is significantly reduced. Finally, the parallel use of the two different CFD codes is decided in order to be able to compare their performance and results in similar test cases.

The ANSYS Fluent and CFX which were used for the final analysis and optimisation are considered among the most accurate CFD codes. The computational methodology used in this paper was based on available recent publications[10,13-17]that model the Pelton runner using Fluent or CFX and compare numerical and experimental results. Due to heavy computational power requirements, the symmetry of the bucket geometry and the periodic symmetry of the case, only two consecutive half buckets of the runner are modelled. The torque on the inner surface was measured on the first bucket and the torque on the outside surface was measured on the following bucket allowing the capture of any backsplash effects. All simulations were applied on four cores Intel Xeon with 3.4 GHz and 16 GB memory RAM computers.

Fig.5 Drawbacks of the initial design investigated using Fluent

3.1 Fluent

The case of the initial runner (Fig.4) was simulated using Fluent to investigate the mesh independency, analyse the computational time cost and fix the settings that were going to be used for the optimisation. The mesh independency was achieved using a mesh with 2.8 M elements with time cost being approximately 8 d. Given that the optimisation process was demanding about 50 simulations, the case of about 1.5 M elements mesh with inviscid flow and ideal jet (constant initial velocity field) was applied. The error introduced by simplification of the case was calculated equal to approximately 0.5% while the time cost of one simulation was reduced down to 2 d. The op-timisation is based on the comparison of the calculated efficiency of the runner geometries and the small systematic error is not expected to affect the results. This assumption was verified in Case 1 as it is presented below.

In addition, the analytical results of the initial runner simulations indicated some obvious drawbacks of the geometry allowing either a straight improvement or a better parameterisation. First of all, an important drawback was the high pressure at the face close to the cut produced when the water is entering the bucket (Fig.5(a)). This phenomenon indicated the modification of the cut and replacement of the face with a sharp edge (Fig.4(b)) with the result being a significant increment in the efficiency. On the other hand, it should be mentioned that the area close to the edge is very vulnerable to erosion[11,18]. Moreover, the fact that the water is exiting through the cut (Fig.5(b)) indicates the modification of its shape and the importance of the radial position parameter. Finally, as it is shown in Fig.5(c) the exiting water from the bucket is hitting the backside of the next bucket at the area close to the rotational point of the runner (back area). So, the exit angle should be higher at the area close to the rotation point and both the exit angle distribution and maximum value should be investigated. These aspects were considered during the FLS optimisation and they had an important influence on the final improvement of the bucket.

The high computational cost did not allow the examination of all the parameters at the same time to safely achieve the geometry that gives the total highest efficiency and examine the interaction between the parameters. Therefore, only the first 4 parameters were investigated simultaneously using the design of experiments technique while the rest parameters were analysed separately. In addition the splitter level (5thparameter) was skipped as the FLS optimisation indicated that there is almost no influence between the parameter and the resultant efficiency. Moreover, the modifications of the splitter tip geometry and the backside of the splitter cannot be numerically described and so just a visual analysis of their influence is provided. The influence of the rest of the parameters is described in comparison with other simulation methods at the next chapter using the appropriate relative values to generalise the results for every Pelton runner. Finally the distribution of the exit angle was investigated separately from the 11 aforementioned parameters.

The ideal exit angle for the performance of one bucket is 0oaccording to Eq.(2)

whereηis the efficiency,η0the maximum efficiency andαthe exit angle, which is valid for the simplified 2-D case and for the case in which the orbit of the exiting water is perpendicular to the edge of the lips. At the initial steps in which the jet is interacting with the bucket, there is a massive amount of water exiting from the back area. The increased thickness of the water film requires a bigger exit angle to reduce the pressure at the backside of the next bucket. The reduced thickness of the water film exiting from the middle and the front area of the bucket allowed much smaller values of the exit angle. At these areas for small exit angles the exiting water slightly touches the backside of the next bucket producing a very small negative pressure (positive torque) caused by the Coandă effect. It should be mentioned that according to the Eq.(1) for small exiting angles (smaller than 5o) the loss caused from the positive angle of the exit angle is almost negligible. A parametric study showed that for small angles at the middle and the front area the efficiency is remaining almost the same and the differences are within the range of the normal computational error (Fig.6).

