Numerical prediction of Pelton turbine efficiency

Transcription

1 IOP Conference Series: Earth and Environmental Science Numerical prediction of Pelton turbine efficiency To cite this article: D Jošt et al 2010 IOP Conf. Ser.: Earth Environ. Sci Recent citations - Hydro-abrasive erosion in hydro turbines: a review Saurabh Sangal et al - Numerical investigation for one badbehaved flow in a Pelton turbine X Z Wei et al View the article online for updates and enhancements. This content was downloaded from IP address on 08/10/2018 at 16:19

2 Numerical prediction of Pelton turbine efficiency 1. Introduction D Jošt 1, P Mežnar 1 and A Lipej 1 1 Turboinštitut, Rovšnikova 7, Ljubljana, 1210, Slovenia dragica.jost@turboinstitut.si Abstract. This paper presents a numerical analysis of flow in a 2 jet Pelton turbine with horizontal axis. The analysis was done for the model at several operating points in different operating regimes. The results were compared to the results of a test of the model. Analysis was performed using ANSYS CFX-12.1 computer code. A k-ω SST turbulent model was used. Free surface flow was modelled by two-phase homogeneous model. At first, a steady state analysis of flow in the distributor with two injectors was performed for several needle strokes. This provided us with data on flow energy losses in the distributor and the shape and velocity of jets. The second step was an unsteady analysis of the runner with jets. Torque on the shaft was then calculated from pressure distribution data. Averaged torque values are smaller than measured ones. Consequently, calculated turbine efficiency is also smaller than the measured values, the difference is about 4 %. The shape of the efficiency diagram conforms well to the measurements. Two phase flow (water, air) in Pelton turbines is turbulent and unsteady. While reasonable results for Francis and Kaplan turbines can be obtained by steady state analysis, that is not possible for Pelton turbines. Besides, free surface flow has to be modeled by a multiphase model. Numerical analysis of flow in a Pelton turbine is therefore much more complex and time consuming. In the last eight years a lot of papers about numerical and experimental analysis of flow in Pelton turbines have been published. A water jet from a Pelton turbine injector was described on the basis of experimental and numerical results in [1]. The influence of secondary velocity fields on jet shape and the influence of jet quality on turbine efficiency were investigated numerically and experimentally [2, 3]. A bucket flow simulation using three adjacent buckets was shown by Mack and Moser in [4]. One of the first attempts to calculate free surface flow in a rotating bucket was done by Kvinsky et al. [5]. Unsteady analysis of a Pelton runner with flow and mechanical simulation was presented by Parkinson et al. [6]; numerical results were validated by pressure measurements on runner buckets, flow visualization and efficiency measurements. The most complete analysis of flow in Pelton turbines was done by Perrig [7]. Flow in buckets was investigated through unsteady onboard wall pressure measurements, high-speed onboard and external flow visualizations, water film thickness measurements and CFD simulations. On the basis of numerical and experimental results mentioned above, a mass of new knowledge about dynamic process in Pelton turbine was obtained. It was also proven that CFD can reproduce free surface flow in Pelton turbines with reasonable accuracy. Numerical prediction of efficiency for Kaplan and Francis turbines has been a part of design procedures for more than twenty years, while usage in design optimization for Pelton turbines remains limited in spite of some encouraging results. Most papers report that the shape of the efficiency curve is well captured, while actual differences between measured and numerically predicted efficiency remain unreported. An exception is [4], where predicted efficiency was quite close to the measurements. The purpose of this paper is not a detailed explanation of flow in Pelton turbines but ruther to numerically analyze flow over different operating regimes and to compare efficiency with experimental values. For this purpose, a 2-jet Pelton turbine with horizontal axis was analyzed. The runner consists of 23 buckets. The turbine was designed for high head. The paper (Jošt et al., [8]) presented a numerical analysis of the same Pelton turbine at three operating points for maximum opening. Obtained data on water jets, evacuating water sheets, and pressure distribution at a single bucket through a 48 degree rotation was quite sensible, but the agreement between numerical and measured efficiency values was rather poor. The analysis was done for a prototype at a jet velocity of about 140 c 2010 Ltd 1

