Numerical optimization of the guide vanes of a biradial self-rectifying air turbine.

Transcription

1 Numerical optimization of the guide vanes of a biradial self-rectifying air turbine. André Ramos Maduro andre.maduro@ist.utl.pt Instituto Superior Técnico, University of Lisbon,Av. Rovisco Pais, Lisboa, Portugal July 2016 Abstract The focus of this paper is the design and optimization of the guide-vane system of a self-rectifying biradial air turbine for use in oscillating water column systems for wave energy conversion. Since the turbine is required to be self-rectifying, there are two sets of guide-vanes, placed symmetrically on both sides of the rotor, instead of a single set (as in the conventional turbines). The efficiency of the biradial turbine with fixed guide-vanes is affected by the losses at the entry to the downstream row of guide-vanes, due to the misalignment between the flow direction and the exit guide-vanes. A problem also shared with other self-rectifying axial-flow impulse turbines. The goal of this paper is to perform a numerical optimization of the guide-vane system geometry so as to increase the turbine efficiency by reducing the losses at the exit guide-vane system, while ensuring the required flow deflection at the inlet guide-vane system. Time dependent flow calculations are performed to assess the performance and to evaluate the pressure losses for each guide-vane system configuration, due to the relatively large separated region at the exit guide-vane system. Finally, the turbine operating curves are obtained by calculating the three-dimensional flow through the complete turbine. Keywords: Biradial turbine, guide-vane design, optimization, blade cascade, wave energy. 1. Introduction 1 Self-rectifying impulse turbines (including the biradial turbine) have severe limitations in efficiency due to low aerodynamic performance of the second stator, positioned downstream of the rotor. Since this type of turbines are required to be selfrectifying, there are two sets of guide-vanes placed symmetrically on both sides of the rotor instead of a single set (as in conventional turbines). The guide-vanes, designed to guarantee the desired flow deflection at the impeller inlet, are the main cause of the poor performance of such turbines in the variant of fixed guide-vanes. When the flow exits the rotor, the existing misalignment between the guide-vanes of the downstream stator and the flow direction makes these behave as bluff-bodies. This effect promotes flow separation on the surface of the blades

2 2 and increases drastically the pressure loss in the exit stator. If this misalignment is too high, the flow passage between the guidevanes is strongly obstructed. Thus further increasing the loss and consequently reducing the turbine performance. In 2009, the Portuguese company Kymaner developed a new system of guidevanes in multiples cascades for the axialflow turbine that bridges the effect of the downstream stator [1, 2]. In this case, the inlet flow deflection is divided in stages by the use of several rows of guide-vanes, strategically positioned at a given rotor distance in order to increase the free space between blades (see Fig. 1). This deflection strategy and preferential positioning between sets of guide-vanes allows to increase the passage of outflow and consequently increase the efficiency of the turbine. Company data shows that this strategy guaranteed an increase of 20% in peak efficiency of the axial turbine in comparison with the case of a single row of guide-vanes. The biradial turbine [3], being also a impulse turbine, has the same problem with the downstream stator. Several projects related with the design of guide-vanes had been developed for this turbine [4, 5], but none had addressed the effect of multiple cascades. Therefore the purpose of this paper is to apply this stator design concept to the biradial turbine and evaluate the benefits of it implementation (see Fig. 2). The aim is to assure the deflection of the inflow, without introducing large losses in the inlet stator, and reduce the outflow obstruction with the introduction of multiple cascades. The number of cascades it is a parameter to determine through the downstream stator performance evaluation for a given rotor operation conditions. The guide-vane sections were designed using a thickness and camber distribution optimization that respects the restriction of preferential position to assure minimum outflow obstruction. This numerical geometry optimization problem is solved with the gradient-free Differential Evolution (DE) method [6]. The DE method is a metaheuristic algorithm that solves the optimization problem by iterative improving a candidate solution through the use of genetic operations. To validate the optimized geometries, time dependent calculations are performed through the downstream stator to evaluate the pressure drop. Finally a 3D flow analysis through the complete turbine will define the performance of this stator concept. Figure 1: Stator of the self-rectifying impulse turbine developed by Kymaner for the ModOndas national project (reproduced from [1]). 1 st row of guide-vanes 2 nd row of guide-vanes Radial flow Rotor Figure 2: Flow representation for exit guide-vane system of the biradial turbine.