Fig.6 Influence of the exit angle at the middle and front area on the efficiency

Fig.7 Effect of the backside surface to the flow

The backside close to the cut area seems to have a significant influence on the final efficiency as the water remains in touch with the backside surface, destroying the form of the cylindrical jet and producing a negative pressure as it has been confirmed both computationally and experimentally[19]. The magnitude of the phenomenon is relative to the angle between the machined front face of the bucket and the jet at the moment the jet initially hits the bucket. So for big radial position and small inclination angle more water remains in touch with the backside and for longer.Usually the water leaves the backside non-uniformly and splashing randomly at the next bucket (Fig.7) while for big values of the radial position it is possible that an amount of water will fail to hit the next bucket as it is being shown in Fig.8.

Fig.8 Water fails to hit the runner

Several modifications of the backside surface close to the cut were made to reduce the adherence of the water and destroy the jet form. In addition, a small improvement of 0.1% to the total efficiency was achieved compared to the bucket with the random backside surface geometry.

Fig.9 Bucket tip extension

Fig.10 Torque in one bucket, modification of splitter tip and reduction of the counter torque

Finally, the use of the sharp edge at the cut can only reduce the positive pressure (negative torque) close to the cut. Further decrease can be achieved by increasing the inclination angle, by decreasing the radial position (increment of the acute angle between the machined front face of the bucket and the jet at the moment of the first interaction) or by modifying the geometry tip angle. The tip of the bucket should be extended closer to the jet to be the first point entering the bucket as it is shown in Fig.9. In addition this modification reduces the aforementioned adherence of the water at the backside surface. After the modification a small efficiency increment of about 0.1% was achieved. Figure 10 provides the torque curves of the original and modified geometries where counter torque is followed by useful torque on the backside of the bucket as the passing jet is pulling the bucket. Very similar phenomena were observed experimentally and presented in Ref.[10].

Fig.11 Surface control curves in Solidworks

3.2 CFX

ANSYS CFX was used for analysis and optimisation of design B used in Case 2. Since design B was an already developed design it was difficult to spot any obvious problems because all such problems would have been corrected through the years of development and application of this design. Therefore the first stage was to parameterise the original design by fitting control curves to it so that the change in geometry can be quantified. Figure 11 shows the control curves of the inner surface. The outside surface was parameterised in a similar fashion and then both surfaces were combined into solid bucket geometry.

Before running the simulations a mesh refinement study was performed and the results were validated against the experimental data of the original design. Mesh independence was achieved at the number of approximately 3×106mesh elements. However, such a simulation was taking 3 d. It was decided to use coarser meshes of approximately 1.5×106elements to reduce the time cost by a factor of two assuming that this coarser mesh would allow reliable back to back comparison. This assumption was made after analysing the torque curves of the same design modelled with different mesh sizing and observing a consistency in coarser meshes simply under predicting the amount of torque but following the same trends. Increased size of cells allowed increasing the timestep and maintaining the same Courant number hence maintaining the same convergence criteria. The duration of such a simplified simulation has dropped to 24 h which was more appropriate for parametric study.

Fig.12 Exiting water sliding on the back of the following bucket

The first group of parameters analysed in Case 2 was named design of experiments (DOE) Study 1 and consisted of four parameters: Bucket length to width ratio (L/ W), Bucket depth to width ratio (H/ W), Splitter Inlet Angle and Exit Angle. The outcome of the DOE Study 1 was an increase in efficiency of design B by 0.9%. The improvement was achieved by extending the bucket length and depth and adjusting the exit angle to match these changes. It should be noted that according to the simulations the optimum exit angle is such that the exiting flow is sliding on the back of the following bucket as showed in Fig.12. The splitter inlet angle had a very small effect on the efficiency. A quantitative analysis of the effect that each parameter has on the efficiency is provided in the following Section 4 Discussion.

