Comparison of the Convergent and Divergent Runners for a Low Head Hydraulic Turbine

Transcription

1 Journal of Energy Power Engineering 9 (2015) doi: / / D DAVID PUBLISHING Comparison of Convergent Divergent Runners for a Low Head Hydraulic Turbine Zbigniew Krzemianowski 1 Romuald Puzyrewski 2 1. Department of Hydropower, The Szewalski Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Fiszera 14, Gdansk , Pol 2. Department of Energy Industrial Apparatus, Mechanical Faculty, Gdansk University of Technology, Narutowicza 11/12, Gdansk , Pol Received: September 29, 2015 / Accepted: October 30, 2015 / Published: December 31, Abstract: The paper presents a method of runner blades design for simple case of hydraulic turbine. Two differently shaped channels of meridional cross section were examined. The quantitative evaluation was performed by means of 3D algorithm. It has been found that, divergent runner prevails convergent from dissipation point of view. Although, influence of draft tube has not been analyzed, presented method is important for designers of low head hydraulic turbines which are not matter of stardization due to variety of environmental conditions. Key words: Hydraulic machinery design, inverse problem, 2D model. 1. Introduction The paper is devoted to design of runner blade cascade supported by 2D 3D computation models, for low head hydraulic turbine. The necessity of developing method of design comes from fact that, planned low head hydraulic turbines are not objects for stardization due to variety of environment, in which such installations are foreseen. Also from oretical point of view, problem is interesting because it insists to develop method of finding geometry of designed blades. Here, attention is focused on two models. Model 2D is presented in version of so-called inverse problem in which boundary conditions do not contain geometry of blades. The model allows creating such geometry. If geometry is determined from 2D model, n 3D computation may be next step to check flow field. The aim of analysis was focused on finding geometry where minimum Corresponding author: Zbigniew Krzemianowski, PhD, research associate, research fields: hydraulic turbines designing analyzing, hydraulic turbines measurement. losses are expected. Besides, influence of turbine meridional shape in domain of runner in two versions: convergent divergent was investigated. It was also interesting to find answer wher function of diffuser behind runner can be partly overtaken by contour as it is in divergent runner. Here, attention is focused on two differently shaped runners. Convergent shape of runner leads to an increase of load of draft tube behind runner if outlet of draft tube is imposed. One has to be aware of that, increase of draft tube load may change final conclusions. The main task of presented paper is method of runner blade design. For 3D computation, commercial code ANSYS/Fluent 15 was applied. 2D model used to calculations was described in Ref. [1]. 2. Geometrical Boundary Conditions The geometry of two contours of flow field discussed here is shown in 1. In both cases, family of lines (y can be treated as a wishful order of stream surfaces flowing by a

2 1038 Comparison of Convergent Divergent Runners for a Low Head Hydraulic Turbine ArcTa (2). The inner line (hub) in both cases is equal to: f inner = 0.06 m. The family of lines is given by function of x (1) in form: an (3) The choice of se function types is based on intuition experience. These functions represent quantitatively factor of a convergence divergence. In authors opinion, this motivation is sufficient for analysis of a convergence divergence upon flow field of turbine stage. The positions of leading trailing edges of runnerr blades were also chosen intuitively. The number of blades in runner domain belongs to geometrical boundary conditions. According to experience, it was taken into account three or four blades in runner domain. The influence of number of blades was investigated by means 3D model. 3. Boundary Conditions for Flow Parameters 1 Two views of flow domain: convergent runner divergent runner. meridional plane) shown in 1, covers domains of guide vanes (DV), gap (DG) runner (DR). These lines were used in non-orthogonal system of coordinates describing outer border denoted as x (1) = 0 internal border denotedd by x (1) = 1. The angular coordinate is denoted by x (2) axial coordinate by x (3). The outer line for converging domain is given as radius function f dependent on axial coordinate x (3) : ArcTan for diverging domain: (1) In front of runner cascade, distributions of flow parameters weree chosen as it follows from previous computation of guide vane domain. The distributionss of axial tangential velocity components in front of runner cascade are shown in 2. The axial velocity distribution in chosen example of geometry gives mass flow rate m = 235 kg/s. 2 Distribution of axial tangential velocity components at inlet to runner domain.

