The Physics of Energy, page 117
Because extraction of wind power at one location reduces the energy in the wind for a substantial distance downstream, wind power does not scale with area as solar power does.
On larger scales, wind can in principle only recover from the shadowing effect of turbines a finite number of times, since the height of the mass of moving air is finite. Thus, in a sense wind may best be thought of as a high-density linear resource, which cannot scale with area over arbitrary regions. This consideration is relevant when estimating the limit to global wind power potential, which we consider next.
28.3.2 Global Wind Power Potential
As mentioned at the beginning of this chapter, roughly 1 PW of wind power is continuously dissipated in the atmosphere. The uncertainty in this number – a factor of two or three – is small compared with the range of current estimates of the wind power available for human use, which vary over roughly two orders of magnitude, from ~1 TW [176] to ~100 TW [189]. Since this subject remains in flux and is controversial, it is not appropriate to review it in depth here. Instead we outline and comment upon the methods of analysis that lead to such divergent estimates.
Estimates of the global wind power potential fall into two categories: top-down approaches that study energy flow in the atmosphere and try to estimate the kinetic energy dissipated in regions of the atmosphere available to human exploitation; and bottom-up approaches that survey wind resources at a local and regional level, make assumptions about accessibility, and then aggregate the harvestable power to obtain a global estimate. Different groups using these two approaches also include different constraints in their estimations. For example, some include the Betz limit, and others factor in the efficiency of existing wind turbines; some exclude the large geographical areas where the wind power resource is below some particular threshold, others consider the entire non-glaciated portion of Earth’s land surface; some include offshore resources, others include only land-based wind power, etc.
A recent top-down analysis by Miller et al. gives a total available resource of 18–68 TW [190]. The higher value can be obtained from what they call a “back of the envelope” calculation: they estimate 900 TW wind power dissipation in the atmosphere, of which about 1/2 is dissipated in the free atmosphere where it is not available. Of the remaining 450 TW, roughly 3/4 is dissipated over ice or deep ocean, leaving 112 TW. Miller et al. then use the Betz limit to further reduce this estimate by a factor of 16/27 to 68 TW; since, however, the Betz limit applies only to single turbines over a given cross-sectional area, and does not imply that the remaining 11/27 of the wind energy in that area is irrevocably lost to downstream turbines, such a reduction may not be warranted. A more detailed analysis of the balance between momentum extraction and dissipation in the lower atmosphere reduces the estimate of Miller et al. – also including the reduction due to the Betz limit – to 21 TW; integration of their methods into a general climate circulation model gives 18–34 TW. Note that this estimate includes all regions where wind energy is dissipated over or near land outside the polar regions. Other workers have combined estimates of the wind power dissipated in the lower atmosphere with more detailed assumptions about viable regions for realistic wind power usage, organization of wind farms, and efficiency of wind turbines, and have obtained even lower estimates of the total available wind resource. de Castro et al., for example, obtain an estimate of global wind-generated electrical power of 1 TW [176].
An example of a bottom-up analysis was performed by Archer and Jacobson in 2005 [189]. They estimated the global onshore wind power resource by combining observations from all seven continents. They took data from a wealth of measurements at 10 m above ground level and extrapolated to 80 m, which is the hub height for a typical modern 1.5 MW wind turbine. One of their wind atlases is shown in Figure 28.24. They selected those sites with mean annual wind speeds at 80 m exceeding 6.9 m/s (corresponding to wind power class 3 and above) and integrated over the regions where these favorable winds were observed. They assumed that favorable areas could be covered with turbines at a density of and estimated the actual power output from the turbines under local conditions. Archer and Jacobson did not exclude inaccessible areas such as Antarctica, but such locations do not dominate their survey. They estimated that if fully exploited, wind power at the onshore locations they identify could provide approximately 72 TW of mechanical power.
Figure 28.24 Measurements of mean annual wind power classes at 80 m for sites in Australia and New Zealand and Tasmania [189].
Global Wind Power Potential
There is no scientific consensus yet on a precise estimate for global wind power potential. Assessments range from 1 TW up to nearly 100 TW. Top-down approaches, which follow the energy budget of the atmosphere and attempt to compute the wind power dissipated in parts of the atmosphere accessible to human technology, are on the low side, and including all relevant constraints seems to push the bound down to a small number of terawatts. Bottom-up approaches, which identify areas with high-quality resources and cover those areas with a high density of wind turbines without considering global constraints give higher estimates. Future research should clarify the matter; at the rate that deployment is currently increasing, wind power production may approach or pass the 1 TW benchmark by mid-century.
One criticism of the bottom-up approach is that it neglects the constraints associated with the limited total energy in any wind system. As we have emphasized, wind is not a two-dimensional resource like solar energy that can be exploited locally in any region on Earth’s surface without affecting nearby regions. Consider for example a prevailing westerly wind sweeping across an area of plains measuring 300 km by 300 km at 8 m/s. The power density in this wind is W/m2. The total power available, however, is not given by multiplying this number by the area of the region ( TW). Rather, since the wind is flowing from west to east, the total power available may be bounded by multiplying the north–south extent of the wind field (300 km) by the height of the wind field, which may be generously taken to be on the order of the height of the troposphere, say 10 km. This reduces the total power available by a factor of 30, to about 1 TW. Furthermore, removal of wind energy from the flow on this scale could affect the wind energy available for many hundreds of kilometers downstream of this area. Note, for example, as shown in Figure 28.3, that the wind speed in the Southern Hemisphere 40–50° latitude belt gradually increases over thousands of kilometers east of the southern tip of South America. Similar effects are seen in the recovery of the wind fields on the lee sides of other continents.