3 m/s. For such high velocity values, the grid near the buckets was not refined enough, therefore the values of y + were much too high. This time analysis was done for the model and the grids were refined. So the values of y + became more acceptable. In order to get a complete efficiency diagram, numerical analysis was performed at 17 operating points (Tab. 1). For operating points with the same opening, the values of discharge were the same. Different values of discharge coefficient ϕ and pressure coefficient ψ were obtained through different rotational speed. Numerical results were compared with the results of the model tests. Table 1 Operating points for numerical flow analysis Point A o / A o,bep ϕ / ϕ BEP ψ / ψ BEP Free surface flow Free surface flows refer to a multiphase situation where the fluids are separated by a distinct resolvable interface. They can be modelled via a homogeneous or inhomogeneous model. The inhomogeneous model should be used when two fluids are being mixed and later separated. For Pelton turbines the homogeneous model is usually used. Slightly better results can be obtained with the inhomogeneous model, but at a higher cost. In this paper homogeneous model was used. The homogeneous model assumes that the transported quantities (with the exception of volume fraction) for the process are the same over all phases. r r Uα = U, p α = p, 1 α N p (1) It is therefore sufficient to solve bulk transport equations for shared fields instead of solving individual transport equations. N p ρ r + ( ρu ) = Γ (2) αβ t β = 1 r r r r r T ( ρ U ) + ( ρu U μ( U + ( U ) )) = SM p (3) t Г αβ in equation 2 is the mass flow rate per unit of volume from phase β to phase α. Density and viscosity are 2

4 calculated from density and viscosity of all phases in the fluid: N p ρ = rα ρ α, = α = 1 N p μ r. (4) α μ α α = 1 A detailed description of multiphase models and modelling of free surface flows can be found in [9]. 3. Flow in the distributor with injectors and jets Optimization of distributors with injectors is extremely important for the efficiency of Pelton turbines. Discharge should be equally distributed between all injectors. Losses of flow energy losses in the distributor have to be as small as possible, but the quality of the jets is even more important. Jets should be compact and velocity within the jet should be uniform. Secondary velocities caused by bends in the distributor are most undesirable as they cause jet dispersion and deviation. Jets lose their circular cross section due to such secondary velocities. A steady state analysis of the flow in the distributor with two injectors was performed over 3 needle strokes in order to check the quality of the distributor. The angle between the injectors was 75 degrees. There were eight piers in each outlet part of the distributor, four of which were rather thick and four were thin. The computational domain consisted of the distributor with injectors and cones behind the injectors (Fig. 1a). The grid has about four million nodes. Jets were formed inside the cones during the calculation. The grid of the distributor remained the same for all calculations, with the exception of parts containing the needle and cones, which were different for each opening. Grids in the cones were refined according to the thickness of the jets. In order to improve the accuracy of numerically obtained jets, the calculation was repeated on a computational domain, which consisted only of the injector and the cone (Fig. 1b). Inlet boundary conditions were obtained from a previous calculation. During the calculation the grid was automatically refined in regions where gradients of water and air volume fraction were the largest. Refinements were carried out in fifteen steps. The initial grid consisted of 1.4 million nodes, while the final grid had 6 million nodes. Table 2 contains the discharge distribution and flow energy losses in the distributor. The difference between discharge through injector 1 and through injector 2 does not exceed %. Flow energy losses in the distributor and injectors are between 1.68 % and 3.39 % of the head. Losses for small openings are greater due to flow in the injectors. a) b) Fig. 1 Computational domain and grid for numerical analysis of flow in the distributor and injectors with jets a) and part for grid refinement b) Table 2 Discharge distribution and flow energy losses in the distributor and injectors, CFD results A o / A o,bep Q 1 /Q*100 (%) Q 2 /Q*100 (%) ΔH/H*100 (%)