3 Comparison of Convergent Divergent Runners for a Low Head Hydraulic Turbine 1039 Model 3D gives pressure field from solution of momentum equation in respect to reference point, where value of pressure is given. The referencee for pressure level was set at outlet to be equal zero. For 2D inverse model, it was necessary to assume at outer contour line of runner domain distribution of pressure from inlet to outlet. The pressure distribution was given as functionn of three parameters:,, where, is non-dimensional coordinate: ; trailing point at outer border of runnerr exponent n b of non-dimensional coordinate. It is worth noting that, p outlet controls pressure drop along runnerr cascade. The examplee of pressure distribution along outer border of runner domain is shown in 3. The parameters are: p in nlet = 8,000 Pa, p o utlet = -5,000 Pa -20,000 Pa, n b = The influence of pressure drop on designed runner blades was thoroughly investigated by means of 2D 3D models. (4) ; p inlet is inlet pressure; p o utlet is pressure in For 3D model, rest of boundary conditions are formulated according to version of turbulence model chosen for computation. This is matter of commercial code (ANSYS/Fluent 15) refore it will not be discussed here. 3 Examples of pressure distribution at outer border of runnerr domain for 2D model. 4. Short Description of 2D Inverse Model The main points of algorithm of inverse 2D model are described in Refs. [1-8]. Therefore details of such approach will be omitted. Method of solution of momentum conservation equation leads to determination of pressure field in domain of runner. Having pressure determined rest of parameters i.e., meridional tangential components of velocity, respectively denoted as, can be computed from mass conservation equation: 1, energy conservation equation: (6) The following notations are used: masss flow rate,, blockage function takes into account space occupied by material of blades, p pressure, ρ density, U rot circumferential velocity of runner, e internal energy, e c total energy, Π potential energy, f radius whichh is stream function dependent on. At runner inlet, along leading edge, values m e c are given from boundary conditions. They are dependent on an x (1) coordinate. Blockage function was chosen as below:, 1 1 (7) (5) Intuitively chosen parameters for blockage factor are: t 1 = 0.2, t 2 = 0.5, t 3 = 1.0, t 4 = It generates distribution of blockage function as it is shown in 4. The arguments x (1) are within range <0-1>. The shape of profile generated by this blockage function is shown in 5. The line crossing profile in middle of thicknesss is skeleton line whichh is determined as trajectory of fluid element at outer contour of runnerr domain. The suction side of profile is given by coordinates:

4 1040 Comparison of Convergent Divergent Runners for a Low Head Hydraulic Turbine 4 Blockage function in runner domain. different: :, (10) In case, parameters were chosen as follows: w 0 = 0.017, w 1 = , w 2 = , w 3 = , w 4 = 1.983, n l = 3.0, n s = 2.0 (11a) in case, parameters were chosen as follows: w 0 = , w 1 = , w 2 = , w 3 = , w 4 = 1.983, n l = 3.0, ns = 2.0 (11b) They lead to loss coefficient distribution shown for both cases in 6. Then increase of internal energy was computed in reference of kinetic energy at inlet of runnerr domain: Δ, The level of inlet kinetic energy was kept constant so it is possible to adjust value of internal energy increase only by loss coefficient distribution. (12) 5. Some Results of 2D Computations 5 The profile of runner at outer contour generated by means of a 2D inverse model. sin 2π1 / cos 2π1 / for pressure side: sin 2π / cos 2π / (8) (9) where, z number of blades, t p division of thickness of profile at both sides of skeleton line. In energy conservation equation, quantity, appears which represents internal energy. From inlet to outlet of runner domain, this function ought to be determined in advance by means of a loss coefficient. It is a matter of intuitionn to prescribe level of loss coefficient. In present paper, authors propose following form of loss coefficient distribution in runner domain for two cases The main goal of 2D inverse problem computation was to generatee geometry of runner blades. After that, generated blade geometry was examined by means of 3D computation model. In order to diminish number of cases in 2D computations, it was assumed scheme computation presented in Table 1 below. 6 Loss coefficient distribution, for two analysed cases.

