Physics and CompSci Research
Experimental Investigation of Wave Energy Conversion in Buoys of Varying Major and Minor Axis Ratios Jessica Lee ABSTRACT The goal of this investigation was to identify the optimal ratio of major to minor axis in an oblate ellipsoid wave buoy to facilitate the greatest power output from a wave energy converter. To create our lab set-up, we constructed an acrylic wave tank with a custom-made wave maker, wave absorber, and wave energy converter. We then printed, using a 3-D printer, buoys of varying shapes but constant volumes and masses. We tested the buoys using waves with frequencies ranging from 0.67 Hz to 1.00 Hz and amplitudes ranging from 1.0 to 3.0 inches; from each trial we gathered voltage and current data. Our findings suggest that the higher the ratio of major to minor axis, the more efficient the buoy at wave energy conversion; of the three buoys tested, the one with the longest major axis produced the most energy per wavefront at all frequencies and amplitudes.
Key Properties of Wave Energy Converters
The growing human energy consumption has increased the stress applied to natural resources and led to the demand for a sustainable solution available to the modern world. To put into perspective the magnitude of energy available from wave power, the 1 TWh of wave energy that enters the coastal waters of the British Isles on the average day is comparable to the average daily energy use in the UK—and wave energy converters (WECs) are the mechanism through which wave energy can be exploited (1). The increased interest in WECs creates the end to which the goals of this investigation are aimed: • first, the construction of a functioning wave tank, wave maker, wave absorber, wave energy converter, and wave buoys; • second, the capture of electrical energy through the heave of an ellipsoid buoy; • third, the identification of the optimal ratio of major to minor axis in an ellipsoid wave buoy in order to facilitate the greatest power output.
Wave energy conversion can be facilitated by a variety of devices, all of which fall under the category of wave energy converters (WEC). One class of WEC, the focus of this paper, is the class of floating devices, which float on the surface of the water and rock back and forth with the incident wave. Rising and falling with the ocean, these devices have buoy or float systems that directly transfer the energy from the wave to the WEC (1). A characteristic specifically applicable to a WEC, and fundamental to the evaluation of its efficiency, is capture width, L_WEC, also termed absorption width or absorption length, which can be related, at a given frequency for an isolated body in three dimensions, to the ratio of the total mean power absorbed by the body to the mean power per unit crest wave width of the incident wave train. For an incident wave:
For a point absorber, or a WEC with dimensions considerably smaller than the wavelength of the wave, that is axisymmetric and operated only in heave, the maximum possible capture width can be represented by the equation (1)
Materials and Methods
Figure 1. Graph of the mean wave energy power across the world in kW/m^2 during the month of January (1)
The wave tank was constructed from cell-cast acrylic, which has the advantages of lighter weight and greater durability than glass, while still retaining high transparency. The dimensions of the tank, as shown in Figure 2, were chosen so that all the pieces can be cut from a commercially available 4’ x 8’ sheet. The thickness of the sheet Volume 3 | 2013-2014 | 89
Physics and CompSci Research is determined by the depth of the water; for this tank, the necessary minimum thickness of the acrylic is 0.95cm (3/8â€?) (2).
The wave maker is of the plunger-type and consists of a wedge that essentially cuts into the surface in order to displace a volume of water and form a wave. The frame of the wave maker was built from plywood and the pieces held together with wood-glue; the entire model was varnished to prevent water damage and sealed with epoxy and silicon sealant to create watertight edges.
Figure 2. Dimensions for the Wave Tank. The wave tank, in this experiment, is filled to 0.3 m, creating a body of water defined as shallow water because the water depth is less than half the wavelength of the wave. The edges of the cut acrylic sheet were checked for abrasions or burrs, and the irregularities smoothed out with an emery cloth. These pieces were then temporarily adhered in place with adhesive tape around the outside of the box. The pieces were permanently welded together through the process of solvent welding; by running a syringe filled with acrylic solvent cement along the insides of the tank, a chemical process occurs that welds the joints together, creating a single piece of acrylic out of two. Solvent welding, and later a silicon sealant along the inside of the joints, ensured the security of the seal.