Next group of parameters analysed in Case 2 was named DOE Study 2 and consisted of three parameters: radial distance, inclination angle and the number of buckets. The outcome of the DOE Study 2 was an increase in efficiency of design B by additional 0.8 %. Even though the radial distance showed to have a huge effect on the efficiency the design B could not benefit much from it since the buckets of design B were already placed at the optimal position. The improvement in efficiency was achieved by reducing the number of buckets and adjusting the inclination angle accordingly. This was an unexpected outcome since turbine manufacturers usually tend to increase the number of buckets. However, it is logical that spacing between buckets requires to be increased after increasing the depth and the length of the bucket in DOE Study 1. Also, longer bucket geometry means that a jet is impinging the bucket usefully for longer time. Since some counter torque is produced every time the bucket meets the jet, reduced number of buckets means reduction in counter torque. Therefore a balance between ensuring smooth transition between buckets and reducing the counter torque is required to achieve optimum efficiency.

4. Discussion

4.1 Bucket length to width ratio (L/ W)

The bucket length to width ratio had an influence of approximately 1% to the efficiency in the chosen area of investigation. It can be seen (Fig.13) that in both cases the sensitivity of the efficiency to the change in L/ Wis almost the same. However, the peak efficiencies were observed at differentL/ Wvalues. This means that the optimum value ofL/ Wis dependent on the design, but its importance is consistent.

Fig.13 Comparison of bucket length to width ratio (L/ W)effect for both designs

Fig.14 Comparison of bucket depth to width ratio (H/ W)effect for both designs

4.2 Bucket depth to width ratio (H/ W)

The bucket depth to width ratio had an influence of less than 1% to the efficiency in the chosen area of investigation. It can be seen (Fig.14) that in both cases the sensitivity of the efficiency to the change in H/ Wis similar. However, the peak efficiencies were observed at different values. This means that the optimum value ofH/ Wis dependent on the design, but its importance is consistent.

4.3 Splitter inlet angle

The efficiency response to the splitter inlet angle was less than 0.4% in the range of 20oas can be seen from the graph in Fig.15. However, the peak angle for both Cases 1 and 2 was identical suggesting that even though this parameter is not of high importance the optimum splitter inlet angle is independent from the design meaning that there is an optimum angle at which the jet needs to be divided.

Fig.15 Comparison of splitter inlet angle effect for both designs

Fig.16 Comparison of exit angle effect for both designs

4.4 Exit angle

The exit angle (Fig.16) proved to have much higher importance than the splitter inlet angle, which is logical since it controls the diversion of the flow in the bucket and hence the amount of the energy extracted. Therefore one must reduce the exit angle to the lowest possible value before it starts to impinge on the back of the following bucket. However, there is no single value for the best efficient exit angle as it is highly dependent on other design parameters like the number of buckets (bucket spacing).

Fig.17 Splitter level effect according to FLS. Design A only

4.5 Splitter level

The level of the splitter was investigated using FLS showing that there is almost no difference in the efficiency by changing the splitter level within normal range as shown in Fig.17. It is expected that FLS is calculating the difference of the efficiency very accurately as this parameter is not related to the backside of the bucket and so it is not investigated with Fluent or CFX.

4.6 Splitter tip angle

The splitter tip angle was investigated only for Case 1 using Fluent showing that there is almost no significant efficiency variation as shown in Fig.18.

