2D Model of Guide Vane for Low Head Hydraulic Turbine: Analytical and Numerical Solution of Inverse Problem

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1 Journal of Mechanics Engineering and Automation 4 (4) 95- D DAVID PUBLISHING D Model of Guide Vane for Low Head Hydraulic Turbine: Analytical and Numerical Romuald Puzyrewski and Zbigniew Krzemianowski. Faculty of Mechanical Engineering, the Gdansk University of Technology, Gdansk, 8-33, Poland. Department of Hydraulic Machinery, the Szewalski Institute of Fluid-Flow Machinery, Polish Academy of Sciences, Gdansk, 8-3, Poland Received: January 3, 4 / Accepted: February, 4 / Published: March 5, 4. Abstract: Low-head hydraulic turbines are the subjects to individual approach of design. This comes from the fact that hydrological conditions are not of a standard character. Therefore, the design method of the hydraulic turbine stage has a great importance for those who may be interested in such an investment. As a first task in a design procedure the guide vane is considered. The proposed method is based on the solution of the inverse problem within the frame of D model. By the inverse problem authors mean a design of the blade shapes for given flow conditions. In the paper analytical solution for the simple cylindrical shape of a guide vane is presented. For the more realistic cases numerical solutions according to the ais-symmetrical model of the flow are also presented. The influence of such parameters as the inclination of trailing edge, the blockage factor due to blade thickness, the influence of loss due to dissipation are shown for the chosen simple geometrical eample. Key words: Hydraulic turbines, inverse problem in a turbomachinery, guide vanes design.. Introduction Contemporary methods of predicting the flow field in turbomachinery blade passages are based on well developed numerical codes of 3D type. This approach is commonly applied for the so-called direct problem when the geometry of blading is given. The situation for an unknown blading, when the geometry has to be designed, imposes to apply simpler models as D or D in order to solve inverse problem i.e., to find the geometry for given boundary conditions. Such an inverse problem is particularly important for low head hydraulic turbines. The variety of conditions for this type of machines is not the subject for standardization. Each case should be treated individually. Therefore, developing of simple models for designing the blading is important task from technical point of view. Corresponding author: Zbigniew Krzemianowski, Ph.D., research associate, research fields: hydraulic turbines designing and analyzing, hydraulic turbines measurement. krzemian@imp.gda.pl. Particularly suitable for the inverse problem is D model based on the assumption that blading influences the change of parameters only in radial and aial directions, neglecting the change in circumferential direction. This idea was conceived a hundred years ago. The first paper having used this approach was presented in 95 by Lorenz []. The idea of stream surfaces crossing along the representative streamlines in turbomachinery passages was presented by Wu [] in 95. Starting from 99 [3] the idea of using D model for designing the blading was developed for different type of turbomachinery blading. The appropriate references are Refs. [4-8]. The main efforts in the last cited papers was laid to include into modeling such factors as distribution of losses in the region of flow, influence of blade thickness and modeling of force acting upon the fluid from blading. The presented method in paper of the blade vane design is based on the solution of inverse problem

2 96 D Model of Guide Vane for Low Head Hydraulic Turbine: Analytical and Numerical within the frame of D model. One can assume the flow surfaces coordinate as: = const, the angular coordinate as () = const and ais coordinate as = const. As the starting point is assumption of a meridional view of ai-symmetrical flow surfaces f(, ) limited by the bounds AB, BD, DC, CA as shown in Fig., where r =.6 m, r =.95 m, =. m, =. m, =., =.. The set of governing equations in a non-orthogonal (in a general case) coordinate system can be presented as follows: Mass flow rate conservation equation f, U m f f Fig. Meridional shape of the analyzed guide vane. where, f = f(, ) assumed function of flow surfaces, U meridional velocity component (that is a vector of aial and radial velocities), ρ density, m( ) mass flow rate function given at inlet edge, τ(, ) blockage factor due to blade thickness. Momentum conservation equation in direction f f sgn U () U f f () f f f f f p p f F f f f f where, p = p(, () ) pressure, U tangential velocity component, = g gravity potential, () g gravitational acceleration, F, F, F force components acting in flow. Momentum conservation equation in () direction U U f f U F f Momentum conservation equation in direction () () () U U U f f f f f f p p f F f f (4)