Figure 3. Constructed acrylic wave tank, the edges welded with solvent cement. The next step was the construction of the wave maker. 90 | 2013-2014 | Volume 3
Figure 4. Plunger-type wave maker, constructed of plywood and supported by metal guides. This wave-maker is operated by hand, and allows for control of both frequency and amplitude. Within the tank, two metal guides support the wave maker, keeping it upright and at the same time providing amplitude control, as markings placed one inch apart indicate the height to which the wave maker is lifted. Frequency control comes in the form of a metronome, which creates the tempo to which the wave maker is operated. Once the wave is created by the wave maker, it travels to the opposite side of the tank, which contains the wave absorber. The model of absorber used in this investigation is of the passive type, defined by an unvaried reaction to an incident wave regardless of the wavelength. The absorber was constructed out of a plywood frame, and supports four sheets of vertical mesh that face the incident wave. A low layer of bricks topped with gravel fills the gaps in the spaces between the vertical mesh. This mesh, and the gravel, diffuse the energy of the incident wave and dampen the refraction of the wave, minimizing interference at the wave energy converter. The next part of this investigation was the construction of a working generator. The generator operates by converting linear heaving motion into electrical power, which is achieved through a magnet and coil generator. The copper coil, wound over a thousand rounds, was suspended over the tank by a series of clamps, and connected to a differential voltage probe and a current probe at the two ends of the wire. As stated by Faradayâ€™s Law of Induction,
Physics and CompSci Research the movement of a magnet through a stationary, tightlywound copper coil, induces a voltage that can be measured by these connected probes.
anchored with slack chains to the side of the wave tank so that the buoy is not pushed by the wave towards the end of the tank. The third hole was drilled at the center of the top of buoy, where a stack of neodymium magnets adhered to a thin metal rod is mounted, pointing straight up. As the incident wave hits the buoy and the buoy begins to move in an oscillating motion, the chains constrain the buoy to move only in heave, pushing the magnets through the coil.
Figure 6. The wave energy converter, consisting of a copper coil and a stack of neodymium magnets inside. The last part of the installment process is the construction of the wave buoys, which, for this investigation, were printed from a 3D printer. To use this printer, a 3D model is first constructed using SolidWorksâ„˘ and uploaded to the printer software. After printing each buoy, an acid bath dissolves any support material used during the printing process.
Major Axis (A) 1 2.5 inch 2 2.73 inch 3 3.03 inch
Minor Axis (B) 2.5 inch 2.39 inch 2.27 inch
Ratio (A:B) 8:8 8:7 8:6
Figure 8. Schematics of a buoy. As the aspect ratio increases, the buoy gets progressively flatter and wider, the value of a increasing in comparison to b. The dimensions of the three buoys created and tested are shown in Figure 8. The volume of the buoys are kept the same, at around 65.4 inch2, or the volume of a sphere with a 2.5 inch radius; however, their proportions are different. The aspect ratio of each buoy varies by 1:8 for each; the first buoy, a sphere, has an aspect ratio of 1:1, the second 8:7, the third 8:6. Full operation of the wavetank can be illustrated in Figure 9. It begins at the wave maker end; the wave maker is manually moved up and down at a specific frequency in a specific amplitude, propagating a wave towards the other end of the tank. The wave hits the buoy, causing it to heave, and pushing the magnets up and down within the coil. This movement generates a voltage and current that is measured by the current and differential voltage probes, and recorded on two graphs: Current vs Time, and Voltage vs Time. Once past the buoy, the wave then hits the wave absorber and is diffused by the mesh and gravel.
Results and Discussion Figure 7. Ellipsoid buoy, slack anchored by two hooks, supporting a stack of magnets within the coil. Three holes were then drilled into the ellipsoid. Two holes were drilled on opposite sides of the buoy for two hooks to be inserted; the hooks allow for the buoy to be
Each trial of this investigation was run as described in the previous section for a 25 second interval. Only the last 15 seconds were used during data analysisâ€”the first 10 seconds are used to stabilize the characteristics of the wave. The raw data gathered from the trials were graphs of Current vs Time and Voltage vs Time. To manipulate these graphs into data that can be used comparatively, the equation: Volume 3 | 2013-2014 | 91
Physics and CompSci Research
Figure 9. Full wave tank schematic, including the wave maker, wave absorber, buoy, and generator.
Figure 10. The data gathered in Trial 3-25 (buy 3, 3 inch amplitude, 1Hz), showing the graphs for as well as the manipulated power graph.current voltage
was used, and the two graphs multiplied to produce a graph of Power vs Time. The integral of the 10-25 second section of the Power vs Time graph then gave the amount of energy, in watts, produced by the multiple incident waves. Figure 10 shows an example trial, with Current recorded in red, Voltage recorded in blue, and the manipulated Power graph in green. Each frequency was expected to yield a specific number of waves for a 15 second interval; this number is confirmed through a visual analysis of the graphs, as a pair of relative maximum and minimum represent the movement of the 92 | 2013-2014 | Volume 3
buoy as it passes the crest and trough of a wave period. The energy dissippated was divided by the number of wavefronts to arrive at the energy per wavefront for each buoy at a given frequency and amplitude. Nine different scenarios were tested: amplitudes of 1, 2, and 3 inches, at 40, 50, and 60 beats per minutes (or 0.67, 0.83, and 1 Hz), with three trials performed at each different amplitude and frequency. Running the analysis shown in Figure 12 for all 27 of the trials associated with the nine amplitudes and frequencies associated with a buoy, allows for the values graphed in Figure 13-15 to be produced.
