Energy harvesting in a fluid flow using piezoelectric materials

M. Curatolo1, M. La Rosa1, P. Prestininzi11University of Roma Tre, Engineering Department, Rome, Italy

Abstract

We present a model able to estimate electrical energy gener-ated by a deformable piezoelectric solid immersed in a fluidflow. This problem involves different physics: solid me-chanics, fluid mechanics, electrostatics and an appropriateelectrical load resistance capable to harvest the electricalenergy developed by the deformation of a piezoelectricmaterial immersed in a fluid flow. The interaction betweendifferent fields of physics makes the problem complex andhighly nonlinear. In our case, since the solid exhibits largedeformations, simulate energy harvesting in a fluid flowrequires multiphysics and a deeply use of advanced featuresof COMSOL.

Keywords: energy harvesting, piezoelectric materials,

fluid-solid interaction.

Introduction

Energy harvesting using piezoelectric materials has beendeeply studied over the last decade [1, 2, 3]. This technol-ogy provides extra sources of electric power which can beused to recharge electronic devices with a limited batteryduration. The concept has ecological implications in re-ducing the chemical waste produced by replacing batteriesand monetary gains by reducing maintenance costs. Forthis reason and not only, in the last years the area of energyharvesting has attracted academic and industry. One ofthe most studied areas is the use of the piezoelectric effectto convert ambient vibration into useful electrical energy.An example of such application is the energy harvestingthrough piezoelectric materials in a flowing fluid [4, 5].Fluid flow has the potential to provide significant mechani-cal energy input for piezoelectric harvesters. However, theefficient conversion of the bulk kinetic energy of a steadyand uniform flow into time-dependent elastic energy in thepiezoelectric structure remains a significant challenge. Themodeling of such multiphysics problem has still to be fullyinvestigated and requires advanced numerical tools whichcan be tackled with COMSOL Multiphysics.

In this work, we focus on the generation of electricalpower through a piezoelectric solid immersed in a channelusing an inverted configuration of the solid with respect tothe flow, similarly to [5]. The solid is a non-homogeneousbilayer solid, clamped at one end on a fixed circular con-straint. The solid is composed of two layers; one is piezo-

electric (PZT 5-A) while the other has a structural functionand enables the generation of electrical power by the piezo-electric layer. In this model, we use the fluid-structure inter-action (FSI) package of COMSOL together with the elec-trostatics and electric circuit packages. In particular, largedeformations of the solid are effectively described throughthe moving mesh and the remeshing features of COMSOLas in [6, 7]. The advantage of numerical simulations eas-ily allows parametric studies for different sizes of the halfcylinder and for different inlet velocities to optimize theenergy harvesting device. Finally, it is worth noting thatin this problem the multiphysics approach is necessary assome relevant informations about the energy harvesting justarise by the coupling between different physics.

Model and Theory

The problem is solved in a bi-dimensional (2D) space usinga orthonormal reference frame (o, e1, e2) and coordinates(X ,Y ), with o the origin and e1 leaning on the horizontalaxis of symmetry of the channel. The computational do-main Ω is the union of a solid domain Ωs, assumed asreference configuration, and a mesh domain Ωm, (see, Fig.1) which represents the domain occupied by the fluid atany instant t of the time interval T . In particular, the soliddomain is composed of a piezoelectric solid Ωsp which,when deformed, develops an electrical field and a struc-tural solid Ωss which has a half cylinder shape to improvevibrations and does not generate any electrical field. Thebalance equations for the solid are written with respect toΩs (material, or Lagrangian formulation), while those forthe fluid are written with respect to Ωm (Arbitrary Eulerian-Lagrangian (ALE) formulation [8]). The channel has heightH and length L, the piezoelectric solid has a length Lp andthickness hp, and the structural solid has length Ls andthickness hs. Both solids are clamped to a circular fixedconstraint of diameter D (the white colored circle in Fig.1). State variables of the model are the material vectorfield us, describing the displacement of the solid, the vectorfield v f , representing the velocity of the fluid, the materialscalar field V , describing the electric potential in the piezo-electric solid and the vector field um, describing the meshdisplacement with respect to the mesh domain Ωmo whichis regenerated at each remeshing. The equations of theproblem consist of two equations for the fluid, the balanceof forces and the conservation of mass, one for the solid,the balance of forces, one additional for the piezoelectric

