82 CHAPTER 4 OPTIMIZATION OF COEFFICIENT OF LIFT, DRAG AND POWER - AN ITERATIVE APPROACH The coefficient of lift, drag and power for wind turbine rotor is optimized using an iterative approach. The coefficient of lift and drag has been optimized using CFD method where as the power coefficient is optimized with Blade Element Momentum (BEM) method using iterative approach that uses convergence of axial and tangential flow factors. The airfoil of NACA 4410 and NACA 2415 has been selected for analysis. The coefficient of lift and drag are predicted using CFD and validated with the available experimental results. The coefficient of power for these profiles has been optimized considering the profile in two different cases. In the first case, the airfoil is considered with drag and varying tip loss correction factor where as in the other case, ignoring the drag and assuming the tip loss correction factor as one. A rotor of one MW capacity as a case study is considered for optimization. The blade used for the rotor is divided into discrete number of sections along its span. At each section, the local tip speed ratio, inflow angle, twist angle, solidity and chord have been found out and used in prediction of axial, tangential flow factors and tip loss correction factor. Iterations are used, till the values of axial and tangential flow factors for two consecutive iterations become closer. Further, the values of the above factors are used in prediction of the optimized power coefficient.
83 The optimization of power coefficient is essential for improving the performance of wind turbines. For a wind turbine blade, the optimum angle of attack and optimum twist angle improve the power coefficient of the wind turbine. The optimum twist of a wind turbine blade is determined using BEM theory. The power coefficient is maximum when the blades are twisted for a specific velocity of wind and rotor. Further, the power coefficient depends on the coefficient of lift and drag corresponding to the discrete blade elements for a particular angle of attack. Glauert (1926) determined the optimum chord and twist distribution for an ideal wind turbine using derived closed form equation with exact trigonometric function method. In the present work, along the blade span, uniform angle of attack with relative wind velocity is obtained from an equation at different section for a specific size of rotor and blade geometry. In the following sections, the selection of parameters and mathematical modeling of turbine blade, CFD analysis of airfoils to find out coefficient of lift and drag and implementation of results of iterative method in optimization of power coefficient are discussed in detail. 4.1 THE MATHEMATICAL MODEL The mathematical modeling of wind turbine blade is performed to study and calculate the power coefficient of turbine rotor. In this modeling, turbine blade is divided into specified elements using BEM theory. The forces acting on the blade can be evaluated using the equations derived based on the principle of conservation of momentum and angular momentum. The power and torque produced by the turbine rotor is also can be evaluated using the forces acting on the blades.
84 The cross section of the rotor blade, the velocities related to airfoil and axial (a) and tangential (a ) flow factors are shown in the Figure 4.1(a), (b) as proposed by Lanzafame and Messina (2007). Figure 4.1(a) shows a single blade with small element of thickness dr at the radius r from the axis of rotation with angular velocity. Figure 4.1 (b) illustrate the wind velocity v, relative velocity W, wind inflow angle ( ), angle of attack ( ), pitch angle ( ), chord length (c), axial flow factor(a), tangential flow factor(a ), lift (L) and drag (D) forces along tangential and normal components. Figure 4.1 Forces and velocities on airfoil with wind velocities (Lanzafame and Messina 2007) The coefficient of lift (C l ) and drag (C d ) depend on the Reynolds number and the angle of attack for an airfoil. The normal forces and torque depend on tangential and axial flow factors which can be evaluated by implementing the momentum and angular momentum conservation equations. The expressions are derived from the principle of conservation of momentum in axial direction between upstream and downstream sections. The axial and tangential forces (dn and dm) acting on the element of thickness dr is
85 calculated using the equations (4.1) and (4.2) as proposed by Lanzafame and Messina (2007). dn = ( ) N (C cos + C sin )cdr (4.1) dm = ( ) ( ) N (C sin C cos )crdr (4.2) The notations used in the above equations are as discussed in the Chapter 1 and the term N represents number of blades in the turbine rotor. Equating the expressions (4.1) and (4.2), the axial flow factor (a) and tangential flow factor (a ) are derived using BEM theory and given in the equations (4.3) and (4.4). = 1 ( ) + 1 (4.3) = 1 ( ) + 1 (4.4) In the above equation, the factor F t is the Prandtl tip loss factor as defined by Hansen (2000) that is given in equation (4.5) and the term chord solidity ( ) is given in the equation (4.6). = ( ) (4.5) = (4.6) The equation (4.3) has a limitation and it yields reliable results between the axial flow factor values of 0 to 0.4. If the axial flow factor is greater than 0.4, an appropriate correction factor (F t ) is to be incorporated as proposed by Glauert (1926). The factor (F t ) is taken as one when the axial
86 flow factor is greater than 0.4 and in other cases the factor (F t ) is derived considering losses at the blade tip using equation (4.5). The equation of axial flow factor (a) is modified by including the factor (F t ) and given in the equation (4.7). = ( ) ( ) (4.7) Glauert (1935) considered an ideal actuator disk model and obtained the relations between axial and tangential flow factors (a and a ), as well as proposed an equation to calculate the inflow angle ( ) by ignoring the secondary effect of drag and tip loss as shown in the equations (4.8),(4.9) and (4.10). = ( ) ( ) (4.8) (1 + ) = (1 ) (4.9) = ( ) ( ) (4.10) In the above equations, is the local tip speed ratio at the r th segment along the blade. The effect of whirl behind the rotor is ignored, the axial flow factor (a) will be 0.33 and the tangential flow factor is zero, then the inflow angle may be determined as shown in the equation (4.11) for the value of >1. = ( ) (4.11) Wilson and Lissaman (1976) performed a local optimization analysis by maximizing the power output at each radial segment along the blade. The axial flow factor was varied until the power contribution became