3 Periodic surface 1 st row of guide vanes 2 nd row of guide vanes Periodic surface Figure 3: Computational domain and blades preferential position for purely radial exit flow. LE1 LE2 1 st row 2 nd row TE2 TE1 Figure 4: Blades parameter for a guide-vane system of multiple cascades. 2. Blade cascades design The positioning of the blades is one of the most important geometrical parameters in this stator design. The guide-vanes are placed in a configuration which maximizes the flow free passage between the blades of the downstream stator (see Fig. 3). This configuration is based on the shadow effect provided by the upstream blades when the flow exit the impeller. Thus, the blades rows are placed within the same angular window β relative to the rotor outflow direction. This strategy ensures that downstream blades are in the wake of upstream ones (see Fig. 2). To control the flow free path in the downstream stator it s defined the blockage factor parameter ζ, given in percentage by ζ(%) = β 100%, (1) Θ where Θ corresponds to the row space an- 3 gle between blades, controlled by the number of blades in the circumferential direction (fixed to 57 blades). The blockage factor ζ represents the obstructed path caused by the exit guide-vanes (symmetrical of the inlet) and it s an indicator of how large is going to be the pressure drop in the exit stator. For a given ζ, the leading and trailing edges of each cascade blade are coincident with the lines of constant β (see Fig. 4). Therefore the blades stagger angle and chord are managed simultaneously by the design parameters R that control the radial position between blades edges, making this two variables implicit in the optimization algorithm. The blades thickness distribution is scaled form the NACA 63A012 [7], using a scale factor parameter for each blade. The camber line is set from Bézier curves, defined with 2 control points, to reduce the number of variables

4 per blade. The first and the last point define the beginning and the end of the curve (see Fig. 5), while the intermediate points coordinates (2 and 3) define the design variables for the camber line. A preliminary exploratory work has shown that the use of 2 intermediate points, with a fixed radial position (r LE, r T E ), generate a wide range of geometries for the camber line curvature. The angular position of intermediate points (δ LE, δ T E ) controls the blade inlet/exit angles (ε LE, ε T E ) when placed in the preferential position. However if the exit flow is not purely radial, as shown in Fig. 3, the first blade s wake does not provide a full shadowing effect to the downstream ones. In this case (α 2 90 o ), the blades are positioned between two logarithmic spirals assuming conservation of angular momentum (see Fig. 6). This means that the stator must be optimized with a given rotation λ between cascades, function of the angle α 2, defined by ) λ = cot α 2 ln ( RLE1 R LE2. (2) This rotation between blade cascades does not interfere with the way the blade sections are parameterized. Since the design method repeats itself at each DE optimization step, the parametrization process should be generic, free of discontinuities and also able to keep the number of design parameters as low as possible. In this way, each cascade blade is designed with the same number of optimization parameters (four). Two design variables for the inlet/exit blade angles, one for the thickness scale factor and another one for the chord and stagger angle ( R). When the number of cascades increases one time there are five more variables to optimize (four from the new cascade and one for 4 the radial relative position between blades). In order to reduce even more the number of parameters, the stator to rotor diameter ratio is fixed to R T E /R rotor = 4.65, where R T E is the radial coordinate of the trailing edge of last stator blade, the closest one to the rotor, and R rotor = 244 mm is the rotor radius. LE LE Camber line Figure 5: Bézier curve with 2 control points used to define the camber line. The parameters δ LE and δ T E are related to ε LE and ε T E respectively, that are use as design variables. Relative rotation between radial cas- Figure 6: cades. TE TE TE Logarithmic spiral 3. Computational domain The computational domain used for the flow simulations is set to bi-dimensional and it represents the stator midplane relative to the axial direction. Periodic boundaries are used to reduce the domain size (see Fig. 3) and consequently the computational time. This domain is used for the inflow and outflow calculations, due to the turbine sym-

