Energy Conv Paper

8/8/2019 Energy Conv Paper

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International Conference on ‘Advances in Energy Research’, (ICAER) 2007

OCEAN WAVES, MECHANICAL IMPULSES AND ELECTRICAL

ENERGY: CONCEPT OF A SIMPLE CONVERSION PROCESS

Surajit Sen1, Adam Sokolow1, Robert Paul Simion1, Diankang Sun1, Robert L. Doney2,

Masami Nakagawa3, Juan H. Agui, Jr.4, and Krishna Shenai5 1 Department of Physics, State University of New York at Buffalo, Buffalo, New York 14260, USA2U.S. Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, Maryland 21005, USA3Department of Mining Engineering, Colorado School of Mines, Golden, Colorado 80301, USA

4 NASA Glenn Research Center, Microgravity Research, 21000 Brookpark Road, Cleveland, Ohio 44135, USA

5Department of Electrical Engineering and Computer Science, University of Toledo, Toledo, Ohio 43606, USA.

1Corresponding author: S. Sen, 239 Fronczak, Dept of Physics, SUNY-Buffalo, Buffalo, NY 14260-1500, USA

Phone: 1-716-645-2017, Fax: 1-716-645-2507, e-mail: [email protected]

AbstractEstimates suggest that a meter of a wave front carries ~ 100 kW of power. A significant amount of renewableenergy is hence dissipated across the time scale of minutes as surface gravity waves dissipate on any beach.Given that the majority of the world population lives near the ocean, it makes sense to explore new technologiesassociated with the conversion of wave energy into electrical energy. Here we consider one such new concept – that of a potentially scalable system that converts wave energy into pulse energy and then to a voltage drop. Theenvisioned system would exploit the well studied physics of nonlinear repulsion between elastic grains to

continuously convert the incident wave energy into dispersionless energy bundles, which can be subsequentlyconverted to stress pulses in a piezo-electric slab. The key points involved in the concept are sketched here.

Keywords: renewable energy in oceans, mechanical to electrical energy, solitary waves, piezo-electric sensors

1. IntroductionOcean waves form because of wind and gravity and provide us with a renewable energy source. In coastalregions, the typical periods associated with waves range between 3 s and 25 s with a dominant frequency in therange of 0.1 Hz (Dean 1991). A detailed discussion on the propagation of gravity waves from deep to shallowwater has been presented in the classic work of Eckart (Eckart 1952).

It is estimated that the world’s energy consumption is < 10 trillion W (TW) per year (Archer 2005). Ocean wave

energy conversion to useful forms of energy is a relatively new field (Muetze 2006). Nevertheless, estimatessuggest that there is 1-10 TW of energy in ocean waves (Boud 2003). A simple search reveals that there are

several existing approaches to extracting wave energy but nearly all of these are in their early stages and noneare able to continuously convert most of the incident energy into tunable and “bite-sized” energy pulses. Herewe present a concept that builds on decades of work on nonlinear dynamics in granular materials and developedand/or commercially available technologies to convert the energy in waves as they come ashore to useful forms

of electrical energy.

We argue that by using simple alignments of elastic spheres (Sen 2003) it is possible to convert the energy in awave that is incident in one end of the alignment to a series of stress pulses at the other end. By placingappropriate piezo-electric slabs at the opposite end of the alignment, the stress pulses can be converted to avoltage drop (Elvin 2001). The system is modular and arrays of alignments may be exploited to scale up theenergy recovery process.

This paper is arranged as follows. Section 2 focuses on the details of converting the ocean wave energy into

mechanical pulses. Section 3 briefly discusses the role of the piezo-electric sensors. Section 4 summarizes thiswork.