87 stationary. Rohrbach and Worobel (1975) investigated the effect of blade number and section lift to drag ratio at the maximum turbine performance. Their results at maximum performance of turbine were yielding lower values than that of results obtained by Wilson and Lissaman (1976). An approximate relationship between the inflow angle ( ) and the local speed ratio ( ) was derived by Nathan (1980) that is a 5 th order polynomial equation and is given in the equation (4.12). = 57.51 35.56 + 10.61 1.586 + 0.114 0.00313 (4.12) In the above equation is in degrees and was derived for a lift to drag ratio ranging from 28.6 to 66.6 by ignoring the effects of secondary flow in the tip and hub regions. The variation of optimum inflow angle ( ) with respect to local tip speed ratio ( ) using the equation (4.10) proposed by Glauert (1935), equation (4.11) proposed by Wilson and Lissaman (1976) and equation (4.12) proposed by Nathan (1980) is shown in the Figure 4.2. From the graph, it is understood that the deviations of inflow angle is more at the hub region at lower r/r ratio for all the equations. The equations proposed by Glauert (1935) and Wilson and Lissaman (1976) have good conformity at tip region and a small variation at hub region. In comparison with Nathan equation, the Glauert s and Wilson and Lissaman equations yielded closer results except at the hub region. As a whole, Nathan s equation is yielding lower values than the other two equations at all regions.
88 Figure 4.2 Variation of optimum inflow angle with radius ratio The optimum twist angle ( ) that changes along the length of the blade can be determined using optimum inflow angle ( ) and angle of attack ( ). 4.2 BLADE SEGMENTATION In the proposed iterative method using BEM theory, a wind turbine blade is divided into discrete number of segments for analysis. The segmentation of a blade is shown in Figure 4.3. A wind turbine of one MW is considered and designed. The blade radius is 32 m, rated wind velocity is10 m/s and rotational speed of 20rpm. The above parameters are selected based on wind turbine design procedure (Burton et al. 2001). The values of radius ratio (r/r), tip speed ratio ( ), inflow angle ( ), chord (c) and twist angle ( ) are calculated for the designed blade and presented in the Table 4.1. Figure 4.3 Segmentation of blade
89 The tip speed ratio is selected as 6.7 corresponding to the wind velocity and rotational velocity of rotor. The inflow angle is determined for this case study and it is designed as per Glauert s equation (4.10). The twist angle is determined by the procedure given in the previous subsection 4.1. The chord distribution is determined from the equation (3.17). The variation of chord and twist angle with respect to radius ratio is seperately shown in the Figures 4.4 and 4.5 repsectively. Table 4.1 Blade design parameters at various segments Segment Number Radius of rotation (r) in m Radius ratio (r/r) Tip speed ratio ( ) Inflow angle( ) in Degrees Chord length (c) in m Twist angle( ) in Degrees 1 4 0.13 0.84 50.08 4.47 41.59 2 6 0.19 1.26 38.55 1.98 30.06 3 8 0.25 1.67 30.86 1.12 22.37 4 10 0.31 2.09 25.55 0.71 17.06 5 12 0.38 2.51 21.72 0.50 13.23 6 14 0.44 2.93 18.85 0.36 10.36 7 16 0.50 3.35 16.63 0.28 08.14 8 18 0.56 3.77 14.87 0.22 06.38 9 20 0.63 4.19 13.44 0.18 04.95 10 22 0.69 4.61 12.26 0.15 03.77 11 24 0.75 5.02 11.26 0.12 02.77 12 26 0.81 5.44 10.42 0.11 01.93 13 28 0.88 5.86 09.69 0.09 01.20 14 30 0.94 6.28 09.05 0.08 00.56 15 32 1.00 6.70 08.49 0.07 00.00
90 Figure 4.4 Chord Distribution Figure 4.5 Twist angle distribution 4.3 CFD ANALYSIS OF AIRFOIL The computational fluid dynamics (CFD) analysis is performed to calculate the coefficient of lift and drag for different airfoils with wide range of angle of attack. The above parameters are used to evaluate the axial and tangential flow factors that are very vital in determining the power coefficient of wind turbine systems. The commercially available software GAMBIT and ANSYS12.0 (FLUENT Module) are used for the computational work. The
91 modeling and meshing is carried out using GAMBIT and the boundary conditions are applied and solved in FLUENT. The airfoil sections NACA 4410 and NACA 2415 are used for the computational analysis that is briefed in the following subsections and the profile of which is shown in the Figures 4.6 (a) and 4.6 (b). The coordinates at upper and lower surfaces of airfoils NACA 4410 and NACA 2415 are shown in the Table A 2.1 and Table A 2.2 in Appendix 2. Figure 4.6(a) NACA 4410 Airfoil Figure 4.6(b) NACA 2415 Airfoil 4.3.1 Modeling and Analysis of Airfoil The modeling of airfoil is done using GAMBIT software. The NACA 4410 airfoil is considered for modeling and the coordinates are developed using cartesian coordinates. 35 different points are located at the upper surface of the airfoil where as 36 points are located at the lower surface. Around the profile, the boundaries are fixed, based on the wind flow area in terms of chord length (c). It is assumed that the boundaries around the airfoil as 9 times of c in front of the leading edge and the 14 times of c behind the trailing edge, 10 times of c from airfoil to far field at top and bottom boundaries. The left side and right side boundaries are termed as velocity inlet and pressure outlet respectively, where as the top and bottom surface of airfoil boundaries are termed as upper and lower walls. The top and bottom boundaries are termed as far field. A profile with boundaries and meshing are illustrated in Figures 4.7 (a) and 4.7 (b).