3 D Model of Guide Vane for Low Head Hydraulic Turbine: Analytical and Numerical 97 Energy conservation equation U () U p U () U R ct ec (5) where, U R runner circumferential velocity, c specific heat, T temperature, ct internal energy changing due to losses (assuming T = const means that no losses in the flow domain eist), e C ( ) total energy function given at inlet edge. Guide vane model condition U F (6) This means that in the flow domain there is no energy etraction from a fluid particle. The additional simplifying assumption concerns F = and behaviour of temperature T due to dissipation in the flow domain allowing to close the system of 5 above presented Eqs. -(5) for the () () unknowns: p, U, U, F, F and search for the analytical and numerical solutions. It can be shown that the set of equations with simplifying additional assumption can be reduced to the hyperbolic type where two families of characteristics eist: I family = const (7) II family f f d (8a) d f or f dr (8b) d f Generally the lines of II family are perpendicular to lines of I family. Along the lines of II family the pressure satisfies the equation: f U dp U () (9a) d f f f f or d or f f f U dp U () f f f (9b) f U dp U () (9c) dr f f Further sections of the paper present the application of the above shown D model. Section presents necessary assumptions that have to be determined to allow finding analytical solution for a cylindrical case of guide vane. If some of these assumptions are abandoned only numerical analytical solution can be found. Therefore, Section 3 presents a few eamples of numerical solution for a cylindrical case of guide vane with no simplifying assumptions. Finally, Section 4 presents relevant conclusions.. Analytical Solution for Cylindrical Case For the simple case of cylindrical vane, the flow surfaces one can assume as follows: f r r r () where, r > r, and coordinate [, ] is constant along cylindrical flow surfaces shown in Fig. as it was aforementioned. This is the first family of characteristics according to Eq. (7). The second family of characteristics following Eq. (8b) has the form dr r r () d Fig. The sketch of streamlines and characteristics.

4 98 D Model of Guide Vane for Low Head Hydraulic Turbine: Analytical and Numerical The characteristics in Fig. are shown as lines starting from the borders AB (A B ) or A C. The integral of Eq. () has the form written in Eq. (). Along the characteristics II family Eq. (9c) has the form dp U () dr r One can integrate the above equation only in case of additional simplifying assumptions, namely (inde denotes parameters at inlet and upper edges): no losses in the flow: ct = const; () no blockage due to profile thickness: τ(, ) = ; uniform distributions of parameters along the leading edge: e C ( ) = const and U = const. From Eq. (5), the following equation can be obtained: p U p U () ec U e C (3) The third assumption allows stating that U E ec const (4) Then Eq. leads to the solution r p E E p (5) r where, p pressure at the starting point for r = r of characteristics II as shown in Fig.. The components of velocity vector are U () r U p ec (6) r A p p* pab p* p AB (8) B A where, p* means boundary pressure at point A (or A ) and p AB may be replaced by p AB and analogically A, B may be replaced by A, B depending on the considered case (Fig. ). The mean value of pressure drop across the guide vane blade is given by r def pbd pbddr r r (9) r r p p p p () * BD BD AB r which allows estimating the outlet angle as follows: m ArcTan r r π rr rln pbd r r where, m is mass flow rate. Moreover, one can find the shape of stream surface representing the skeleton of the designed blade in the form of analytical function: 3 r pab z z z CU, r, p AB () 3 U r B r where, r, φ, z are the cylindrical coordinates (z ). The eample of such a shape is shown in Fig. 3. U U const (7) The above solution depends on boundary condition given as a function p ( ) along the border AB (or A B ) or A C as shown in Fig.. For the case of linear change of pressure along the border AB (or A B ), it can be written as Fig. 3 The eample of the skeleton shape according to Eq. (). 3. Numerical Solutions for Cylindrical Case Analytical solution may be used as the check for numerical solutions according to the algorithm presented

5 D Model of Guide Vane for Low Head Hydraulic Turbine: Analytical and Numerical 99 in the introduction. Among very large possible number of parameters defining the geometry of designed blade, only a few factors were chosen to show their influence on the blade shape. They are listed in Table. Table The chosen factors to investigate the influence on the blade shape. Factor Trailing edge inclination Parameters kept constant Pressure drop at outer radius, no losses Influenced parameter Outlet angle Fig. 5 Changes of the outlet angle for different inclination of the trailing edge. Blockage factor (blade thickness) Loss coefficient distribution in the flow domain Pressure drop at outer radius, inclined trailing edge, no losses Pressure drop at outer radius, inclined trailing edge Eistence of solution Eistence of solution 3. Trailing Edge Inclination Fig. 4 presents below the eample of cylindrical flow through a guide vane with different inclined trailing edge marked by distances: a (=. m), a, a, a 3 at outer radius. In Fig. 5, the distribution of the outlet angle along the trailing edge is shown. It is noteworthy how the inclination uniforms the distribution of outlet angle. The mean values of outlet angle for different angles width are shown in Fig. 6. The mean pressure drop also changes as it is shown in Fig. 7 when inclination increases. The relative pressure drop falls down as outlet angle increases due to trailing edge inclination. Fig. 6 The mean value of outlet angle for different values of inclination a. Fig. 7 The mean relative pressure drop across a blade for different values of inclination a. 3. Blockage Factor Influence Fig. 4 The meridional sketch of the inclined trailing edge. The influence of blade thickness is introduced by blockage factor τ >, which appears in Eq.. Factor ( τ) < causes the increase of meridional velocity U what results in lower pressure and tangential () velocity U in energy conservation Eq. (5). This may lead to the lack of solution. The blockage factor τ may be introduced in the form of function as follows: t t3, t t4