5 Y Z X metry condition. The only difference between the two domains is the radial size of the upstream block. In the outflow simulation, the exit block is extended due to the large wake and vortices release associated to the flow separation at the blades surfaces. The discretization scheme used for this computational domain is hybrid (see Fig. 7). Around each blade there is a structured block of mesh, in "O" shape, whose construction is based on the surface curvature of the blade [8]. This type of construction assures that the density of points on the surface of the blade is higher in the zones of high curvature, witch is perfect to define the blade trailing edge (see Fig. 8). Involving all the "O" shape meshes there is one unstructured block of mesh whose construction is base on Delaunay triangulations [9]. This block leaves plenty of room for the blade sections to change shape every optimization step, and it s responsible for making the connection between the upstream and the downstream blocks of structured mesh. Figure 7: Mesh blocks. Structured mesh Unstructured mesh Figure 8: Detail of the computational mesh near the blade trailing edge. Y Z X 4. Viscous flow computation The viscous flow simulations were performed using the commercial software FLUENT, that solves the Reynolds- Averaged Navier-Stokes equations (RANS) coupled with a transitional flow k ω SST turbulence model. In this calculations the flow is considered incompressible (Ma<0.3 at T amb = 300K), and the equations are solved using the pressure-based coupled algorithm. The pressure is computed with the 2 nd order scheme and the other equations with the 2 nd order upwind scheme. The inflow conditions are assumed to be steady, while the outflow conditions are considered to be unsteady, due to the relatively large separated flow area at the downstream region of the blades, making this calculations time dependent. The inlet boundary condition for the inflow and outflow calculations is the same. It is imposed as mass flow with turbulence intensity and hydraulic diameter as closure quantities. The turbulence intensity is different for each case, being a higher value estimated for the exit stator due to the passage through the rotor. For both calculations an outflow condition is imposed at the domain exit with the target mass flow equal to the inlet. The blade walls are set to walls with no-slip boundary condition. The working fluid is air with ρ = kg/m 3 and viscosity µ = Pa s. The reference pressure is the standard atmospheric pressure p ref = Pa. The design mass flow rate used for this calculations results from the design conditions at the rotor entrance (V 1t = 2U and α 1 = 30 o ). Combining this conditions with model size Reynolds number Re = ΩR 2 rotor /ν = , where rotation speed Ω = 500 rpm, results the design mass flow rate of 1.49 kg/s. 5

6 5. Optimization strategy The inlet stator blades are optimized through the DE algorithm for a given blockage factor parameter ζ (Eq. 1). This parameter is not a design optimization variable because it is intended, in this paper, to evaluate how it influences the multiple cascades. Therefore blades geometry and position parameters are optimized to guarantee the minimum angle at the rotor entrance (α 1 ) and low pressure losses at the inlet stator. This multi-objective problem can be solved using an objective function F (z) with a weight factor, as the one used in [5]. This weight factor choice is always a difficult decision due to the parameters different nature and because it influences directly the final optimization result. To avoid using a weight factor the pressure drop is estimated through the blade s boundary layer separation. If the flow separates from the surface of any blade, the pressure losses in the inlet stator are considered too high and the tested geometry is discarded from the optimization process. If the flow does not separate, the pressure loss is considered too small to be optimized, and the algorithm is responsible to compute the mean value of the flow angle at the impeller entrance (α 1 ). This pressure losses evaluation strategy allows to set a single term objective function, defined by F (z) = α 1. (3) Since the blades have a rounded trailing edge there is always flow separation in that region. Therefore the flow separation is evaluated in 90% of the line length that defines the suction side of the blade, starting from the leading edge. This means that separation can occur in the last 10% of the 6 line. An acceptable value that assures low pressure losses in the inlet stator. After each geometry calculation the flow separation is evaluated before computing the objective function. To evaluate this, the velocity profiles are analyzed using a local coordinate system on the suction side of the blade (see Fig. 9). Inflection point Flow separation point Reversed flow Zero velocity line Figure 9: Local coordinate system (s x, s y ) on the suction side of a blade. 6. Results In order to determine the effect of the number of cascades Z R in the blockage factor ζ, several optimizations were performed, see Fig. 10. It is chosen to evaluate this effect for three configurations (Z R = 1, 2 and 3 cascades) with the design outflow angle α 2 = 90 o. This means that the outflow is purely radial and it corresponds to the minimum kinetic energy at the rotor exit. For each optimized geometry (point in the graphic), the angle at the rotor entrance α 1 represents the maximum flow deflection that respects the preferential position and separation condition on the surface of the blades, guaranteeing low pressure losses in the inlet stator. As predicted, the blades blockage factor decrease with the increasing number of cascades. For all configurations the flow deflection decreases with the decreasing blockage factor due to the blade load reduction. To check the performance of each configuration, time dependent calculations are performed to evaluate the pressure drop in

