The Physics of Energy, page 125
31.1.1 Physics of Hydropower
The basic physics of hydropower is very simple. A mass m of water at a height h has gravitational potential energy
(31.1)
from eq. (2.9). Hydropower is produced by the conversion of this gravitational potential energy into useful mechanical energy. In a hydroelectric plant, the force of the water is used to turn a turbine. The turbine in turn drives a generator that produces electromagnetic power, as described in §3 and §38.
In a typical hydropower setup, a dam is built to hold water in a reservoir (Figure 31.3). An enclosed channel carries water from the bottom of the reservoir through a turbine (Figure 31.4), and releases it into a river or other outflow channel. The difference in height Z between the top of the reservoir and the position of the turbine is called the hydraulic head. When a small volume of water having mass density kg/m passes through the turbine, the potential energy lost to the reservoir is , from eq. (31.1), where . Thus, the power output of the turbine can be described by the equation
(31.2)
where is the flow rate of the water and ε is the efficiency of the turbine.
Figure 31.3 Schematic of a hydroelectric dam showing the head Z. When a small volume of water flows through the turbine, the reduction in total potential energy of the water in the reservoir is given by . (Credit: Tomia reproduced under CC-BY-SA 3.0 license via Wikimedia Commons)
Figure 31.4 Schematic of a modern hydro turbine connected to an electric generator (see §38 (Electricity generation and transmission)). (Credit: top-alternative-energy-sources.com)
Because the power is proportional to the hydraulic head Z, the power output of the dam is essentially proportional to its height. Note that the power of the dam depends upon the potential energy of the water at the top of the reservoir, even though the water passing through the turbine is flowing from the bottom of the reservoir. This follows directly from conservation of energy; when a volume of water is removed from the reservoir the drop in water depth shows that the loss in potential energy is , assuming that the volume of water removed is small enough that the change in depth is much smaller than the depth (). This result can be understood physically from the increased pressure at the bottom of the reservoir that drives the turbine (Problem 31.1).
Hydropower
The power that can be extracted from a hydroelectric dam is given by
where ρ is the mass density of water, Q is the water flow rate (volume/time), g is the gravitational constant, Z is the hydraulic head (the height differential between the top of the reservoir and the turbine), and ϵ is the turbine efficiency. Turbine efficiencies for hydropower are not bounded by the Betz limit and can reach around 95% in practice.
Another feature of hydropower dams is that they can be used for energy storage; by pumping water into the reservoir, energy can be efficiently converted from electromagnetic to potential form. We discuss the storage aspect of hydropower further in §37.
31.1.2 Hydropower Technology
Unlike a wind or marine current turbine that extracts kinetic energy from a moving fluid, a hydropower turbine extracts gravitational potential energy from a volume of water by releasing deep water that initially has no velocity from a region of high pressure to a region of much lower (atmospheric) pressure. The Betz limit is not relevant under these circumstances, since the force from the water’s pressure is associated with mechanical potential energy and not with the bulk kinetic energy of a moving fluid, so there is no theoretical bound on the efficiency of a hydropower turbine.
Large turbines used in modern hydroelectric facilities such as the Three Gorges Dam on the Yangtze (Chang Jiang) River in Hubei, China can achieve efficiencies of 96% or greater. As of 2012, the Three Gorges power plant (Figure 31.5) was the largest hydroelectric facility in the world. With a dam height of 101 m, a hydraulic head of 80.6 m, a flow rate of up to 900 m3/s for each generator, and an average turbine efficiency near 95%, each turbine has a maximum power of around MW. With 32 turbines, the facility has a capacity of 22.5 GW.
Figure 31.5 Left: Three Gorges Dam on the Yangtze (Chang Jiang) river in Hubei, China; in 2012 this was the largest hydroelectric facility in the world with capacity of 22.5 GW. Right: One of 32 Francis turbines used in the Three Gorges Dam, with an average efficiency of nearly 95% and capacity 700 MW. (Credit: Voith Gmbh)
Large-scale hydropower facilities raise a number of practical and environmental concerns. Dams disrupt local ecosystems; for example, the presence of a dam reduces the downstream flow of sediment carrying valuable nutrients, can lead to excess methane production by rotting vegetation in the reservoir area, and can interfere with fish spawning. In recent years many dams in the American West have been removed in order to restore natural riverine ecosystems. Also, large-scale hydropower installations flood large land areas, leading to land use efficiency that is comparable to large-scale solar power installations (see Problem 31.2); the riverine ecosystems disrupted are generally both ecologically richer and rarer than desert locations where solar power can be harvested. Accumulation of blocked sediment behind the dam can interfere with water flow through the turbines and cause practical difficulties with plant operation. The water weight in large reservoirs may lead to a shift in land mass. In some situations there is concern that such an effect may increase the chances of an earthquake in seismically active areas and/or increase the chances of dam failure. Catastrophic dam failures can result in significant human casualties and loss of property. On the positive side, the role of dams in flood control and irrigation can have significant social and economic value. Furthermore, in some situations such as the Three Gorges dam, dams can play an important role in expanding transport channels through water-borne navigation.
