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Contribution of lift-to-drag ratio on power coefficient of HAWT blade for different cross-sections

  • Muhammad A. R. Yass EMAIL logo , Raghad Majeed Rasheed and Amer Hamad Muhiesen
Published/Copyright: November 24, 2022
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Abstract

The aim of this study is to integrate the best lift-to-drag ratio zone to chief the highest power coefficient for horizontal axis wind turbine (HAWT) blade. Different cross-section, symmetrical, unsymmetrical, and supercritical airfoils (NACA 0012, NACA 4412, and Eppler 417) are used. FORTRAN code (f.90) was built to calculate aerodynamic data and the power coefficient based on Blade Element Momentum theory. This article deals selection of the most effective zone from the lift-to-drag ratio versus blade radius curve that gives the best incidence angle distribution. The results show a good performance that leads to approximated equal lift-to-drag distribution along the blade radius that indicates the highest power coefficient of at least 15% increases. The highest values of the power coefficient of NACA 0012, NACA 4412, and Eppler 417 were 0.476, 0.4966, and 0.482, respectively. The lift-to-drag ratio distribution zones were the most specific method of generating the maximum power coefficient for the HAWT blade. Important results and conclusion were found for further blade design.

Symbols Description Unit
a Axial factor
á Angular factor
C D Drag coefficient
C L Lift coefficient
C P Power coefficient
P Power (W)
r Blade radius (m)
U Flow velocity (m/s)
λ Tip speed ratio
ϕ Angle between wind and quarter chord of a vertical axis airfoil (deg)
Ω Angular velocity of the wind turbine rotor (rad/s)
ρ Air density (kg/m3)
σ Rotor solidity
α Angle of attack (deg)
η Efficiency
φ Angle of relative wind (deg)
υ kinematic viscosity (m2/s)
ω Angular velocity of the wind (rad/s)
θ Pitch angle (deg)
V Flow velocity in Y axis (m/s)
F Force N
I Incidence angle (deg)
R Rotor radius (m)

1 Introduction

The aerodynamic efficiency of an airfoil is defined by highest lift-to-drag ratio which is created by the specific angle of attack and the value of this angle differs from airfoil to airfoil depending on its behavior [1]. Lift-to-drag ratio depends on zero drag, aspect ratio, and span efficiency and is independent of weight. The usage of airfoil in wind turbine has not been much restricted than in airplane wing because wind turbine operates at low speed than airplane [2].

