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International Journal of Aerospace Engineering
Volume 2011, Article ID 635750, 16 pages
http://dx.doi.org/10.1155/2011/635750
Research Article

Optimum Operational Parameters for Yawed Wind Turbines

Department of Mechanical Engineering and Materials Science, Washington University in St. Louis, St. Louis, MO 63130, USA

Received 8 March 2011; Accepted 7 April 2011

Academic Editor: Paul Williams

Copyright © 2011 David A. Peters and Xi Rong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A set of systematical optimum operational parameters for wind turbines under various wind directions is derived by using combined momentum-energy and blade-element-energy concepts. The derivations are solved numerically by fixing some parameters at practical values. Then, the interactions between the produced power and the influential factors of it are generated in the figures. It is shown that the maximum power produced is strongly affected by the wind direction, the tip speed, the pitch angle of the rotor, and the drag coefficient, which are specifically indicated by figures. It also turns out that the maximum power can take place at two different optimum tip speeds in some cases. The equations derived herein can also be used in the modeling of tethered wind turbines which can keep aloft and deliver energy.

1. Introduction

Wind turbines have been around for centuries, from the simple water pump in Arabia to the current widely used HAWT (horizontal axis wind turbine). With more and more public attention drawn to wind energy for its ease of development and environmental friendliness, wind farms that consist of hundreds of wind turbines at the same location are now being constructed rapidly all around the world [1]. Some of the wind farms are even built offshore by using large floating platforms due to the stronger and steadier wind at sea. However, the potential impact of wind farms on the biological, physical, and human environments is still under inspection, and the economics of offshore wind farms is still being demonstrated [2, 3]. Recently, much interest has arisen in new types of wind turbines like tethered wind turbines which can fly at high altitudes while delivering energy from wind [46]. Since the captured wind energy is proportional to the cube of the wind speed, a tethered wind turbine at high altitude may have 500 times as much available power as a wind turbine on the surface.

Although wind turbines vary in configuration, most of them incorporate the same mechanism of extracting wind energy by using rotor blades. To determine the optimum conditions when the maximum power is generated by the individual rotor, CFD (computational fluid dynamics) is a widely used method due to its predictive accuracy. Other methods like FVM (free vortex models) have also been developed. However, these approaches become computationally intensive for the study on wind farms that involve hundreds of individual wind turbines [7], and they are of limited use in the modeling of new types of wind turbines such as tethered wind turbines. Therefore, BEM (blade element momentum) is still widely used because of its simplicity and overall accuracy.

BEM theory was formulated by Glauert and gives both the force in the axial direction and the tangential direction [7]. It analyzes the rotor blade in sections and adds up the forces on all sections to get the resultant force of the rotor [8]. However, wind direction is usually not considered in the BEM, and the wind is generally assumed to be perpendicular to the rotor. Therefore, it is still unknown what the true available power and the optimum operational condition would be for wind turbines in yawed flows. This gap has created obstacles in the consideration of wind turbines in yawed flows.

The purpose of the work described herein is to use momentum and blade-element considerations to determine the optimum operating conditions for a yawed wind turbine and to determine the maximum power that the turbine can generate under various wind directions. The work culminates in the design figures which can be used to design wind turbines. The derivation described herein also paves the way for our further study on tethered wind turbines.

2. Physical System

Figure 1 shows the physical system that will be analyzed in this paper. It should be noted that the yawed wind turbine herein is considered to be a rotor, the axis of which is fixed to some support. The wind is assumed to be horizontal, and it impinges on the rotor at a wind direction angle (between the wind direction and the axis of rotor disk), denoted as 𝛾.

635750.fig.001
Figure 1: Physical system model. 𝑊: wind speed and 𝛾: wind direction angle which is between the wind and the rotor axis.

It is further assumed that the forces on the blades are determined solely by the lift and drag characteristics of the airfoil shape of the blades. Since momentum theory holds, the induced flow will be normal to the rotor disk. Swirl velocity is neglected in this study. No in-plane net loading is considered. Our desire is to determine the optimum operating conditions of the wind turbine under various wind directions in order to generate the most power possible. The results are shown in the figures at the end of the paper.