5 Comparison of Convergent Divergent Runners for a Low Head Hydraulic Turbine 1041 Table 1 Variants of 2D model computations. Convergent contour Pressure p outle t at upper corner of runner blades (reference p outlet = (-20,000)-5,000 Pa parameter) (-5,000, -6,000, -7,000, -8,000, -11,000, -13,000, -15,000, -20,000, -23,000, -28,000, -30,000) The reference parameter p outlet willl be plotted as an abscissa in most figures presented below. For convergent contour, two examples of geometry are shown in 7. For divergent contour, similar geometries are shown in 8. To compare computed cases, mean values of parameters averaged over inlet outlet surfaces were used. The mean pressure drop along runner blades versus p outle et is shown in 9. No difference in pressure drop is visible for both contours. Divergent contour p outlet = (-26,000)-5,000 Pa (-5,000, -6,000, -7,111, -12,000, -18,000, -23,426,-28,000, -26,000) 8 Two examples of geometry for divergent contour (: p outle et = -5,000 Pa, : p outlet = -20,000 Pa). 7 Two examples of geometry for convergent contour (: p outlet = -5,000 Pa, : p outlet = -20,000 Pa). p out tlet (Pa) 9 Pressure drop along runner blades for two contours (convergent divergent).

6 1042 Comparison of Convergent Divergent Runners for a Low Head Hydraulic Turbine 2 L Δ 2 (18) The kinetic energy related to meridional velocity 10 The outlet angle of absolute velocity for two contours (convergent divergent). The outlet angle of abso lute velocity is shown in 10. The most interesting cases are those in whichh outlet angle value 2 is close to 90 what means axial outflow. The argument for axial outflow comes from a discussion of energy equation. Let us definee energy at inlet outlet to runner (assuming that, all parameters are averaged), which are available to conversion into mechanical energy in runner domain: ( 13a) 2 2 ( 13b) Thus, energy conservation equation along runner domain can be written as: (14) component /2 is tied up with masss conservation equation cannot be minimized for given mass flow rate. The kinetic energy of tangential velocity component /2 can be treated as part of lost energy (Le). If re is no means to convert this amount of energy into pressuree energy, it inevitably will be dissipated. Then, we can treat sum: Δ (19) as losses, which has to be minimized. This is reason why behind runner, axial velocity is preferable. The existence of tangential velocity diminishes amount of work converted by runner. 11 showss sum Le as a function of p outlet. On basis of 11, conclusion can be drawn that, minimum Le places in neighborhood of p out let = -5,000 Pa. This points out geometry whichh can be considered as an advisable. Neverless, such conclusionn should be checked by means of 3D Energy conservation equation can be rewritten as follows: where: Δ Δ (15) (16) is amount of energy converted by runner Δ 0 (17) is increase of internal energy due to dissipation now can be presented in form: 11 Lost energy Le as a function of pressure p outlet for two contours (convergent divergent).