6 D Model of Guide Vane for Low Head Hydraulic Turbine: Analytical and Numerical where, t, t, t 3, t 4 the coefficients and ( ) aial dimensionless coordinate (Eq. (8)) in the domain of the blade. Such a function gives sufficient freedom to predict the blade distribution thickness of designed blade. The question how to distribute the blade thickness along the skeleton line remains open. Let us consider two eamples. In the first one the parameters were introduced as follows: t =.5; t =.; t 3 =.5; t 4 = -. The shape of blockage factor function in coordinates (, ) is shown in Fig. 8. The maimum value of τ is.4. The blade shape in two views is shown in Fig. 9. Fig. Second eample of the blockage factor distribution according to Eq.. Fig. Two views of the calculated blade shape with white spots close to leading edge. Fig. 8 First eample of the blockage factor τ distribution according to Eq.. Fig. 9 Two views of the calculated blade shape. In the second eample, we choose the following set of numbers as follows: t =.5; t =.; t 3 =.5; t 4 = -. The shape of blockage factor has the higher value of maimum τ =.9 as it can be seen in Fig.. Too high thickness causes a lack of solution in the area close to the blade inlet. It is shown in Fig Loss Coefficient Distribution The dissipation is the factor, which has to be introduced into the model. Let us assume that the losses are defined as a part of an inlet kinetic energy, and they are distributed in the flow domain according to the coefficient: nl, w w w w3 (4) As an eample we can consider the set of coefficients as follows: w =.3; w = -.5; w =.; w 3 =.5, n l =.; n s = 3. The above coefficients result in the shape of function ζ as it is shown in Fig.. If we repeat the computation for the conditions represented in Fig. 9 the presented above dissipation coefficients lead to the blade shape shown in Fig. 3. The mean loss coefficient related to isentropic velocity of pressure drop is about.5 (~5%). ns

7 D Model of Guide Vane for Low Head Hydraulic Turbine: Analytical and Numerical Fig. The eample of the loss coefficient in blade domain according to Eq. (4). Fig. 4 The eample of the increased losses coefficient distribution according to Eq. (4). Fig. 3 The shape of the blade as in Fig. 9 corrected due to included losses. Let us now increase the losses according to function ζ shown in Fig. 4. The isentropic loss coefficient in this case has the high value of level.48 (48%). It causes the lack of solution at outlet part of blade as it is shown in Fig. 5. Too high losses do not allow calculating the flow in the shape of cylindrical surfaces. Fig. 5 The shape of blade with empty eit part due to lack of solution. 4. Conclusions Presented above solutions of inverse problem by means of D model refers to the most simple case of assumed cylindrical stream function. Nevertheless the results can be helpful for understanding the physical features of the flow through the blade. The following

8 D Model of Guide Vane for Low Head Hydraulic Turbine: Analytical and Numerical main points can be summarized as follows: For the simplest case of cylindrical flow field, analytical solution in form of Eq. () gives the geometrical shape of stream surface which is representative to the blade or channel of nozzle; () Numerical solution of more general set of conservation equations shows how the inclination of nozzle trailing edge uniforms of outlet angle along the radius; If the epected blade thickness is too big, there is no solution in the form of cylindrical shape of stream surfaces. The limits can be established by means of presented method; (4) Too high dissipation losses lead to the lack of solution for assumed cylindrical distribution of stream surfaces. Acknowledgments The work was supported by National Science Committee under Grant No. 6694/B/T//4 for the Szewalski Institute of Fluid-Flow Machinery Polish Academy of Sciences in Gdansk (Poland). Paper was presented on the XXth Polish National Fluid Mechanics Conference XX KKMP on Sept. 7-,, held in Gliwice, Poland. References [] H. Lorenz, Theorie und Berechnung der Vollturbinen und Kreisel Pumpen, Zeitschrift des Vereines Deutscher Ingenieure 49 (4) (95) [] C.H. Wu, A general theory of three-dimensional flow in subsonic and supersonic turbomachines of aial-, radial-, and mied-flow types, Trans. of ASME 74 (95) [3] R. Puzyrewski, Lectures on Theory of Turbomachinery Stage Two Dimensional Model (D), Gdansk University of Technology, Gdansk, Poland, 998. (in Polish) [4] R. Puzyrewski, K. Namieśnik, Comparison between conical and paraboloidal turbine stages: Inverse problem, in: 3rd International Symposium on Eperimental and Computational Aerothermodynamics of Internal Flows (ISAIF), Beijing, Sept. -6, 996. [5] M. Banaszek, D, D and 3D Models of flow through a hydraulic kaplan turbine: Improving turbine installations by mathematical simulation and physical modeling methods, in: Inter. Scientific and Engineering Conference, Beloe Ozero, Ukraine, Oct. 9-, 6. [6] R. Puzyrewski, Radial force influence upon nozzle blade flow: Inverse problem, in: Proc. of International Conference SYMKOM 95, Scientific Bulletin of Lodz University of Technology, No. 8, Lodz, Poland, 995. [7] P. Flaszyński, Comparison of two dimensional and three dimensional models based on turbomachinery flow passages, Ph.D. Thesis, Gdansk University of Technology,. (in Polish) [8] Z. Krzemianowski, Blade shape design method for the reversible hydraulic turbines, Ph.D. Thesis, Gdansk University of Technology, 3. (in Polish)