chapter 4 optimization of coefficient of lift,...
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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.
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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.
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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 (Cl) and drag (Cd) 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
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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 ‘Ft’ 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 (Ft) is to be incorporated as
proposed by Glauert (1926). The factor (Ft) is taken as one when the axial
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flow factor is greater than 0.4 and in other cases the factor (Ft) 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 (Ft) 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 rth
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
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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 5th 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.
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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
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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)
Tipspeed
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
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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
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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).
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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 5o of angles of attack as
linear region, due to greater reliability of both experimental and computed
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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
2
i b2i j j j
1 v v( v) ( vx ) G ( v) C Y St x x x x
The transported variable in the Spalart-Allmaras model, v , is identical
to the turbulent kinematic viscosity except in the near-wall (viscosity-
affected) region. where Gv is the production of turbulent viscosity, and Yv is
the destruction of turbulent viscosity that occurs in the near-wall region due to
wall blocking and viscous damping. v and Cb2 are the constants and Sv 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.
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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 5o of angle of attack
No of Grid cells
Wind Velocity (v)
in m/s
Lift Coefficient
(Cl)
DragCoefficient
(Cd)
Maximum Pressure
Coefficient (Cpr)
15150 5 0.6593 0.0401 0.952 20253 5 0.6654 0.0425 0.95435566 5 0.680 0.0434 0.96442364 5 0.680 0.0434 0.964
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4.3.3 Validation of Coefficient of Lift and Drag
The values of coefficient of lift (Cl) and drag (Cd) 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 Cl and Cd 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 attack in Degrees
Wind tunnel CFD Analysis
Cl Cd Cl Cd
-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 0.01580 0.0702 0.0124 0.1329 0.01641 0.1355 0.0157 0.1783 0.01772 0.1871 0.0207 0.2290 0.01983 0.2511 0.0265 0.2818 0.02274 0.3082 0.0331 0.3322 0.02655 0.3660 0.0397 0.3826 0.03106 0.4093 0.0455 0.4301 0.03647 0.4678 0.0537 0.4758 0.04258 0.5049 0.0587 0.5205 0.04949 0.5517 0.0661 0.5635 0.0570
10 0.5682 0.0736 0.6029 0.065211 0.6287 0.0827 0.6399 0.074112 0.6652 0.0901 0.6735 0.083713 0.6886 0.0950 0.7055 0.094114 0.6989 0.1041 0.7351 0.105215 0.7533 0.1298 0.7606 0.117116 0.7650 0.1389 0.7219 0.124117 0.7842 0.1595 0.7220 0.135518 0.7154 0.2232 0.7005 0.151319 0.6570 0.2529 0.6451 0.1886
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Figure 4.8(a) Validation of coefficient of lift of NACA 2415
Figure 4.8(b) Validation of coefficient of drag of NACA 2415
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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 5o. The results of Cl and Cd 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
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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 105th iterations. The
results of coefficient of lift (Cl) 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 (Cd) 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 (Cpr) 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 ‘Cpr’ is given as
(4.16)
The pressure coefficient (Cpr) at lower and upper surfaces of NACA
4410 airfoil with wind velocity of 5 m/s and 5o 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
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The pressure coefficient (Cpr) 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 (Cl), drag (Cd) and maximum
pressure coefficient (Cpr) of NACA 4410 and NACA 2415 at various wind
velocities with 5o 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
5o Angle of attack – NACA 4410
Wind Velocity(v)
in m/s
Lift Coefficient
(Cl)
DragCoefficient
(Cd)
Maximum Pressure
Coefficient (Cpr)5 0.680 0.0434 0.964
10 0.686 0.0426 0.95815 0.691 0.0420 0.95420 0.695 0.0416 0.95025 0.697 0.0413 0.949
Table 4.4 (b) Coefficient of lift, drag & Maximum pressure coefficient at
5o Angle of attack – NACA 2415
Wind Velocity (v)
in m/s
Lift Coefficient
(Cl)
DragCoefficient
(Cd)
Maximum Pressure
Coefficient (Cpr)5 0.384 0.036 0.882
10 0.386 0.033 0.87015 0.388 0.032 0.86720 0.390 0.031 0.86525 0.393 0.030 0.860
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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 (Cl)
5 m/s 10 m/s 15 m/s 20 m/s 25 m/s
0 0.404 0.409 0.412 0.416 0.4191 0.464 0.470 0.473 0.480 0.4822 0.519 0.524 0.527 0.533 0.5353 0.574 0.580 0.583 0.589 0.5914 0.629 0.634 0.636 0.642 0.6445 0.680 0.686 0.691 0.695 0.6976 0.731 0.737 0.741 0.745 0.7487 0.780 0.785 0.790 0.794 0.7988 0.826 0.832 0.837 0.842 0.8449 0.872 0.877 0.881 0.885 0.88710 0.914 0.920 0.922 0.926 0.92811 0.953 0.959 0.962 0.964 0.96512 0.988 0.995 0.997 0.997 0.99813 1.020 1.026 1.026 1.026 1.02714 1.047 1.052 1.052 1.051 1.04915 1.072 1.076 1.062 1.048 1.04716 1.095 1.095 1.073 1.054 1.05517 1.114 1.110 1.084 1.060 1.06118 1.129 1.064 1.063 1.062 1.02719 1.140 1.073 1.026 0.979 0.979
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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 (Cd)
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
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Table 4.5(c) Maximum coefficient of pressure (Cpr) of NACA 4410 at
various angles of attack for different wind velocities
Angle of attack in Degrees
Maximum Coefficient of pressure (Cpr)
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 (Cl) and drag (Cd) 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 17o 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
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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 OPTIMISINGPOWER 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 Performance parameters
Rotor diameter
Wind velocity
Tip speed ratio
Angle of attack
Inflow angle
Twist angle
Chord
Tip speed ratio
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 (Ns), coefficient of lift (Cl) and drag (Cd) 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)
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of axial and tangential flow factors. The optimized value of power coefficient will be yielded as output after termination.