Various experimental and theoretical research studies have been carried out on the performance of wind turbine blades. ref. [3] studied a system for evaluating aerodynamic performance characteristics using two groups of NACA airfoils and airfoils used were five-digit series NACA (63-221, 65-415; 23012,23021) and four-digit series NACA (2,421, 2,412, 4,412, 4,424) for three blades HAWT. The airfoils used from root to tip in each blade were the same. A computer program was created to automate the entire procedure. Their results show that the airfoil elementary power coefficient of NACA 4412 and NACA 23012 was higher than other airfoils. Ref. [4] suggested a stable and aerodynamic design using NACA 4412 profile of (800 mm) long blade with a power of (600 Watt) with mini-HAWT. The distributions of the length chord and twist angle of the preliminary blade model are calculated. A reasonable compromise was provided between high efficiency and good stared. The blades were developed using MATLAB programming. The optimized blade chord decreases by 24%, and the thickness decreases by 44%. The optimized blade’s power level was increased significantly to 30% relative to that of the standard blade. Ref. [5] explained a design and optimization of a small blade of HAWT using self-code. The blades were fabricated using NACA 4412, NACA 2412, and NACA 1812 with wind speeds of 5 m/s, which was the most prominent wind speed prevailing in the Indian peninsula. The self-created code based on Blade Element Momentum (BEM) theory was generated an optimum blade profile that operates at high efficiency using multiple airfoils. Twist angle distribution, chord distribution, and other parameters for different airfoil sections along the blade are determined through the proposed code. The 4.46 m rotor diameter was used to achieve a power coefficient of 0.490 and produced a power output of 0.56 kW. The result of the blade analysis achieved using Q-blade software showed a reliable agreement with the proposed code and performance analysis of the wind turbine. The power coefficient acquired through MATLAB code was 0.490, and this value was very close to that obtained by using Q-blade (0.514). In addition, the difference in the output power between the two values was only 28.58 W. The goal of the study investigated by ref. [6] was to compare the aerodynamic behavior of the lift coefficient, drag coefficient, and lift–drag ratio and to observe the airfoil efficiency at different angles of attack (−15° to 15°). The contrast of six airfoils S809, S835, NACA 63415, NACA 63215, FX 63-137, and FX76-100 was performed with softwares XFOIL and FLUENT to select the maximum lift-to-drag ratio and to determine their aerodynamic coefficients by BEM theory. The FLUENT results showed that the maximum C l/C d for FX63-137 was 115.08 and occurred at an optimum angle of attack (AOA) of 4°. The maximum C l was 1.66, which at optimum AOA 14° for the FX63-137 airfoil. In addition, the lift coefficient was 0.893 at an optimum AOA of 0°. The other airfoils have 0.18–0.35 lift coefficient. The XFOIL results showed that the maximum C l/C d for FX63-137 was 102.5 and occurred at an optimum AOA of 4°. The maximum C l = 1.81 at AOA of 14° for the same airfoil. The highest lift coefficient was 0.903 at 0°. The FLUENT and XFOIL results proved the best aerodynamic performance possess by FX63-137 airfoil at high values of Reynolds number. ‎Ref. [7] presented a way for the definition of aerodynamic performance characteristics of HAWTs. The twist of the blade was calculated based on BEM theory. The optimal power coefficient determined at the blade was twisted according to a program that achieves upon the variation of the l lift and drag coefficients with AOA. The results show that the optimum angle of attack and optimum twist angle of the blade enhances the performance of the wind turbine. The airfoils NACA 4410 and NACA 2415 were taken into study for the valuation of this proposed approach.

Ref. [8] the optimum AOA for (NACA0012, NACA2412) airfoils to find the lift-to-drag ratio was numerically considered. The lift coefficient varies linearly at the same range of angle of attack. Optimum values of lift coefficient were reached if the angle of attack increases. There was a region of lift coefficient where the drag coefficient has its lowest value referred to as stall. Good agreement data were get with the experimental AOA between −5 and 5°. NACA 2412 has a higher power output than the NACA 0012 [9]. A calculation model was presented based on the SCADA data and the aerodynamic theory. Two methods were proposed for the calculation of the power coefficient: one was based on statistical data and the other was based on real-time data. The calculation result for a two-wind turbine showed that the maximum power coefficient was 0.593 if the wind speed directly measured was used during the Maximum Power Point Tracking (MPPT). The power coefficient was reduced to 0.397 after wind speed correction. The power coefficient was time-varying even in the region of MPPT When using the real-time data. The wind rotor rotational speed regulation was delayed due to the wind rotor moment of inertia, and the wind speed was time-varying. Ref. [10] studied and analyzed lift and drag performances of NACA 0015 airfoil numerical and experimentally by measuring the forces every two degrees from 0° to 20° at low Reynolds numbers (Re). The numerical analysis was performed using the CFD program, which was FLUENT. The experiment test was led in low-speed wind tunnel. The numerical results were compared with the experiment and show that the stall angle consisted of turbulence occurring next to the airfoil [11]. The performances and behaviors of the multi-cross-section HAWT blade design with and without fences were addressed. The supercritical airfoils (FX66-S-196 V, FX63-137 S, and SG6043) were used. The same dimensions of the single-cross-section NACA4412 blade were used to compare the behaviors and overall performances. Numerical analyses were performed with a self-code (F.90) and CFD based on BEM theory. The multi-cross-section blades show an approximately 8% increase in power coefficient compared with the single-cross-section blade. The boundary layer theory was used to design the fences, and their positions were determined experimentally. The increase in total power coefficient was about 16% when using fences with high flutter stability.

