- Email: [email protected]
Energy harvesting from transverse ocean waves by a piezoelectric plate
International Journal of Engineering Science 81 (2014) 41–48
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Energy harvesting from transverse ocean waves by a piezoelectric plate X.D. Xie a,b, Q. Wang b,⇑, N. Wu b a b
School of Mechanical and Automotive Engineering, Hubei University of Arts and Science, Xiangyang, Hubei 441053, China Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, Manitoba R3T 5V6, Canada
a r t i c l e
i n f o
Article history: Received 19 February 2014 Received in revised form 24 March 2014 Accepted 9 April 2014
Keywords: Energy harvesting Ocean waves Transverse wave motion Piezoelectric harvester Root mean square (RMS)
a b s t r a c t An ocean wave energy harvester from the transverse wave motion of water particles is developed by the piezoelectric effects. The harvester is made of two horizontal cantilever plates attached by piezoelectric patches and fixed on a vertical rectangular column. To describe the energy harvesting process, a mathematical model is developed to calculate the output charge and voltage from the piezoelectric patches according to the Airy linear wave theory and the elastic beam model. The influences on the root mean square (RMS) of the generated power from the piezoelectric patches, such as the ocean depth, the harvester location under the ocean surface, the length of the cantilevers, the wave height, and the ratio of wave length to ocean depth, are discussed. Results show that the RMS increases with the increase in the length of cantilevers and the wave height, and decrease in the distance of the ocean surface to the cantilevers and the ratio of the wave length to ocean depth. As a result, an optimum ocean depth is obtained to achieve a maximum RMS at different harvester locations under the ocean surface. A value of the power up to 30 W can be realized for a practical transverse wave with the values of the ocean depth, wave length, wave height and harvester location under the ocean surface to be 10.6 m, 21.2 m, 4 m, and 2 m, respectively. This research develops a novel technique leading to efficient and practical energy harvesting from transverse waves by piezoelectric energy harvesters that could be easily fixed on an offshore platform. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Energy crisis and environmental problems such as global warming and atmospheric pollution have prompted people to study technologies of exploiting new energy. In the past decade, applications of piezoelectric materials in energy generation and harvesting have received much attention owing to their unique characteristics (Anton & Sodano, 2007; Cook-Chennault, Thambi, & Sastry, 2008; Kim, Kim, & Kim, 2011). A piezoelectric material is a type of smart materials exhibiting a direct piezoelectric effect of the internal generation of electrical charge resulting from an applied mechanical force and a reverse piezoelectric effect of the internal generation of a mechanical strain resulting from an applied electrical field http://en.wikipedia.org/wiki/Piezoelectricity, (Duan, Quek, & Wang, 2005; Wang & Quek, 2000, 2002; Wang, Quek, Sun, & Liu, 2001). Since early 2000, an amount of energy generators and harvesters by using piezoelectric effects, such as piezoelectric coupled cantilevers, shells, cymbals and stacks, with various designs of electrical circuits have been developed ⇑ Corresponding author. Tel.: +1 204 474 6443. E-mail address: [email protected] (Q. Wang). http://dx.doi.org/10.1016/j.ijengsci.2014.04.003 0020-7225/Ó 2014 Elsevier Ltd. All rights reserved.
