Abstract
The optimized chord and twist angle of the preliminary blade design through Blade Element Momentum theory are nonlinear distributions, which adds to the complexity of blade manufacture and does not always guarantee the best aerodynamic performance. In this paper, the effect of the linearization on aerodynamic performance using PrandtlGlauert correction model was investigated through four cases: case 1 and case 2 and case 3, where the chord and the twist angle are linearized and case 4, where sole chord is linearized. The effect of the linearization using Shen correction model while making a comparison to the linearization using PrandtlGlauert correction model was also studied. The simulation is conducted for S809 wind turbine blade profile. The results show that case 4 using Shen correction model represents the best technique of linearization in terms of higher aerodynamic performance and easy manufacturing process.
1 Introduction
The energy issue is at the heart of international concerns. The global economy is indeed facing a major challenge: meeting the growing need for energy while reducing greenhouse gas emissions. The use of renewable energies remains the most effective solution to this challenge. Great hope is placed in wind energy, which represents one of the world's most responsive renewable energy resources. Most wind turbines aim to capture the maximum energy from the wind. The blade is the main component that converts the kinetic energy of the wind into mechanical energy, so the design of this component has a big impact on the energy efficiency of a wind turbine. The design of a wind turbine blade is primarily based on aerodynamic modeling. In order to maximize power output and minimize costs, many researchers are trying to find methods to optimize blade design [1].
Benini and Toffolo [2] presented a biobjective optimization method to maximize the annual energy production and minimize the energy cost for the design of a horizontal axis wind turbine. A multiobjective evolutionary algorithm and an aerodynamic model based on blade element theory were used to achieve this goal. Kale and Varma [3] have identified two main objectives to optimize the design of a wind turbine blade which are the improvement of power performance and system startup. They showed an increase in power coefficient and a reduction in starting speed. Bottasso et al. [4] have treated an aero structural approach to optimize the design of a wind turbine blade. This approach is based on giving the profiles an appropriate position along the length of the blade according to their structural or aerodynamic role. Close to the hub zone are placed airfoils that respond to structural considerations, while in the tip zone are placed airfoils that respond to aerodynamic considerations. Mohammadi et al. [5] presented an optimization process where the goal of the work is to maximize the output torque. This optimization is firstly performed with two variables: profile type and angle of attack; and secondly with three variables by adding the chord variable to the previous variables. They have shown that the output torque has increased very significantly when the threevariable optimization is used. Wang et al. [6] optimized the blade design of a Fixed Pitch Variable Speed wind turbine while taking into account the Reynolds number effect. The objective of this approach is to improve the energy performance of the wind turbine through optimization of the chord and twist angle for each element of the blade. Thumthae [7] sought to find optimal chord and twist angle distributions and adequate rotor speed variation to maximize the energy efficiency of a variable speed horizontal axis wind turbine.
Blade Element Momentum (BEM) theory is one of the most widely used methods in wind turbine aerodynamics as it provides an acceptable and efficient approach to wind turbine blade design and analysis [8]. However, BEM theory has failed to match the experience. To improve this theory, several studies have made corrections to this model which mainly concern the correction of tip loss. Certainly, a wind turbine with a finite number of blades is different from a wind turbine with an infinite number of blades. To account for this difference, Prandtl introduced the phenomenon of tip loss. Based on the simple onedimensional theory that allows the prediction of the wind turbine performance, Glauert was able to develop the BEM theory. To obtain results consistent with reality, Glauert added Prandtl's tip loss correction to the calculations of the BEM theory. According to Glauert, only the induced velocities were affected by tip loss. According to de Vries, the correction includes the induced velocities as well as the mass flow. Shen et al. showed that existing tip loss corrections are not coherent and do not predict a correct physical behavior near the tip. To give better predictions of the load in the tip region, a new model for tip loss correction has been developed by Shen et al. [9, 10]. Besides, the nonlinearization of the chord and twist angle of the BEM theory poses problems in the manufacturing of the wind turbine blade. Several researchers have worked on the linearization of the chord and twist angle to overcome the difficulty of manufacturing and to reduce the cost of manufacturing wind turbine blades. Manwell [11] used two constants to linearize the expression of the chord and one constant to linearize the expression of the twist angle. Maalawi et al. [12] estimated a linear distribution of the chord and an exponential distribution of the twist angle. The linearized chord is presented by a line tangential to the theoretical distribution of the chord at the blade position of