In this article, we studied the behaviors and performances of different cross-sections blade. The symmetrical, unsymmetrical, and supercritical airfoils (NACA 0012, NACA 4412, and Eppler 417) were taken into consideration for evaluating this proposed approach. The study deals selection of the most effective zone from the lift-to-drag ratio versus blade radius curve that gives the best incidence angle distribution. The lift-to-drag ratio versus the angle of attack curve is divided into 3, 4, and 5 zones and then studied the distribution of each zone along the blade radius for all selected airfoils. The number of zones is dependent on the distribution of lift-to-drag ratio through a range of angles of attack. The study was based on choosing the lift-to-drag distribution zone along the blade radius to achieve the highest power coefficient. The calculation of design and optimization was performed using the F.90 code and QBlade software based on BEM theory.

1.1 Best lift-to-drag ratio

Blade element efficiency occurs at r and r + dr, and this ratio can be defined [12]

(1) η = d P u d P t = ω d M l V d F v = U d F u V d F v .

The aerodynamic force dR and its resultants dF u and dF v where it rotate or rotor axis, dP u is power contribution of element by rotor, and d P t is wind contribution to blade element [13].

(2) d F u = d R l sin I d R d cos I ,

(3) d F v = d R l cos I + d R d sin I ,

(4) and cot I = U V ,

(5) η = d R l sin I d R d cos I d R l cos I + d R d sin I cot I .

By putting, tan ε = d R d d R l = C d C l , we can write the previous expression under the form:

(6) η = 1 tan ε . cot I cot I + tan ε cot I = 1 tan ε . cot I 1 + tan ε . tan I .

The aerodynamic efficiency is as much higher as tan ε is lower. At the limit, if tan ε was equal to zero, the efficiency would be equal to the unity. Actually, the value of tan ε depends on the incidence angle value [14].

1.2 Local power coefficient

The maximum power able of being extracted from the wind flow passing inside the annulus (r, r + dr) is given by the equation [11]:

(7) dP u = ω d M = ρ π r 3 d r ω 2 ( h 1 ) ( 1 + k ) .

This value corresponds to a local power coefficient:

(8) C p = d p ρ π r d r V 1 2 = ω 2 r 2 V 1 2 ( h 1 ) ( 1 + k ) = λ 2 ( h 1 ) ( 1 + k ) .

λ Being equal to ωr /V 1.

For maximum value of power coefficient or ideal turbine, consider CD = 0 then [15]

tan = CD CL = 0 [ 15 ]

(9) G E = ( 1 k ) ( h + 1 ) ( h 1 ) ( 1 + k ) = cot 2 I = λ 2 ( h + 1 ) 2 ( 1 + k ) 2 .

After simplification the above equation, there follows:

(10) λ 2 = 1 k 2 h 2 1 .

From which, it is found that:

(11) h = 1 + 1 k 2 λ 2 .

Putting this value of h in the equation of the power coefficient Cp leads to:

(12) C p = λ 2 ( 1 + k ) 1 + 1 k 2 λ 2 1 .

For a given value of λ, the power coefficient has a maximum when: d C p d k = 0

The calculations show that the maximum is obtained for a value of k which satisfies the equation [16].

(13) λ 2 = 1 3 k + 4 k 3 3 k 1 .

This equation can be written as:

(14) 4 k 3 3 k ( λ 2 + 1 ) + λ 2 + 1 = 0 ,

(15) Let k = λ 2 + 1 cos θ .

Substituting for k value, in the previous equation gives, after dividing by ( λ 2 + 1 ) 3 / 2 :

(16) 4cos 3 θ 3 cos θ + 1 λ 2 + 1 = 0 ,

(17) As : 4 cos 3 θ 3 cos θ = cos 3 θ ,

(18) cos 3 θ = 1 λ 2 + 1 ,

(19) cos ( 3 θ π ) = 1 λ 2 + 1 .

From which there follows:

(20) θ = 1 3 cos 1 1 λ 2 + 1 + π 3 = 1 3 tan 1 λ + π 3 .

At each value of λ , it is possible to determine θ then k, and therefore the maximum value of Cp.