92 Figure 4.7(a) Airfoil boundaries Figure 4.7(b) Meshing around the airfoil The meshed geometry of airfoil is imported in ANSYS from GAMBIT and analyzed using the FLUENT module. Inlet velocity for the simulation is fixed as 10 m/s and turbulence viscosity ratio is taken as 10. A fully turbulent flow solution called as Spalart-Allmaras model used by Laursen et al. (2007) and Thumthae and Chitsomboon (2006) is used in ANSYS FLUENT for the analysis using the procedure suggested by them. The Spalart-Allmaras model is a relatively simple one-equation model that solves a modeled transport equation for the kinematic eddy (turbulent) viscosity. The calculations are performed for up to 5 o of angles of attack as linear region, due to greater reliability of both experimental and computed
93 values in this region and apart from this value it is assumed as non linear as used by Thumthae and Chitsomboon (2006). The Spalart-Allmaras model was designed specifically for aerospace applications involving wall-bounded flows and has been shown to give good results for boundary layers subjected to adverse pressure gradients. It is also gaining popularity in the turbomachinery applications. In its original form, the Spalart-Allmaras model is effectively a low-reynolds-number model, requiring the viscosity-affected region of the boundary layer to be properly resolved. In ANSYS FLUENT, however, the Spalart-Allmaras model has been implemented to use wall functions when the mesh resolution is not sufficiently fine. This might make it the best choice for relatively crude simulations on coarse meshes where accurate turbulent flow computations are not critical. The governing equation of Spalart-Allmaras model is given as 1 v v ( v) ( vx ) G ( v) C Y S t x x x x i b2 i j j j 2 The transported variable in the Spalart-Allmaras model, v, is identical to the turbulent kinematic viscosity except in the near-wall (viscosityaffected) region. where G v is the production of turbulent viscosity, and Y v is the destruction of turbulent viscosity that occurs in the near-wall region due to wall blocking and viscous damping. v and C b2 are the constants and S v is a user-defined source term. The model constants are assumed as default values. (Spalart and Allmaras, 1992) The convergence criteria used for this analysis is 1e -4 as it is followed by several researchers for the analysis of airfoils. In FLUENT software, the solution method is selected as simple. A simple solver is utilized and the operating pressure is set to zero.
94 4.3.2 Grid Independent Analysis One critical parameter when using CFD is the grid size. In general, a larger, more refined grid provides a better solution at the expense of computational time. The first step in performing a CFD simulation should be to investigate the effect of the mesh size on the solution results. Generally, a numerical solution becomes more accurate as more nodes are used, but using additional nodes also increases the required computer memory and computational time. The appropriate number of nodes can be determined by increasing the number of nodes until the mesh is sufficiently fine so that further refinement does not change the results. In order to eliminate errors due to grid refinement, a grid independence study was conducted for the wind velocity of 5m/s. Table 4.2 shows the effect of number of grid cells in coefficient of lift, drag and maximum pressure coefficient at 5 of angle of attack. This study revealed that a C-type grid topology with 35566 quadrilateral cells would be sufficient to establish a grid independent solution. Table 4.2 Grid independent analysis at 5 o of angle of attack No of Grid cells Wind Velocity (v) in m/s Lift Coefficient (C l ) Drag Coefficient (C d ) Maximum Pressure Coefficient (C pr ) 15150 5 0.6593 0.0401 0.952 20253 5 0.6654 0.0425 0.954 35566 5 0.680 0.0434 0.964 42364 5 0.680 0.0434 0.964
95 4.3.3 Validation of Coefficient of Lift and Drag The values of coefficient of lift (C l ) and drag (C d ) are computed using Computational Fluid Dynamics (CFD) at wind velocity of 20 m/s and the results are compared with the published experimental results of Mehrdad Ghods (2001) performed using wind tunnel at 20 m/s for NACA 2415 profile. The values of C l and C d obtained from CFD analysis and wind tunnel experiments are presented in Table 4.3 and the values are shown graphically in Figure 4.8(a) and 4.8(b). The Figure shows there is good conformity of the CFD results. Table 4.3 Comparison of results of wind tunnel and CFD analysis Angle of Wind tunnel CFD Analysis attack in Degrees C l C d C l C d -5-0.2167 0.0008-0.1011 0.0214-4 -0.1610 0.0017-0.0549 0.0188-3 -0.1039 0.0041-0.0069 0.0170-2 -0.0406 0.0066 0.0417 0.0160-1 0.0193 0.0091 0.0864 0.0158 0 0.0702 0.0124 0.1329 0.0164 1 0.1355 0.0157 0.1783 0.0177 2 0.1871 0.0207 0.2290 0.0198 3 0.2511 0.0265 0.2818 0.0227 4 0.3082 0.0331 0.3322 0.0265 5 0.3660 0.0397 0.3826 0.0310 6 0.4093 0.0455 0.4301 0.0364 7 0.4678 0.0537 0.4758 0.0425 8 0.5049 0.0587 0.5205 0.0494 9 0.5517 0.0661 0.5635 0.0570 10 0.5682 0.0736 0.6029 0.0652 11 0.6287 0.0827 0.6399 0.0741 12 0.6652 0.0901 0.6735 0.0837 13 0.6886 0.0950 0.7055 0.0941 14 0.6989 0.1041 0.7351 0.1052 15 0.7533 0.1298 0.7606 0.1171 16 0.7650 0.1389 0.7219 0.1241 17 0.7842 0.1595 0.7220 0.1355 18 0.7154 0.2232 0.7005 0.1513 19 0.6570 0.2529 0.6451 0.1886