2. Conversion of Ocean Waves to Controlled Mechanical PulsesLet us consider an axially aligned set of elastic spheres encased in a cylinder. We assume that there is a movable piston at one end of the cylinder and a wall at the other end. We consider the wall to be a piezo-electric slab that




can generate an electric field in response to a stress. When an impulse is incident at one end of such analignment, it ends up propagating as a non-dispersive energy bundle or as a solitary wave (Nesterenko 1983,Sinkovits 1995, Sen 2001). The solitary wave would produce a stress pulse through the piezo-electric slab whenit reaches the other end and hence would generate an electric field. This simple idea, extended to incorporateenergy transport through granular chains for long-lived pulses, is the underlying principle behind the presentconcept for energy recovery from ocean waves.

We now ask two fundamental questions – first, how do we know that the concept is workable and second, if

there is room for a concept such as this one when it comes to ocean energy recovery. There are presently severaltechnologies that are potentially available to recover energy carried by ocean waves – these include floating buoys that sway as waves pass by and the energy associated with the swaying motion is converted to electricity,turbine wheels turned by wave power and more. Here we envision systems that are made out of a large number

of cylinders comprised of spherical grains in contact and between walls, placed in close-packing and encased inchambers. At one end of each cylinder there would be some piston-like structure that can respond to the forceexerted by the waves. At the other end there would be appropriately designed piezo-electric slabs or tiles with

Figure 1: Normalized total kinetic (black) and potential (red) energies for 12 different tapered chain systems are shown.

Total system energy is constant to approximately 1 part in 1012. Each subplot element is characterized by a specific tapering

q and number of particles in the chain, N. Panels (a,d,g,j) represent a monodisperse chain of increasing length such that

panel (j) is the longest tapered chain and panel (c) is the shortest. For clarity, each subplot only shows the first 20% of the

total simulation time.




associated circuitry that would convert the mechanical pulses into electrical voltage. The chambers themselvesare envisioned to be fastened to some stable rock structure such that they would not be easily affected by theconstant onslaught of the waves. The present concept may be most useful in places where there is an abundanceof rocky beaches and strong waves. An operating version of the system is expected to be inexpensive tomanufacture and install, require low maintenance, not in need of a significant input power supply, and not likelyto pollute the environment. We now discuss the underlying physics of the work.

We first consider the dynamical behavior of the individual alignments of equal sized spherical elastic grains

placed within the confining walls of a cylinder. The centers of the grains should lie along the axis of the cylinder to minimize frictional losses. The grains can be metallic, glassy or even polymeric. Most of the studies,

theoretical and simulational as well as experimental, have been carried out for metallic grains which tend to offer low dissipation and are hence best suited for our purpose. We first summarize what happens when a deltafunction impulse is incident at one end of the alignment. Initially, of course, all energy received by the edge

grain would be kinetic. This edge grain will press against the next grain, which would then press against the oneadjacent to it and push back the edge grain, etc. In the process, the initial kinetic energy will end up being

converted to part kinetic energy and part potential energy. When the pulse reaches the piezo-electric slab at theother end, both potential and kinetic energies are transferred to the piezo-electric slab and hence the pulse can beconverted to an electrical voltage. The chain is a way to convert a sharp impulse into a specific form of energy pulse or bundle. It has been discussed elsewhere that the properties of the energy bundle can be tuned by

manipulating the chain geometry (Sen 2001).

Elastic grains repel upon intimate contact via the Hertz law (Hertz 1881). Suppose there are two grains with radii

R i and R i+1. Let these grains be a distance xi,i+1 apart upon intimate contact. The overlap between the grains is

defined as δi,i+1 = R i + R i+1 – xi,i+1 ≥ 0. According to Hertz law, the repulsive potential is V(δi,i+1) = ai,i+1δi,i+1n,

where ai,i+1=(2/5D)(R iR i+1/[R i+R i+1])1/2

, D=(3/2)(1-σ2)/Y, where σ is the Poisson ratio for the material and Y is

the bulk Young’s modulus and for spheres, n=5/2. The Hertz potential is softer than the harmonic potential (n=2case) at small enough compressions but quickly exceeds the magnitude of the harmonic potential upon increasedcompression. Energy transport between the grains interacting only via the Hertz potential is known to be

approximately “ballistic” (Nesterenko 1983, Sen 2001). Careful analysis shows that energy transport betweenspherical grains is mediated via traveling energy bundles or solitary waves that are about 5 grain diameters wide(Sen 2001). When n=2, the average kinetic and average potential energies of the system are equal to half the total