2 Analyses and discussion

The aerodynamic point of view for all HAWT blades is the selection of the airfoil that is capturing the wind energy more effectively. The L/D ratio was the technical point of selecting the airfoil and it depend on the value of angles. The equal distribution of the L/D ratio along the wind turbine blade gives the maximum power coefficient that can be established by the selection of different angles at the blade section.

Three airfoils were selected symmetrical (NACA-0012) (Figures 1 and 2), unsymmetrical (NACA-4412) (Figures 3 and 4), and super critical (Eppler-417) (Figures 5 and 6).

Figure 1 
               NACA 0012 airfoil geometry.
Figure 1

NACA 0012 airfoil geometry.

Figure 2 
               Lift & drag for NACA 0012.
Figure 2

Lift & drag for NACA 0012.

Figure 3 
               NACA 4412 airfoil geometry.
Figure 3

NACA 4412 airfoil geometry.

Figure 4 
               Lift & drag for NACA 4412.
Figure 4

Lift & drag for NACA 4412.

Figure 5 
               Eppler 417 airfoil geometry.
Figure 5

Eppler 417 airfoil geometry.

Figure 6 
               Lift & drag for Eppler 417.
Figure 6

Lift & drag for Eppler 417.

Different zones of angle of attack were selected for each airfoil to know which zone gives approximately equal L/D distributed with the angle of attack along the blade section (Figures 79). All these angle-of-attack zones were selected and analyzed to calculate the power coefficient as shown in Figures 1017 and Table 1 for NACA-0012, Figures 1827 and Table 2 for NACA-4412, and Figures 2837,38,39 and Table 3 for Eppler-417. The comparison between the current study and the previous study is shown in Table 4.

Figure 7 
               NACA 0012 zone selected.
Figure 7

NACA 0012 zone selected.

Figure 8 
               NACA 4412 zone selected.
Figure 8

NACA 4412 zone selected.

Figure 9 
               Eppler 417 zone selected.
Figure 9

Eppler 417 zone selected.

Figure 10 
               Zone (A) for NACA 0012.
Figure 10

Zone (A) for NACA 0012.

Figure 11 
               Zone (B) for NACA 0012.
Figure 11

Zone (B) for NACA 0012.

Figure 12 
               Zone (C) for NACA 0012.
Figure 12

Zone (C) for NACA 0012.

Figure 13 
               Zone (D) for NACA 0012.
Figure 13

Zone (D) for NACA 0012.

Figure 14 
               Power coefficient (CP) for zone (A) NACA 0012.
Figure 14

Power coefficient (CP) for zone (A) NACA 0012.

Figure 15 
               power coefficient (CP) for zone (B) for NACA 0012.
Figure 15

power coefficient (CP) for zone (B) for NACA 0012.

Figure 16 
               Power coefficient (CP) for zone (C) for NACA 0012.
Figure 16

Power coefficient (CP) for zone (C) for NACA 0012.

Figure 17 
               Power coefficient (Cp) for zone (D) for NACA 0012.
Figure 17

Power coefficient (Cp) for zone (D) for NACA 0012.

Table 1

NACA 0012 data

r LD zone (A) L/D zone (B) L/D zone (C) LD zone (D) ALO CP zone (A) CP zone (B) CP zone (C) CP zone (D)
0.107 43.89952 68.64226 78.18008 77.73189 1 0.0208 0.0292 0.0176 0.0132
0.214 39.50274 66.48779 77.63682 78.17361 2 0.0651 0.1206 0.0683 0.0507
0.321 34.79645 64.08402 76.92907 78.52108 3 0.0911 0.327 0.1785 0.1277
0.428 29.77832 61.42086 76.04659 78.76679 4 0.1045 0.4423 0.3555 0.2826
0.535 24.44765 58.48864 74.97893 78.90287 5 0.1181 0.4625 0.4399 0.4073
0.642 18.80558 55.27829 73.71545 78.92104 6 0.1258 0.4595 0.4705 0.4608
0.749 12.85522 51.78145 72.24538 78.81268 7 0.125 0.4439 0.47 0.4764
0.856 6.60174 47.99068 70.55791 78.56881 8 0.1112 0.411 0.4549 0.4576
0.963 0.05249 43.89952 68.64226 78.18008 9 0.0745 0.3543 0.4288 0.3971
1.07 −6.78295 39.50274 66.48779 77.63682 10 0.0225 0.2687 0.3924 0.3641
Figure 18 
               Zone (A) for NACA 4412.
Figure 18

Zone (A) for NACA 4412.