Smaller-scale hydropower systems based on water wheels or underwater turbines like those discussed in §31.4 have fewer environmental impacts. Microhydro facilities with generating capacity from a kilowatt to several megawatts can be built on dams with a low hydraulic head and can produce renewable energy with reduced environmental impact. While not practical for large-scale power generation, such implementations provide a carbon-free renewable energy resource that is particularly useful in remote locations with regular water flow.
31.1.3 Extent of Resource
As shown in Figure 31.6, global hydroelectricity generation has been growing steadily for many years. By 2013, world hydropower capacity passed 1 TW and hydropower actually provided about 3750 TWh/y (13.5 EJ/y), or about 430 GW of electric power averaged over the year, close to 1/6th of world electricity. This corresponds to a capacity factor of ~43% [168]. The recent relatively rapid growth of hydropower has been dominated by new installations in Asia and, to a lesser extent, Europe and Latin America. Hydropower deployment in North America has been relatively static for many years. The present distribution of hydroelectric power capacity over the world’s major geographical regions is summarized in Figure 31.7.
Figure 31.6 After a brief hiatus at the beginning of the century, hydroelectricity generation has been growing more rapidly than its earlier historical average. After [168].
Figure 31.7 Hydroelectric generating capacity in 2012 by region [12].
The total potential for hydropower is limited. The World Energy Council and the IPCC estimate that ~10 000 TWh/y (36 EJ/y) of hydropower remains that could in principle be exploited, potentially increasing world hydropower by a factor of 3.7 to an annual average of about 1.6 TW [168, 199]. The greatest potential for new hydropower deployment is in Asia, Africa, and Latin America. In Asia alone, the theoretical undeveloped hydropower potential has been estimated at 7000 TWh/y (25 EJ/y), and 92% of potential hydropower resources in Africa are undeveloped. Whether these resources will be developed depends upon many economic, environmental, and political considerations, particularly those related to competition for land use and water resources.
Because hydropower arises as a third-order effect from solar power, it is natural that the total power available from gravitational potential energy of water is substantially less than the total amount of power available in wind, which is itself a second-order effect of solar power. Indeed, 1.6 TW is substantially less than the ~500 TW of wind energy dissipated in the lower atmosphere, but may not be too different from the amount of wind energy that can be successfully harvested (see §28.3.2). The difference in the fraction of recoverable energy between these resources lies in the fact that, as discussed above, hydropower is concentrated by natural forces from a diffuse two-dimensional resource to an essentially pointlike resource.
31.2Wave Power
Wind blowing across the surface of the ocean magnifies small disturbances on the sea surface, giving rise to waves that increase in size with the strength and duration of the wind. Waves produced at one point in the ocean propagate outward, carrying energy often as far as thousands of kilometers until the waves break on the shore of a continent, island, or shallow undersea feature such as a reef. Along many ocean coastlines, incident wave energy provides a potential renewable energy source with relatively high power density. In this section we describe the propagation of ocean waves, the life cycle of a wave from production to breaking, and some technologies for extracting ocean wave energy.
The basic equations that govern surface waves and simple surface-wave solutions are described in §31.2.1. The reader more interested in energy applications may skip this section on a first reading; the only result needed from this section is the final form of the propagating wave (31.9). In §31.2.2 we use conservation of energy to characterize the dispersion relation between frequency and wavelength for propagating surface waves, and to describe the transport of energy in surface waves. The life cycle of a wave from its initial production by wind to breaking on the shore is described in §31.2.3, and the extent of wave energy resources and practical mechanisms for harvesting wave energy are described in §31.2.4 and §31.2.5.
For a more detailed introduction to surface wave physics see [200]. A nice description of the current understanding of wave processes is given in [172]. The physics of some of the basic principles for wave energy capture are described in [201], and a more recent overview of wave energy conversion technologies is given in [202]. Finally, for a concise introduction to water waves in an introductory physics text, see [203].
31.2.1 Propagating Surface Waves
A complete analysis of surface waves at the ocean–air boundary is very complicated. The fluid equations are nonlinear, and the boundary conditions themselves depend upon the solution, since the position of the air–water interface depends upon the motion of the wave. Fortunately, many of the key physical properties of propagating ocean waves can be understood in a simplified analysis in which water is assumed to be an incompressible, ideal fluid. Recall that an ideal fluid is one in which viscous losses can be ignored and vorticity is conserved (see §29.2). We further make the approximation that the displacement of the water from its equilibrium position is small. We describe the consequences of the first two assumptions below. The last approximation allows us to ignore terms that are quadratic (or of higher order) in the water’s displacement. The result is a linearized description of propagating surface waves that captures many of their most important properties from the point of view of waves as an energy resource. Note that during the processes of wave production and breaking, nonlinear effects become relevant.
