Wave Energy

Emmanuel Bonifacio October 24, 2010

Submitted as coursework for Physics 240, Stanford University, Fall 2010

 Fig. 1: Schematic of an OWC Onshore Converter System

Origin and Quantification of Wave Energy

In the ever increasing quest for renewable energy sources, energy from oceans and more specifically from its waves or currents can be considered a large and mostly untapped reservoir. Harnessing the power of waves breaking on the shore, or that of the phenomenal mass displacement of the gulf stream could potentially bring a significant amount of renewable energy in mankind's portfolio.

Waves are generally generated by wind blowing over the ocean, although tsunamis can be generated by large water displacements such as those occurring during earthquakes or major landslides, we will however limit ourselves to the study of wind generated waves, as harnessing the power of tsunamis is beyond current technical reach. Seeing as wind is itself indirectly generated by solar power (differential heating leads to differential in pressures, generating wind) waves can be seen as a derivative of solar power.

The study of the propagation of waves can be traced back to D'Alembert who formulated the first linear wave equation. Adequate first order descriptions of water waves was initially obtained thanks to Sir George Biddell Airy and what is now known as the Airy wave theory. Further developments were analyzed by Sir George Stokes to adequately describe waves with increased steepness ratio (the ratio of a wave's height to its length), and a new theory the Cnoidal wave theory was developed to accurately describe shallow water waves.

With such in depth study of ocean waves, one can accurately describe the energy transported by a wave in deep water given its main characteristics : its height, between crest and trough, and its period : the time necessary for the wave to propagate along the distance between two crests.

Given the velocity potential (that can be obtained through linear wave theory) one can derive the kinetic and potential energy of a wave per meter of crest and unit of surface.

The sum of these two energies can be found to be [1]

where ρ is the density of water g the gravitational acceleration and A is the amplitude of the wave (that is to say half its height H).

Schematic of a Wave and Its Characteristics

One can be more interested in determining the power that a meter of crest holds, and this can be obtained by multiplying the amount of energy transported by the group velocity. In deep water, extrapolating from Airy wave theory, one can find the dispersion relation to be k = ω2/g. [2] This yields a group velocity of cg=g/2ω. Thus since T (the period of our wave) = 2π/ω we obtain our total power

With this formula, for a given wave period and height, we can compute the power that can be extracted per meter of crest of that wave.

One can then use this formula, along with measurements of the average wave length and height to determine the available worldwide (or local) resources in terms of wave power. It must be taken into account that as waves are locally generated by wind, but travel over long distances, the state of the sea in a given location is rarely homogenous in wave height and direction. It is an aggregate of multiple waves travelling through that location, originated all over the sea, and statistical treatment of the measurements is thus necessary.

Potential and Limitations

Such studies have been conducted and yield, for the US seashore, that the available incident yearly energy influx through waves offshore is situated in the 2,100 TWh/yr range. [3] The breakdown of incident energy over the different coastlines can be seen in the map below. The incoming energy is a function both of the available seashore length and of the energy density per meter of wave crest in a given region. As this energy density can vary greatly from one region to another, depending on whether the waves ending up on that shore are originated in regions with stronger winds, it is essential to thoroughly analyze the positioning of a wave energy converter if a greater power output-to-size ratio is desired.

Potential Wave Energy Per Region of US Seashore

Given this potential, one would assume that 2,100 TWh/yr of power could be harnessed for commercial use in the USA. However the conversion of wave power, much like that of any other form of power is inefficient. Thereof we must take into account the potential efficiencies of our wave energy capturing device before being able to describe the full energetic potential of wave capture.

One of the most common and well mastered type of device used to harness wave energy is an OWC (oscillating water column) system. The working principle of this system is quite simple : as the wave progresses under a fixed housing chamber, it displaces air when the crest passes, or creates a vacuum when the trough passes, forcing air flow in and out of the chamber at a regular pace, provided regularity of the waves themselves. This air flow can be directed to a turbine - a bidirectional well's turbine in most cases - to generate power.

This system is similar to wind turbines in that it uses a gas flow, generated by the waves, to generate power. Thus it must obey Betz' law, and the efficiency converted from the air stream can be a maximal 59% of the stream's energy. That stream however does not in itself represent the total energy of the wave. As wave energy has to be converted to air movement, there is an additional step (with its associated inefficiencies) in the process. These inefficiencies can be hard to compute as they depend on the design of the device. What's more, these devices cannot harness the full potential of wave power as some of that power is contained further in depth than the part of the wave that can impact the device. However, for shoreline devices in shallow water, this is not an issue and, for off shore devices, the fraction of energy contained in depth decreases exponentially rapidly with depth. This issue should therefore not be a major concern.

An example of this type of device is the Limpet, a shoreline based OWC co-developed by Queen's University of Belfast and Wavegen. Specific attributes of the Limpet contained in A brief review of wave energy [2] enable us to compute its efficiency. The device is said to have an approximate width of 21 m, thus it can potentially harness the power of 21m of wave crest. The average wave power level on site is assessed to be 20kW/m of wave crest, and the average output of the device 206kW. A simple calculation leads to a 49% efficiency in wave power capture. We can see that this efficiency is not too far from the 59% maximal efficiency of our turbine, thus suggesting that more than 80% of the wave's power is converted to air movement with this design.

Another factor to note when considering the total amount of power on US shores is that, to harness it fully, one would require a complete coverage of wave crests along the coastline. This is of course highly impractical as space must be freed to enable commercial and leisurely navigation, as well as to limit visual pollution.

Thus if we were to assume that a sufficient area could be equipped with wave energy converters to tap a third of the potential wave power - this would necessitate thousands of miles covered, one can see the crucial point of scalable systems here - with the aforementioned 49% efficiency we could potentially generate 343 TWh/yr.

Conclusions

We have seen that wave energy, indirectly generated by solar energy, is a potentially large source of renewable power and can readily be harnessed through energy wave converters. We have also seen the caveats to tapping into this potential, especially with regards to the area requirements and efficiency limitations. A fast calculation with yields that on a reasonably feasible scale, this power would only amount for 3% of the total US consumption (estimated at 11200 TWh/yr). [4] Thus we can see that even though the potential energy of waves is very large, it cannot represent more than a fraction of the US portfolio of energy in the near future.

© Emmanuel Bonifacio. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] S.M. Calisal, "A Note on the Derivation of Potential Energy for Two Dimensional Water Waves," Ocean Eng. 10, 133 (1983).

[2] T.W. Thorpe, "A Brief Review of Wave Energy," UK Department of Trade and Industry, ETSU-R120, 25 May 99.

[3] R. Bedard et al., "Final Summary Report, Project Definition Study, Offshore Wave Power Feasibility Demonstration Project," Electric Power Research Institute, E2I EPRI Global WP 009 - US Rev 2, 22 Sep 05.

[4] "Annual Energy Review 2009, U.S. Energy Information Administration.