5 Velocity vectors on plane y = 0 and the shapes of jets are presented in Fig. 2. A contraction of the jet and wake behind the needle is clearly visible. Grids are sufficiently refined, therefore the interphase between water and air is very thin. Due to the well designed distributor, no dispersion or deviation of the jet is present. The cross-section of the jet is a nearly perfect circle, only the smallest opening upon very close scrutiny shows an influence of eight piers in the distributor. In order to get all three components of jet velocity, we defined a local cylindrical coordinate system in such a way that the axis is situated in the direction of the injector. The axial velocity component is therefore the velocity of the jet in the direction of the injector, while the radial and circumferential velocity components were used to calculate the magnitude of secondary velocity. The thickness and velocity of jets obtained numerically on refined grids were compared to theoretical values obtained from turbine head and discharge (see Table 3). Numerically obtained jets for three openings were % thinner than theoretical ones. Velocity in the jet is not uniform, therefore averaged and maximal values of jet velocity divided by theoretical values must be presented. Calculated jet velocity is greater than the theoretical one due to a reduced thickness of the jets. Averaged values of secondary velocity do not exceed 0.35 %, while maximum values do not exceed 1.7 % of jet velocity. The influence of the grid on jet shape and thickness is shown in Fig. 3. The top part of the image shows the grid and the water volume fraction on a cross-section of the cone before and after grid refinement. The coarse grid clearly shows a deformation of the jet shape. After grid refinement, the deformation is no longer present. It was thus obviously not a consequence of a bend or piers in the distributor. Grid refinement is even more important in the location where the jet is exiting the injector. The lower part of Fig. 3 shows how the angle and thickness of the jet became altered after grid refinement. At A o / A o,bep = 0.5, where the effect of grid refinement was most significant, the thickness of the jet obtained on a coarse grid was 5 % greater and shows an averaged velocity of 9.4 % less than after grid refinement. A o / A o,bep Velocity vector plot at plane y = 0 Water volume fraction, plane y = 0 and cross-section at runner inlet water - red, air -blue Fig. 2 Velocity vector plot on plane y = 0 and thickness and shape of jets for different openings, refined grids 4

6 Table 3 Comparison of theoretically and numerically obtained values of jet thickness and velocity A o / A o,bep D CFD /D t (-) v CFD, aver /v t (-) v CFD, max /v t (-) Fig. 3 The effect of grid refinement on the cross-section shape of the jet (top) and on the thickness of the jet (bottom), A o / A o,bep = Flow in the runner The second step was a numerical analysis of flow in the runner. The computational domain consisted of a runner and a thin ring around it (Fig. 4). Due to the symmetry of the turbine, the computational domain was reduced by half. A symmetry boundary condition was prescribed at symmetry plane y = 0. The unstructured grid consisted of about 25.7 million elements and 11 million nodes. The velocity components and water and air distribution at the two inlets were prescribed. In order to get a more accurate shape of the jet, the grid at the inlet was refined separately for each jet diameter. The outlet boundary condition was prescribed at the remaining part of the outer surface of the ring. The opening boundary condition at constant pressure and air as a fluid was defined for the other boundaries of the domain. Transient calculations were performed with a time step of 0.2 degrees of runner rotation. 20 iterations were performed per each time step. At first as inlet boundary conditions we prescribed the values of jet velocity and water volume fraction from previous analysis of the flow in distributor with injectors obtained on basic grids. After verifying the results it became clear that these results were too inaccurate. Next calculations were performed with theoretical values of jet velocity and thickness prescribed at the inlets. Meanwhile, analysis of flow in injectors and cones on refined grids were carried out for three openings. Differences in thickness and velocity of the jets were significant. Since we have already obtained some results for the runner with theoretical jets, we continued that way. The results presented in this paper were therefore obtained with theoretical jets, only the results for maximal opening were obtained also with jets calculated on refined grids. We plan to repeat the analysis with calculated jets for other openings for at least some operating points. Approximately 600 time steps (120 degrees of runner rotation) were done at each operating point. Fig. 5 shows the value of torque on the shaft during calculation at point 9. At the beginning of calculation the whole domain was filled with air and torque was equal to zero. When the jet was approaching the runner, the torque increased and after some time we were able to observe periodic behaviour. The runner has 23 buckets so the period is degrees of runner rotation. The angle between the injectors is 75 degrees, therefore the two jets 5

