Energy Harvesting from Flows Using Piezoelectric Patches

Abstract

The highly nonlinear phenomenon of fluid–structure interaction is discussed, including examples drawn from nature and early work on aircraft flutter. Recent work on extracting the energy in a fluid stream by piezoelectric elements is reviewed, including some of the underlying physics. Whilst the energy extracted from fluttering elements is low, it is a subject of interest for powering Ultra-Low Power (ULP) devices and systems since this method of energy extraction is thought to offer a quiet alternative to conventional wind turbines. Researchers have investigated the use of thin piezoelectric patches coupled to a geometrically shaped, polymeric membrane (via a revolute hinge) which can amplify the bending, strain and hence power. Such systems respond via flutter induced by resonant bending instability of the system, or by the utilisation of time-varying external pressure gradients formed around the system. Key factors that influence performance are examined, such as critical flutter speed, mass ratio, position of revolute hinge, aspect ratio and type of piezoelectric material. The chapter concludes with a discussion of the practical implications of such systems in the future.

List of Symbols

$A$

Cross-sectional area

$a$

Acceleration

$AR$

Aspect ratio

$b$

Leaf base length

$C$

Capacitance

$c$

Compliance or damping

$C_{1,2,3,\ldots,8}$

Constants in the general solutions of the space functions,Y₁ Y₁

$D$

Characteristic dimension of bluff body or electric displacement

$d$

Piezoelectric coupling coefficient

$E$

Electric field strength or elastic modulus$\left(\frac{1}{c}\right)$

$EI$

Bending stiffness

$F$

Input forcing function

$f$

Vortex shedding frequency

$h$

Beam thickness or leaf height

$I$

Moment of inertia

$I_{b}$

Moment of inertia of an object rotating about its base axis

$k$

Stiffness

$L$

Beam length

$\mathcal{L}$

Laplacian operator

$M$

Bending moment

$m$

Mass per unit length of the beam, or mass

$N$

Number of discretisation points

$P$

Power

$P_{\text{ave}}$

Average power

$P_{i}$

Instantaneous power in an i th second interval

$R_{L}$

Load resistance

$R_{L_{\text{opt}}}$

Optimum load resistance

$r$

Radius

$r_{v}$

Rankine vortex radius

$Re$

Reynolds number

$R_{i}$

i th analytical natural frequency ratio

$R_{i_{\,\text{comp}}}$

i th computational natural frequency ratio

$S$

Mechanical strain

$s$

Distance from beam leading-edge to upstream bluff body, or Laplacian coordinate

$St$

Strouhal number

$T$

Mechanical stress

$t$

Time, or when superscript, denotes the transpose of a

$T_{g}$

Glass transition temperature

$T_{i}\,(t)$

i th time function in variable separable method

$u$

Fluid stream-wise velocity component

$u_{c}$

Critical flutter speed

$u_{q}$

Quenching speed

$V _{\text{RMS}}$

Root-mean-square voltage

$v$

Velocity or cross-stream velocity component

$w$

Beam width

$x,\, y,\, z$

Cartesian coordinates

$x_{i},\,y_{1},\,z_{i}$

i th local coordinates

$Y _{i}\,(x)$

i th space function in variable separable method

$Y (s),\,Z(s)$

Laplace transform of a function

$z$

Distance from beam neutral axis to point of interest

$\beta _{i}$

$\root{4}\of{\rho _{s}A\omega _{i}^{2}/EI}$

$\delta$

Damping ratio

$\epsilon$

Electric permittivity

$\eta$

Non-dimensional hinge position, x/L

$\lambda$

Eigenvalue

$\lambda _{i}$

i th mode shape eigenvalue

$\mu$

Mass ratio

$\nu$

Kinematic viscosity or normalised critical flutter speed

$\varphi$

Phase difference

$\rho _{ f}$

Fluid density

$\rho _{s}$

Beam density

$\omega$

Angular frequency or angular velocity

$\omega _{n}$

nth natural frequency

$i$

Index

$\mbox{ Hinge}$

Hinged beam

$\mbox{ Uniform}$

Uniform beam

$\mbox{ Comp}$

Computational