42
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48
(Cook-Chennault et al., 2008; Kim et al., 2011; Shu, Lien, & Wu, 2007; Wickenheiser, Reissman, Wu, & Garcia, 2010; Xie, Wang, & Wu, 2014; Xie, Wu, Yuen, & Wang, 2013). In addition, many research works were conducted on optimizing designs of piezoelectric coupled structures (Shenck & Pradiso, 2001; Wang & Wang, 2000; Yang, Yeo, & Priya, 2012) for more effective energy harvesting. These devices aimed to achieve practical portable microelectromechanical systems (MEMS) Cook-Chennault et al., 2008 via collecting energy of human actives, such as men’s working (Wang & Wu, 2012) and the bikes’ motions (Friswell & Adhikari, 2010). Recently, many research studies were conducted on piezoelectric energy harvesting from ambient vibrations (Khaligh, Zeng, & Zheng, 2010; Li, Yuan, & Lipson, 2011; Nader, Zhu, & Cooper, 2012; Vatansever, Hadimani, Shah, & Siores, 2011; Wu, Wang, & Xie, 2013) by natural energies such as solar energy, wind energy, and ocean-wave energy. Vatansever et al. (2011) evaluated energy harvesting from ceramic based piezoelectric fiber composite structures subjected to different wind loadings. Li et al. (2011) introduced a bio-inspired piezo-leaf architecture, which was in dangling cross-flow stalk that converted wind energy into electrical energy, by wind-induced flutter motion. Wu et al. (2013) developed an effective and compact wind energy cantilever harvester subjected to a cross wind. Sufficient electrical energy output as high as 2 watts was realized by tuning the resonant frequency of the harvester with a proof mass on the tip of the cantilever. As is indicated, wind energy has a larger energy density than the solar energy, and ocean-wave energy is more persistent and spatially concentrated than the wind energy (Falnes, 2007). Generally, the density of the ocean wave-energy is 4–30 times of that of wind energy (Zhang & Lin, 2011). Therefore, it is more efficient and effective to develop new technologies in harvesting the ocean wave-energy with piezoelectric materials. In view of considerable large powers from ocean wave motions, which can easily exceed 50 kW per meter of wave front (Murray & Rastegar, 2009), harvesting energy from ocean waves to electrical energy by piezoelectric effects has long been pursued as an alternative or self-contained power source. Zurkinden, Campanile, and Martinelli (2007) designed an ocean wave energy converter consisting of a flexible foam substrate attached by piezoelectric layers and simulated the efficiency of converting wave energy to electric energy by considering influences from different aspects such as the free surface wave, the fluid–structure–interaction, the mechanical energy input to the piezoelectric material, and the electric power output using an equivalent open circuit model. Taylor, Burns, Kammann, Powers, and Welsh (2001) developed a flexible polyvinylidene fluoride (PVDF) ‘‘eel’’ device to convert the mechanical energy in ocean or river water flows into electricity using the regular traveling vortex behind the bluff body to strain the piezoelectric elements. Using a similar principle, Li and Lipson (2009) explored a ‘‘piezo-leaf’’ energy-harvesting system where the PVDF strip of the ‘‘eel’’ system was replaced by a PVDF cantilever with a large triangular plastic ‘‘leaf’’ attached to the free end of the cantilever to improve the power generation. Burns (1987) provided a piezo device consisted of a buoy floating on the ocean surface, a few anchor chains fixed on the ocean-bed, and an array of piezoelectric micro thin films between the buoy and chains, and showed that the device can generate electric power when the piezoelectric films bear tension and compression alternatively duo to the up and down motion of the buoy. Taylor (1980) developed a piezoelectric device made of a buoy, the supporting structure, and piezoelectric layered sheets floating on the surface and anchor chains. The dimensions of each piezoelectric layered sheet were designed according to the wave length to improve the efficiency of wave energy conversion. Murray and Rastegar (2009) presented a two-stage electrical energy generators with two decoupled systems using the mechanism of strumming a guitar, in which low-frequency heaving-buoy can successively excite an array of vibratory elements (secondary system) into resonance and wave energy can be efficiently harvested using piezoelectric elements from the resonant secondary system. Previous studies on the source of the ocean wave energy harvesting with piezoelectric materials can be classified into two main categories: (1) vibrations caused by small longitudinal wave motions near the seabed or the vortex caused by the bluff body fixed on the seabed, and (2) vibrations of heaving-based buoy on the sea surface. In all these cases, only limited electric energy could be generated since the wave motions were not used directly and efficiently. Meanwhile, there has been very little amount of work dedicated to the theoretical framework of the energy harvesting from the transverse ocean wave motions using piezoelectric coupled structures. Therefore, a comprehensive model is indispensable in designs of the piezoelectric harvesters. In the monochromatic linear plane waves in deep water, particles near the surface move in circular paths, making water waves a combination of longitudinal and transverse wave motions. As the depth below the free surface increases, the radius of the circular motion decreases. At a depth equal to half the wavelength, the orbital movement has decayed to less than 5% of its value at the surface (http://en.wikipedia.org/wiki/Ocean_wave). In other words, in deep oceans, the energy from the longitudinal wave motion near the surface decays exponentially, and only transverse wave motions could be used efficiently by a harvester under the ocean. Therefore, it is anticipated that a more efficient energy harvesting technique can be developed to fully utilize the transverse wave motion in contrast to the aforementioned techniques that benefit only from the vertex, longitudinal wave vibration near the ocean-bed or the heaving buoy. In order to achieve this goal, a mathematical model is developed to investigate and calculate the energy harvesting from a simple and portable piezoelectric cantilever that can be easily fixed on a fixed structure under ocean or floating structures such as an offshore platform (see Fig. 1(b)). The collected electrical energy is realized by the electromechanical coupling effect of the piezoelectric patches from the transverse wave motion near the surface of ocean according to the linear Airy wave theory and the Euler–Bernoulli beam theory. The energy harvesting under different conditions, such as the cantilever location under ocean surface, the ocean depth, the wave height, the length of the cantilevers and the ratio of the wave length to the ocean depth, is calculated and discussed. It is expected that the device directly makes full use of the energy from the transverse wave motion near the surface of ocean to generate higher electric power that is sufficient for small power electric appliances of offshore platform.