7 90% 80% Z R =1 Z R =2 Z R =3 4.0 ζ 70% 60% K p Z R =1 Z R =2 Z R =3 50% 40% α 1 [ ] Figure 10: Blockage factor for systems of 1, 2 and 3 cascades optimized for α 2 = 90 o Re Figure 11: Outflow pressure loss for systems of 1, 2 and 3 cascades optimized for α 2 = 90 o. the downstream stator. The selected geometries penalizes this turbine s stator performance. for this calculations match the ro- In this configuration the pressure drop is al- tor entrance operating condition (α 1 = most twice the verified in the other tested 30 o ). The outflow angle is set to the stators design condition (α 2 = 90 o ). Fig. 11 presents the dimensionless pressure drop (Eq. 4) for each selected configuration as function of the Reynolds number (Eq. 5), configurations. Since the 2 and 3 cascades configurations present almost the same exit losses and both fulfill the inlet stator operating condition, it is preferred to adopt the Z R = 2 configuration. This one brings more K p = p benefits from the economical and manufacturing point of view , (4) 2 R TE However, in order to increase the range Re stat = cv of stator candidates the 2 cascade configuration is optimized for two new different R TE. (5) υ angles: α The dimensionless pressure drop K p is 2 = 65 o and 115 o. Which means a rotation of ±λ between the two cascades defined as the total pressure variation across (see Fig. 6). The optimization process is the blades rows p 01 normalized by the dynamic pressure based on the radial compo- replicated from the previous one, used for the design angle α nent of the flow velocity at the last blade 2 = 90 o, and is presented in Fig. 12. In this two new configurations trailing edge radial section (the same for all the blockage factor presents the same evolution as before, but in the particular case configurations). In this case, the Reynolds number is based on the chord c of the last of the α blade and the radial velocity at the same 2 = 65 o configuration, the blockage factor is significantly smaller for the blade trailing edge radial section. same rotor entrance condition (α Due to the extremely turbulent flow 1 ). The reason behind this huge reduction relays on verified at the downstream section of the the way the blades are placed within the blades, the pressure losses are completely lines of constant β. Where the configurations optimized for design angles α independent from the Reynolds number. The high blockage factor observed for 2 closer to the rotor entrance design angle (α the Z R = 1 configuration (70%) severely 1 ) are 7

8 the most benefited. Finally, a set of calculations for different outflow angles were performed to verify if the design angle α 2 of each configuration fulfills the purpose of it optimization (see Fig. 13). This result shows that each optimized configurations assures the best performance for their own design angle (α 2 = α 2 ), and on the other hand shows that all configurations are very sensitive to variations of the outflow angle. ζ 90% 80% 70% 60% 50% 40% 30% 20% α 2 = 65 α 2 = 90 α 2 = % α 1 [ ] Figure 12: Blockage factor for Z R = 2 optimized for α 2 = 90 ± 25 o. K p α 2 = α 2 = 90 α 2 = α 2 [ ] Figure 13: Influence of the angle α 2 in the outflow pressure loss, for Z R = 2 optimized for α 2 = 90 ± 25 o. 7. Turbine performance 7.1. Modeling the three-dimensional flow To have more reliable values of each stator performance, three-dimensional flow calculations through the complete turbine were performed. The aim of this calculations is the validation of the rotor operating conditions, namely the angle at the rotor entrance, and find out which configuration provides more benefit to the turbine. The rotor used for this calculation, as well as the respective meshes, are reported [10]. This rotor was optimized to maximize the total-static efficiency for the design condition α 1 = 30 o, the same angle used to analyze the turbine stators. All the 5 optimized stator configurations were simulated with this rotor. The computational domain is defined by: 2 stators (inlet and exit) and 1 rotor, assuming periodic boundaries (see Fig. 14). The periodicity of the rotor and stators domains are different due to number of blades (11 rotor blades to 57 circumferential stator blade sets). The Mixing Plane model [11] was then used to compute the quantities between domains interfaces. Two rotor connection block were used to avoid intense gradients at the mixing plane. A moving reference frame [11] was used to compute the flow through the rotor. The rotor rotation speed Ω was set to rpm, which respects the full scale Reynolds number of Re = for a rotor diameter of 1.5 m [12] and turbine flow rate Q = 37.1 m 3 /s. The atmospheric static pressure is imposed as boundary condition at the exit surface of the domain. All computations are performed assuming steady incompressible flow. 8