3. Case of a Horizontal Axis Wind Turbine

The first set of equations to be derived and solved are for the case of rotor with horizontal axis, 𝛾=0. This case provides a good baseline and insight for the more complicated yawed case that will be derived later.

By breaking a blade down into several small sections, we derive the forces on each of these small blade elements by analyzing the airfoil of the blade (Figure 2). These forces are then integrated along the entire blade. Thus, the resultant force on the rotor is obtained from the forces on all blades [8]. This is achieved by the following.

635750.fig.002
Figure 2: Airfoil for one blade element. 𝛼: angle of attack; 𝜙: angle between the relative wind and the plane of rotation; 𝜃: pitch angle; 𝐿: lift force normal to the wind direction; 𝐷: profile drag parallel to the wind direction; 𝑑𝐹1: thrust force on the blade element normal to the plane of rotation; 𝑑𝐹2: drag force on the blade element tangential to the circle swept by the rotor; 𝑊: relative wind speed; 𝑊𝑃: component of 𝑊 normal to the plane of rotation; 𝑊𝑇: component of 𝑊 tangential to the circle swept by the rotor; 𝑐: chord length; 𝑥: radial location of a blade element.

All the forces can be normalized on 𝜌𝜋𝑅2𝑊2/2:1𝐿=2𝜌𝜋𝑅2𝑊2𝐶𝐿,1𝐷=2𝜌𝜋𝑅2𝑊2𝐶𝐷,𝐶𝐿[],=𝑎sin(𝛼)(1) where 𝜌 is the air density, 𝑅 is the radius of the rotor, 𝑊 is the relative wind speed, 𝑎 is the slope of the lift curve, 𝛼 is the angle of attack between the chord line and the relative wind, 𝐷 is the profile drag parallel to the wind direction, 𝐶𝐷 is the drag coefficient, 𝐿 is the lift force normal to the wind, and 𝐶𝐿 is the lift coefficient.

Since we are interested only in the force normal to and tangential to the plane of rotation, the lift and drag forces are projected in these directions𝐹1=𝐿cos𝜙+𝐷sin𝜙,(2)𝐹2=𝐿sin𝜙𝐷cos𝜙,(3)sin(𝜃𝜙)=sin𝜃cos𝜙cos𝜃sin𝜙,(4)𝑊cos𝜙=𝑇𝑊2𝑃+𝑊2𝑇,(5)𝑊sin𝜙=𝑃𝑊2𝑃+𝑊2𝑇,(6) where 𝜃 is the pitch angle between the chord line and the plane of rotation, 𝜙 is the angle between the relative wind and the plane of rotation, 𝐹1 is the thrust of each blade normal to the plane of rotation, 𝐹2 is the drag of each blade tangential to the circle swept by the blade, 𝑊𝑃 is the component of the relative wind speed normal to the plane of rotation, and 𝑊𝑇 is the component of the relative wind speed tangential to the circle swept by the blade.

In this theoretical development, it is assumed that we are finding the optimum loading condition for a turbine in uniform air flow. For that reason, it is assumed that the airfoil sections (i.e., the chord and airfoils) have been chosen to give a reasonable life-to-drag ratio. Thus, this paper assumes that all airfoils are in the unstalled region such that the lift coefficient is a constant lift-curve slope multiplied by the angle of attack. Naturally, for a final wind turbine design, the optimum condition would have to be checked to make sure that the airfoils are not stalled and that the tip speeds are in a reasonable Mach range. From this and (1)–(6), it is possible to determine the thrust of each blade element normal to the plane of rotation, denoted as 𝑑𝐹1, and the drag of each blade element tangential to the circle swept by the blade, denoted as 𝑑𝐹2, as follows (in the appendix):𝑑𝐹1=12𝜌𝑎𝑐𝑑𝑥𝑊2𝑇𝜃+𝑊𝑝𝑊𝑇,𝑑𝐹2=12𝜌𝑎𝑐𝑑𝑥𝑊2𝑇𝐶𝐷𝑎𝑊𝑝𝑊𝑇𝜃+𝑊2𝑝,(7) where 𝑐 is the chord length and 𝑥 is the radial location of a blade element.

From the flow geometry,𝑊𝑃𝑊=𝑊𝑣,𝑇=Ω𝑥,(8) where 𝑣 is the induced flow and Ω is the angular velocity of the blade.