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48
43
Fig. 1. Set-up of a piezoelectric energy harvester.
2. Energy harvesting from ocean waves by a piezoelectric harvester A mathematical model is developed to describe the principle of a piezoelectric energy harvester by exploring the transverse motions of water particles in ocean waves. Fig. 1 illustrates schematically a piezoelectric energy harvester, consisting of two cantilever substrates attached by piezoelectric patches and fixed on a vertical rectangular column, subjected to a transverse wave pressure from water particles. From this configuration, it is clearly seen that the design only uses the transverse motions for energy harvesting. Piezoelectric patches with equal length, denoted by a, are mounted one by one on both faces of the two cantilevers with a length of l (see Fig. 1). The distance from the bottom of the cantilever to the end of the mth piezoelectric patch is l1; and h1 and h are the thicknesses of the piezoelectric patch and the cantilever, respectively. The width of the cantilever and the piezoelectric patches is b. According to the Airy linear wave theory, the transverse wave pressure applied to the harvester at height z, counting from the bottom of the ocean, can be modeled as (Zhu, 1991):
fV ¼ (
1 @wðx; tÞ @wðx; tÞ @ 2 wðx; tÞ uz C D qw b uz þ C M qw bhaz C m qw bh ; 2 @t @t @t 2 sinh kðzþdÞ sinðkx sinh kd 02 H sinh kðzþdÞ cosðkx 2 sinh kd
uz ¼ x0 H2 az ¼ x
x0 tÞ ; x0 tÞ
ð1Þ
ð2Þ
where uz and az are the transverse velocity and acceleration of water particles in the ocean, respectively; x is the position in the longitudinal direction of ocean waves; z is the position in the transverse direction from the ocean surface; CD, CM, and Cm are coefficients of the drag force and inertia forces of the cantilever and the added mass, respectively; qw is the density of the ocean water; w(x, t) is the transverse displacement function of the cantilever (0 6 x 6 l); H, k and x0 are the height, wave number and angular frequency of the ocean wave motion, respectively; d is the ocean depth. In practice, it is reasonable to assume uz Ew(x, t)/Et at z = z1, so Eq. (1) can be simplified as follows:
fV ¼
1 @ 2 wðx; tÞ C D qw b uz juz j þ C M qw bhaz C m qw bh 2 @t2
¼ K 1 ðzÞ sinðkx x0 tÞj sinðkx x0 tÞj þ K 2 ðzÞ cosðkx x0 tÞ K 3
@ 2 wðx; tÞ ; @t2
ð3Þ
44
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48 2
kðzþdÞ kðzþdÞ where K 1 ðzÞ ¼ 18 C D qw bx02 H2 sinh , K 2 ðzÞ ¼ 12 C M qw bhx02 H sinh , and K3 = Cmqwbh. It is worth noting that the transverse sinh kd sinh2 kd
pressure, fv, is a complex function of the space coordinates (x, z) and time coordinate (t), which includes a drag force (the first item on the right side of Eq. (3)) and an inertial force (the last two items on the right side of Eq. (3)) that are induced by the water particles flowing around a body. Therefore, the mathematic solution of the piezoelectric coupled cantilever loaded by the transverse wave pressure cannot be obtained from a standard procedure and requires the following investigation. According to the Euler–Bernoulli beam theory, the governing equation of the piezoelectric energy harvester subjected to the transverse wave pressure is expressed as:
EI
@4w @2w þ ðK þ q AÞ ¼ f ðx; tÞ ¼ K 1 ðzÞ sinðkx x0 tÞj sinðkx x0 tÞj þ K 2 ðzÞ cosðkx x0 tÞ; 3 1 @x4 @t2
ð4Þ