7 Comparison of Convergent Divergent Runners for a Low Head Hydraulic Turbine 1043 computation. Two reasons will influence on differences between 3D 2D models: (1) The finite number of blades which sts in opposition to infinite number of blades in 2D model; (2) 3D computation provides more realistic estimation of dissipation effect than it is usually assumed in 2D computation. 12 shows level of dissipation energy introduced in 2D computation for two cases of loss coefficient distribution. One can notice from 12 that, dissipated energy for two contours is of same value. The loss coefficient identically distributed in domain of runner for both contours leads to same level of dissipation. The energy equation is valid also for an arbitrary point of runner domain. One can rewrite this equation into following form: 2 2 Left h side of above equation contains given values at runner inlet. Right h side can be treated as quadratic equation for if rest of parameters are given. Pressure field in runner domain can be computed according to method described in Ref. [1]. The solution of above equation has form: + Π Δ (20) Δ (21) where, Δ is discriminant of quadratic equation. It is possible to have situation when re is no real solution for due to negative discriminant. This may happen when blockage factor increase of internal energy due to dissipation are too high. Then we may have situation where at some points in runner domain no solution exists. The grids of points wheree solution was sought for are shown in 13 for p outlet = -5,0000 Pa. p outlet t (Pa) 12 The level of dissipation energy assumed in computation by means of 2D model. 13 Convergent divergent blades obtained by means of 2D model for p outlet = -5,000 Pa. The 13a presents convergent case 13b presents divergent case. In case of convergent contour, empty region of blade ( white spot ) close to leading edge may be observed in which no solution for

8 1044 Comparison of Convergent Divergent Runners for a Low Head Hydraulic Turbine was obtained. In case of divergent contour ( 13b), solution was obtained in whole runner domain for comparable set of parameters. In convergent contour, decrease of cross section due to convergence increased meridional velocity. The decrease of cross section manifests increase of blockage factor. To fill up white spot area, an additional assumption should be introduced. To create surface wheree no solution is obtained, geometrical extrapolation from neighboring area may be applied or if 1,, approximation is justified. In this way, geometry based on 2D inversee model for 3D computation may be obtained. 6. The 3D Computation of Flow Domain Estimation of Internal Energy Increase by Means of Kinetic Dissipation Energy Value The 3D computation with version of turbulence model k-ε provides with distribution of kinetic dissipation energy ε in runner domain. In order to estimate increase of internal energy, due to turbulence let us examinee two relations. The first one is relation between entropy production s m rate of dissipation ε (m 2 /s 3 ): where, T absolute temperature, ρ density. The second relation is energy conservation equation written for internal energy: Δ (22) (23) One can simplify above equation for incompressible fluid neglecting heat conduction. Combining se equations it can be written: (24) integrating increase of internal energy Δe (m 2 /s 2 ) is estimated as follows: Δ The mean value of let us estimate as: The interval of time Δt is related to trajectory of representative fluid element which passes runnerr domain. As a representative length Δlaxial, we can choose axial length of runner at mean diameter mean value of axial velocity: n one can estimate: d Δ Δ The foregoing considerations formulate estimation of increase of internal energy due to turbulent dissipation, which is prevailing factor of dissipation. 7. Some Results of 3D Computations d d 14 showss level of internal energy increasee due to dissipation according to results of 3D computation by means of turbulence model k-ε. The level of energy dissipation assumed for 2D computation, as it is shown in 12, was overestimated for case compared to values in 14. The case shows a good guess comparing with 3D results. p outlet t (Pa) 14 The level of internal energy increase due to dissipation computed by means of turbulence model k-εε (three four blades runner for divergent contour). Δ (25) (26) (27) (28)