96 Figure 4.8(a) Validation of coefficient of lift of NACA 2415 Figure 4.8(b) Validation of coefficient of drag of NACA 2415
97 4.3.4 Results of the CFD Analysis The airfoil NACA 4410 is selected for prediction and analysis of coefficient of lift and drag. The iterative method is used for the analysis. The selected profiles are tested for finding coefficient of lift and drag separately with wind velocity in the range of 5 m/s to 25 m/s in steps of 5 m/s for the angle of attack of 5 o. The results of C l and C d corresponding to wind velocity of 5 m/s are presented in Figures 4.9(a) and 4.9(b). Figure 4.9(a) Iterations for coefficient of lift at 5 m/s Figure 4.9(b) Iterations for coefficient of drag at 5 m/s
98 The analysis is performed until the values of coefficient of lift and drag reaches stable in iterations at various angle of attack. In this present study, the coefficient of lift and drag becomes stable after 105 th iterations. The results of coefficient of lift (C l ) at wind velocities of 10 m/s, 15 m/s, 20 m/s and 25 m/s are shown in Figures A 3.1 A 3.4 in Appendix 3. The results of coefficient of drag (C d ) at wind velocities of 10 m/s, 15 m/s, 20 m/s and 25 m/s are shown in Figures A 3.5 A 3.8 in Appendix 3. The Pressure coefficient (C pr ) at any point over the airfoil surface is an important parameter as it affects the coefficient of lift and drag. It is the ratio of the difference in pressure at a point on the airfoil surface with free stream pressure to the kinetic energy possessed by the wind. The equation to calculate C pr is given as (4.16) The pressure coefficient (C pr ) at lower and upper surfaces of NACA 4410 airfoil with wind velocity of 5 m/s and 5 o of angle of attack at various points of the airfoil surface is predicted and its variation is shown as a graph in Figure 4.10. Figure 4.10 Pressure coefficient at upper and lower surfaces of NACA 4410
99 The pressure coefficient (C pr ) at various wind velocities of 10 m/s, 15 m/s, 20 m/s and 25 m/s for the airfoil NACA 4410 is predicted as explained in the section and shown as Figures A 3.9 A 3.12 in Appendix 3. The prediction of coefficient of lift (C l ), drag (C d ) and maximum pressure coefficient (C pr ) of NACA 4410 and NACA 2415 at various wind velocities with 5 o of angle of attack is shown in the Tables 4.4(a) and 4.4(b). Table 4.4 (a) Coefficient of lift, drag & Maximum pressure coefficient at 5 o Angle of attack NACA 4410 Wind Velocity (v) in m/s Lift Coefficient (C l ) Drag Coefficient (C d ) Maximum Pressure Coefficient (C pr ) 5 0.680 0.0434 0.964 10 0.686 0.0426 0.958 15 0.691 0.0420 0.954 20 0.695 0.0416 0.950 25 0.697 0.0413 0.949 Table 4.4 (b) Coefficient of lift, drag & Maximum pressure coefficient at 5 o Angle of attack NACA 2415 Wind Velocity (v) in m/s Lift Coefficient (C l ) Drag Coefficient (C d ) Maximum Pressure Coefficient (C pr ) 5 0.384 0.036 0.882 10 0.386 0.033 0.870 15 0.388 0.032 0.867 20 0.390 0.031 0.865 25 0.393 0.030 0.860
100 The variation of coefficient of lift, drag and maximum pressure coefficient of NACA 4410 airfoil for various angle of attack at wind velocities of 5 m/s, 10 m/s, 15 m/s, 20 m/s and 25 m/s are shown in Tables 4.5(a), 4.5(b) and 4.5(c) respectively. Table 4.5 (a) Coefficient of lift of NACA 4410 at various angles of attack for different wind velocities Angle of attack in Degrees Coefficient of lift (C l ) 5 m/s 10 m/s 15 m/s 20 m/s 25 m/s 0 0.404 0.409 0.412 0.416 0.419 1 0.464 0.470 0.473 0.480 0.482 2 0.519 0.524 0.527 0.533 0.535 3 0.574 0.580 0.583 0.589 0.591 4 0.629 0.634 0.636 0.642 0.644 5 0.680 0.686 0.691 0.695 0.697 6 0.731 0.737 0.741 0.745 0.748 7 0.780 0.785 0.790 0.794 0.798 8 0.826 0.832 0.837 0.842 0.844 9 0.872 0.877 0.881 0.885 0.887 10 0.914 0.920 0.922 0.926 0.928 11 0.953 0.959 0.962 0.964 0.965 12 0.988 0.995 0.997 0.997 0.998 13 1.020 1.026 1.026 1.026 1.027 14 1.047 1.052 1.052 1.051 1.049 15 1.072 1.076 1.062 1.048 1.047 16 1.095 1.095 1.073 1.054 1.055 17 1.114 1.110 1.084 1.060 1.061 18 1.129 1.064 1.063 1.062 1.027 19 1.140 1.073 1.026 0.979 0.979
101 Table 4.5 (b) Coefficient of drag of NACA 4410 at various angles of attack for different wind velocities Angle of attack in Degrees Coefficient of drag (C d ) 5 m/s 10 m/s 15 m/s 20 m/s 25 m/s 0 0.0166 0.0157 0.0142 0.0145 0.0141 1 0.0200 0.0191 0.0193 0.0178 0.0175 2 0.0247 0.0237 0.0243 0.0226 0.0223 3 0.0301 0.0292 0.0284 0.0281 0.0278 4 0.0363 0.0355 0.0348 0.0344 0.0341 5 0.0434 0.0426 0.0420 0.0416 0.0413 6 0.0514 0.0505 0.0490 0.0497 0.0494 7 0.0601 0.0593 0.0581 0.0584 0.0580 8 0.0697 0.0689 0.0682 0.0677 0.0675 9 0.0801 0.0792 0.0786 0.0781 0.0779 10 0.0912 0.0902 0.0896 0.0891 0.0889 11 0.1025 0.1018 0.1013 0.1008 0.1006 12 0.1146 0.1141 0.1134 0.1128 0.1127 13 0.1274 0.1267 0.1259 0.1252 0.1251 14 0.1407 0.1397 0.1391 0.1385 0.1384 15 0.1550 0.1539 0.1528 0.1518 0.1516 16 0.1705 0.1689 0.1671 0.1653 0.1651 17 0.1867 0.1850 0.1828 0.1805 0.1803 18 0.2038 0.1996 0.1992 0.1988 0.1933 19 0.2227 0.2238 0.2226 0.2215 0.2262
