The Physics of Energy, page 123
Darrieus mechanical systems are at ground level, convenient for maintenance
constant chord and no twist in some VAWTs such as giromills make blade fabrication easier
rapid rotation rate allows high efficiency.
Disadvantages of VAWTs compared to HAWTs include:
lower rotor height (Darrieus) prevents VAWTs from reaching stronger winds at greater height
greater blade mass per unit swept area increases costs
cables are required to stabilize the rotor shaft against torques that would cause wobble
torque on blades varies periodically with rotation, causing periodic stresses that reduce efficiency and fatigue components
without pitch control on blades, stall control must be used to limit power in high winds
no torque at zero wind speed, so they are not self-starting
greater mass at largest distance from shaft requires more robust mechanical components
VAWTs experience more drag than HAWTs since the drag must be averaged over all angles of attack as the blade rotates.
At present HAWTs are favored especially at large scale, but research on VAWTs remains active and future developments may change the situation.
Wind turbine blades are critical dynamical components in wind power systems, and are as central to wind power as photodiodes are to solar PV power. The way that the blades are oriented relative to the direction of the airstream, together with a blade design optimized for a very large lift-to-drag ratio, enable the blades to capture power efficiently. In this section we study how lift and drag forces act on a rotating blade to harvest power from the wind. After establishing the basic mechanism that drives the blades forward, we introduce a simple model related to axial-momentum theory that gives insight into some of the main features of blade design, in particular the way that blades must change their shape and orientation as a function of distance from the hub about which they rotate.
30.2.1 Geometry
We assume that the turbine is oriented so that it faces into the wind, which blows in the -direction. As shown in Figure 30.8, the vertical defines the y-axis, and the z-axis is horizontal and perpendicular to the wind direction. The wind velocity at the turbine is , where v is the windspeed seen by a stationary observer away from the turbine and a is the axial induction factor (30.6). With this orientation, the airstream seen by an observer at rest on the blade is independent of the blade’s angle in the yz-plane, so for convenience we analyze a blade at the instant that it is vertical, pointing in the -direction. The blade is assumed to be rotating uniformly with angular velocity Ω. Thus, a point on the blade at radius r is moving in the -direction with velocity .
Figure 30.8 Coordinate system for analyzing a HAWT. The wind defines the x-axis, the vertical defines the y-axis. The turbine is shown at a point where one blade is vertical, the configuration depicted in Figure 30.9.
Example 30.2 Power Coefficient of a Drag Machine
The panemone mentioned at the outset of this chapter and shown in Figure 30.1 is one example of a windpower device powered by drag forces. Such machines rely on the drag force of wind bouncing off their surfacesto power their rotors. The analysis of Box 28.3 suggests that such devices will still be subject to a Betz-typelimit, although they are not as closely related to the actuator disk model as a horizontal-axis wind turbine.
To illustrate the analysis of a drag-powered machine, we consider the simple case of a flat rectangular plate of height h and width (area ) rotating at angular velocity Ω about an axis perpendicular to a wind stream with windspeed v– a model for the panemone with two vanes shown in the figure. Note that although the machine harvests wind power from an area , the power in an equal area is unavailable due to the shielding required for the wind to exert a net torque on the machine. We estimate the power at the instant that the plate is perpendicular to the wind stream, when the power is a maximum. Consider an element of area at a distance r from the axis. As we learned in §2.3.1, drag pressure can be parameterized by a drag coefficient , with , where is the speed of the fluid relative to the surface on which it acts. The relative velocity is , which we assume is positive for all , and the torque due to the drag is . The total power is obtained by multiplying the torque by Ω (see eq. (2.44) and integrating over r,
where is the power in the airstream, and is the tip-speed ratio . (Note that the factor of one half to account for the shielded area is included in .) r100pt
The power coefficient for this drag machine is maximum at and takes the value . To give an upper bound on , we take a drag coefficient of corresponding to air reflecting ballistically from a flat plate oriented perpendicular to an airstream. Thus the maximum power coefficient in the most favorable configuration of this machine is , which is 20% of the Betz limit. Furthermore, the power drops off as the plate’s orientation changes. A panemone can be made more efficient by removing the material close to the axis of rotation, since the drag pressure produces little torque in this region. Indeed, Figure 30.1 indicates that this effect was known to the designers of that machine.
Other designs of drag-based machines have higher efficiencies. The Savonius rotor shown in the figure is one example. Savonius machines are simple, inexpensive, and robust. The “trough-shaped” rotor not only has a high drag coefficient, but also generates lift when the rotor turns out of the wind, with the result that the power coefficient increases [179]. Figure 30.7 shows for a tip-speed ratio of order one for a Savonius rotor. (Figure credit: Adapted from The Renewable Energy Website (REUK))
A cross section through the blade at this instant is shown in Figure 30.9, along with other aspects of the geometry needed in the analysis below. An observer at rest on the blade sees an airstream with velocity w given by the vector sum of the wind and the headwind due to rotation,
(30.10)
Figure 30.9 Lift and drag forces on an element of a rotor blade at a distance r out from the hub. (a) Components of the airstream seen by an observer at rest on the blade. (b) Angles defining orientation of the blade with respect to the airstream and the blade’s direction of rotation. (c) Motion of the blade viewed from above in an Earth-fixed reference frame. The lift and drag force vectors are not drawn to scale.