7 impact different parts of the buckets at a given moment. Torque oscillations are therefore smaller and the shape of the curve is more irregular. The efficiency was calculated from torque averaged over time with only the last three periods (bold part of the curve in Fig. 5) used. 250 Torque (M) angle of runner rotation Fig. 4 The computational domain and grid for runner analysis Fig. 5 Torque on the shaft during calculation Fig. 6 shows the pressure distribution on runner buckets and water jets and evacuating sheets for three operating regimes. High pressure on regions where the jets impact the buckets and low pressure at the backsides of the buckets can be clearly distinguished. At small openings with thin jets the variation of the pressure is smaller, also the evacuating water sheets are smaller and there are less small water jets and drops in the region around the runner. A detailed explanation of pressure distribution, jets and formation of evacuating sheets during a single period is included in [8], therefore it is not repeated here. A o / A o,bep = , Point 2 A o / A o,bep = 1.0, Point 8 A o / A o,bep = 0.5, Point 15 Fig. 6 Pressure distribution on buckets and jets and evacuating water sheets for different operating conditions Fig. 7 shows the calculated and measured values of efficiency divided by measured efficiency at BEP. Results obtained with theoretical jets were calculated for six openings and shapes of the efficiency curves are quite well captured. The numerically obtained values are however approximately 4 % smaller than the measured ones. We can not explain why the discrepancy between calculated and measured results is the largest at the optimal opening. For the smallest two openings only two operating points were analyzed, the calculations for additional points are in progress. For the maximum opening, analysis of the runner was also carried out with the inlet conditions obtained from analysis of the injectors on a refined grid and the efficiency values are much closer to the measured ones. The difference does not exceed 2%. The results are very promising, but we do not know whether the results for other openings will have the same accuracy. 6

8 One of the reasons for discrepancies between numerical and experimental results is the grid, which is still not refined enough. The effect of the grid on jet shape and thickness can be seen in Fig. 8. The grid part near the inlets is refined and the surface of the jet in this part is smooth with a thin interphase between water and air. However, in the rotating part with buckets the grid is not fine enough. The interphase between water and air is thick, the surface of the jet is not smooth and the structure of the grid is easily seen. The difference between calculated kinetic energy values at the inlet and values slightly nearer to the buckets in the rotating part is between %. Also, the maximum value of y + at the buckets is still 80. These are the key reasons for insufficient torque on the shaft and consequently insufficient values of calculated efficiency η/ηbep (-) Experiment CFD, theoretical jets CFD, numerically obtained jets ϕ/ϕ BEP ( - ) Fig. 7 Turbine efficiency: measured and numerically obtained values 5. Computational effort and hardware for Pelton turbine analysis Numerical analysis of a flow in a Pelton turbine is very time consuming. Analysis is unsteady and free surface flow has to be modeled using one of the multiphase models. In our case, where the grids for runner analysis had 25.7 million elements and 11 million nodes, calculation at one operating point took five days on 128 processors. Analysis was carried out on LSC Adria supercomputer cluster with 2048 processor cores, so calculations for several operating points ran simultaneously but we still found the analysis of 17 operating points much too long. The situation would be even worse if an inhomogeneous model was used or the grids were additionally refined. Excessive CPU time is perhaps the most important reason for limited use of CFD in the design of Pelton turbines. 6. Conclusion a b c Fig. 8 Deformation of the jet due to overly coarse grid in a region with buckets, (red - water, blue - air) a) inlet, b) offset of inlet in direction to the region with buckets, c) symmetry plane 7