102 Table 4.5(c) Maximum coefficient of pressure (C pr ) of NACA 4410 at various angles of attack for different wind velocities Angle of attack in Degrees Maximum Coefficient of pressure (C pr ) 5 m/s 10 m/s 15 m/s 20 m/s 25 m/s 0 0.957 0.951 0.946 0.941 0.941 1 0.919 0.917 0.915 0.913 0.913 2 0.944 0.933 0.933 0.932 0.932 3 0.938 0.932 0.929 0.926 0.926 4 0.949 0.947 0.945 0.943 0.941 5 0.964 0.958 0.954 0.950 0.949 6 0.951 0.939 0.935 0.931 0.930 7 0.961 0.958 0.958 0.958 0.954 8 0.967 0.964 0.960 0.955 0.955 9 0.963 0.962 0.962 0.962 0.962 10 0.978 0.976 0.976 0.975 0.975 11 0.977 0.974 0.971 0.968 0.967 12 0.977 0.976 0.975 0.975 0.975 13 0.983 0.981 0.980 0.979 0.978 14 0.977 0.974 0.972 0.971 0.971 15 0.978 0.976 0.974 0.971 0.971 16 0.979 0.976 0.976 0.975 0.975 17 0.973 0.970 0.973 0.975 0.974 18 0.974 0.971 0.970 0.969 0.971 19 0.974 0.970 0.972 0.974 0.974
103 The variations of coefficient of lift (C l ) and drag (C d ) of NACA 4410 for various angles of attack at wind velocity of 10 m/s are shown using CFD analysis and Correlation in Figures 4.11(a) and 4.11(b) respectively. It is observed that the coefficient of lift is maximum at 17 o of angle of attack and the stall occurs beyond this limit. The coefficient of drag increases with the increase in angle of attack and it is not linear. The CFD analysis has good conformity with the developed correlations. Figure 4.11(a) Coefficient of lift of NACA 4410 for various angles of attack at 10 m/s Figure 4.11(b) Coefficient of drag of NACA4410 for variousangles of attack at 10m/s
104 This CFD analysis is used to predict the optimal angles of attack and it is validated using the experimental results and the correlations. The results of CFD analysis made a good agreement with methods described in the literature. Further, the analysis is extended to predict the coefficient of lift, drag and pressure coefficient of various airfoils by varying the wind velocities and angle of attack. The above developed methodology is useful in predicting the above parameters for any airfoil at various working conditions even if there are no experimental results. The flow is attached up to the maximum lift point and beyond that point stall occurs. Under typical design conditions, the coefficient of lift and drag is proved theoretically and confirmed by the computation. The coefficient of lift and drag obtained from the CFD analysis is useful in optimizing the power coefficient using BEM method. 4.4 NUMERICAL CALCULATIONS FOR OPTIMISING POWER COEFFICIENT The optimization of power coefficient of wind turbines is essential to maximize the power output. The following design and performance parameters are listed below that will be useful in optimization of power coefficient based on BEM method using iterative procedure. Design parameters Rotor diameter Wind velocity Tip speed ratio Angle of attack Inflow angle Twist angle Chord Tip speed ratio Performance parameters Coefficient of lift Coefficient of drag Pressure coefficient Axial flow factor Tangential flow factors Tip loss correction factor Power coefficient Power output
105 The above parameters are discussed in the previous chapters and different sections of this chapter as well in detail and very much essential for the prediction and optimization of power coefficient of a wind turbine at specified working conditions. The calculation of power coefficient for a given blade using equations is not accurate and time consuming. The recent development of computer software leverages the use of numerical methods that involves many iterations and yields better results in shorter duration. The present numerical analysis in optimization of power coefficient is carried out as two different cases as explained below. Case (i): The axial and tangential flow factors for a blade is calculated by considering the Coefficient of drag and Tip loss correction factor. The power coefficient for this case is calculated by the equation (4.17). (1 ) (4.17) Case (ii): The axial and tangential flow factors for a blade are calculated by neglecting the Coefficient of drag and assuming Tip loss correction factor as one. The power coefficient for this case is calculated by the equation (4.18). = 4 (1 ) (4.18) The effect of ignoring the coefficient of drag and usage of tip loss correction factors are briefly explained in the section 4.1. The procedure adopted for both the cases are illustrated as the flow chart in Figure 4.12. The programming for the flow diagram is coded in MATLAB software. The parameters like rotor radius (R), blade segment radius (r), wind velocity (v), tip speed ratio (, angle of attack ( ), rotor speed in rpm (N s ), coefficient of lift (C l ) and drag (C d ) corresponding to the selected airfoil is given as input. The optimum power coefficient is calculated for above mentioned two cases for blade with airfoils NACA 4410 and NACA 2415 by iterative procedures. The iterations will terminate after the convergence (attaining the stable value)