Figure 19 
               Zone (B) for NACA 4412.
Figure 19

Zone (B) for NACA 4412.

Figure 20 
               Zone (C) for NACA 4412.
Figure 20

Zone (C) for NACA 4412.

Figure 21 
               Zone (D) for NACA 4412.
Figure 21

Zone (D) for NACA 4412.

Figure 22 
               Zone (E) for NACA 4412.
Figure 22

Zone (E) for NACA 4412.

Figure 23 
               Power coefficient for zone (A) NACA 4412.
Figure 23

Power coefficient for zone (A) NACA 4412.

Figure 24 
               Power coefficient for zone (B) NACA 4412.
Figure 24

Power coefficient for zone (B) NACA 4412.

Figure 25 
               Power coefficient for zone (C) NACA 4412.
Figure 25

Power coefficient for zone (C) NACA 4412.

Figure 26 
               Power coefficient for zone (D) NACA 4412.
Figure 26

Power coefficient for zone (D) NACA 4412.

Figure 27 
               Power coefficient for zone (E) NACA 4412.
Figure 27

Power coefficient for zone (E) NACA 4412.

Table 2

NACA 4412 data

r L/D zone (A) L/D zone (B) L/D zone (C) L/D zone (D) L/D zone (E) ALO CP zone (A) CP zone( B) CP zone (C) CP zone (D) CP zone (E)
0.107 127.0555 132.9032 127.8518 115.6428 84.72221 1 −0.0111 −0.0112 −0.0125 −0.0133 −0.0148
0.214 125.1248 131.8051 129.9954 119.1474 88.41953 2 0.0486 0.0312 0.0121 0.0062 −0.0039
0.321 123.0828 130.4361 131.7064 122.3853 92.24059 3 0.2461 0.187 0.1028 0.0845 0.0566
0.428 120.9619 128.8391 132.9705 125.3025 96.15593 4 0.4047 0.3631 0.2931 0.2672 0.2089
0.535 118.7907 127.0555 133.7857 127.8518 100.129 5 0.473 0.4514 0.4234 0.3961 0.3689
0.642 116.5936 125.1248 134.1619 129.9954 104.1164 6 0.4878 0.4863 0.4758 0.472 0.4631
0.749 114.3915 123.0828 134.1196 131.7064 108.0684 7 0.49 0.4927 0.4966 0.4964 0.4913
0.856 112.2018 120.9619 133.6881 132.9705 111.93 8 0.487 0.4901 0.4807 0.4683 0.4139
0.963 110.0385 118.7907 132.9032 133.7857 115.6428 9 0.4736 0.4837 0.4269 0.3896 0.2253
1.07 107.9129 116.5936 131.8051 134.1619 119.1474 10 0.4438 0.4713 0.4153 0.3214 0.1774
Figure 28 
               Zone (A) for Eppler 417.
Figure 28

Zone (A) for Eppler 417.

Figure 29 
               Zone (B) for Eppler 417.
Figure 29

Zone (B) for Eppler 417.

Figure 30 
               Zone (C) for Eppler 417.
Figure 30

Zone (C) for Eppler 417.

Figure 31 
               Zone (D) for Eppler 417.
Figure 31

Zone (D) for Eppler 417.

Figure 32 
               Zone (E) for Eppler 417.
Figure 32

Zone (E) for Eppler 417.

Figure 33 
               Zone (F) for Eppler 417.
Figure 33

Zone (F) for Eppler 417.

Figure 34 
               Power coefficient for zone (A) Eppler.
Figure 34

Power coefficient for zone (A) Eppler.

Figure 35 
               Power coefficient for zone (B) Eppler.
Figure 35

Power coefficient for zone (B) Eppler.

Figure 36 
               Power coefficient for zone (C) Eppler 417.
Figure 36

Power coefficient for zone (C) Eppler 417.