Since the resulting airstream varies in strength and direction with r, it is necessary to consider the forces acting on each section of the blade as a function of r. It is conventional to characterize each section of the blade by the lift and drag coefficients of an infinite cylinder with the same cross section as the blade at that value of r. This set of approximations defines blade-element theory [177, 179]. To simplify the analysis still further, we ignore the fact that the turbine imparts angular momentum to the airstream, as in the axial-momentum analysis of §30.1.1. Without wake rotation, blade-element theory is consistent with axial-momentum analysis and yields the same total power (eqs. (30.7). A more thorough treatment including wake rotation refines but does not qualitatively change the results of this analysis [179].
The blade is free to move only in the -direction, but its chord line can be twisted with respect to this direction in order to better orient the airfoil with respect to the airstream. As shown in Figure 30.9, the angle of attack α is determined by the difference between the twist angle θ and the angle of the apparent wind direction ϕ, so . Trigonometry relates ϕ to the windspeed , angular velocity Ω, and radius r,
(30.11)
where is the tip-speed ratio introduced earlier in this chapter.
To maximize the lift, which powers the turbine, and to reduce drag, it is desirable to keep the angle of attack small, positive, and approximately constant along the length of the blade. Under fixed conditions (constant v and Ω), eq. (30.11) then requires that for α to stay constant, θ must increase as r decreases. In other words, the blade must twist into the wind as r decreases. The twist angle (as a function of r) is determined by the tip-speed ratio for which the blade design has been optimized. The optimal tip-speed ratio determines a relationship between the angular frequency of rotation and the wind speed. To maintain the ideal tip-speed ratio, as the wind velocity increases the rate of rotation of the blade must increase correspondingly. If this is not possible, then the angle of attack all along the blade grows, reducing the efficiency of the blade. We discuss this issue in more detail below.
30.2.2 Forces on a Blade Element
From Figure 30.9 it is clear that the lift force drives the blade forward, powering the turbine, while the drag force opposes the turbine’s motion. The key to an efficient blade design is therefore to maximize the lift and minimize the drag on the blade. Figures 29.20 and 29.21 show that these wind turbine airfoils have been designed to have large lift and a large lift-to-drag ratio at angles of attack that are small and positive. In practice, ratios of order 100:1 can be obtained in the vicinity of 6–8°. This enables us to make the additional approximation of ignoring compared to .
The lift force per unit length has components tangential to the direction of rotation, , and normal to the direction of rotation, . The normal force or thrust is opposed by the forces exerted by the structure that supports the turbine and is responsible for removing momentum from the axial flow of the wind. The tangential force creates the torque that keeps the turbine blades rotating, and the corresponding reaction on the wind produces wake rotation. We can compute the power harvested by the wind turbine, which is equal to the power delivered to the blades through torque, by using the thrust to determine the momentum removed from the wind.
The tangential and normal components of the force per unit length on a blade element (ignoring drag, as discussed above) can be obtained from the components of defined in eq. (29.34),
(30.12)
where we have used , K denotes the chord length of the turbine blade, and we have multiplied by the number of blades B to obtain the net force per unit radial length on the blade assembly. From axial-momentum analysis of the pressure difference across the turbine, we can compute the thrust per unit area, , on the turbine blades,
(30.13)
Applying this force on a radial section of area gives the axial-momentum approximation to the normal force per unit length along the blade,
(30.14)
Equating this result to from eq. (30.12), we obtain a constraint between the axial induction factor a and the design parameters K, B, , and ϕ,
Blade Design
Wind turbine blades must twist and thicken toward the rotor hub in order to maintain a fixed, small angle of attack and to generate sufficient thrust as the airstream direction seen by the blade changes with radius r. In a simple model which assumes axial-momentum flow and ignores drag forces, at each value of radius the twist θ and chord K are related to the axial induction factor a, the lift coefficient , and the angle of attack α by
(30.15)
where we have used eq. (30.11) to introduce the tip-speed ratio.
Consistency of axial-momentum theory requires that the tangential component of the force per unit length must generate the torque that powers the wind turbine. Indeed, combining the expression for in eq. (30.12) with the constraint (30.15), it can be shown that the power generated by the torque is the same as the power (30.7) harvested from the wind stream (Problem 30.7). Thus the energy harvested by the wind turbine can be viewed either as the kinetic energy removed from the axial flow of the wind or as the work done turning the rotor. Note that in this analysis we have essentially assumed that the turbine blades are dense enough in the area of the actuator disk that all air passing through the disk encounters a blade. This is discussed in further detail below.