106 of axial and tangential flow factors. The optimized value of power coefficient will be yielded as output after termination. Start Calculate r/r & r Enter r, R, v, Calculate r and c Enter Calculate Enter C l, C d Calculate F t Calculate a and a Calculate =tan -1 r (1+a )/(1-a) Calculate a new Calculate a new No Proceed until a,a Yes Compute C p C p 3 a (1-a)d Stop Figure 4.12 Flow diagrams for optimization of power coefficient
107 4.4.1 Validation of the BEM Analysis Tool The above BEM method based on iterative procedure to predict the optimum power coefficient is to be validated with experimental results before application. Validation of the proposed procedure is carried out with the experimental results published by Schepers (2002) for the twisted and tapered blades Risoe Wind Turbine. Its specifications are shown in Table 4.6 and the geometrical characteristics like twist, chord and thickness at various radii are shown in Figure 4.13. The C l and C d of the airfoil for different angle of attack is presented by them are shown in the Figure 4.14. Table 4.6 Specification of Risoe wind turbine (Schepers 2002) Number of Blades Turbine diameter Rotational Speed Cut-in wind speed Control Rated power Root extension Blade set angle Twist Root Chord Tip Chord Airfoil 3 19.0 m 35.6 and 47.5 rpm 4 m/s Stall Control 100 KW 2.3 m 1.8 degrees 15 degrees (max) 1.09 m 0.45 m NACA63-2xx Series
108 Figure 4.13 Geometry characteristics of Risoe wind turbine (Schepers 2002) Figure 4.14 C l and C d of NACA 63-2xx airfoil (Schepers 2002) The same profile used by Schepers (2002) is modeled and analyzed using the BEM method and the power coefficient has been calculated separately for two cases mentioned in the previous section. The power output of the turbine is evaluated using the equation (1.8) in Chapter I, section 1.4.
109 The outcome of the results of two cases and the experimental results are shown in the Figure 4.15. It is observed that the values of BEM analysis with case (i) has good conformity with experimental work whereas the BEM analysis with case (ii) is closer with experimental values at lower wind velocities and starts deviating at higher wind velocities as the drag force is completely ignored. Thus the proposed BEM analysis is validated with case (i) at all velocities. The results of case (ii) show the higher power as the drag forces are ignored. In actual working conditions the drag forces will be present. Hence, the two cases will show the effect of drag forces on power generation of the turbine at specified working conditions. Figure 4.15 Comparison of BEM result with experimental values 4.5. RESULTS AND DISCUSSION 4.5.1 Axial and Tangential Flow Factors The axial and tangential flow factors (a and a ) are calculated for the airfoils NACA 4410 with wind velocities of 10 m/s, 15 m/s and 20 m/s at
110 angle of attack of 5 o and shown in the Table 4.7. The results are illustrated graphically for comparison in Figures 4.16(a) and 4.16(b). Table 4.7 Axial and Tangential flow factors of NACA 4410 with various wind velocities Iterations Axial flow factor (a) Tangential flow factor (a') 10m/s 15m/s 20m/s 10m/s 15m/s 20m/s 1 0.2418 0.1724 0.1419 0.1855 0.2581 0.3295 2 0.1998 0.1146 0.0828 0.1779 0.2363 0.2974 3 0.1847 0.1032 0.0745 0.1748 0.2316 0.2928 4 0.1797 0.1011 0.0735 0.1737 0.2307 0.2922 5 0.1780 0.1008 0.0733 0.1733 0.2306 0.2921 6 0.1775 0.1007 0.0733 0.1732 0.2306 0.2921 7 0.1773 0.1007 0.0733 0.1732 0.2305 0.2921 8 0.1773 0.1007 0.0733 0.1732 0.2305 0.2921 9 0.1773 0.1007 0.0733 0.1732 0.2305 0.2921 10 0.1773 0.1007 0.0733 0.1732 0.2305 0.2921 Figure 4.16(a) Iterations of axial flow factor (a) for NACA 4410
111 Figure 4.16(b) Iterations of tangential flow factor (a ) for NACA 4410 The values of axial and tangential flow factors reduce as the iterations are increased and attain the optimum value at different iterations that are highlighted in the Table 4.9. It is observed that the optimum axial flow factor (a) decreases with increase in wind velocity. The optimum axial flow factor at wind velocities of 10 m/s, 15 m/s and 20 m/s are 0.1773, 0.1007 and 0.0733 respectively. The tangential flow factor (a ) increases with increase in wind velocity and the optimum values have been obtained at different iterations. The optimum tangential flow factor at wind velocities of 10 m/s, 15 m/s and 20 m/s are 0.1732, 0.2305 and 0.2921 respectively. The above BEM analysis has been performed for another airfoil NACA 2415 and the results are presented below in Table 4.8 and in Figures 4.17(a) and (b).
