Feasiblity of using Piezoelectricity in the Conversion of Ocean Wave Energy Amadou G. Thiam and Allan D. Pierce presented at 1st Annual MREC Technology Conference (New England Marine Renewable Energy Consortium) Fall River, Massachusetts October 15, 2009 A small group at Boston University is seeking to better understand the fundamentals of electromechanical energy conversion in the context of its applications to ocean wave energy capture and conversion The present paper is concerned with piezoelectricity, which has been relatively neglected in the literature on ocean wave energy conversion The paper is a “progress report” and is intended for stimulation of further efforts. First question addressed: Given that one desires a certain rate of energy conversion, what is the minimal amount of piezoelectric material required for a “suitable designed” device? First prototype: piezoelectric disk Equations for a piezoelectric disk Representative piezoelectric material constants (PZT-4): (lead zirconate titanate) Electrical energy extracted from a single piezoelectric disk Premise: all amplitude quantities oscillate with the same frequency as the waver waves Premise: average time rate of energy extraction is proportional to the average of the square of the electrical current drawn. A brief derivation (given in next slide) shows that best choice for R is Premise: There is sufficient latitude in the design of the external electronics for the energy capture that one can choose the constant R. Derivation of optimum choice for equivalent resistance R Begin with first piezoelectric constitutive relation: Power extracted: Assume harmonic oscillations: After some algebra: Maximum when: etc. Omit the overcarats, and set Solve for V: Maximum possible power: Analysis for piezoelectric disk Estimation of how much piezoelectric material it would take to produce 1 Watt (Based on maximum strain of 0.001) Resulting rough estimate is Principal limitations exist because cubic centimeters PZT is a ceramic, so it is stiff and per Watt very brittle. Second prototype: piezoelectric strip Equations for a piezoelectric strip Representative piezoelectric material constants (PVF-2): (polyvinylidene fluoride) Derivation of optimum choice for equivalent resistance R for case of piezoelectric strip Begin with first piezoelectric constitutive relation: Power extracted: Assume harmonic oscillations: After some algebra: etc. Omit the overcarats, and set Maximum when: Solve for V: Maximum possible power: Analysis for piezoelectric strip Estimation of how much piezoelectric material it would take to produce 1 Watt (Based on maximum strain of 0.03) Resulting rough estimate is Principal limitations exist because the coupling coefficient is relatively cubic centimeters small. per Watt A cube, 24 cm on a side for one watt. Ceramic and polymer piezoelectric materials are manufactured, and the material parameters are not necessarily cast in stone. For energy conversion, there are two principal desires: One or the other of the two parameters or should be as large as possible The maximum tensile strain before fracture occurs should be as large as possible Are there better materials already out there? Can better materials be developed? Preceding estimates are based on the premise that the angular frequency ω is the same as that of the water wave. Could one design a device that steps up ω by a respectable factor? Possibly….. Possible method for stepping up the frequency I. Partially transform translational motion to rotational motion Slider-crank mechanism Oscillatory motion of inertial mass drives connecting rod causes wheel with fixed axle to rotate with same rotational frequency as oscillation frequency Possible method for stepping up the frequency II. Use succession of meshed gears to step up the frequency Gear with smaller radius rotates faster. III. Have wheel with substantially larger radius than the gear that is rotating with the highest frequency on the same shaft as that gear IV. Use slider crank mechanism to drive oscillating force on a piezoelectric element (many, many design hurdles, but…..) Having a great piezoelectric material is not enough. You have to achieve a design where the strains in the piezoelectric materials are large under the circumstances of representative water waves. Simplified model for initial exploration of strain amplification concepts hollow buoy Wavelength much larger than buoy diameter Wave-induced slope of surface unimportant still water level Buoy heaves up and down with the water wave Motion of internal parts has negligible effect on the motion of buoy casing Any mooring to the bottom has negligible = wave height effect Buoy motion is stable Assumptions are easily modified for any refined analysis. Simplified dynamical model internal oscillating mass In the first approximation, the coupling device behaves as a massless spring. coupling device If water is moving up and ballast and down with constant ancillaries angular frequency: (Substantial modifications are Tensile force in coupling needed if the mass is driven device: near resonance. Size of buoy is constraint on how much amplification you can have.) The dynamic force is intrinsically small because of the quadratic dependence on ω, and ω is only of the order of 1. The angular frequency in this equation cannot be stepped up. Either one amplifies F or one amplifies the pressure it causes, or one does both. A possible approach is to achieve pressure amplification by insertion of piezoelectric disks Voltage drop: Electrical power out: Make the cross-sectional area A small. internal If oscillating mass coupling device then a substantial pressure amplification has been achieved You are doing much better than just holding a piezoelectric disk at a fixed distance from the bottom, but ……. This was the principal reason for the pessimistic assessment of piezoelectricity in McCormick’s book. Electrical power output in terms of tensile force For a single piezoelectric disk: From piezoelectric equations: After some algebra: (This number is for PZT, which is a very stiff ceramic.) Interim result for electrical power output of the device with inserted disks Derived in preceding slide for a single disk: Derived from Newton’s second law: For a device with N elements: Questionable if one could make area A to be small enough for this power to be comparable to a watt. Force amplification internal resulting from having oscillating coupling mass coupling device function device as a cantilever beam. Magnification factor: ballast and ancillaries (could be as large as 100) Idealize cantilever beam as I-beam Upper and lower flanges are piezoelectric strips Tensile force in a flange varies linearly with distance along beam length --- approximately taken into account with Electrical power output by a piezoelectric strip on a cantilever beam flange (number is for PVF-2) Concluding Remarks internal For any piezoelectric material oscillating there are fundamental limitations mass on power output per volume of material Presently available piezoelectric materials preclude piezo devices coupling being viable contenders for major device energy conversion Outlook may change with development of new materials ballast and ancillaries Mechanical design techniques may radically improve power output With present state of the art, piezoelectric devices can be a viable contender for energy storage. Small power, but respectable accumulation of energy for intermittent retrieval.