The BEM theory gives nonlinear distributions of the optimized chord and twist angle. The nonlinearization of these distributions poses problems in the manufacturing process of a wind turbine blade and does not always guarantee the best aerodynamic performance. In this paper, the effect of the linearization on aerodynamic performance using PrandtlGlauert correction model was investigated through four cases. The effect of the linearization on the aerodynamic performance using Shen correction model was also studied. The results have determined the best technique of linearization in terms of higher aerodynamic performance and easy manufacturing process.
2 Materials and methods
2.1 Aerodynamic design
The aerodynamic design of a wind turbine is based on the BEM theory, which is the composition of the momentum theory and the blade element theory.
One element of a multiple stream tube
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
One element of a multiple stream tube
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
One element of a multiple stream tube
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
2.2 Preliminary blade design
Lift and drag coefficients of S809 profile
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Lift and drag coefficients of S809 profile
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Lift and drag coefficients of S809 profile
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
2.3 Design linearization method
The chord and twist angle of the preliminary blade design through optimum rotor theory are nonlinear distributions, which adds to the complexity of blade manufacture. The use of linear chord and twist angle distributions remains the most effective solution to avoid this complexity. On the other hand, it can also increase the aerodynamic performance. Figures 4 and 5 present the chord and twist angle distributions of the linearized blades and the preliminary blade design.
Chord distribution of the linearized blades and the preliminary blade
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Chord distribution of the linearized blades and the preliminary blade
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Chord distribution of the linearized blades and the preliminary blade
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Twist angle distribution of the linearized blades and the preliminary blade
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Twist angle distribution of the linearized blades and the preliminary blade
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Twist angle distribution of the linearized blades and the preliminary blade
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Case 1 which combines
Case 2 which combines
Case 3 which combines
Case 4 which combines
3 Results and discussion
3.1 Linearization using PrandtlGlauert correction model
The comparison between the thrust curves of the preliminary blade and the linearized blades is given by Fig. 6a. The latter indicates that case 1, which combines
Thrust and torque of the linearized blades and the preliminary blade using PrandtlGlauert correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Thrust and torque of the linearized blades and the preliminary blade using PrandtlGlauert correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Thrust and torque of the linearized blades and the preliminary blade using PrandtlGlauert correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
The curves of the torque, the power, the power coefficient of the linearized blades, and the preliminary blade are presented in Figs 6b and 7 respectively. Compared to the preliminary blade design, case 1 and case 2 and case 3, where the chord and the twist angle are linearized, decrease the torque, the power and the power coefficient along the blade. We see that the decrease of case 1 is more important than those of case 2 and case 3. Case 4, where sole chord is linearized at
Power output and power coefficient of the linearized blades and the preliminary blade using PrandtlGlauert correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Power output and power coefficient of the linearized blades and the preliminary blade using PrandtlGlauert correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Power output and power coefficient of the linearized blades and the preliminary blade using PrandtlGlauert correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Thus, it appears that case 4 using PrandtlGlauert correction model has the best aerodynamic performance among all four cases. Table 1 shows the maximum torque, the maximum power, and the maximum power coefficient of the linearized blades, and the preliminary blade using PrandtlGlauert correction model. The results show that the singlechord linearization has a significant effect on the peak torque, power and power coefficient, which allows case 4 to increase these peaks, while the linearization of the two: chord and twist angle leads to a decrease in these peaks along the blade.
