Optimization Computational Model for Piezoelectric Energy Harvesters Considering Material Piezoelectric Microstructure Agostinho Matos, José Guedes, K. Jayachandran, Hélder Rodrigues Contact: ago.matoz@gmail.com 11/09/2014 Instituto Superior Técnico
Motivation Nowadays there are many sources of free energy: a) Natural Energy wind, waves, solar, etc b) Human Technology engines, industrial machines, etc Many of the energy sources cause mechanical vibrations. A piezoelectric material can convert vibrations to power Real world applications can have various types of loadings
Motivation Applications & More...
Motivation To deliver power it is not enough... It is necessary to deliver the required power... A piezofiber composite plate of 2.2 cm 3 produces 120 mw Now in 2014 it can be done 1.73e10 computations per mwh.
Piezoelectric Constitutive Equations & Others S = S E T + d T E k D = d T + ε T E k The electric current goint out the electrode (S φ ) is: I = Q e Q e = S φ n i D i ds For a Resistor, the harvested power: P a = 1 2 R I 2
Piezoelectric Problem Equations Constitutive Equations T ji,j = ρu i D i,i = 0 S ij = u i,j+u j,i ; E 2 i = φ,i Electric Machine Equations, for a Resistor V=RI Boundary Conditions: φ = φ on S φ (electroded part) D j n j = 0 on S D (not electrodes) T ij n i = t j on S T u i = u i on S u S = S φ S D = S u S T
Piezoelectric Harvester Setup Longitudinal Generator Unimorph Cantilever Transverse Generator Bimorph Cantilever i) Yellow and Vi surfaces are electrodes; ii) Dark blue is substrate and light blue is a piezoelectric iii) Orange vector P indicates polarization or z-direction
Non-Ressonance Results The electrical power of one resistance is P a Harvester P a Loading Longitudinal Generator Transverse Generator Cantilever Unimorph 1 2 R wd 3,3 σ lpa 2 Pressure 1 2 R wd 3,2 σ tpa 2 Pressure 1 2 R wd 3,2 σ apa 2 Tip Bending Moment For the bimorph similar expressions to unimorph;
Piezo Materials Piezo Materials : PZT-5H and BaTiO3 - are transversely isotropic (IEEE format) S E in 1e- 12 m^2/n S11 S12 S13 S33 S44 S66 ε T in 8.85e- 12 F/m ε 11 ε 33 PZT-5H 16.5-4.78-8.45 20.7 43.5 42.6 BaTiO3 7.38-1.39-4.41 13.1 16.4 7.46 PZT-5H -274 593 BaTiO3-33.7 93.9 d in 1e-12 C/N d31 d33 d15 For substrate it is used Brass PZT-5H -274 593 741 BaTiO3-33.7 93.9 561
FEM Validation It is compared the power results of the developed equations and ANSYS FEM results; power relative error is inferior to 8.5% Configuration P a 0 (pw) P a Theory_0 (pw) RE (%) L.G. 3.92e-3 3.92e-3 0.00 T.G. 5.05e-4 5.05e-4 0.00 Unimorph 3.23e-4 3.52e-4 8.24 Bimorph Series 4.79e-4 5.14e-4 6.81 Bimorph Parallel 1.92e-3 2.06e-3 6.80
Optimization Algorithm The objective function : Max P a The design variables : (φ, θ, ψ) [313] for each piezoelectric material layer Constraints: (φ, θ, ψ) ε [ 180, 180] degrees Optimization method: simulated annealing
Setup Loadings L.G. And T.G All the loadings are harmonic 1Hz Load Cases for Longitudinal & Transverse Generators: Load Cases P: 10 MPa 10 MPa Load Cases PS: 10 or 40 MPa Shear Maximizing P a is the same as maximizing piezoelectric constants Max d in 1e-12 C/N d31 d33 d34 d35 BaTio3 186 224 166 561 PZT 5H 274 593 48.5 741
Configura tion Plus Shear Load (MPa) Results L.G. And T.G Piezo Mat P a 0 Time (min) N eval φ max θ max ψ max P a max P a max P a 0 Loading Condition (pw) (deg) (deg) (deg) (pw) P.1 L.G. ---- BaTiO 3 3.92e-3 46.2 253-70 50-115 2.15e-2 5.5 P.2 L.G. ---- PZT-5H 1.56e-1 46.7 253 90 180 130 1.56e-1 1.0 P.3 T.G. ---- BaTiO 3 5.05e-4 36.5 190-120 -125 5 1.45e-2 28.7 P.4 T.G. ---- PZT-5H 3.33e-2 47.4 253-10 0-40 3.33e-2 1.0 Configura tion Plus Shear Load (MPa) Piezo Mat P a 0 Time (min) N eval φ max θ max ψ max P a max P a max P a 0 Loading Condition (pw) (deg) (deg) (deg) (pw) PS.1 L.G. 10 BaTiO 3 3.92e-3 45.0 235 50 55 50 6.35e-2 16.2 PS.2 L.G. 10 PZT-5H 1.56e-1 49.1 253-80 180-40 1.56e-1 1.0 PS.3 T.G. 10 BaTiO 3 5.05e-4 48.8 253-180 55-35 3.53e-2 70.0 PS.4 T.G. 10 PZT-5H 3.33e-2 48.6 253-145 180-110 3.33e-2 1.0 PS.5 L.G. 40 BaTiO 3 3.92e-3 47.7 253-140 -55-135 3.35e-1 85.4 PS.6 L.G. 40 PZT-5H 1.56e-1 48.8 253 65 20-45 1.58e-1 1.0 PS.7 T.G. 40 BaTiO 3 5.05e-4 26.5 145 160 50 130 2.25e-1 445.7 PS.8 T.G. 40 PZT-5H 3.33e-2 48.5 253-180 40 50 5.34e-2 1.6
Conclusion & Future Work Non-ressonance with a resistance connected what is desired to increase in the case of a constant stress loading is the piezoelectric constants d ij ; It is necessary to investigate if in ressonance the power will increase too as for out of ressonance When choosing a piezoelectric material for a specific application the loading type must be accounted The piezo material can be modelled as a polycrystallyne one
Homogenization & Future Work A piezoelectric material has a crystalline microstructure. Each crystal or grain has its own orientation with its grain boundaries; the 3D orientation of each single crystal can be knowed using X-ray diffraction contrast tomography; Homogenization theory allows to calculate overall material properties based in the microstructure 3D grains reconstruction
Homogenization & Future Work The homogenization calculates overall material properties of a composite microstructure Optimizing overall material d33 varying material orientation increases d33 114%
Acknowledgements: This work is supported by the Project FCT PT DC/EME-PME /120630/2010? Questions?
Bimorph Series and Parallel Connections