A more definitive estimate of world wind power potential awaits further research. Ongoing work aimed at understanding the wake of a wind turbine (which can have measurable impact extending over many kilometers) and the total energy lost to the wind field through the combined effect of the wind turbine and turbulence in the wake, combined with a more detailed understanding of the total energy flow and availability in specific wind patterns and wind locations, may make a more precise estimate of wind power limitations possible in the coming years. In the meantime we quote Emeis’s conclusion [180] that “Probably a single-digit number given in terawatts is a realistic estimate for the wind energy available from ... Earth’s atmosphere.” Recent projections of world wind power capacity in 2050 by the IEA [191] range from 2300–2700 GW. At an anticipated 31% capacity factor, this corresponds to an average of 0.70–0.83 TW of electric power generated from the wind. Thus the lowest estimates of Earth’s wind power resource may be tested in practice before the end of this century.
Discussion/Investigation Questions
28.1 What is the most promising site for wind power within a 100 kilometer radius of your location? Can you identify the meteorological origin of its favorable position? If you can find data, make an estimate of its potential.
28.2 The Bahrain World Trade Center is a prominent example of building-integrated wind power. What are the parameters of this installation? What are its pros and cons? What is your opinion of building-integrated wind power in general?
28.3 Investigate the history of the Cape Wind project in the state of Massachusetts and discuss the arguments made in favor of and against this project.
Problems
28.1 [T] Assume that the pressure in a volume of air can be expressed to first order in a local coordinate system as . Show from the balance of forces on a small volume element dV that the force per unit volume is f in the -direction. Using the fact that the coordinate system can be chosen arbitrarily, show that more generally .
28.2 [T,H] The forces on a parcel of moving air in the Ekman layer include the pressure force , the Coriolis force (where ), and a frictional force directed opposite to the wind velocity , where arises from the velocity gradient and turbulent viscosity. The coefficient function decreases with increasing height z through the Ekman layer and vanishes at the top of the layer. Find the wind velocity by demanding that the sum of the forces on the parcel vanishes. Express υ in terms of components along and . Show that your result reproduces qualitatively the wind pattern (both in direction and magnitude) shown in Figure 28.10.
28.3 Evaluate the Coriolis factor at latitude. How large a pressure gradient (in millibars/km) is necessary to sustain a geostrophic wind at 30 m/s at this latitude?
28.4 Suppose a gradient wind (see Box 28.7) is flowing along a circular isobar with radius R at a latitude of 45°. Compare the strength of the Coriolis and centrifugal forces on this flow as a function of wind speed. At what speed (as a function of R) are the two inertial forces equal?
28.5 [T] The wind speed at any site can be written as the sum of its average plus a fluctuating term that averages to zero, . Show that the effect of the fluctuations is always to make greater than .
28.6 Compute the energy density and power density in a steady wind at 8 m/s and at 16 m/s.
28.7 What is the maximum power that could be extracted by a mechanical device of cross-sectional area 10 m in a steady wind at 10 m/s?
28.8 [T] Suppose the wind speed obeys a Gaussian frequency distribution in the two components of the horizontal velocity ,
(28.15)
Show that the wind speed distribution, , where , is a Rayleigh distribution (Weibull parameter ). What is the scale ?
28.9 [T] Show that for integer n.
28.10 [T] Derive the formula for in eq. (28.12).
28.11 A steady wind is blowing at 8 m/s at 200 m above ground level. Estimate the wind power at a height of 10 m above a large open cropped field. Approximately what height is needed for a turbine in a small town to reach wind flowing with the same power?
28.12 Consider building a wind turbine at a wind power class 5 site. Assume that the hub height is 50 m and the blade length is 15 m, and that the turbine operates at 50% of the maximum theoretical efficiency. Estimate the average power output of the turbine.
28.13 Reference [192] studies wind power prospects for several locations in Saudi Arabia, among them Yenbu (Y) and Al Qaysumah (A). They report Weibull parameters measured at a height of 10 m: m/s, ; m/s, . Estimate the maximum available wind power density and the wind power class for each site at a height of 80 m assuming a roughness length (sand). Can you explain (on the basis of geography) why the winds at Yenbu are so much more variable than at Al Qaysumah?
28.14 Use the data given in the wind rose Figure 28.19 to make a crude estimate of the average and root mean cube wind speed at Klamath Lake, Oregon. From these, estimate the Weibull parameters for this distribution. Is your result close to a Rayleigh distribution ()? Is this a promising wind power location?
