D Zakharov

H Zhang

RS Langley

Y Zi

English

J. Gosliga

D. J. Wagg

Crossref

A Theoretical Description of a Multi-source Energy Harvester

Gosliga, J.

Wagg, D.J.

White Rose Research Online

This is a repository copy of A theoretical description of a multi-source energy harvester.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/140265/Version: Accepted VersionProceedings Paper:Gosliga, J. and Wagg, D.J. orcid.org/0000-0002-7266-2105 (2018) A theoretical description of a multi-source energy harvester. In: Sensors and Instrumentation, Aircraft/Aerospace and Energy Harvesting. 36th IMAC, A Conference and Exposition on Structural Dynamics 2018, 12-15 Feb 2018, Orlando, USA. Conference Proceedings of theSociety for Experimental Mechanics Series, 8 . Springer , pp. 41-47. ISBN 978-3-319-74641-8 https://doi.org/10.1007/978-3-319-74642-5_5© 2019 The Society for Experimental Mechanics, Inc. This is an author produced version of a paper subsequently published in Sensors and Instrumentation, Aircraft/Aerospace andEnergy Harvesting , Volume 8. Uploaded in accordance with the publisher's self-archiving policy.eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. The theory of a prototype multi-source energy harvesterJ. Gosliga∗and D. J. WaggDepartment of Mechanical Engineering, University of Sheffield, S1 3JD, UKOctober 19, 2017AbstractBy harvesting energy from more than one source, it is possible to improve the power output froman energy harvester. In this paper we present an analysis that allows the maximum power absorbedby a multi-source harvester to be found. This is based on an extension to analysis that was previouslyused to derive a power bound for a single source mechanical energy harvester driven by stochasticvibration. Firstly, a single source power bound is derived for a system with thermo-electrical coupling,driven by stochastic time-varying temperature gradients. This power bound is verified using numericalsimulations carried out in MATLAB. This analysis is then extended to a system with thermo-electro-mechanical coupling, driven by both fluctuating temperature gradients and mechanical vibration. Itis shown that the resulting power bound is the sum of the maximum power absorbed by the thermalsystem and the mechanical system. As this power bound is greater than that for a single source system,it demonstrates that a system driven by multiple sources has the potential to absorb more power thana system driven by a single source.1 IntroductionEnergy harvesting is a process for converting energy from one form into another, more useful form. Previouslythis was done with harvesters that could only convert from one form of energy. However, devices that canconvert from multiple sources are now being explored as they can achieve a greater power output for thesame size device. There are two main types, harvesters that use multiple methods to enhance the outputfrom a single source and harvesters that harvest from multiple distinct sources. This paper will define atrue multi-source harvester as one that utilises multiple distinct sources. As there are a huge number ofpossible theoretical combinations, in this paper the scope is restricted to devices using some combination ofthe piezoelectric and pyroelectric effects.One of the first hybrid devices of this type used shape-memory alloys in combination with dielectricmaterials so that piezoelectric and pyroelectric effects could be combined to enhance the amount of energyharvested from a single thermal source [3]. Piezoelectric and pyroelectric effects were combined to enhancethe output from a triboelectric generator in [5]; despite using multiple effects, this device still only harvestsenergy from a single mechanical source. The combination of tribo/piezo/pyro-electric effects was also usedin a device [4] to harvest energy from independent mechanical and thermal sources at the same time, makingthis a true multi-source harvester. In this paper we will explore a device which harvests from both mechanicaland thermal source by utilising the piezoelectric and pyroelectric effects.The upper limit on the power that an energy harvester can absorb is a useful method for benchmarkingdevices, as this will be the limit to how much useful power they can generate. This number does notdepend on the efficiency of the device, and merely represents how much energy