Fig.18 The splitter tip angle effect for Case 1

4.7 Splitter tip geometry, backside of the splitter

The splitter tip geometry and the backside were investigated only for Case 1 using Fluent. These geometrical characteristics influence only the flow at the backside and they could not be optimised using FLS. The optimisation of these parameters was based on the observation of the flow from the analytic results and after trying 8 different geometries a 0.2% increment in the efficiency was succeeded. Due to the small influence of these parameters and because the bucket used for Case 2 was an already developed design, no investigation was performed using CFX.

Fig.19 Comparison of bucket inclination angle effect for both designs

4.8 Inclination angle

The effect of bucket inclination angle as shown in Fig.19 was approximately 1.1% in the tested area. Both designs reacted similarly to the variation of theinclination angle. The difference between the optimum inclination angles of both designs was approximately 2o.

Fig.20 Comparison of bucket radial position effect for both designs

4.9 Radial position

The study of this parameter showed consistent efficiency response to the variation of the radial position expressed as the ratio of tip circle diameter/pitch circle diameter. It can be seen in Fig.20 that both Cases 1 and 2 provide similar response of approximately 0.5% in efficiency over the variation of 0.2 in tip circle diameter/pitch circle diameter. However, the peak efficiency is observed at noticeably different radial positions. The explanation for these different optimum positions is in the essential differences of the designs. Specific speed of the design used for Case 2 was approximately 2.5 times higher than the specific speed of the design used for Case 1. It means that the relative size of the bucket is noticeably bigger in Case 2. Therefore the distance from the pitch circle diame- ter to the tip circle diameter is also bigger when the bucket is placed at the optimum position.

Fig.21 Comparison of number of buckets effect for both designs

4.10 Number of buckets

The number of buckets varies in Pelton runners accordingly to the literature[10,20-23]but there is a tendency to put as many buckets as possible up to 22 or 23[20,21]. The CFD investigation was performed after fixing all aforementioned parameters and it was concluded that less buckets should be used. It is believed that the correct position and shape of the bucked allow the use of fewer buckets while the loss caused by the separation of the jet from the buckets cannot be avoided. Moreover, the specific speed and the relative size of the buckets influence the optimum number of buckets as it is shown in Fig.21.

5. Conclusions

For the optimisation of the Pelton runner shape a detailed study was performed using three different CFD codes: FLS, ANSYS Fluent and ANSYS CFX. In Case 1 a basic runner was optimised successfully within two steps. At the first step it was optimised using the FLS code producing a very efficient runner and then a further optimisation study was performed using Fluent. In Case 2 an already developed runner with much higher specific speed was optimised using CFX. The results of the numerical simulations were consistent with each other and the error of the calculations relatively small to affect the results. The influence of the most important geometric parameters of the runners against their relative efficiency value, were investigated and it was concluded that it is generally independent of the model design specific speed, i.e., similar inclination on the graphs of parameter against efficiency for the two specific speeds. On the other hand, the optimum value i.e., the value at the point on the graph of maximum efficiency, for each one of the most important parameters appears to be strongly depended on the specific speed. The construction of the new runners and the experimental confirmation of the results are planned as the next steps to this computational study.

Acknowledgements

The authors would like to thank Lancaster University Renewable Energy Group and Fluid Machinery Group, the Laboratory of Hydraulic Turbo Machines at the National Technical University of Athens, Gilbert Gilkes and Gordon Ltd and the EU ERASMUS programme for the financial support.

[1] WILLIAMS D. Environment protection requirements for small hydro schemes[C]. European Hydropower Summit. Bucharest, Romania, 2012.

[2] AGGIDIS G. A., ŽIDONIS A. Hydro turbine prototype testing and generation of performance curves: Fully automated approach[J]. Renewable Energy, 2014, 71(11): 433-441.

[3] PATEL K., PATEL B. and YADAV M. et al. Development of Pelton turbine using numerical simulation[C].IOP Conference Series: Earth Environment Science. Timişoara， Romania， 2010.

[4] BENZON D., ŽIDONIS A. and PANAGIOTOPOULOS A. et al. Impulse turbine injector design improvement using computational fluid dynamics[J]. Journal of Fluids Engineering, 2015, 137(4): 041106.