Thus, the thrust force on the entire rotor, denoted as 𝑇, can be obtained from integrated forces on all blades and the extracted power, denoted as 𝑃, can be obtained from the work done by the drag force:𝑇=𝑏𝑅0𝑑𝐹1,𝑃=𝑏𝑅0𝑑𝐹2Ω𝑥,(9) where b is the number of blades.

Substituting (7) and (8) into (9), we obtain1𝑇=𝜌𝑎𝑏𝑐6Ω2𝑅31𝜃+4Ω𝑅2(𝑊𝑣),(10)1𝑃=𝜌𝑎𝑏𝑐8Ω3𝑅4𝐶𝐷1𝑎16Ω2𝑅3+1(𝑊𝑣)𝜃4Ω𝑅2(𝑊𝑣)2.(11)

The above two equations are derived from blade-element theory. It is obvious that the thrust and the power are both relative to the blade number and the angular velocity of the rotor.

We can also obtain the thrust 𝑇 from momentum theory:𝑇=𝜌𝜋𝑅2(𝑊𝑣)2𝑣.(12)

Combining (10) and (12) and normalizing all velocities on 𝑊, we obtain:1𝑤=2+1116𝑎𝜎𝐽21𝑎642𝜎2𝐽21+141𝑎𝜎𝐽+3𝑎𝜃𝜎𝐽2,(13)𝜎=𝑏𝑐𝜋𝑅,(14) where 𝜎 is the rotor property parameter, 𝑤 is the induced flow coefficient, which is 𝑣/𝑊 and should be less than 0.5, 𝐽 is the normalized tip speed, which is Ω𝑅/𝑊.

For (11), we normalize power on 𝜌𝜋𝑅2𝑊3/2 and all velocities on 𝑊:𝐶𝑃=𝑎𝜎𝐶𝐷1𝐽4𝑎3𝜃3(1𝑤)𝐽2+12(1𝑤)2𝐽,(15) where 𝐶𝑃 is the normalized power coefficient extracted from wind.

Equations (13) and (15) are used to determine optimum operational conditions for the horizontal axis rotor. It is useful to firstly assign practical values to a and σ, both of which are only determined by the shape of blades:𝐶𝑃=0.1528𝜃(1𝑤)𝐽2+0.2292(1𝑤)2𝐽0.02𝐶𝐷𝐽3,(16)𝑤=0.5+0.0287𝐽0.50.0033𝐽2+0.1528𝜃𝐽20.1146𝐽+1,(17) where 𝑎=5.73 and 𝜎=0.08.

Equations (16) and (17) consist of five variables: 𝐶𝑃, 𝑤, 𝐽, 𝜃 and 𝐶𝐷. It is difficult to find the direct relationship among five variables with only two equations. To find the optimum operating conditions, we fix one of the two variables (𝜃 or 𝐽) at a certain value. By varying 𝐶𝐷 from 0.01 to 0.09 discretely, we use (16) and (17) to find the relationships among the three remaining variables. Thus, the discussion can be divided into two parts: 𝜃 fixed and 𝐽 fixed.

3.1. Numerical Analysis of 𝜃-Fixed Case

Figures 4(a), 4(b), 4(c), 5(a), 5(b), and 5(c) show how the produced power interacts with the tip speed and the induced flow when the pitch angle is fixed at practical values. For each value of 𝜃 and 𝐶𝐷, we use (17) to find each corresponding 𝐽 with 𝑤 varied from 0 to 0.5. Then, (16) is used to find each corresponding 𝐶𝑃. It is common in these figures that each curve has one peak which indicates optimum condition, except for the one case with 𝐶𝐷=0 in Figure 4(b), which has two clear peaks. That special curve will be specifically discussed later in our yawed wind turbine. As 𝜃 becomes relatively larger, the phenomenon of the curve twisting is seen in Figures 5(b) and 5(c), which indicate the possibility of one induced flow value corresponding to two different powers produced.