is available to a device. Bycomparing the upper limits on the power absorbed, it can be shown that there is more power available whenharvesting from multiple sources. There already exists a power bound for a piezoelectric harvester driven bymechanical vibration [2]. Since no such bound exists for a pyrorelectric harvester, in Section 2, it is necessaryto derive an expression for the power absorbed by a pyroelectric harvester driven by stochastically varyingtemperatures. In Section 2.1, this expression will be verified through numerical simulations. Finally, in∗Address all correspondence to this author jsgosliga1@sheffield.ac.uk.1Section 3, we will derive an expression for the power absorbed by the full harvester from both a mechanicaland a thermal source.2 Deriving a bound on the power absorbed by a pyroelectric sys-temThe theoretical power bound for a mechanical harvester using piezoelectric materials with stochastic vibra-tional input has been shown to beP =πS0xm2. (1)as demonstrated in [2]. P represents the power absorbed by the device; S0x is the amplitude of the powerspectral density of the input vibrations; and m is the total mass of the device. This bound has beenshown to be valid for piezoelectric devices in [1], however it has not been extended to pyroelectric devices.Since pyroelectric devices are driven by a temperature gradient which creates a potential rather than adisplacement, the analysis will differ from that presented by [2] or [1]. Furthermore, the coupling between thevoltage and temperature gradient is not the same as the electromechanical coupling found in piezoelectrics.The state equations for a pyroelectric device in an equilibrium state are as followsCsθ̈ +Rsθ̇ − pu̇ = 0 (2)Cvü+R−1v u̇+ pθ̈ = 0, (3)where Cv is the electrical capacitance; Rv is the electrical resistance; Cs is the analogous thermal capaci-tance; Rs is the analogous thermal resistance; θ is the temperature difference between the device and thesurroundings with reference to an equilibrium state; u̇ = v where v is the instantaneous voltage; and prepresents the pyrolectric coupling coefficient.To write these equations in a state-space form we need to eliminate the second-order term θ̈ in Eq. (3).So it is necessary to substitute in Eq. (2) into (3). This givesCvü+ (R−1v + p2C−1s )u̇− pRsC−1s θ̇ = 0. (4)We can now introduce c which represents the temperature difference between the heat source and thesurroundings. This difference is assumed to fluctuate in such a way that it is equivalent to a stationarywhite-noise input such that E[c̈(t)c̈(t + τ)] = πS0θδ(τ), where S0θ is the amplitude of the power spectraldensity of the temperature fluctuations of the heat source, and δ is the value of the Dirac delta function attime τ . This gives us a pyroelectric system which is driven by a stochastically varying thermal gradient asrepresented byCsθ̈ +Rsθ̇ − pu̇ = Csc̈. (5)where the term on the right-hand side represents an entropy flow into the system.These can be combined and expressed in a form describing the state of the coupled system and theoutputsẋ = Ax +Gζ(t), (6)where the coefficients for the state-space variables areA =0 1 0 00 −C−1s Rs 0 −pC−1s0 0 0 10 pC−1s C−1s Rq 0 −C−1v (R−1v + p2C−1s ). (7)If we assume τ = 0, then E[c̈(t)2] = πS0θE[ζ(t)2] and so c̈(t) =√πS0θ ζ(t). This results in the input matrixG =0√πS0θ00. (8)If we introduce an augmented mass matrix,M =0 0 0 00 Cs 0 00 0 0 00 0 0 Cv, (9)then we can express the power absorbed by the coupled system in a similar fashion to [2], which givesP = (1/2)Tr[MGGT], (10)where Tr is the trace of the resultant matrix. ThereforeP = (1/2)Tr0 0 0 00 πS0θCs 0 00 0 0 00 0 0 0, (11)which in this case leads to the following expression for the maximum power absorbed by the pyroelectricsystemP =πS0θCs2. (12)2.1 Numerical simulation of the pyroelectric systemIn order to verify the analytical results, a numerical simulation of the system was carried out. The systemdescribed by Eqs. (4) and (5) was excited by a fluctuating temperature gradient, such as that shown inFig. 