Imagine starting with a body of water at rest, where the surface is smooth and horizontal. We describe the fluid by following the motion of each packet of water relative to its position in the equilibrium rest configuration. When a disturbance arises, a small packet of water initially at the point x is displaced to a new position , where
(31.3)
The variation of s over time describes the dynamical motion of the water (see Figure 31.8), where tracks the displacement of the packet of water that would be at position in the original equilibrium configuration. This is the Lagrangian approach to fluid dynamics, in which the relevant variables move along with the fluid, as discussed in §29.1.2. The velocity of the packet of water with respect to fixed (Eulerian) coordinates is . The essence of the linearized description is to retain only terms linear in s or v in the equations of motion. This approximation is valid as long as any given packet of water does not migrate too far from its equilibrium position. This would not work, of course, if we were attempting to describe the bulk motion of the fluid, as in a flowing river or blowing wind. Appearances notwithstanding, water waves do not result in cumulative bulk motion over time, and the linearized description is a good starting point.
Figure 31.8 A slice in the xz-plane through a water surface wave moving in the x-direction with height above the resting surface given by the coordinate z. The idealized waveform is independent of the third coordinate y in the direction out of the page. The configuration of the water is described by the displacement vector of an infinitesimal packet of water from its rest position at time t. The magnitude of the z-displacement is exaggerated.
In our analysis we consider idealized waves moving in a direction , with the vertical direction and the undisturbed water’s surface at (see Figure 31.8). For such waves, there is no displacement of the fluid in the y-direction, and the motion is invariant under translation in this direction, so that all physical phenomena of interest can be described in terms of the two coordinates x and z. In the linear approximation, the assumptions that water is incompressible, inviscid, and irrotational lead to two constraints on the displacement s
(31.4)
(31.5)
These constraints are derived in Box 31.1, and simplify the description of water waves enormously. In particular, eq. (31.5) allows us to write the displacement s as the gradient of a scalar function, ,
(31.6)
in exactly the same way that the vanishing of in electrostatics (see §3.1) allowed us to write E as the gradient of the electrostatic potential. Furthermore upon substituting eq. (31.6) into eq. (31.6), we obtain a single, linear, second-order differential equation for ,
(31.7)
where we have simplified the Laplacian defined in eq. (B.17), using the fact that is independent of y.
Box 31.1 Constraints on Water Waves in the Linear Approximation
We assume that water is an incompressible, ideal fluid. In §29 we showed that incompressibility implies ,where v is the velocity of the fluid at any point. As discussed in §29.2.3, vorticity is conserved in the flow of an ideal fluid – a vortex once formed would never dissipate – so if an ideal fluid begins at rest, then it will not develop vorticity when set in motion by wind blowing in a straight line. Thus, for the water waves we study, it is a good approximation to take the flow to be irrotational with zero vorticity, . Roughly speaking, these two conditions, and , together with the linear approximation, are the origins of the constraints (31.4) and (31.5).
Another approach exposes the importance of the linear approximation. Since the fluid is incompressible a small packet of fluid should have the same volume in the equilibrium coordinates and in the physical coordinates defined by eq. (31.3) . (For simplicity of notation we have suppressed the third coordinate y.) In other words, . But we can regard the transformation from equilibrium to physical coordinates as an ordinary change of variables. According to the rules of calculus,
So the volume is preserved if and we ignore terms quadratic in s. A similar analysis yields .
A final comment: incompressibility precludes the possibility of waves where the density of the bulk fluid oscillates sinusoidally in time. These are, of course, sound waves, and while sound certainly propagates in water, these are not the waves that we see traveling along the ocean’s surface.
An immediate consequence of these constraints is that there are no waves in an infinite volume of water without an interface. This is the analog of the statement that in electrostatics, the potential is constant if there is no charge anywhere in space. Electrostatics is governed by eq. (3.12), , so eq. (31.7) corresponds to electrostatics with . Thus, the water–air interface is crucial to the dynamics of water waves, and we should expect that the effect of a wave propagating on the surface will die out with depth.
We thus seek oscillatory solutions of eq. (31.7) for that vanish as . Since we expect a wave-type solution with some wave number k and (angular) frequency ω, we anticipate that a propagating wave will be described by a function of the general form
(31.8)
where is a function of depth that vanishes as . Substituting eq. (31.8) into eq. (31.7), we find , so the solution that decreases as is , where is a constant. From eq. (31.6) we find the displacement vector ,
(31.9)
The characterization of the wave in terms of the single function gives a convenient description of the problem and solution, though it is also possible to check directly that the displacement given in (31.9) solves the constraint equations (31.4) and (31.5) (Problem 31.4).
The solution described by eqs. (31.9) has a simple geometric structure (see Figure 31.9). The wave form has wavelength , and travels to the right at velocity . This velocity is the phase velocity (see §4.1.2), the velocity at which the phase of the oscillation advances, as distinguished from the group velocity, which is discussed in the next section. Note that each small packet of water moves in a circle, confirming the assertion that there is no bulk motion of the fluid and supporting the use of the linear approximation. The radius of the circle for each particle is , decreasing exponentially with depth.
