Feasiblityof using Piezoelectricity in the Conversion of Ocean Wave

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					 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.

				
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