Figure 2: (Top panel) The monodisperse granular chain is shown with a piston on the left side of the system

and with a wall on the right end of the syste, the piezo-electric sensor may be placed in lieu of the wall;

(bottom panel) The total energy (Etot), Kinetic Energy (KE) and Potential Energy (PE) of a granular chain

with N=20, q=14% is shown as a function of time.




energy. When n>2, the average kinetic energy exceeds the average potential energy and this sets the time scalefor pulse propagation through the chain and also the width of the pulse. Thus an ability to control the potentialenergy can potentially allow us to control pulse propagation. In addition to examining pulse propagation insystems where all grains have the same radii, we have also studied cases where the grains progressively shrink along the chain such that R i+1=(1-q)R i, where q is the tapering percentage. When q>0, the energy pulseaccelerates as it propagates down the chain and the solitary wave breaks down. Figure 1 presents the details of

energy propagation through various chains consisting of N=20 grains placed between rigid end walls andsubjected to a delta function perturbation from one end. It can be seen that for typical metallic systems (e.g.,

titanium-aluminum alloys which have been used in the simulations reported in Figure 1) a pulse takes ~ 1millisecond to travel along an N=20 chain. We consider chains with and without restitutional losses. Therestitutional loss w is defined in our system as follows: Funload/Fload = 1 – w, where w is the restitution constant,typically between 0.01 and 0.1. A typical restitution coefficient may lie between 0.04 and 0.08 or so. We havetalked only about delta function impulses. Below we consider what happens when one edge is subjected tolonger-lived perturbations, i.e., we address whether one can one convert a long lived wave pulse into a finite setof energy bundles that can easily generate a significant voltage drop. Such a step would mean that the power of the wave can be efficiently packaged and harvested.

An incident ocean wave would be expected to generate ~ several seconds of acceleration on the edge grain of an

appropriately placed granular alignment (see Section 1). This would be the equivalent of several hundred round

trips of a single pulse. We therefore assume that an incident ocean wave would generate an approximatelyconstant acceleration on the edge grain for a “large enough” length of time so that for practical purposes we do

not have to worry about when the acceleration might end. We now ask how the confined granular alignmentwould behave. To this end we numerically solve the coupled Newton’s equation of motion for every grain in the

chain using a velocity Verlet algorithm (Allen 1987) by assuming that one wall is a piston and the other is a piezo-electric slab. We calculate the dynamical variables such as position, velocity, and acceleration of everygrain in the system as functions of time. The kinetic, potential, and total energies of the system are henceobtained. Figure 2 presents our calculations for the case in which one end of the chain is driven by a constant

force. The results are robust enough such that small scale force fluctuations do not significantly affect our results.

The underlying physics associated with the time dependent behavior of the total energy and the kinetic and

potential energies can be described as follows. A constant acceleration is felt by an edge grain at all times. Thisacceleration leads to over-compression of the system resulting in the peaks in system potential energy and theassociated dips in the system kinetic energy in Figure 2 (lower panel). The over-compression eventually leads tostrong enough outward acceleration of the grains, including the edge grain, such that the system subsequentlyundergoes a dilation phase until the applied constant acceleration of the edge grain begins to dominate and theabove process repeats itself. If F is the average magnitude of the applied constant force and T is the breathing period we find that TF1/6=constant for fixed q (see Figure 3). By tuning q, one can hence control T.