112 Table 4.8 Axial and Tangential flow factors of NACA 2415 with various wind velocities Iterations Axial flow factor (a) Tangential flow factor (a') 10m/s 15m/s 20m/s 10m/s 15m/s 20m/s 1 0.2891 0.2098 0.1740 0.2237 0.3165 0.4071 2 0.2717 0.1582 0.1150 0.2213 0.2999 0.3788 3 0.2632 0.1442 0.1037 0.2201 0.2950 0.3731 4 0.2591 0.1407 0.1017 0.2195 0.2937 0.3721 5 0.2572 0.1399 0.1013 0.2192 0.2934 0.3720 6 0.2564 0.1396 0.1013 0.2191 0.2933 0.3719 7 0.2560 0.1396 0.1013 0.2190 0.2933 0.3719 8 0.2558 0.1396 0.1013 0.2190 0.2933 0.3719 9 0.2557 0.1396 0.1013 0.2190 0.2933 0.3719 10 0.2557 0.1396 0.1013 0.2190 0.2933 0.3719 Figure 4.17 (a) Iterations of axial flow factor (a) for NACA 2415
113 Figure 4.17 (b) Iterations of tangential flow factor (a ) for NACA 2415 The values of axial and tangential flow factors of wind turbine blade with airfoil NACA 2415 reduce as the iterations are increased and attain the optimum value at different iterations that are highlighted in the Table 4.7. It is observed that the optimum axial flow factor (a) decreases with increase in wind velocity. The optimum axial flow factor at wind velocities of 10 m/s, 15 m/s and 20 m/s are 0.2557, 0.1396 and 0.1013 respectively. For axial flow factors greater than 0.4 the BEM theory does not yield reliable results (Lanzafame and Messina, 2007). Hence, the axial flow factor values are within 0.4, the BEM theory yields reliable results. The tangential flow factor (a ) increases with increase in wind velocity and the optimum values have been obtained at different iterations. The optimum tangential flow factor at wind velocities of 10 m/s, 15 m/s and 20 m/s are 0.2190, 0.2933 and 0.3719 respectively.
114 4.5.2 Power Coefficient (C p ) The power coefficient of two wind turbines with NACA 4410 airfoil and NACA 2415 airfoil for various wind velocities is calculated for two cases using the axial and tangential flow factors and the results are compared. The power coefficient of wind turbine with NACA 4410 airfoil at various wind velocities at 5 o of angle of attack for two cases is shown in Table 4.9 and the comparison of the results are presented in Figure 4.18. Table 4.9 Power coefficient of wind turbine - NACA 4410 at 5 o of AOA Wind velocity (v) in m/s Power Coefficient -NACA 4410 Case (i) Case (ii) 3 0.420 0.520 4 0.432 0.530 5 0.473 0.570 6 0.476 0.558 7 0.442 0.496 8 0.419 0.445 9 0.392 0.404 10 0.366 0.370 11 0.344 0.342 12 0.325 0.319 13 0.309 0.299 14 0.294 0.282 15 0.282 0.267 16 0.271 0.255 17 0.261 0.244 18 0.252 0.234 19 0.245 0.225 20 0.238 0.217
115 Figure 4.18 Comparison of power coefficient NACA 4410 The power coefficient of wind turbine with NACA 2415 airfoil at various wind velocities at 5 o of angle of attack is shown in Table 4.10 and comparison of the results are presented in Figure 4.19. Table 4.10 Power coefficient of wind turbine - NACA 2415 at 5 o of AOA Power Coefficient - NACA Wind velocity (v) 2415 in m/s Case (i) Case (ii) 3 0.303 0.515 4 0.430 0.575 5 0.452 0.538 6 0.415 0.476 7 0.387 0.432 8 0.355 0.390 9 0.327 0.356 10 0.304 0.327 11 0.284 0.304 12 0.267 0.284 13 0.253 0.267 14 0.241 0.253 15 0.230 0.241 16 0.220 0.230 17 0.212 0.221 18 0.205 0.213 19 0.198 0.205 20 0.192 0.199
116 Figure 4.19 Comparison of power coefficient NACA 2415 From the above study, the following inferences are obtained. The power coefficient attains the maximum at a particular wind velocity and it drastically reduces at other wind velocities for both the cases. The maximum power coefficient for case (ii) is higher as the drag forces are neglected for both the airfoils. The power coefficient is closer at higher wind velocities for both the airfoils. At lower wind velocities, the power coefficient in case (ii) is higher than case (i). The power coefficient of the airfoil NACA 4410 reaches the maximum value of 0.570 at 5 m/s in case (ii) that is closer to Betz s limit of 0.593.