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Excerpt from the Proceedings of the 2018 COMSOL Conference in Lausanne

solid, the Gauss’s Law and a last one for the mesh displace-ment in the mesh domain usually taken as a Poisson’s likeequation [9]:

ρ f v f +ρ f (∇v f )(v f − um) = divΓ in Ωm ×T ,

divv f = 0 in Ωm ×T ,

ρsus = divS in Ωs ×T ,

divD = 0 in Ωsp ×T ,

div(A∇um) = 0 in Ωmo ×T .

(1)

The upper dot indicates the time derivative, ∇ the nablaoperator, div the divergence operator, ρ f the fluid massdensity, Γ the fluid stress, ρs the solid mass density, S thesolid reference stress, D the electric displacement of thepiezoelectric solid and A the Winslow operator [9]:

(A∇um)i j = Ai jhk(um)h,k. (2)

The fluid is assumed incompressible and linearly viscouswith dynamics viscosity µ f ; its stress Γ is given by:

Γ =−pI+2µ f (sym∇v f )−23

µ f (divv f )I, (3)

where p is the fluid pressure and I is the identity matrix.Both structural and piezoelectric solid are assumed to belinear elastic:

S = FSe, (4)

with F = I+∇us the deformation gradient and:

Se =Ys

1+νsE+

Ys νs

(1+νs)(1−2νs)tr(E)I, (5)

for the structural solid Ωss and:

Se = CE− eT Eel , (6)

for the piezoelectric solid Ωsp. In the structural solid,tr is the trace operator, Ys and νs are respectively theYoung’s modulus and the Poisson ratio of the material,E = 1/2(FT F− I) is the nonlinear strain measure (Green-Lagrange strain) with FT indicating the transpose of F. Inthe piezoelectric solid C is the stiffness tensor, e is thepiezoelectric coupling tensor and Eel =−∇V is the elec-tric field. Parameters of the geometry and the fluid areindicated in Table (1), while parameters of the solids andthe non-zero components of C and e are indicated in Table(2). The electric displacement in the piezoelectric solid isgiven by:

D = eE+ ko kEel , (7)

where ko is the vacuum permittivity and k is the second-order tensor of dielectric relative permittivity measured atconstant applied stress. System of equations (1) is supple-mented with boundary and initial conditions; ∂Ωsm is the

interface between the solid body and the fluid, where FSIboundary conditions are posed:

Tn =−Γn , v f = um = us on ∂Ωsm ×T , (8)

with n normal to the solid boundary and T = S(F∗)−1 theCauchy stress in the solid; on the boundaries of the channel∂Ωm we assign a no-moving condition for the mesh:

um = 0 on ∂Ωm ×T , (9)

Nevertheless, on the channel boundaries we assign addi-tional conditions. In particular, on the top and bottomboundaries of the channel ∂Ωmtb we assign a no slip wallcondition:

v f = 0 on ∂Ωmtb ×T (10)

and on the left boundary of the channel ∂Ωml we assign ahorizontal inlet fluid velocity condition:

v f = (1− exp(−t/τ))U f (Y )e1 on ∂Ωml ×T , (11)

with τ a characteristic time, U the steady inlet velocity andf (Y ) the vertical profile of the inlet flow:

f (Y ) =(y−H/2)(y+H/2)

(−H2/4). (12)

Finally, on the right boundary of the channel ∂Ωmr weassign a outlet condition:

p = 0 on ∂Ωmr ×T . (13)

At the interface between the piezoelectric solid and thestructural solid ∂Ωspb we assign a null reference electricalpotential V = 0, while on the top boundary of the piezo-electric solid ∂Ωspt we assign a potential V = V such thatit holds: ∫

∂Ωspt

D ·n ds =VR

(14)

where R is the external electrical load resistance. The leftand right boundaries of the piezoelectric solid ∂Ωspl areassumed to be insulated from an electrical point of view:

D ·n = 0 on ∂Ωspl ×T . (15)

On the interface between the solid and the fixed circularconstraint ∂Ωsc the displacement of the solid is null:

us = 0 on ∂Ωsc ×T . (16)

Finally, we assign homogeneous initial conditions. All pa-rameters of the model are listed in Tables (1, 2). Reynoldsnumber of simulations is approximately 105.