The breathing process of a system under driving could eventually cease because the energy pumped into the

system due to the driving is less than the energy lost due to restitutional losses. Under such circumstances, our exploratory studies suggest that the number of breaths of the system is inversely proportional to the restitution

400

450

500

550

600

650

700

750

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0 10 20 30 40

q (in %)

T F 1 / 6

0.25

0.5

0.75

1

5

10

15

30

45

60

75

90

Figure 3: Here we show the dependence of the scaled quantity6/1

TF versus tapering q for a variety of

values of the applied force F (in kN) at the left edge. The time period T is measured in microseconds.




coefficient w. We find that the product of the number of breaths and the restitution coefficient w appears to be a

constant independent of the tapering in the chain, i.e., independent of the details of energy transport within thechain. The behavior of these systems when the energy input exceeds energy lost is not well understood at thistime. The breathing would generate periodic stresses on the piezo-electric slab. In Section 3 below we brieflyoutline how such stresses can be converted to voltage drops.

3. Pulses onto Piezo-Electric Sensors to Voltage Drops In piezo-electric materials such as the slab mentioned above, an applied stress results in an electrical voltage(Ikeda 1996). The process is reversible in the sense that an applied voltage also generates a mechanical stress.Due to the space restrictions here we refrain from developing what would have to be a formidable mathematicaldiscussion of the process of conversion of the periodic mechanical stresses on the piezo-electric slabs intovoltage that will be generated where the crystal is placed in an appropriate circuit. We focus instead on how the piezo-electric slab can be used to harvest power.

.A well known way to convert the applied stresses on a piezo-electric slab to a voltage is as follows. The

approach involves a typical power harvesting circuit as shown in Figure 4 below (Horowitz 1989, Elvin 2001).Here a half-diode-bridge is connected to a charging capacitor. The resistor R across the charging capacitor isadded to the circuit to take into account voltage leakage. The voltage generated on the capacitor can be modeled

using electric circuit simulation software. Results of a typical simulation (Elvin 2001) are displayed in the V vs tsketch in Figure 4. The model calculations reveal that the voltage on the capacitor increases on each positivevalue of the load.

4. Summary and ConclusionsTo summarize, we have argued that ocean waves release a tremendous amount of energy as they come ashore.This energy is largely wasted. If the energy in ocean waves can be harvested, it could solve at least a significant

fraction of the world’s energy needs. To harvest ocean energy it would be important to efficiently andinexpensively convert the energy into useful forms of energy such as into a electrical energy. Here we show thatit may be possible to design assemblies of granular chains that would be able to convert wave energy into bite-

sized chunks or bundles of mechanical energy. The mechanical energy bundles can be next converted via a half- bridge power harvester with a piezo-electric slab into a potential drop. An assembly of granular chains with theappropriate circuitry and casing may be firmly embedded on rock surfaces and/or beaches where the waves comeashore. The system could potentially be a relatively inexpensive one to construct and maintain.

This work has been supported by the US Army Research Office.

References

Allen, M.P. and D.J. Tildesley, 1987, Computer Simulation of Liquids, Clarendon, Oxford.Archer, C. L. and M. Z. Jacobson, 2005, Evaluation of global windpower. J. Geophys. Res.-Atm., 110, D12110

Boud, R, 2003, Status and Research and Development Priorities, Wave and Marine Accessed Energy, UK Dept.of Trade and Industry (DTI), DTI Report # FES-R-132, AEAT Report # AEAT/ENV/1054.

Figure 4: The top panel shows a half-bridge power harvester with a leakage resistant R. The lower panel

sketches the typical form of the voltage generated in the capacitor as a function of time under sinusoidal

loading.




Dean, R.G. and R.A. Dalrymple, 1991, Water Wave Mechanics for Engineers and Scientists, World Scientific,Teaneck, New Jersey.Eckart, C., 1952, The propagation of gravity waves from deep to shallow water, Nat’l Bur of Stds., Circular 521,Washington, DC., 165-173.Elvin, N.G., A.A. Elvin and M. Spector, 2001, A Self-Powered Mechanical Energy Strain Sensor, Smart Mater.

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