Figures 6, 7, and 8 show the optimum operating conditions when the greatest power is produced. Each of these plots is produced by starting with a value of 𝐶𝐷, stepping through values of 𝐽, and getting each corresponding 𝑤 and 𝐶𝑃. Then, it is possible to determine the maximum 𝐶𝑃, the corresponding optimum 𝐽 and 𝑤. It is obvious that the maximum power, the optimum tip speed, and the optimum induced flow decrease as the drag coefficient gets larger. This can be verified by comparing peak positions of each curve in Figures 4(a), 4(b), 4(c), 5(a), 5(b), and 5(c). Although the optimum induced flow coefficient is 1/3 for an ideal rotor in axial flows, the optimum values in Figure 8 are not always 1/3.

3.2. Numerical Analysis of 𝐽-Fixed Case

Figures 9(a), 9(b), 9(c), 9(d), 10(a), 10(b), 10(c), and 10(d) show how the power interacts with the pitch angle and the induced flow when the tip speed is fixed at practical values. The plotting method is the same numerical process as that for Figures 4(a), 4(b), 4(c), 5(a), 5(b), and 5(c). As 𝐽 increases from Figures 9(a) to 9(d), the curves in each figure move apart from each other and the peak of each curve gradually comes out and moves rightward. However, the movement of the peak is unclear in Figures 10(a), 10(b), 10(c), and 10(d). These optimum conditions can be seen in Figure 13. It should be noticed that the two curves with 𝐶𝐷 being 0.08 and 0.09 in Figure 10(d) totally lie below zero and the turbine is unable to produce power in these cases.

Figures 11, 12, and 13 show the optimum operating conditions when the greatest power is produced. The plotting method is the same as that of Figures 6, 7, and 8. It is clear from Figure 11 that the maximum power decreases linearly as the drag coefficient gets larger with the tip speed drastically affecting the velocity of decreasing. In Figures 12 and 13, however, the drag coefficient does not affect the optimum pitch angle and the induced flow, both of which are only determined by the tip speed.

4. Case of a Yawed Wind Turbine

When the wind direction angle 𝛾 is an arbitrary value (Figure 3), we can divide the initial wind speed into components normal and parallel to the rotor axis and obtain the thrust and the power based on momentum and blade-element derivations. We generalize the momentum equations in the following:𝑊1=𝑊cos𝛾,(18)𝑊2=𝑊sin𝛾,(19)̇𝑚=𝜌𝜋𝑅2𝑊1𝑣2+𝑊22,(20)𝑇=2̇𝑚𝑣,(21) where 𝑊 is the initial wind speed, 𝑊1 is the component of 𝑊 normal to plane of rotation, 𝑊2 is the component of 𝑊 parallel to the plane of rotation, 𝑣 is the induced flow, ̇𝑚 is the mass flow rate at the rotor, and 𝑇 is the thrust on the rotor.

635750.fig.003
Figure 3: Physical system model in case of yawed wind turbine. 𝑊: initial wind speed; 𝑊1: component of 𝑊 normal to the plane of rotation; 𝑊2: component of 𝑊 parallel to the plane of rotation; 𝑣: induced flow; 𝛾: wind direction angle between the wind and the rotor axis; 𝜓𝑖: angle between 𝑊2 and the 𝑖th blade; Ω: angular velocity of rotor; 𝑑𝐹𝑖1: thrust on the element of the 𝑖th blade, which is normal to the plane of rotation; 𝑑𝐹𝑖2: drag on the element of the 𝑖th blade, which is tangential to the circle swept by the 𝑖th blade.
fig4
Figure 4: (a) 𝐶𝑃 versus 𝐽 for 𝜃=0, (b) 𝐶𝑃 versus 𝐽 for 𝜃=2, and (c) 𝐶𝑃 versus 𝐽 for 𝜃=3.
fig5
Figure 5: (a) 𝐶𝑃 versus 𝑤 for 𝜃=0, (b) 𝐶𝑃 versus 𝑤 for 𝜃=2, and (c) 𝐶𝑃 versus 𝑤 for 𝜃=3.
635750.fig.006
Figure 6: Max 𝐶𝑃 versus 𝐶𝐷.
635750.fig.007
Figure 7: Optimum 𝐽 versus 𝐶𝐷.
635750.fig.008
Figure 8: Optimum 𝑤 versus 𝐶𝐷.
fig9
Figure 9: (a) 𝐶𝑃 versus 𝜃 for 𝐽=1, (b) 𝐶𝑃 versus 𝜃 for 𝐽=3(c) 𝐶𝑃 versus 𝜃 for 𝐽=5, and (d) 𝐶𝑃 versus 𝜃 for 𝐽=7.
fig10
Figure 10: (a) 𝐶𝑃 versus 𝑤 for 𝐽=1, (b)𝐶𝑃 versus 𝑤 for 𝐽=3, (c) 𝐶𝑃 versus 𝑤 for 𝐽=5, and (d) 𝐶𝑃 versus 𝑤 for 𝐽=7.
635750.fig.0011
Figure 11: Max 𝐶𝑃 versus 𝐶𝐷.
635750.fig.0012
Figure 12: Optimum 𝜃 versus 𝐶𝐷.
635750.fig.0013
Figure 13: Optimum 𝑤 versus 𝐶𝐷.