Figure 37 
               Power coefficient for zone (D) Eppler.
Figure 37

Power coefficient for zone (D) Eppler.

Table 3

NACA 0012 data

r L/D zone (A) L/D zone (B) L/D zone (C) L/D zone (D) L/D zone (E) L/D zone (F) ALO CP zone (A) CP zone (B) CP zone (C) CP zone (D) CP zone (E) CP zone (F)
0.107 120.6863 148.6024 149.7731 142.473 132.5301 107.9224 1 −0.0135 −0.0119 −0.0117 −0.0118 −0.012 −0.0123
0.214 113.5424 142.6 153.9455 149.7731 142.473 120.7183 2 0.0618 0.0297 0.0159 0.0134 0.0111 0.0069
0.321 106.9214 135.59 154.8998 153.9455 149.7731 132.5301 3 0.216 0.1685 0.1544 0.1461 0.1385 0.124
0.428 100.9702 128.1397 152.9302 154.8998 153.9455 142.473 4 0.2581 0.2395 0.2499 0.2394 0.2295 0.2106
0.535 95.80402 120.6863 148.6024 152.9302 154.8998 149.7731 5 0.4288 0.3166 0.262 0.312 0.2606 0.31
0.642 91.54546 113.5424 142.6 148.6024 152.9302 153.9455 6 0.4456 0.4539 0.4112 0.401 0.3871 0.3746
0.749 88.37979 106.9214 135.59 142.6 148.6024 154.8998 7 0.4636 0.4797 0.4645 0.4639 0.3991 0.4583
0.856 86.65945 100.9702 128.1397 135.59 142.6 152.9302 8 0.4778 0.4784 0.482 0.4667 0.4647 0.4589
0.963 87.16521 95.80402 120.6863 128.1397 135.59 148.6024 9 0.4606 0.4709 0.47 0.4676 0.4635 0.4455
1.07 91.96774 91.54546 113.5424 120.6863 128.1397 142.6 10 0.4274 0.4562 0.4629 0.4558 0.4497 0.4178
Table 4

Compare the proposed results with previous works

Comparison Airfoils Reynolds Number Power coefficient cp
Present study NACA 4412 1 × 106 0.4966
NACA 0012 1 × 106 0.476
EPPLER 417 1 × 106 0.482
[17] NACA 4412 1 × 105 0.49
[18] NACA 0012 1.76 × 106 0.48

For NACA-0012, the highest value of power coefficient was at zone (D) (CP = 0.476 at λ = 7), for NACA-4412 the highest power coefficient was at zone (c) (CP = 0.04966 at λ = 7), and for Eppler-417 the highest value of power coefficient was at zone (c) (CP = 0.482 at λ = 7).

Figure 38 
               Power coefficient for zone (E) Eppler 417.
Figure 38

Power coefficient for zone (E) Eppler 417.

Figure 39 
               Power coefficient for zone (F) Eppler 417.
Figure 39

Power coefficient for zone (F) Eppler 417.

3 Conclusion and future works

  1. The symmetrical, unsymmetrical, and supercritical airfoils (NACA 0012, NACA 4412, and Eppler 417) were taken into consideration for evaluating this proposed approach. The lift-to-drag ratio was distributed along the blade radius after choosing an effective zone that achieved an optimum power coefficient. The theoretical analyses yield the following conclusions.

  2. Best windmill blade airfoil selection should be at a high L/D ratio and lowest angle of attack.

  3. Popper distribution of L/D ratio generates the equal force on the blade section which integrates a stable blade and reduces variation.

  4. Equal distribution of L/D ratio on blade radius increased power coefficient by about 15% of other distribution.

  5. Proper distribution gives the best selection of working angles of attack at the blade section.

  6. Minimize the overall loss of wind turbine power.

The future work could be extended to several other NACA and NREL profiles. The evaluation of profiles could be extended to an entire blade. The blade could be evaluated with a mixture of multi sections profiles over the entire blade.

  1. Conflict of interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this study.

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Received: 2022-03-08
Revised: 2022-03-25
Accepted: 2022-03-29
Published Online: 2022-11-24

© 2022 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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