30.2.3 Blade Design
Under the simplifying assumptions of axial-momentum flow and no drag forces, eqs. (30.11) and (30.15) relate the chord length K and twist angle θ as functions of r to the tip-speed ratio , the axial induction parameter a, and the angle of attack α,
(30.16)
(30.17)
The power coefficient (30.8) depends only on a in the axial-momentum analysis, and reaches the Betz limit at . An ideal design, therefore, would adjust θ and K as functions of r subject to the constraint that , while keeping α in the range where is large and is small. In practice, eqs. (30.16) and (30.17) must be solved iteratively since an optimal value of α cannot be chosen without knowing the lift and drag coefficients, which in turn depend on the chord length. A designer typically works within a given family of blade profiles (see, for example, Figure 30.11) with the aim of keeping a as close to 1/3 as possible.1 To illustrate the general result that both the twist and the chord length grow with decreasing radius, θ and are graphed in Figure 30.10 for , and , and . A sketch of a blade profile based on this model is shown in the insert in Figure 30.10. Fabricating light and durable blades with such complex profiles is one of the significant challenges of wind turbine design.
Figure 30.10 Variation of a wind turbine blade’s ideal chord length and twist with distance from the hub for the simple model described in the text. The analysis breaks down in the red shaded region. See the text for parameter values. The inset shows a blade profile based on this optimal design (after [197]). After [179].
Figure 30.11 Variation of a wind turbine blade profile as a function of distance from the hub, showing increasing chord length and twist closer to hub. From [194].
When the effects of wake rotation and drag are included, the relations among K, , r, α, and a become more complex but their qualitative features are preserved. An example of the variation of chord length and twist with distance from the hub for a realistic wind turbine blade is shown in Figure 30.11.
The choice of K and ϕ depends on the tip-speed ratio. Therefore if a turbine has been designed for optimum power coefficient at one tip-speed ratio, it will be suboptimal at both larger and smaller values. An example is shown in Figure 30.12. This turbine has been designed to function optimally at , where it reaches approximately 78% of the Betz limit. In addition to vanishing as vanishes as becomes large (v small) because the wind is not blowing strongly enough to activate the turbine (see Question 30.1).
Figure 30.12 Variation in the power coefficient with tip-speed ratio for a blade optimized for . (Credit: Prof. dr. Gerald J.W. van Bussel, TU Delft, NL)
30.2.4 Wind Turbine Power
A HAWT blade can maintain an optimal power coefficient if the tip-speed ratio can be held fixed at the design value. Since changes with the wind speed, can be held fixed only if Ω is adjusted as v changes. Wind turbines with variable Ω are known as variable-speed turbines. Historically, however, most wind turbines have operated at a fixed value of Ω, dictated by the frequency of the electric grid to which the turbine is connected (see §38). Such fixed-speed turbines only reach their design power coefficient at one value of wind speed. Modern developments in power electronics have made it possible to convert the power generated by variable-speed turbines to the desired line frequency, and deployment of this type of turbine has increased in recent years.
Wind Turbine Operation and Power
Wind turbines can be operated at fixed or variable speed and fixed or variable pitch. Fixed-speed, fixed-pitch turbines reach their design efficiency only at a single wind speed. Variable-speed turbines adjust their angular velocity to keep the tip-speed ratio near optimal as the wind speed changes. They shed power at high wind speeds by pitching their blades away from the wind.
The wind power potential of a given turbine at a given site is obtained by integrating the product of the turbine’s power coefficient and the site’s wind frequency distribution ,
The power output of a specific wind turbine design at a given location can be predicted from measurements of the power density in the wind and a measured power coefficient similar to that shown in Figure 30.12,
(30.18)
where the dependence of on the tip-speed ratio has been displayed explicitly. The power output of an ideal variable-speed turbine in a steady wind is shown in Figure 30.13(a). At speeds below the cut-in speed , typically 3–5 m/s, the torque is assumed to be insufficient to turn the blades and they are held fixed. Above is adjusted to give the optimal tip-speed ratio, and the power output grows proportional to . At the rated wind speed , typically 12–15 m/s, the power output reaches the power rating of the generator. At higher wind speeds, adjustments are made to prevent the power growing further (see below) and the power remains fixed at the rated value until the wind speed reaches the cut-out speed , typically around 25 m/s, above which braking mechanisms stop the turbine rather than risk damage to the rotor. Thus, in the interval , we have, effectively, . The actual power curve of the Vestas V90 3 MW turbine shown in Figure 30.13(b) is reasonably well approximated by the ideal case.
Figure 30.13 The power output and power coefficient for HAWTs: (a) ideal variable-speed HAWT, with constant power coefficient between and rated speed; (b) actual Vestas V90 3 MW variable-speed, variable-pitch wind turbine; (Credit: (b) Wind Power Program, www.wind-power-program.com) (c) actual Windera S fixed-speed, fixed-pitch wind turbine rated for 3.2 kW at 11 m/s. (Credit: (c) Ennera)
The power output of a fixed-speed turbine differs significantly from that of a variable-speed turbine. In particular, the rated power is achieved only at the design wind speed. At higher or lower speeds the falloff of shown in Figure 30.12 reduces the power significantly. The smallest HAWTs, which employ both fixed speed and fixed pitch to minimize cost and maximize simplicity of operation must sacrifice efficiency at wind speeds both below and above their rated speed.