9 1 st cascade 2 nd cascade Inlet stator Outlet stator Rotor Periodic surface Rotor connection block turbine torque), Ω is the rotation speed, D is the rotor diameter and p is the pressure head (difference between the turbine inlet total pressure and the exit average static pressure). Rotor blade Figure 14: Turbine computational domain. 80% 75% 70% Z R =1 Z R =2 Z R = Results As expected, Fig. 15 shows that boundary layer on the nozzle walls has a significant effect on the axial distribution of the angle of the absolute flow at the rotor entry. Nevertheless the results indicate that flow incidence to the rotor blades is small in the approximately 60% of the channel height. η 65% 60% 55% 50% Φ Figure 16: Turbine efficiency versus dimensionless flow rate, for systems of 1, 2 and 3 cascades optimized for α 2 = 90 o. 50% 40% 80% 75% 70% α 2 = 65 α 2 = 90 α 2 = 115 Channel height 30% 20% 10% η 65% 60% 55% 50% Φ 0% α 1 [ ] Figure 15: Profile of angle α 1, for the system of 2 cascades optimized for α 2 = 90 o. The turbine total-static efficiency η and the dimensionless flow coefficient Φ plotted in Figs are defined as η = P Q p, (6) Φ = Q ΩD. (7) 3 Here, Q is the volume flow rate, P is the turbine power output (P = ΩT, T as the 9 Figure 17: Turbine efficiency versus dimensionless flow rate, for systems of 2 cascades optimized for α 2 = 90 ± 25 o. For all configurations, the peak efficiencies are very close to the rotor design condition (Φ = 0.19). The large efficiency drop verified for smaller flow rate coefficients is caused by the effect of high negative flow incidence to the rotor blades, and due to the large misalignment between the flow direction and the downstream stator blades (see Fig.18). The smooth efficiency drop for the higher flow rate coefficients is caused by the combined effect of the positive flow

10 incidence to the rotor blades, and the pressure loss in the downstream stator, which is proportional to the square of the flow rate coefficient. α 2 [ ] Z R =1; α 2 = 90 Z R =2; α 2 = 90 Z R =3; α 2 = 90 Z R =2; α 2 = 65 Z R =2; α 2 = Φ Figure 18: Rotor exit angle versus dimensionless flow rate, for all configurations. 8. Conclusions A new stator blades design was presented for the biradial turbine. Several stator configurations were tested in order to evaluate the effect of the number of cascades. The implementation of multiple cascades increased the turbine performance due to the reduction of the downstream stator flow obstruction. Results of numerical flow calculations indicate a total-static peak efficient of 76% for the Z R = 2 configuration, optimized for α 2 = 65 o. This may be compared with the corresponding numerical results for the axially movable guide-vanes biradial turbine [12], which gave 83%. References [1] Luis Trigo. Kymaner presentation - The OWC option. files/04_luis_trigo_kymaner.pdf. Accessed: [2] WavEC - Offshore Renewables. Seminar and B2B meetings "Powering the future - Marine energy opportunities". energy_2009#.vxknuvyrlck. Accessed: [3] AFO Falcao, LMC Gato, and EPAS Nunes. A novel radial self-rectifying air turbine for use in wave energy converters. part 2. results from model testing. Renewable Energy, 53: , [4] J.C.C. Henriques and L.M.C. Gato - Optimization of the guide-vanes of a self-rectifyng biradial turbine, Technical report, IST, [5] T. M. Q. Candeias. Numerial optimization of the upstream stator blades of a self-rectifying biradial turbine. Master s thesis, IST - Technical University of Lisbon, [6] Rainer Storn and Kenneth Price. Differential evolution a simple and efficient heuristic for global optimization over continuous spaces. Journal of global optimization, 11(4): , [7] Ira Herbert Abbott and Albert Edward Von Doenhoff. Theory of wing sections, including a summary of airfoil data. Courier Corporation, [8] Harry L Morgan Jr. A computer program for the analysis of multielement airfoils in twodimensional subsonic, viscous flow. In Aerodynamic Analyses Requiring Advanced Computers, volume 347, page 713, [9] J-D Muller, Philip L Roe, and H Deconinck. A frontal approach for internal node generation in Delaunay triangulations. International Journal for Numerical Methods in Fluids, 17(3): , [10] D. N. Ferreira - Rotor optimization for the self-rectifying biradial air turbine, Technical report, IST/IDMEC, [11] Fluent-Inc. Fluent 6.3 users guide. Fluent documentation, [12] A F O Falcao, LMC Gato, and EPAS Nunes. A novel radial self-rectifying air turbine for use in wave energy converters. Renewable Energy, 50: ,