## Abstract

The optimized chord and twist angle of the preliminary blade design through Blade Element Momentum theory are non-linear 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 Prandtl-Glauert 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 Prandtl-Glauert 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 bi-objective optimization method to maximize the annual energy production and minimize the energy cost for the design of a horizontal axis wind turbine. A multi-objective 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 start-up. 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 three-variable 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 one-dimensional 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 non-linearization 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 non-linear distributions of the optimized chord and twist angle. The non-linearization 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 Prandtl-Glauert 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.

### 2.2 Preliminary blade design

### 2.3 Design linearization method

The chord and twist angle of the preliminary blade design through optimum rotor theory are non-linear 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.

Case 1 which combines

Case 2 which combines

Case 3 which combines

Case 4 which combines

## 3 Results and discussion

### 3.1 Linearization using Prandtl-Glauert 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

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

Thus, it appears that case 4 using Prandtl-Glauert 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 Prandtl-Glauert correction model. The results show that the single-chord 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 Prandtl-Glauert correction model

Maximum torque | Maximum power | Maximum power coefficient | ||

Preliminary blade | 71.7131 | 540.1787 | 0.0298 | |

Linearized blades using Prandtl-Glauert 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 Prandtl-Glauert 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 Prandtl-Glauert correction model in all cases (case 1, case 2, case 3, case 4) as shown in Figs 8 and 9a.

Close to the tip positions, the thrust and the torque and the power of Shen correction model are higher than using Prandtl-Glauert correction model in all cases, except for the thrust from case 3 and case 4, which is smaller than using Prandtl-Glauert 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 Prandtl-Glauert 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 Prandtl-Glauert 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 non-linear 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 Prandtl-Glauert correction model was investigated. The effect of the linearization using Shen correction model while making a comparison to the linearization using Prandtl-Glauert 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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