Physics and CompSci Research Trial Buoy Amplitude (inches) Frequency (bmp) Time Interval Energy (J) Number of Peaks Energy Per Wavefront (J) Average (J) 25 3 3 60 10 s - 25 s 0.008736 15 5.824E-04 5.748E-04 26 3 3 60 10 s - 25 s 0.008629 15 5.753E-04 27 3 3 60 10 s - 25 s 0.0085 15 5.667E-04 Figure 12. Trials 25-27 of Buoy 3, illustrating the data analysis process. The integrated power over the 15 second time interval is divided by the 15 waves that hit the buoy during that period and the energy per wavefront found
Figure 13: This graph shows the data points of 0.67 Hz, graphing Energy captured per wavefront against amplitude. The colors indicate the buoy; red for Buoy 1, green for Buoy 2, and blue for Buoy 3.
Figure 14. This graph shows the data points of 0.83 Hz, graphing Energy captured per wavefront against amplitude. The colors indicate the buoy; red for Buoy 1, green for Buoy 2, and blue for Buoy 3.
Figure 15: This graph shows the data points of 1.00 Hz, graphing Energy captured per wavefront against amplitude. The colors indicate the buoy; red for Buoy 1, green for Buoy 2, and blue for Buoy 3. Closely comparing Figure 13 and 14, for many of these frequencies and amplitudes the energy absorbed by the buoy for a wavefront of a wave in a 0.67 Hz trial or 0.83 Hz trial differs little: Buoy 2, for example, at a two inch amplitude, shows less than a 2% difference between the average values for a 0.67 Hz and 0.83 Hz. There is also no trend for which tends to be greater; for the three amplitudes of the three different buoys, six of the nine points have a higher energy absorbed value for 0.67 Hz, differences ranging from <2% in Buoy 2, at 2 inches, to >60% in Buoy 3, at 3 inches. However, an outlier for all three graphs would be the prominent jump between the relatively similar 0.67 and 0.83 Hz values in Figures 13 and 14 and the much higher 1.00 Hz values in Figure 15. During the investigation it was determined that the wave tank had likely been structered in a way that it resonanted at 1.00 Hz, and a standing wave formed within the tank because of inconsistences in the wave damper. The presence of a standing wave increased the amplitude of the wave and thus the heave of the buoys, heightening the energy output of each buoy at 1.00 Hz beyond that of the 0.83 Hz and a 0.67 Hz. Looking at each individual graph, at all nine points, the average values show a progressive increase from Buoy 1 to Buoy 2, then from Buoy 2 to Buoy 3. Two of these points are inconclusive for two of the associated buoys: the error bars of Buoys 2 and 3 cross at 2 inches and 0.83 Hz, as Volume 3 | 2013-2014 | 93
Physics and CompSci Research with Buoys 1 and 2 at 3 inches and 0.63 Hz. A noticeable trend across all three buoys is the progression of values from Buoy 1 to Buoy 2 to Buoy 3. As the buoy ratio changes, becoming wider and flatter, the values indicate that it absorbs more and more energy for each indicate wavefront. The blue points on the graph, representing Buoy 3, are higher than those of the green, representing Buoy 2. The same can be same of the green points being higher than the red, representing Buoy 1, and creating a progession upward of Buoy 1, Buoy 2, and Buoy 3, for all nine combinations of amplitude and frequency.
Conclusions and Future Work The data gathered through this investigation indicates a probable trend between buoy aspect ratio and captured electrical energy; the two are seen to be positively correlated, in that as the ratio of major to minor axis in an oblate buoy increases, the electrical energy captured through heave increases. Further work to be pursued in this study includes the testing of buoys with aspect ratios of 8:5, 8:4, and so forth to see if the trends indicated in this paper continue. Another possible extension of this study would focus on prolate rather than oblate ellipsoids: Figure 8 indicated an oblate buoy, where the cross section of the buoy would be circular when cut along the major axis, rather than a prolate buoy, where the cross section would be circular along the minor axis.
Works Cited  Cruz, João. Ocean Wave Energy. Springer: SpringerVerlag Berline Heidelberg, 2008. Print.  “How to Build Your Own Acrylic Aquarium.” Hubpages. 24 June 2011. Web. 20 May 2013. <http://tehgyb. hubpages.com/hub/How-to-Build-Your-Own-AcrylicAquarium>
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