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Excerpt from the Proceedings of the 2018 COMSOL Conference in Lausanne

0 2 4 6 8 10

0

1

2

3·103

t (s)

P (W/m)

Figure 1: Fig:ref

0 2 4 6 8 10

2

0

2

4·102

t (s)

v (m)

Figure 2: Fig:ref

Channel

H

L

Ls

Lp

D

m @m = @mtb [ @ml [ @mr

@mtb

@ml @mr

a)

b)

Figure 1: a) Initial geometry of the model with the boundariesof the channel where some boundary conditions are applied. Thegray colored domain represents the fluid. b) Zoom of the initialgeometry close to the solid. The green domain is the structuralsolid and the orange domain is the piezoelectric solid which isconnected to an external electrical load resistance not shown inthe figure.

COMSOL settings

The model is solved through a time dependent solver witha time interval from 0 to 10 s. In the time stepping nodewe use the backward differentiation formula (BDF) witha maximum step of 0.01 s, maximum BDF order 2 andminimum BDF order 1. Moreover, we use the PARDISOsolver with the automatic (Newton) nonlinear method. Theautomatic remeshing feature has a distortion condition typecomp1.fsi.I1isoMax< 2. The model has about 3 ·105 de-grees of freedom and takes about 2 days to run a simulationwith a processor 2.3 GHz and RAM 16 GB.

Results

By solving system eq. (1) together with boundary andinitial conditions, using data of Tables (1, 2), we obtainlarge deformations of the solid due to the fluid kineticsenergy which is converted into elastic energy of the solid.In Fig. 2, fluid pressure and vorticity fields over a region ofthe domain are observed at different instants. The vorticityfield is evaluated by taking the third component of the curlof the fluid velocity:

ω = (∇×v f ) · e3. (17)

In particular, a large circular vortex (see Fig. 3) is createdby the presence of the half cylinder. The latter is essential

pf (Pa)

1500 500

!3 (1/s)

500 500

t = 4.4 s

t = 4.5 s

t = 4.6 s

t = 4.7 s

Figure 4: Fig:refFigure 2: Pressure and vorticity fields at different instants. Thesolid exhibits large deformations which are described by the mov-ing mesh and the remeshing features.

to increase vibrations of the solid and harvest more elec-trical power in such configuration. The pressure profileover a horizontal cut line below the solid shows a travelingwave which is not compensated by the pressure above thesolid itself. This pressure difference induces a deformationwhich can be measured through the vertical displacementat the tip of the solid (see Fig. 4a). The displacement, aftera transient state due to the evolution of the inlet profileover time, reaches a stationary oscillatory behavior. Thefrequency of oscillations is evaluated through a frequencyspectrum showed in Fig. 4b for different inlet fluid veloci-ties. The vibration amplitude increases with the inlet fluidvelocity as showed in experimental results [5]. Moreover,also the vibration frequency reasonably increases with theinlet fluid velocity; in particular, we obtain f = 1.4 [Hz]and f = 1.6 [Hz], respectively for U = 0.5 [m/s] blue curveand U = 1.0 [m/s] red curve.

The elastic energy of the piezoelectric solid is trans-formed into electrical energy and harvested through anexternal electrical load resistance. The harvested electricalpower over time is showed in Fig. 5 and evaluated by:

P =V 2

R. (18)

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Excerpt from the Proceedings of the 2018 COMSOL Conference in Lausanne

Figure 3: Top: Zoom of vorticity field and cut line where thepressure is evaluated at different instants. Bottom: Pressure isevaluated at different instants on the cut line showed in the topimage. The pressure moves forward like a wave and induces largedeformations of the solid.