Substituting (18)–(20) into (21), we obtain𝑇=2𝜌𝜋𝑅2𝑣𝑊2+𝑣22𝑣𝑊cos𝛾.(22)

We then use the blade-element derivations to obtain the thrust and the drag on the element of the 𝑖th blade:𝑊𝑖𝑇=Ω𝑥+𝑊2sin𝜓𝑖,𝑊(23)𝑖𝑃=𝑊1𝑣,(24)𝑑𝐹𝑖1=12𝑊𝜌𝑎𝑐𝑑𝑥𝑖𝑇2𝜃+𝑊𝑖𝑃𝑊𝑖𝑇,(25)𝑑𝐹𝑖2=12𝑊𝜌𝑎𝑐𝑑𝑥𝑖𝑇2𝐶𝐷𝑎𝑊𝑖𝑃𝑊𝑖𝑇𝑊𝜃+𝑖𝑃2,(26) where 𝜓𝑖 is the angle between 𝑊2 and the 𝑖th blade, 𝑊𝑖𝑇 is the component of the relative wind speed which is tangential to the circle swept by the 𝑖th blade, 𝑊𝑖𝑃 is the component of the relative wind speed which is normal to the plane of rotation, 𝑑𝐹𝑖1 is the thrust on the element of the 𝑖th blade which is normal to the plane of rotation, and 𝑑𝐹𝑖2 is the drag on the element of the 𝑖th blade which is tangential to the circle swept by the 𝑖th blade.

Substituting (23) and (24) into (25) and (26), we find𝑑𝐹𝑖1=12×Ω𝜌𝑎𝑐𝑑𝑥𝜃2𝑥2+𝑊2sin2𝛾sin2𝜓𝑖+2Ω𝑥𝑊sin𝛾sin𝜓𝑖+Ω𝑥𝑊cos𝛾Ω𝑥𝜈+𝑊2sin𝛾cos𝛾sin𝜓𝑖𝜈𝑊sin𝛾sin𝜓𝑖,𝑑𝐹𝑖2=12×𝐶𝜌𝑎𝑐𝑑𝑥𝐷𝑎Ω2𝑥2+𝑊2sin2𝛾sin2𝜓𝑖+2Ω𝑥𝑊sin𝛾sin𝜓𝑖𝜃Ω𝑥𝑊cos𝛾Ω𝑥𝜈+𝑊2sin𝛾cos𝛾sin𝜓𝑖𝜈𝑊sin𝛾sin𝜓𝑖+𝑊2cos2𝛾+𝜈2.2𝜈𝑊cos𝛾(28)

Again, the thrust force on the entire rotor T can be obtained from integrated forces on all blades, and the extracted power P can be obtained from the work done by the drag forces:𝑇=𝑏𝑖=1𝑅0𝑑𝐹𝑖1,𝑃=𝑏𝑖=1𝑅0𝑑𝐹𝑖2Ω𝑥,(29) where 𝑏 is the number of blades.