117 The airfoil NACA 2415 also reaches the maximum value of power coefficient which is 0.575 at the wind velocity of 4 m/s in case (ii). In case (i), the maximum power coefficient for NACA 4410 airfoil is 0.48 at 6 m/s and NACA 2415 is 0.45 at 5 m/s as the effect of coefficient of drag and tip loss correction factors is considered. This iterative method yields the optimum values of flow factors and thereby the prediction of power coefficient at various wind velocity will be optimum. 4.5.3 Power Developed by Wind Turbine The power developed by the wind turbine is calculated using the equation (1.8) given in the Chapter I, section 1.4. The power coefficient predicted from the previous section is used in that equation and the area of the rotor is calculated from the rotor diameter 64m and the designed wind velocity (v) is taken as 10 m/s. Hence the power developed by the turbine with airfoils NACA 4410 and NACA 2415 is calculated for the two cases separately and the corresponding power coefficient is given in the Tables 4.11 and 4.12 respectively. The results are shown in Figures 4.20 and 4.21.
118 Table 4.11 Power developed for various wind velocities NACA 4410 Wind velocity (v) in m/s Power Coefficient (C p ) Power (MW) Case (i) Case (ii) Case (i) Case (ii) 3 0.420 0.520 0.06 0.03 4 0.432 0.530 0.02 0.07 5 0.473 0.570 0.12 0.15 6 0.476 0.558 0.21 0.25 7 0.442 0.496 0.31 0.35 8 0.419 0.445 0.45 0.47 9 0.392 0.404 0.59 0.61 10 0.366 0.370 0.76 0.77 11 0.344 0.342 0.95 0.95 12 0.325 0.319 1.17 1.14 13 0.309 0.299 1.41 1.36 14 0.294 0.282 1.68 1.61 15 0.282 0.267 1.97 1.87 16 0.271 0.255 2.30 2.16 17 0.261 0.244 2.66 2.48 18 0.252 0.234 3.05 2.83 19 0.245 0.225 3.48 3.20 20 0.238 0.217 3.94 3.60 Figure 4.20 Comparison of power for case (i) and (ii) NACA 4410
119 Table 4.12 Power developed for various wind velocities NACA 2415 Wind velocity (v) in m/s Power Coefficient (C p ) Power (MW) Case (i) Case (ii) Case (i) Case (ii) 3 0.303 0.515 0.04 0.03 4 0.430 0.575 0.02 0.08 5 0.452 0.538 0.12 0.14 6 0.415 0.476 0.19 0.21 7 0.387 0.432 0.28 0.31 8 0.355 0.390 0.38 0.41 9 0.327 0.356 0.49 0.54 10 0.304 0.327 0.63 0.68 11 0.284 0.304 0.78 0.84 12 0.267 0.284 0.96 1.02 13 0.253 0.267 1.15 1.22 14 0.241 0.253 1.37 1.44 15 0.230 0.241 1.61 1.69 16 0.220 0.230 1.87 1.96 17 0.212 0.221 2.16 2.25 18 0.205 0.213 2.48 2.57 19 0.198 0.205 2.82 2.92 20 0.192 0.199 3.19 3.30 Figure 4.21 Comparison of power for case (i) and (ii) NACA 2415
120 However, the wind turbine was designed for developing one MW at 10 m/s wind velocity, the power developed by the wind turbine in all other wind velocities ranging from 3 to 20 m/s was calculated and presented for both the cases. The turbine cannot reach 1MW power generation at the design condition (10 m/s), it is able to reach 1MW power production at 12 m/s for NACA 4410 airfoil at 13 m/s for NACA 2415 airfoil. From the Table 4.10, it is found that for the wind turbine with NACA 4410 airfoil the coefficient of power at 10 m/s is 0.366 for case (i) and 0.370 for case (ii) and the corresponding power developed is 0.76 MW and 0.77 MW for respectively. From the Table 4.11, it is identified that for the wind turbine with NACA 2415 airfoil the coefficient of power at 10 m/s is 0.3039 for case (i) and 0.3272 for case (ii) and the corresponding power developed was 0.63MW and 0.68 MW respectively. This is because while designing, the power coefficient is considered to be in ideal condition. 4.6 SUMMARY A one MW of horizontal axis wind turbine is designed at the wind velocity of 10 m/s. The chord and twist angle distributions are indentified using BEM method. The coefficient of lift, drag and power for the wind turbine is optimized using an Iterative approach. The two airfoils NACA 4410 and NACA 2415 have been selected for analysis. The CFD method is used to optimize the coefficient of lift and drag and the power coefficient is optimized with BEM method using Iterative approach that uses convergence of axial and tangential flow factors. The coefficient of lift, drag and pressure for these airfoils are predicted at various wind velocities and angle of attack using CFD and results are validated with the available experimental results and developed correlations. The power coefficient of wind turbine with NACA 4410 and NACA 2415 has been optimized by studying the two different cases. The effect of drag and tip loss correction factor is considered for
121 finding optimum power coefficient and the results are presented for two cases. The result of the two cases based on BEM method is validated with the experimental work. The power developed by the wind turbine for two airfoil sections are also computed and presented in this chapter.