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.

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):

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

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
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

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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