Substituting (28) into (29), we obtain1𝑇=21𝜌𝑎𝑐𝜃3𝑏Ω2𝑅3+𝑊2𝑅sin2𝛾𝑏𝑖=1sin2𝜓𝑖+Ω𝑅2𝑊sin𝛾𝑏𝑖=1sin𝜓𝑖+12𝑏Ω𝑅21𝑊cos𝛾2𝑏Ω𝑅2𝜈+𝑊2𝑅sin𝛾cos𝛾𝑏𝑖=1sin𝜓𝑖𝜈𝑊𝑅sin𝛾𝑏𝑖=1sin𝜓𝑖,1𝑃=2𝐶𝜌𝑎𝑐𝐷𝑎14𝑏Ω3𝑅4+12Ω𝑅2𝑊2sin2𝛾𝑏𝑖=1sin2𝜓𝑖+23Ω2𝑅3𝑊sin𝛾𝑏𝑖=1sin𝜓𝑖1𝜃3𝑏Ω2𝑅31𝑊cos𝛾3𝑏Ω2𝑅31𝜈+2Ω𝑅2𝑊2sin𝛾cos𝛾𝑏𝑖=1sin𝜓𝑖12Ω𝑅2𝜈𝑊sin𝛾𝑏𝑖=1sin𝜓𝑖+12𝑏Ω𝑅2𝑊2cos21𝛾+2𝑏Ω𝑅2𝜈2𝑏Ω𝑅2.𝜈𝑊cos𝛾(30)

To simplify the equation, we consider the most widely used rotor which has three blades: 𝑏=3,3𝑖=1sin𝜓𝑖=0,3𝑖=1sin2𝜓𝑖=32.(31)

Thus,1𝑇=2Ω𝜌𝑎𝑐𝜃2𝑅3+32𝑊2𝑅sin2𝛾+32Ω𝑅23𝑊cos𝛾2Ω𝑅2𝜈,1(32)𝑃=2×𝐶𝜌𝑎𝑐𝐷𝑎34Ω3𝑅4+34Ω𝑅2𝑊2sin2𝛾Ω𝜃2𝑅3𝑊cos𝛾Ω2𝑅3𝜈+32Ω𝑅2𝑊2cos23𝛾+2Ω𝑅2𝜈23Ω𝑅2.𝜈𝑊cos𝛾(33)

Then, we combine (22) and (32) and normalize all power on 𝜌𝜋𝑅2𝑊3/2, and all velocities on 𝑊:4𝑤𝑎𝜎1+𝑤212𝑤cos𝛾+21𝐽𝑤+𝜃3𝐽2+12sin2𝛾12𝐽cos𝛾=0,𝐶𝑃𝐶=𝑎𝜎𝐷𝐽4𝑎3+𝐽sin2𝛾𝜃3(cos𝛾𝑤)𝐽2+12(cos𝛾𝑤)2𝐽,(34) where 𝑤 is the induced flow coefficient, 𝐽 is the normalized tip speed, and 𝜎 is the rotor property parameter, which is 𝑏𝑐/(𝜋𝑅).

Again, we assign a and σ the practical values:𝑤1+𝑤212𝑤cos𝛾+0.0573𝐽𝑤+0.1146𝜃3𝐽2+12sin2𝛾0.0573𝐽cos𝛾=0,𝐶𝑃=0.1528𝜃(cos𝛾𝑤)𝐽2+0.2292(cos𝛾𝑤)2𝐽0.02𝐶𝐷𝐽3+𝐽sin2𝛾,(35) where 𝑎=5.73 and 𝜎=0.08.

Figures 14(a), 14(b), 15(a), 15(b), 16(a), 16(b), 17(a), 17(b), 18(a), 18(b), 19(a), and 19(b) show how the produced power interacts with the tip speed and the induced flow when the pitch angle and the wind direction angle are fixed at practical values. All plots are produced by starting with a value of 𝐶𝐷, stepping through values of 𝐽, and getting each corresponding 𝑤 and 𝐶𝑃 based on (35). Most figures in the yawed rotor are similar to the figures in the horizontal axis rotor: as the tip speed or the induced flow gets larger, the power increases to a global maximum followed by monotonous decreases; the phenomenon of twisting comes out in 𝐶𝑃 versus 𝑤 at the relatively large pitch angle and wind direction angle.