[5] BARSTAD L. F. CFD analysis of a Pelton turbine[D]. Master Thesis, Trondheim, Norway: Norwegian University of Science and Technology, 2012.

[6] SOLEMSLIE B. W., DAHLHAUG O. G. A reference Pelton turbine design[C]. IOP Conference Series Earth Environment Science. Beijing, China, 2012.

[7] ŽIDONIS A., AGGIDIS G. A. State of the art in numerical modeling of Pelton turbines[J]. Renewable and Sustainable Energy Reviews, 2015, 45: 135-144.

[8] ANAGNOSTOPOULOS J. S., PAPANTONIS D. E. Flow modelling and runner design optimization in turgo water turbines[J]. Proceedings of World Academy of Science Engineering and Technology, 2007, 22: 206-211.

[9] ANAGNOSTOPOULOS John S., PAPANTONIS Dimitris E. A fast Lagrangian simulation method for flow analysis and runner design in Pelton turbines[J]. Journal of Hydrodynamics, 2012, 24(6): 930-941.

[10] PERRIG A. Hydrodynamics of the free surface flow in Pelton turbine buckets[D]. Doctoral Thesis, Lausanne, Swizerland: EPFL, 2007.

[11] GIANNAKOGLOU K. C. Design of optimal aerodynemic shapes using stochastic optimization methods and computational intelligence[J]. Progress in Aerospace Science, 2002, 38(1): 43-76.

[12] THAKE J. The micro-hydro Pelton turbine manual[M], London, UK: ITDG Publishing, 2000.

[13] KLEMETSEN L. A. An experimental and numerical study of the free surface Pelton bucket flow[D]. Master Thesis, Trondheim, Norway: Norwegian University of Science and Technology, 2010.

[14] JOŠT D., MEŽNAR P. Numerical prediction of Pelton turbine efficiency[C]. IOP Conference Series: Earth Environment Science. Timişoara， Romania， 2010.

[15] XIAO Y., WANG Z. and ZHANG J. et al. Numerical and experimental analysis of the hydraulic performance of a prototype Pelton turbine[J]. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 2014, 228(1): 46-55.

[16] GUPTA V., PRASAD V. and KHARE R. Effect of jet shape on flow and torque characteristics of Pelton turbine runner[J]. International Journal of Engineering Research and Applications, 2014, 4(1): 318-323.

[17] GUPTA V., KHARE R. and PRASAD V. Performance evaluation of Pelton turbine: A review[J]. Hydro Nepal: Journal of Water, Energy and Environment, 2013, 13: 28-35.

[18] PADHY M., SAINI R. Effect of size and concentration of silt particles on erosion of Pelton turbine buckets[J]. Energy, 2009, 34(10): 1477-1483.

[19] PERRIG A., AVELLAN F. and KUENY J. L. et al. Flow in a Pelton turbine bucket: Numerical and experimental investigations[J]. Journal of Fluids Engineering, 2006, 128(4): 350-358.

[20] NECHLEBA M. Hydraulic turbines: Their design and equipment[M]. Prague, Czechoslovakia: Penerbit Artia, 1957.

[21] ATTHANAYAKE I. U. Analytical study on flow through a Pelton turbine bucket using boundary layer theory[J]. International Journal of Engineering and Technology, 2009, 9(9): 241-245.

[22] EISENRING M. MHPG series: Harnessing water power on a small scale. Volume 9: Micro Pelton turbines[M]. Gallen, Switzerland: SKAT, Swiss Centre for Appropriate Technology, 1991.

[23] NASIR B. A. Design of high efficiency Pelton turbine for micro hydropower plant[J]. International Journal of Electrical Engineering and Technology, 2013, 4(1): 171-183.

*Corresponding author: AGGIDIS George A.,

E-mail: g.aggidis@lancaster.ac.uk