1. The results in Fig. 2, show that the absorbed by the system increases linearly with Cs the thermalcapacitance. The numerical results for the power absorbed match the power bound predicted by Eq. (12).Also included in the figure is total power dissipated. This is the sum of the power dissipated by the resistorand as entropy through the thermal resistance. Since the power dissipated can never exceed the powerabsorbed, the total power lost by the system is limited by Eq. 12.The parameter values used in the simulation are S0θ = 2 × 10−3π−1 ◦K2/s; Cv = 1 × 10−6 F ; Rv =1×106Ω; p = 1×10−10 Rs = 1×106 ◦K2/W ; and Cs was varied between 1×10−2J/◦K2 and 1×10−4J/◦K2.In the thermal simulation, the expression for the power absorbed from the varying temperature gradient isP = E[Csc̈ θ̇] (13)and the amount of power that is dissipated is given byP = E[R−1s θ̇2] + E[R−1v u̇2]. (14)3 Deriving a power bound for a multi-source electro-mechanicalsystemNow that we have found the state equations for a pyroelectric system, we can combine these with theequations for a piezoelectric system. In this way, we can derive a power-bound for a system driven by boththermal and mechanical energy. This will also allow us to examine a system with multiple outputs ratherthan a single output.We will consider a system in which a single piece of dielectric material—displaying both pyroelectricand piezoelectric effects—experiences both varying mechanical displacement and fluctuating thermal gradi-ents. This dielectric material is then connected to a simple resistor-capacitor (RC) circuit, representing theharvesting circuitry. The system can be described using the following system of coupled equationsmÿ + γẏ + ky + du̇ = −mb̈, (15)Csθ̈ +Rsθ̇ − pu̇ = Csc̈ (16)1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2Time (s)-15-10-5051015Temperature gradient (°K)Figure 1: A 0.1s sample of the time history of the stochastically varying input temperature gradient cCvü+ (R−1v + p2C−1s )u̇− pRsC−1s θ̇ − dẏ = 0. (17)where Eq. (15) represents a mechanical system with mass m, damping coefficient γ and stiffness k. Theseequations were derived in a similar fashion to Eqs. (5) and (4).This system is being driven not only by vibration, but also by some form of stochastically varyingtemperature difference. Since these equations represent a dielectric system, there is an piezoelectric elec-tromechanical coupling term d and a pyroelectric thermoelectrical coupling term p. Therefore thermal energyinto the system can either remain as thermal energy, or convert into electrical or mechanical energy. Thesame goes for any mechanical energy into the system. The flow of energy in the system is demonstrated inFig. 3.Energy in the system can be dissipated through mechanical damping or as entropy or dissipated throughthe resistor. The amount of energy dissipated in each case is determined by the damping coefficient γ,the thermal resistance Rs, or the electrical resistance Rv, respectively. In this system, we consider thatthe resistor and other circuitry are separated from the dielectric material. This means that heat dissipatedthrough the resistor can be considered lost to the environment and does not feed back into the thermalsystem. We also consider that the mechanical damping dissipates energy is such a fashion that this is alsocompletely lost to the environment.It is assumed that both the motion of the base, b, and the fluctuation of the temperature gradient, c,can be approximated as stationary white-noise processes, such thatE[b̈(t)b̈(t+ τ)] = πS0xδ(τ), (18)E[c̈(t)c̈(t+ τ)] = πS0θδ(τ). (19)The separate coupled Eqs. (15), (16) and (17), can be combined using generalised state-space variable10-6 10-5 10-4 10-3 10-2Thermal capacitance (J / °K2)10-1010-910-810-710-610-510-4Power (W)Figure 2: The solid line (orange) shows the linear increase in power from increasing thermal capacitance,where the power spectral density of the fluctuating temperature S0θ and other system parameters are heldconstant. It can be seen that the power absorbed (marked by crosses) from the stochastically varyingtemperature gradient c is equal to the total power dissipated by the system (marked by circles)MechanicalThermalElectricalPiezoelectricPyroelectricVibrationFluctuatingtemperature EntropyDampingResistanceFigure 3: A schematic of the system showing arrows into the dashed box representing energy absorbed bythe system from both vibrations and fluctuating temperatures. Energy is transferred within the system viathe piezoelectric and pyroelectric effect. Arrows out of the system (dashed box) represent energy dissipatedthrough the damping, and electrical and thermal resistancex to describe the