No
Yes
Figure 4.12 Flow diagrams for optimization of power coefficient
Calculate r and c
Calculate
Calculate Ft
Calculate a and a’
Calculate =tan -1 r (1+a’)/(1-a)
Calculate anew
Proceed until a,a’
Compute Cp
Cp3a’(1-a)d
Stop
Start
Calculate a’new
Calculate r/R & rEnter r, R, v,
Enter
Enter Cl, Cd
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 Cl and Cd 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
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Figure 4.13 Geometry characteristics of Risoe wind turbine (Schepers 2002)
Figure 4.14 Cl and Cd 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.
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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 5o 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
IterationsAxial 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 0.29216 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
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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).
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Table 4.8 Axial and Tangential flow factors of NACA 2415 with various
wind velocities
IterationsAxial 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
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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.
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4.5.2 Power Coefficient (Cp)
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 5o 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 5o of AOA
Wind velocity (v) in m/s
Power Coefficient -NACA 4410 Case (i) Case (ii)
3 0.420 0.5204 0.432 0.5305 0.473 0.5706 0.476 0.5587 0.442 0.4968 0.419 0.4459 0.392 0.40410 0.366 0.37011 0.344 0.34212 0.325 0.31913 0.309 0.29914 0.294 0.28215 0.282 0.26716 0.271 0.25517 0.261 0.24418 0.252 0.23419 0.245 0.22520 0.238 0.217
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Figure 4.18 Comparison of power coefficient – NACA 4410
The power coefficient of wind turbine with NACA 2415 airfoil at
various wind velocities at 5o 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 5o of AOA
Wind velocity (v) in m/s
Power Coefficient - NACA 2415
Case (i) Case (ii) 3 0.303 0.5154 0.430 0.5755 0.452 0.5386 0.415 0.4767 0.387 0.4328 0.355 0.3909 0.327 0.356
10 0.304 0.32711 0.284 0.30412 0.267 0.28413 0.253 0.26714 0.241 0.25315 0.230 0.24116 0.220 0.23017 0.212 0.22118 0.205 0.21319 0.198 0.20520 0.192 0.199
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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.
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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 (Cp) Power (MW)
Case (i) Case (ii) Case (i) Case (ii)
3 0.420 0.520 0.06 0.034 0.432 0.530 0.02 0.075 0.473 0.570 0.12 0.156 0.476 0.558 0.21 0.257 0.442 0.496 0.31 0.358 0.419 0.445 0.45 0.479 0.392 0.404 0.59 0.6110 0.366 0.370 0.76 0.7711 0.344 0.342 0.95 0.9512 0.325 0.319 1.17 1.1413 0.309 0.299 1.41 1.3614 0.294 0.282 1.68 1.6115 0.282 0.267 1.97 1.8716 0.271 0.255 2.30 2.1617 0.261 0.244 2.66 2.4818 0.252 0.234 3.05 2.8319 0.245 0.225 3.48 3.2020 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 (Cp) Power (MW) Case (i) Case (ii) Case (i) Case (ii)
3 0.303 0.515 0.04 0.034 0.430 0.575 0.02 0.085 0.452 0.538 0.12 0.146 0.415 0.476 0.19 0.217 0.387 0.432 0.28 0.318 0.355 0.390 0.38 0.419 0.327 0.356 0.49 0.5410 0.304 0.327 0.63 0.6811 0.284 0.304 0.78 0.8412 0.267 0.284 0.96 1.0213 0.253 0.267 1.15 1.2214 0.241 0.253 1.37 1.4415 0.230 0.241 1.61 1.6916 0.220 0.230 1.87 1.9617 0.212 0.221 2.16 2.2518 0.205 0.213 2.48 2.5719 0.198 0.205 2.82 2.9220 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.