The average harvested electrical power is approximately 0.5[mW/m] which is in agreement with classical experimentalresults [2, 3]. It is worth noting, that the optimal electricalresistance load, which maximize the harvested electricalpower, was not investigated in this work. Indeed, in thiscase, the optimal electrical resistance load depends on thevibration frequency of the solid and therefore changes withthe inlet fluid velocity as demonstrated in Fig. 4. Thisis quite important in order to design a resistive matchingcircuit which optimize the energy harvesting [10].

Conclusions

We presented a model able to simulate a piezoelectric en-ergy harvester immersed in a fluid flow. Results of simu-lation include: the fluid velocity and the pressure field inthe channel, deformation of the solid, electrical potentialin the piezoelectric solid and harvested electrical poweron the external electrical load resistance. A further im-

8

0 2 4 6 8 10

0

1

2

3·103

t (s)

P (W/m)

Figure 1: Fig:ref

0 2 4 6 8 10

2

0

2

4·102

t (s)

v (m)

Figure 2: Fig:ref

Channel

Inlet flowFixed constraintVibrating bilayer beam

Structural layer

Piezoelectric layer

Figure 3: Fig:ref

1

R

P

higher

frequen

cy

lower

frequ

ency

R

f

P

higher

frequen

cy

lower

frequ

ency

0 0.5 1 1.5 2 2.5 3

0

10

20

30

40

f (Hz)

FourierCoeff.

U = 0.5 m/sU = 1 m/s

a)

b)

Figure 4: Vertical displacement of the solid evaluated at thefree tip and its frequency spectrum. a) A stationary oscillatorymovement is obtained after a transient state. b) The frequencyspectrum of the vertical displacement for two different inlet fluidvelocities U = 0.5 [m/s] blue curve and U = 1.0 [m/s] red curve.The peaks represent the frequencies of solid vibration.

0 2 4 6 8 10

0

1

2

3·103

t (s)

P (W/m)

Figure 1: Fig:ref

0 2 4 6 8 10

2

0

2

4·102

t (s)

v (m)

Figure 2: Fig:ref

Channel

Inlet flowFixed constraintVibrating bilayer beam

Structural layer

Piezoelectric layer

Figure 3: Fig:ref

Figure 5: Harvested electrical power on load resistance over time.After the transient state, the harvested average electrical power issteady and not null.

provement of the model may be represented by the use ofRANS-based turbulence models already implemented inCOMSOL which could describe more effectively the fluidflow and its interaction with the solid.

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Table 1: Geometry of the model and physical fluid parameters

L 2 [m] channel lengthH 0.65 [m] channel heightLs 0.11 [m] length of structural solidLp 0.061 [m] length of piezoelectric solidhs 5 ·10−4 [m] structural layer thicknesshp 2.5 ·10−4 [m] piezoelectric layer thicknessD 0.05 [m] fixed constraint diameterρ f 1000

[kg∗m−3

]fluid mass density

µ f 10−3 [Pa∗ s] dynamic fluid viscosityU 1

[m∗ s−1

]fluid inlet velocity

τc 1 [s] dynamic fluid viscosity

Table 2: Piezoelectric and structural solid physical parameters inthe global cartesian base.

ρs 7800[kg∗m−3

]solid mass density

Ys 1011 [Pa] Young’s modulusνs 0.3 Poisson ratio

C11 =C33 1.20∗1011[Pa] stiffness comp. 11C12 =C23 7.51∗1010[Pa] stiffness comp. 12

C13 7.52∗1010[Pa] stiffness comp. 13C22 1.11∗1011[Pa] stiffness comp. 11

C44 =C66 2.10∗1010[Pa] stiffness comp. 44C55 2.25∗1010[Pa] stiffness comp. 55

e16 = e34 12.29[C/m] coupling comp. 16e21 = e23 −5.35[C/m] coupling comp. 21

e22 15.78[C/m] coupling comp. 22k11 = k33 919.1 perm. comp. 11

k22 826.6 perm. comp. 22

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Excerpt from the Proceedings of the 2018 COMSOL Conference in Lausanne