fig14
Figure 14: (a) 𝐶𝑃 versus 𝐽 for 𝜃=0 and 𝛾=15 and (b) 𝐶𝑃 versus 𝐽 for 𝜃=0 and 𝛾=60.
fig15
Figure 15: (a) 𝐶𝑃 versus 𝐽 for 𝜃=2 and 𝛾=15 and (b) 𝐶𝑃 versus 𝐽 for 𝜃=2 and 𝛾=60.
fig16
Figure 16: (a) 𝐶𝑃 versus 𝐽 for 𝜃=3 and 𝛾=15 and (b) 𝐶𝑃 versus 𝐽 for 𝜃=3 and 𝛾=60.
fig17
Figure 17: (a) 𝐶𝑃 versus 𝑤 for 𝜃=0 and 𝛾=15 and (b) 𝐶𝑃 versus 𝑤 for 𝜃=0 and 𝛾=60.
fig18
Figure 18: (a) 𝐶𝑃 versus 𝑤 for 𝜃=2 and 𝛾=15 and (b)𝐶𝑃 versus 𝑤 for 𝜃=2 and 𝛾=60.
fig19
Figure 19: (a) 𝐶𝑃 versus 𝑤 for 𝜃=3 and 𝛾=15 and (b) 𝐶𝑃 versus 𝑤 for 𝜃=3 and 𝛾=60.

However, the curve with 𝐶𝐷=0 in Figure 15(a) has two global maxima, which is the same as those in Figure 4(b). For 𝜃 being 2° and 𝛾 being 0°, the maximum 𝐶𝑃 is 0.59 when 𝐽 is 8.13 and 20.52; for 𝜃 being 2° and 𝛾 being 15°, the maximum 𝐶𝑃 is 0.58 when 𝐽 is 10.43 and 16.54. It is somewhat surprising to get two optimum values of 𝐽 for only one maximum 𝐶𝑃. To study this further, we also fixed 𝜃 at values around 2° and computed the maximum 𝐶𝑃 and its corresponding 𝐽 when 𝐶𝐷 is 0. The numerical results are shown in Table 1.

tab1
Table 1: Optimum condition at 𝐶𝐷 = 0.

It can be concluded from the table that, at some pitch angle value around 1.8°, 1.9°, 2.0°, and 2.1°, two optimum tip speeds may coexist provided that the drag coefficient equals 0 and the wind direction angle is small.

Figures 20(a), 20(b), 20(c), 21(a), 21(b), 21(c), 22(a), 22(b), and 22(c) show the optimum operating conditions when the greatest power is produced. All plots are generated by starting at a value of 𝛾, stepping through values of 𝐽, and getting each corresponding 𝑤 and 𝐶𝑃. Then, it is possible to determine the maximum 𝐶𝑃, the corresponding optimum 𝐽 and 𝑤. It is clear that the maximum power goes down monotonously as the wind direction angle gets larger. For the optimum tip speed and the optimum induced flow, most of them also decrease monotonously as the wind direction angle increases except for the ones with 𝐶𝐷=0. It should be noted that the 𝐶𝐷=0 cases are numerically sensitive and so behave differently from the 𝐶𝐷0 cases. As a verification of the coexisting optimum tip speeds mentioned above, the optimum 𝐽 jumps upside and down when 𝐶𝐷=0 in Figure 21(b).

fig20
Figure 20: (a) Max 𝐶𝑃 versus 𝛾 for 𝜃=0, (b) Max 𝐶𝑃 versus 𝛾 for 𝜃=2, and (c) Max 𝐶𝑃 versus 𝛾 for 𝜃=3.
fig21
Figure 21: (a) Optimum 𝐽 versus 𝛾 for 𝜃=0, (b) Optimum 𝐽 versus 𝛾 for 𝜃=2, and (c) Optimum 𝐽 versus 𝛾 for 𝜃=3.
fig22
Figure 22: (a) Optimum 𝑤 versus 𝛾 for 𝜃=0, (b) Optimum 𝑤 versus 𝛾 for 𝜃=2, and (c) Optimum 𝑤 versus 𝛾 for 𝜃=3.

5. Conclusion

Momentum-energy and blade-element-energy equations have been derived to find optimum conditions for both yawed wind turbines and new types of wind turbines using rotor blades. They are solved numerically by fixing some parameters at practical values. Then, the interactions between the produced power and the influential factors of it are generated in the figures.

It is a general principle that, under various wind direction angles, the maximum possible power produced is strongly affected by the tip speed, the pitch angle of the rotor, and the drag coefficient from the wind, which are specifically indicated by figures. For each of optimum operating conditions, most maximum powers correspond to unique optimum tip speed except for some special cases in which two optimum tip speeds coexist. As the wind direction angle increases to right angle, the maximum power decreases monotonously to zero.