9 Comparison of Convergent Divergent Runners for a Low Head Hydraulic Turbine 1045 The difference of Δe between threee four blades in runner is negligible, covered by scatter of numerical results. The slight increase of dissipation for lower values of p outlet can be explained by increasing of profiles length, as it can be seen in Figs The sum of lost energy Le from 3D computation is shown in 15. Comparing to data in 12, it can be concluded that, minimum is shifted to left, giving recommended solution at p outlet = -23,600 Pa for three blades p outlet = -16,150 Pa for four blades. 16 shows averaged pressure drop along 3-blade runner domain. Comparing to values of pressure at inlet of runner, one can notice that, outlet pressure is closer to zero. This is in accordance to p out tlet (Pa) 16 Distribution of pressure p inlet at inlet to runnerr domain pressure drop Δpp along runner blades (3-blade runner with divergent contour). p out tlet (Pa) 17 Distribution of pressure p inlet at inlet to runnerr domain pressure drop Δpp along runner blades (4-blade runner with divergent contour). 15 Sum of Le energy showing minimum for three four blades for divergent contour convergent contour. boundary condition for pressure in computation of 3D model. Comparing values in 16 to dataa in 10, it became evident that, in 2D model, we have a higher pressure drop due to fact of infinite number of blades. For given mass flow rate, cascade is more dense, higher pressure drop is obtained as it can be seen comparing 16 to 17 for two different numbers of runner blades (3 4-blade runner). It must be emphasized that, velocity conditions at inlet to runner was kept at same level same was also reference pressure behind runner. In 18, pressure drop for divergent contour 3 4-blade runner is confronted. Table 2 presentss losss energy, pressure at outlet pressure differencee for 3 4-blade runners.

10 1046 Comparison of Convergent Divergent Runners for a Low Head Hydraulic Turbine channel shaping. Especially in case when maximumm power output (not efficiency) is main target. This problem exceeds volume of present paper. Acknowledgments The work was supported by Polish National Science Centre under grant no. 6694/B/ /T02/2011/400 for The Szewalski Institute of Fluid-Flow Machinery in Gdansk, Polish Academy of Sciences. References 18 Comparison of pressure drops for two different blade numbers of runner with divergent contour convergent contour. Table 2 Comparison of 3D computations results for three four blades runners. Divergent contour Convergent contour Parameters Three blades Four blades Three blades Four blades Le min (m 2 /s 2 ) p outlet min (Pa) , , , ,8888 Δp stat (Pa) 10,345 10,155 16,547 17,045 10,000 10,000 17,000 17, Conclusions (1) 2D inverse method is effective to obtain geometry of blades for runner; (2) 3D computation for two differently shaped runners shows that, from dissipation point of view, divergent runner is more effective thatt convergent one; (3) The increase of draft tube load for convergent runner may change final decision concerning [1] [2] [3] [4] [5] [6] [7] [8] Puzyrewski, R Two Dimensional Inverse Method of Turbomachinery Stage Design, Developments in Mechanical Engineering. Gdansk: Gdansk University of Technology Publishers, ISBN Puzyrewski, R., Krzemianowski, Z D Model of Guide Vane for Low Head Hydraulic Turbine: Analytical Numerical Solution of Inverse Problem. Journal of Mechanical Engineering Automation 4 (3): Puzyrewski, R., Krzemianowski, Z Two Concepts of Guide Vane Profile Design for a Low Head Hydraulic Turbine. Journal of Mechanical Engineeringg Automation 5 (4): Krzemianowski, Z., Puzyrewski, R D Computations of Flow Field in a Guide Vane Blading Designed by Means of 2D Model for a Low Head Hydraulic Turbine. Journal of Physics: Conference Series 530 (1): 1-8. doi: / /530/1/012031, ISSN (Online). Puzyrewski, R., Krzemianowski, Z Impact of Blade Quality on Maximum Efficiency of Low Head Water Turbine. Journal of Mechanics Mechanical Engineering 17 (3): ISSN Puzyrewski, R Lectures on Theory of Turbomachinery Stage Two Dimensional Model (2D). Gdansk: Gdansk University of Technology. Puzyrewski, R., Namieśnik, K Comparison between Conical Paraboloidal Turbine Stages: Inversee Problem. Presented at 3rd International Symposiumm on Experimental Computational Aerormodynamics of Internal Flows (ISAIF), Beijing, China. Flaszyński, P Comparison of Two Dimensional Three Dimensional Models Based on Turbomachinery Flow Passages. Ph.D. sis, Gdansk University of Technology.