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 𝑑𝑛 3 produces 120 mW Now in 2014 it can be done 1.73e10 computations per mWh.
Piezoelectric Constitutive Equations & Others 𝑈 + 𝑒 𝑈 𝐹 𝑙 𝑇 = 𝑇 𝐹 𝐸 = 𝑒 𝑈 + 𝜁 𝑈 𝐹 𝑙 The electric current goint out the electrode ( 𝑇 𝜚 ) is: 𝐽 = −𝑅 𝑓 𝑅 𝑓 = −𝑜 𝑗 𝐸 𝑗 𝑒𝑇 𝑇 𝜚 1 2 𝑆 𝐽 2 For a Resistor, the harvested power: 𝑄 𝑏 =
Piezoelectric Problem Equations Constitutive Equations 𝑈 𝑘𝑗,𝑘 = 𝜍𝑣 𝑗 𝐸 𝑗,𝑗 = 0 𝑣 𝑗,𝑘 +𝑣 𝑘,𝑗 𝑇 𝑗𝑘 = ; 𝐹 𝑗 = −𝜚 ,𝑗 2 Electric Machine Equations, for a Resistor V=RI Boundary Conditions: 𝑝𝑜 𝑇 𝜚 (electroded part) 𝜚 = 𝜚 𝐸 𝑘 𝑜 𝑘 = 0 𝑝𝑜 𝑇 𝐸 (not electrodes) 𝑝𝑜 𝑇 𝑈 𝑈 𝑗𝑘 𝑜 𝑗 = 𝑢 𝑘 𝑣 𝑗 = 𝑣 𝑗 𝑝𝑜 𝑇 𝑣 𝑇 = 𝑇 𝜚 ∪ 𝑇 𝐸 = 𝑇 𝑣 ∪ 𝑇 𝑈
Piezoelectric Harvester Setup Longitudinal Generator Unimorph Cantilever Bimorph Cantilever Transverse Generator 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 𝑄 𝑏 𝑄 Harvester 𝑏 Loading Longitudinal Generator 1 2 2 𝑆 𝑥𝑒 3,3 𝜏 𝑚𝑞 𝐵 Pressure 1 2 2 𝑆 𝑥𝑒 3,2 𝜏 𝑢𝑞 𝐵 Transverse Generator Pressure 1 2 2 𝑆 𝑥𝑒 3,2 𝜏 𝑏𝑞 𝐵 Cantilever Unimorph Tip Bending Moment For the bimorph similar expressions to unimorph;
Piezo Materials Piezo Materials : PZT-5H and BaTiO3 - are transversely isotropic (IEEE format) 𝑻 𝑭 in 1e- 𝜻 𝑼 in 8.85e- 𝜻 𝟐𝟐 𝜻 𝟒𝟒 12 m^2/N S11 S12 S13 S33 S44 S66 12 F/m PZT-5H 16.5 -4.78 -8.45 20.7 43.5 42.6 PZT-5H -274 593 BaTiO3 7.38 -1.39 -4.41 13.1 16.4 7.46 BaTiO3 -33.7 93.9 d in 1e-12 d31 d33 d15 C/N PZT-5H -274 593 741 BaTiO3 -33.7 93.9 561 For substrate it is used Brass
FEM Validation It is compared the power results of the developed equations and ANSYS FEM results; power relative error is inferior to 8.5% 𝑸 𝒃𝟏 𝑸 𝒃𝑼𝒊𝒇𝒑𝒔𝒛_𝟏 |RE Configuration (𝒒𝒙) (%)| (pw) 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 4.79e-4 5.14e-4 6.81 Series Bimorph 1.92e-3 2.06e-3 6.80 Parallel
Optimization Algorithm The objective function : Max 𝑄 𝑏 The design variables : (𝜚, 𝜄, 𝜔) [313] for each piezoelectric material layer Constraints: (𝜚, 𝜄, 𝜔) 𝜗 [−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 PS : Load Cases P : 10 or 40 MPa 10 MPa 10 MPa Shear Maximizing 𝑄 𝑏 is the same as maximizing piezoelectric constants Max d in 1e-12 d31 d33 d34 d35 C/N BaTio3 186 224 166 561 PZT 5H 274 593 48.5 741
Results – L.G. And T.G Configura Shear 𝑸 𝒃𝟏 𝝔 𝒏𝒃𝒚 𝜾 𝒏𝒃𝒚 𝝎 𝒏𝒃𝒚 𝑸 𝒃𝒏𝒃𝒚 𝑸 𝒃𝒏𝒃𝒚 tion Plus Piezo Time 𝑶 𝒇𝒘𝒃𝒎 Load 𝑸 𝒃𝟏 Mat (min) Loading (deg) (deg) (deg) (pw) (pw) (MPa) Condition 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 Shear 𝑸 𝒃𝟏 𝝔 𝒏𝒃𝒚 𝜾 𝒏𝒃𝒚 𝝎 𝒏𝒃𝒚 𝑸 𝒃𝒏𝒃𝒚 𝑸 𝒃𝒏𝒃𝒚 tion Plus Piezo Time 𝑶 𝒇𝒘𝒃𝒎 Load 𝑸 𝒃𝟏 Mat (min) Loading (deg) (deg) (deg) (pw) (pw) (MPa) Condition 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 𝑒 𝑗𝑘 ; 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
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