Appendix

Derivation for the Differential Thrust and Drag

Substituting (1), (4), (5), and (6) into differential forms of (2) and (3), we obtain𝑑𝐹1=12𝑎𝜌𝑐𝑑𝑥cos𝜃𝑊𝑝𝑊𝑇sin𝜃𝑊2𝑇+12𝑊𝜌𝑐𝑑𝑥𝑃𝑊2𝑇+𝑊2𝑃𝐶𝐷,𝑑𝐹2=12𝑎𝜌𝑐𝑑𝑥cos𝜃𝑊2𝑃sin𝜃𝑊𝑃𝑊𝑇12𝑊𝜌𝑐𝑑𝑥𝑇𝑊2𝑇+𝑊2𝑃𝐶𝐷.(A.1)

Since 𝑊𝑃 is far less than 𝑊𝑇, the pitch angle 𝜃 is very little, and the drag coefficient 𝐶𝐷 is far less than the lift curve slope 𝑎 in practical situation, we find approximations𝑊2𝑇+𝑊2𝑃=𝑊𝑇,𝐶sin𝜃=𝜃,cos𝜃=1,𝐷𝑎=0.(A.2)

Thus,𝑑𝐹1=12𝜌𝑎𝑐𝑑𝑥𝑊2𝑇𝜃+𝑊𝑝𝑊𝑇,𝑑𝐹2=12𝜌𝑎𝑐𝑑𝑥𝑊2𝑇𝐶𝐷𝑎𝑊𝑝𝑊𝑇𝜃+𝑊2𝑝.(A.3)

Nomenclature

𝑎:Slope of lift curve
𝑏:Number of blades
𝑐:Chord length (m)
𝐶𝐷:Drag coefficient (𝐷/(𝜌𝜋𝑅2𝑊2/2))
𝐶𝐿:Lift coefficient (𝐿/(𝜌𝜋𝑅2𝑊2/2))
𝐶𝑃:Normalized power coefficient (𝑃/(𝜌𝜋𝑅2𝑊3/2))
𝑑𝐹1:Thrust on blade element, which is normal to the plane of rotation (N)
𝑑𝐹2:Drag on blade element, which is tangential to the circle swept by rotor (N)
𝑑𝐹𝑖1:Thrust on element of the 𝑖th blade, which is normal to the plane of rotation (N)
𝑑𝐹𝑖2:Drag on element of the 𝑖th blade, which is tangential to the circle swept by the 𝑖th blade (N)
𝐷:Profile drag parallel to the wind direction (N)
𝐹1:Thrust on blade, which is normal to the plane of rotation (N)
𝐹2:Drag on blade, which is tangential to the circle swept by the rotor (N)
𝐽:Normalized tip speed (Ω𝑅/𝑊)
𝐿:Lift force on the rotor (N)
̇𝑚:Mass flow rate (kg/s)
𝑃:Power extracted from wind (Watts)
𝑅:Radius of blade (m)
𝑇:Thrust on the rotor (N)
𝑣:Induced flow (m/s)
𝑤:Induced flow coefficient (v/W)
𝑊:Wind speed (m/s)
𝑊1:Component of the initial wind speed which is normal to the plane of rotation (m/s)
𝑊2:Component of the initial wind speed which is parallel to the plane of rotation (m/s)
𝑊𝑃:Component of the relative wind speed which is normal to the plane of rotation (m/s)
𝑊𝑇:component of the relative wind speed which is tangential to the circle swept by the rotor (m/s)
𝑊𝑖𝑃:Component of the relative wind speed which is normal to the plane of rotation (m/s)
𝑊𝑖𝑇:Component of the relative wind speed which is tangential to the circle swept by the 𝑖th blade (m/s)
x:Radial location of a blade element (m)
𝜌:Density of air (kg/m3)
𝛼:Angle of attack (rad)
𝛾:Wind direction angle, which is between the wind direction and the rotor axis (rad)
𝜃:Pitch angle (rad)
𝜙:Angle between the relative wind and the plane of rotation (rad)
Ω:Blade angular velocity (rad/s)
𝜓𝑖:Angle between 𝑊2 and the 𝑖th blade (rad)
𝜎:Rotor property parameter (𝑏𝑐/(𝜋𝑅)).

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