Maximums of torque, power and power coefficient of linearized blades, and preliminary blade for PrandtlGlauert correction model
Maximum torque  Maximum power  Maximum power coefficient  
Preliminary blade  71.7131  540.1787  0.0298  
Linearized blades using PrandtlGlauert correction model  Case 1  44.5459  335.5413  0.0192 
Case 2  49.9239  376.0513  0.0202  
Case 3  65.8494  496.0101  0.0270  
Case 4  74.6161  562.0450  0.0308 
3.2 Linearization using Shen correction model
This part is interested in examining the effect of the linearization using Shen correction model on the aerodynamic performance while making a comparison to the linearization using PrandtlGlauert correction model. There is not much difference in the root and middle of the blade between the linearization using Shen correction model and the linearization using the PrandtlGlauert correction model in all cases (case 1, case 2, case 3, case 4) as shown in Figs 8 and 9a.
Thrust and torque of the linearized blades and the preliminary blade using Shen correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Thrust and torque of the linearized blades and the preliminary blade using Shen correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Thrust and torque of the linearized blades and the preliminary blade using Shen correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Power output and power coefficient of the linearized blades and the preliminary blade using Shen correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Power output and power coefficient of the linearized blades and the preliminary blade using Shen correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Power output and power coefficient of the linearized blades and the preliminary blade using Shen correction model
Citation: International Review of Applied Sciences and Engineering 2022; 10.1556/1848.2022.00439
Close to the tip positions, the thrust and the torque and the power of Shen correction model are higher than using PrandtlGlauert correction model in all cases, except for the thrust from case 3 and case 4, which is smaller than using PrandtlGlauert correction model in the range
Table 2 presents a comparison of the maximum torque, power and power coefficient values obtained using the Shen correction model and those obtained using the PrandtlGlauert correction model. The result shows that in all cases, the peaks of torque, power and power coefficient increased significantly when using the Shen correction model, while maintaining case 4 as the best linearization case compared to the other cases studied.
Maximums of torque, power and power coefficient of linearized blades, and preliminary blade for two correction models
Maximum torque  Maximum power  Maximum power coefficient  
Preliminary blade  71.7131  540.1787  0.0298  
Linearized blades using PrandtlGlauert correction model  Case 1  44.5459  335.5413  0.0192 
Case 2  49.9239  376.0513  0.0202  
Case 3  65.8494  496.0101  0.0270  
Case 4  74.6161  562.0450  0.0308  
Linearized blades using Shen correction model  Case 1  47.9381  361.0937  0.0199 
Case 2  52.4661  395.2002  0.0222  
Case 3  69.8250  525.9560  0.0326  
Case 4  78.7702  593.3359  0.0376 
4 Conclusion
The Blade Element Momentum theory gives nonlinear distributions of the optimized chord and twist angle, which adds to the complexity of blade manufacture and does not always guarantee the best aerodynamic performance. To avoid this complexity and achieve easy manufacturing, we used linear chord and twist angle in the design of wind turbine blades. In this paper, the linearization of the chord and twist angle consists first of drawing two straight lines: one passes through the positions
Case 1 which combines
Case 2 which combines
Case 3 which combines
Case 4 which combines
The effect of the linearization on aerodynamic performance using PrandtlGlauert correction model was investigated. The effect of the linearization using Shen correction model while making a comparison to the linearization using PrandtlGlauert correction model was also studied. The results showed that case 4 using Shen correction model represents the best technique of linearization in terms of higher aerodynamic performance and easy manufacturing process.
Although the presented BEM method has advantages, it also has some limitations. In particular, we find that this method does not take into account structural considerations and cannot increase aerodynamic performance in the inner positions of the blade.
Nomenclature

Axial induction factor 

Tangential induction factor 

Chord length 

Number of blades 

Lift coefficient 

Drag coefficient 

Relative wind speed 

Wind velocity 

Inflow angle 

Twist angle 

Angle of attack 

Tip loss factor 

Local radius 

Angular velocity of rotor 

Radius of rotor 

Air density 

Tip speed ratio 

Local speed ratio 

Solidity of rotor 

Aerodynamic thrust 

Aerodynamic torque 
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