Wind Turbine Aerodynamics

AERODYNAMICS OF WIND TURBINE

A wind turbine is a device that extracts kinetic energy of the wind and converts into useful energy. The power produced by a wind turbine depends on the interaction between the wind turbine rotor and the wind. The wind turbine aerodynamics is important, to design a blade and to analyze aerodynamic performance of the rotor. A number of scientists have derived various methods for aerodynamic analysis of wind turbine rotors [BSJB 2001] [MMR 2002] [Pat 2006] [Wood 2011].

1. One dimensional momentum theory

The aerodynamic analysis carried out here is based on momentum theory. For this analysis following assumptions are made,

- Homogenous flow,
- incompressible flow,
- steady state flow,
- No frictional drag,
- A non-rotating wake,
- Infinite number of blades,
- Uniform thrust over rotor area,
- Ambient static pressure at far upstream and far downstream

The control volume to analyze wind turbine aerodynamics is shown in Fig. 1. Surface of stream tube and two cross-sections 1 and 3 of stream tube are the volume boundaries of control volume. The wind flow is across the ends of the stream tube only. To analyze energy extraction process a uniform ‘actuator disc’ representing wind turbine is considered. This disc creates a pressure discontinuity in the air stream tube flowing through it.

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Fig. 1: Actuator disc model of a wind turbine without wake [MMR 2002]

The upstream cross-section area is smaller than that of the disc, while the downstream cross-section area is larger than the disc. The stream-tube is expanding to keep the mass flow rate same everywhere. The mass flow rate along the stream-tube is given as,

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where [illustration not visible in this excerpt] is the air density, A is the cross sectional area, U is the air velocity and subscripts indicate values at named cross-sections in Fig. 1.

The actuator disc induces a velocity variation that of the free stream velocity. The stream wise component of this induced flow at the disc is given by aU1, where a is called the axial flow induction factor, or the inflow factor. The stream wise velocity at the disc is,

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When the air passes through the actuator disc, it undergoes an overall change in velocity, (U1 - U2). The rate of change of momentum is the product of the overall change of velocity and mass flow rate through the stream-tube,

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The force causing change of momentum comes completely from the pressure difference across the actuator disc, because the stream-tube is otherwise surrounded by air at atmospheric pressure, which gives zero net force. Therefore,

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The pressure difference in the above equation can be determined by applying Bernoulli’s equation. The total energy in upstream and downstream is different. Hence, Bernoulli’s equation should be applied separately to the upstream and Therefore, for upstream section,

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For incompressible horizontal flow, above equation becomes,

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Similarly, for downstream section,

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Subtracting these equations, we get,

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Substituting pressure difference from Equation 8 in Equation 4,

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Simplifying and rearranging,

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The quantity, U1a, the induced velocity at the rotor, is a combination of the free stream velocity and the induced velocity. As the axial induction factor increase from zero, the wind velocity behind the rotor slows more and more. If a = 1/2, the wind has slowed to zero velocity behind the rotor and the simple theory is no longer applicable.

1.1 Power coefficient

The thrust, T can determined by, the net sum of the forces on each side of the actuator disc,

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As this thrust is concentrated at the disc, the rate of work done by the thrust is TU1. Hence, the power extraction from the air is given by,

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Replacing the control volume area at disc Ad with rotor area A and the free stream velocity U1 by U the above equation can be written as,

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The power available in the wind for rotor area [illustration not visible in this excerpt]

The wind turbine performance is usually described by the ratio of rotor power and power in the wind and known as power coefficient, CP

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The fraction of power in the wind that extracted by the rotor is also represented by the non-dimensional power coefficient. From Equation 12, the power coefficient is,

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The maximum value of power coefficient is given by taking the derivative of the power coefficient with respect to a and equating to zero. Therefore

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Hence, the maximum power coefficient,

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This maximum achievable power coefficient is named as the Betz limit (1926, Albert Betz, the German aerodynamicist). Till date, the wind turbine has not been designed that exceeded the Betz limit [MMR 2002].

Similar to power, the thrust coefficient, CT, is the ratio of thrust force to dynamic force and mathematically expressed as,

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A non-dimensional thrust coefficient is also represented as,

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The thrust coefficient is maximum (CT =1.0), when a = 0.5 and the downstream velocity is zero. At maximum power output (a = 1/3), CT has a value of 8/9. A graph of CP and CT for an ideal Betz turbine and non-dimensional downstream wind velocity is plotted in Fig. 2.

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Fig. 2: Variation of CP and CT with axial induction factor a [MMR 2002]

This graph shows, if the axial induction factor is greater than 0.5, this idealized model is not valid. In practice, the thrust coefficient can go as high as two, when the axial induction factor exceeds 1/2. In this case complicated flow patterns obtained and cannot be represented by this simple model.

The Betz limit, 0.593 is the maximum value of the theoretical achievable power coefficient. In practice, this value is further decreased because of wake rotation, finite blade number and related tip losses and aerodynamic drag.

The overall wind turbine efficiency is also dependent on various efficiencies such as mechanical efficiency, electrical efficiency of generator, transmission efficiency, and ineffective area in addition to rotor power coefficient. Hence, overall efficiency of wind turbine is given by,

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2. Ideal horizontal axis wind turbine with wake rotation

In practice, wind turbine consists of a rotor with a finite number of blades rotating at an angular velocity  about axis parallel to wind direction. In this case the downstream rotates in the opposite direction to the rotor as a result of reaction to the torque exerted by the flow on the wind turbine rotor. Fig. 3 illustrates the annular stream tube model with wake rotation.

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Fig. 3: Stream tube model with wake rotation [MMR 02]

In this case less energy is extracted by the rotor compared to energy extraction without wake rotation. This is the result of generation of rotational kinetic energy in the wake.

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