28.15 The following wind frequency distribution data were obtained in the month of January at a height of 10 m above an airport runway in Buffalo, New York. Calm: 2.0%; 0–5 kt: 5.3%; 5–12 kt: 40.1%; 12–20 kt: 39.3%; 20 kt: 13.3% (1 knot 0.514 m/s). Assume that the wind above 20 kt has an average value of 25 kt. Make a crude estimate of the power in the wind at 10 m. Is this a promising location for wind power based on the January data alone?
* * *
1 Jet streams originate in the vertical as well as horizontal temperature and pressure differences at the boundaries between primary atmospheric circulation cells, but an explanation of their direction and intensity is beyond the level of this book.
2 Although it has little theoretical justification, another model for the height dependence of the wind that is often used, especially when surface roughness information is not available, assumes that the wind speed varies as the 1/7th power of the height, .
CHAPTER 29
Fluids: The Basics
Wind and flowing water are moving fluids that carry kinetic energy in the motion of their constituent molecules. Wind turbines capture this energy for human use. Other systems attempt to capture the kinetic energy in ocean waves or marine currents. Up to this point, it has not been necessary to go beyond a colloquial description of fluids, but in order to explain how some of these systems work, we must describe some basic properties of fluids in more detail. Although the behavior of fluids can become very complicated– indeed, the analysis of turbulent flow in many situations requires tremendous computational resources – the basics are relatively simple and give considerable insight into how it is possible to harvest energy efficiently from wind and moving water.
Two key attributes of fluids have been introduced in earlier chapters. In the previous chapter we found that the power density in a fluid flow is , and is a good measure of the quality of a wind power resource. The notion that fluids exert a drag force on objects that move relative to the fluid was introduced in §2, where we studied air resistance,which consumes much of the energy expended by automobiles. One principal aim of this chapter is to introduceand explain the physical origins of lift, another force that a flowing fluid can exert on an object. Lift plays an essentialrole in the physical mechanism that powers wind turbines and therefore it merits careful consideration. One otherforce that even a static fluid can exert on an object is the buoyant force, mentioned in §5 and §6.
We begin (§29.1) by defining a fluid and describing how a flowing fluid can be characterized in terms of local variables such as velocity, energy density, mass and energy flux, in addition to the familiar thermodynamic variables of pressure, density, and temperature. In the following section (§29.2) we explore the implications of conservation of mass and energy in fluid flow. We specialize to steady flows of fluids, which are simpler than general fluid flow, yet general enough to describe most situations ofinterest to us. Putting aside frictional losses for the moment, we use conservation of energy to derive Bernoulli’s equation, an important basic result of fluid dynamics that relates the pressure and flow velocity in a moving fluid. This section ends with adiscussion of fluid circulation and vorticity. The next section (§29.3) is devoted to friction in fluids, which is parameterized by viscosity. After defining viscosity and explaining how it is measured, we introduce Reynolds number, a dimensionless parameterthat determines the relative importance of viscous and inertial forces on objects immersed in fluid flows. With these ingredients inhand, we turn to the discussion of lift in §29.4. We explain the connection between lift and circulation embodied in the Kutta–Zhukovskiĭ theorem, and apply it to airfoils, where viscous forces establish a steady fluid flow in whichviscous losses are small. Finally, we discuss the importance of vorticity in the performance of wings and wind turbineblades.
As mentioned at the outset, fluid dynamics is a large and complex subject. A basic introduction from a physics perspective can be found in [193]. Issues in fluid dynamics are treated from the perspective of energy science in [7].
Reader’s Guide
This chapter develops some of the basic principles of fluid mechanics. Conservation laws, Bernoulli’s principle, vorticity, circulation, viscosity, and Reynolds number are introduced. The phenomenon of lift is described in some detail. Lift plays a central role in understanding the operation of wind turbines.
Prerequisites: §2 (Mechanics), §5 (Thermal energy), §28 (Wind energy).
This material is used in the following chapter (§30) to describe aspects of wind turbines and their design.
29.1Defining Characteristics of a Fluid
29.1.1 What is a Fluid?
Fundamentally and intuitively, a fluid is a large collection of molecules that are free to move. More precisely, a fluid can be defined as a substance that cannot support a shear stress in mechanical equilibrium. Shear stress is a concept similar to pressure. Both are forces per unit area. Pressure is a force per unit area that acts perpendicular to a surface; shear is a force per unit area that acts in a direction parallel to the surface on which it is applied. When the person in Figure 29.1 pushes on the crate held stationary by friction, the force applied at an angle θ to the horizontal has a downward component that generates a pressure on the floor and a horizontal component that exerts a shear stress on the floor. Like pressure, shear stress is measured in pascals (1 Pa = 1 N/m2) in SI units.
Figure 29.1 When a person pushes with a force F on a crate, which is held motionless on the floor by static friction, they are exerting both pressure and shear stress.
While a solid object like the floor in Figure 29.1 stays fixed when acted upon by the shear stress exerted by the crate, the same is not true of water or honey. These materials respond to a shear stress by flowing. Fluids can withstand pressure in equilibrium without motion, but not shear stress.1 Water and gases such as air and steam, which are of primary interest to us, are classic examples of fluids.