9 The numerical analysis of flow in the Pelton turbine was performed for several operating points. On the basis of the results it can be concluded: - Accurate calculation of jet thickness and velocity is crucial for prediction of Pelton turbine efficiency. - Too coarse grid greatly influences the results and is one of the most important reasons for discrepancies between numerical and experimental values of efficiency. - Numerical results are sufficiently accurate to be used for efficiency prediction in a design process. - A numerical analysis of a Pelton turbine demands large computing capacities. Its limited use in design process is mostly due to excessive CPU time. Acknowledgments The research is partially financed by the Slovenian Research Agency ARRS Contract No. L and Contract No Nomenclature A 0 A 0, BEP D CFD D t H p Q Q 1 Q 2 U r v CFD,aver v CFD,max v t ΔH Relative opening [-] Relative opening at BEP [-] Jet diameter obtained by CFD [mm] Theoretical jet diameter [mm] Head [m] Pressure [N/m 2 ] Discharge [m 3 /s] Discharge through injector 1 [m 3 /s] Discharge through injector 2 [m 3 /s] Velocity vector Averaged velocity in jet obtained by CFD [m/s] Maximal velocity in jet obtained by CFD [m/s] Theoretical jet velocity Flow energy losses in distributor and injectors [m] η η BEP η rel φ Ψ ρ BEP CFD Turbine efficiency Turbine efficiency at BEP Relative efficiency [-], ( = η / η BEP ) Discharge coefficient [-] Pressure coefficient [-] Density Best Efficiency Point Computational Fluid Dynamics References [1] Parkinson E, Garcin H, Vullioud G, Zhang Z, Muggli F and Casartelli E 2002 Experimental and Numerical Investigation of Free Jet Flow at a Model Nozzle of a Pelton Turbine Proc. of the XXI IAHR Symp. on Hydr. Machin. and Syst. (Lausanne, Switzerland) [2] Veselý J and Pochylý F 2003 Stability of the flow through Pelton Turbine Nozzles Hydro 2003 (Dubrovnik, Croatia) [3] Staubli T, Abgottspon A, Weibel P, Bissel C Parkinson, E Leduc and J Leboeuf 2009 Jet Quality and Pelton Efficiency Proc. of Hydro 2009 Progress - Potential - Plans (Lyon, France) [4] Mack R and Moser W 2002 Numerical Investigation of the Flow in a Pelton Turbine Proc. of the XXI IAHR Symp. on Hydr. Machin. and Syst. (Lausanne, Switzerland) [5] Kvicinsky S, Kueny J L, Avellan F and Parkinson E 2002 Experimental and Numerical Analysis of Free surface Flows in Rotating Buckets Proc. of the XX1 IAHR Symp. on Hydr. Machin. and Syst. (Lausanne, Switzerland) [6] Parkinson E, Neury C, Garcin H, Vullioud G and Weiss T 2005 Unsteady Analysis of a Pelton Runner with Flow and Mechanical Simulations Hydro 2005 (Beljak, Austria) [7] Perrig A 2007 Hydrodynamics of the Free Surface Flow in Pelton Turbine Buckets PhD Thesis (Ecole polytechnique federale de Lausanne, Lausanne, Switzerland) [8] Jošt D, Lipej A and Mežnar P 2006 Numerical Analysis of Dynamic Behaviour of a Flow in a Pelton turbine IAHR Int. Meet. of WG on Cavit. and Dynam. Probl. in Hydr. Machin. and Syst. (Barcelona, Spain) [9] ANSYS CFX 2010 ANSYS CFX Solver Theory Guide (Release 12.1) 8