displacement y, the temperature gradient θ and voltage u wherex =yẏθθ̇uu̇. (20)The new equation describing the state of the coupled system and the outputs then becomesẋ = Ax +Gζ(t), (21)where the coefficients for the state-space variables areA =0 1 0 0 0 0−m−1k −m−1γ 0 0 0 −m−1d0 0 0 1 0 00 0 0 −pC−1s Rs 0 −C−1s0 0 0 0 0 10 pC−1v d 0 C−1v C−1s Rs 0 −p2C−1v (R−1v + C−1s )(22)and the input matrixG =0−√πS0x0√πS0θ00. (23)Again we can introduce an augmented mass matrixM =0 0 0 0 0 00 m 0 0 0 00 0 0 0 0 00 0 0 Cs 0 00 0 0 0 0 00 0 0 0 0 Cv. (24)Using this augmented matrix M, we can express the power absorbed by the coupled system in a similarway to that derived previouslyP = (1/2)Tr[MGGT], (25)where Tr is the trace of the resultant matrix. ThereforeP = (1/2)Tr0 0 0 0 0 00 πS0xm 0 0 0 00 0 0 0 0 00 0 0 πS0θCv 0 00 0 0 0 0 00 0 0 0 0 0, (26)which in this case leads to the following expression for the combined maximum powerP =πS0xm2+πS0θCv2. (27)This theoretical limit for a system with multiple sources appears to be the sum of Eqs. (1) and (12), whichdescribe the thermal power bound and the mechanical power bound respectively.4 ConclusionThe aim of this paper was to explore devices which utilise multiple sources of energy in order to improveoutput. Upper limits on the power that a harvester could absorb were used as it provides a measure withwhich to benchmark devices. If the upper limit of a theoretical device that harvests from multiple sourcesis greater than that for a single source, then it represents a potential improvement. In order to comparethe upper limits for a multi-source harvester, it was necessary to find limits for different energy types. Inthis case, a limit on the amount of thermal energy that could be harvested using the pyroelectric effect wasfound. This showed that the same theoretical framework used to find the limit for a mechanical harvestercould be extended to different types of harvester. Once the limit for a pyroelectric harvester had beenfound, the expression was validated using a numerical simulation. The equations of state for a system withinputs from multiple sources were then derived. This system used both the piezoelectric and the pyroelectriceffect in combination. This allowed for both mechanical and thermal energy to be converted into electricalenergy. A similar state-space analysis was carried out on these equations. This combined system was foundto have an upper limit on the power absorbed that was the sum of the power-bounds for the mechanicaland thermal systems individually. This demonstrates that there is more power available to this type ofmulti-source harvester than the single source harvesters.It is believed that this theoretical framework can be extended to harvesters of other forms of energy, sothat the maximum possible power available in a given environment can be quantified. As a result it wouldbe possible to make a systematic comparison of which combination of energy sources would provide the mostpotential power in a given system.References[1] D. Hawes. Nonlinear Stochastic Vibration Analysis for Energy Harvesting and Other Applications. PhDthesis, University of Cambridge, 2016.[2] R. S. Langley. A general mass law for broadband energy harvesting. Journal of Sound and Vibration,333(3):927–936, 2014.[3] D. Zakharov, B. Gusarov, E. Gusarova, B. Viala, O. Cugat, J. Delamare, and L. Gimeno. Combined Py-roelectric, Piezoelectric and Shape Memory Effects for Thermal Energy Harvesting. Journal of Physics:Conference Series, 476(1):012021, 2013.[4] H. Zhang, S. Zhang, G. Yao, Z. Huang, Y. Xie, Y. Su, W. Yang, C. Zheng, and Y. Lin. SimultaneouslyHarvesting Thermal and Mechanical Energies based on Flexible Hybrid Nanogenerator for Self-PoweredCathodic Protection. ACS Applied Materials and Interfaces, 7(51):28142–28147, 2015.[5] Y. Zi, L. Lin, J. Wang, S. Wang, J. Chen, X. Fan, P. K. Yang, F. Yi, and Z. L. Wang. Triboelectric-pyroelectric-piezoelectric hybrid cell for high-efficiency energy-harvesting and self-powered sensing. Ad-vanced Materials, 27(14):2340–2347, 2015.

A theoretical description of a multi-source energy harvester

https://eprints.whiterose.ac.uk/140265/1/Experimental-Multi-source-Energy-HarvesterV3.pdf

A theoretical description of a multi-source energy harvester

Abstract

Similar works

Full text

Available Versions

Crossref

White Rose Research Online