where q1 is the material density of the cantilever substrate; EI and A are the flexural rigidity and cross section area of the cantilever substrate, respectively. It is assumed that the piezoelectric patches are mounted tightly on the surface of the cantilever. Due to the bending motion of the cantilever, the two faces of the beam exhibit positive and negative axial strains alternatively. As a result, the generated charge Q ig and voltage V ig on the surface of the ith piezoelectric patch is hence provided as (Lee & Moon, 1990):
Q ig ðtÞ
! e31 bðh þ h1 Þ @wðx; tÞ @wðx; tÞ ; ¼ 2 @x x¼ia @x x¼ði1Þa
V ig ðtÞ ¼
! e31 ðh þ h1 Þ @wðx; tÞ @wðx; tÞ ; @x x¼ia @x x¼ði1Þa 2C 0V
ð5Þ
ð6Þ
where e31 is the piezoelectric constant; CV is the electrical capacity of the piezoelectric patches; and C 0v is the electrical capacity per unit width of the piezoelectric patches (e.g. C 0V = CV/b). To obtain the displacement field, w(x, t), in Eq. (4), the vibration governing equation should be solved. First, the free vibration solution of the piezoelectric energy harvester in Eq. (4) can be obtained:
WðxÞ ¼ C 1 cosh sx þ C 2 sinh sx þ C 3 cos sx þ C 4 sin sx 0 6 x 6 l;
ð7Þ
where W(x) is the modal function of the cantilever of a particular mode; C1, C2,. . . , C4 are the corresponding unknown coefficients and s are given as
s¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 ðK 3 þ q AÞx 1 ; EI
ð8Þ
and x is the angular frequency of the structure. The boundary conditions of the piezoelectric energy harvester are given as:
WðxÞ ¼ 0;
dWðxÞ dx
2 WðxÞ EI d dx 2
3 WðxÞ EI d dx 3
¼ 0;
¼ 0;
if x ¼ 0;
¼ 0; if x ¼ l:
ð9Þ
Substituting Eq. (7) into these boundary conditions of the piezoelectric energy harvester leads to four linear equations, from which we can easily obtain the solution of the nth normal mode of the cantilever, Wn(x), corresponding to the nth natural frequency, xn, of the piezoelectric energy harvester. The forced vibration of the piezoelectric energy harvester subjected to the transverse wave pressure can thus be provided as follows:
wðx; tÞ ¼
1 X W n ðxÞ qn ðtÞ;
ð10Þ
n¼1
where qn(t) is the generalized coordinate, which is determined by the distributed transverse wave pressure applied to the harvester given in Eq. (3).
qn ¼
1 ðK 3 þ q1 AÞBn xn
Z
t
Q n ðsÞ sin xn ðt sÞds;
ð11Þ
0
Rl Rl where Q n ðsÞ ¼ 0 f ðx; sÞW n ðxÞdx and Bn ¼ 0 W 2n ðxÞdx. Finally, by substituting Eqs. (10) and (11) into Eqs. (5) and (6), the generated charge Q ig ðtÞ and voltage V ig ðtÞ from the ith piezoelectric patch bonded on the piezoelectric energy harvester subjected to transverse wave pressure at time t can be solved:
Q ig ðtÞ
! 1 e31 bðh þ h1 Þ X dW n ðxÞ dW n ðxÞ qn ðtÞ; ¼ 2 dx x¼ia dx x¼ði1Þa n¼1
ð12Þ
45
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48
V ig ðtÞ ¼
! 1 e31 ðh þ h1 Þ X dW n ðxÞ dW n ðxÞ qn ðtÞ; dx x¼ia dx x¼ði1Þa 2C 0V n¼1
ð13Þ
where 1 6 i 6 m, and m is the number of the piezoelectric patches mounted on one face of one of the cantilevers. Finally, the expression of RMS of the electric power is provided. When the transverse wave oscillates for a period of T, the RMS of the generated electric power from time 0 to T can be given as:
prms e
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 T ¼ ½pe ðtÞ2 dt ; T 0
ð14Þ
where pe(t) is the total generated electric power of all the piezoelectric patches mounted on the cantilevers at time t P dQ ig ðtÞ i (0 < t < T) and it is given by pe ðtÞ ¼ 4 m i¼1 dt V g ðtÞ. To compute the RMS of the generated electric power, the duration can be partitioned into j time steps with a sufficiently short time interval Dt. As a result, the expression in Eq. (14) can be rewritten in a discrete form for computation:
prms e
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u j u X Dt ¼ 4t ð½Pe ðt i Þ2 þ ½Pe ðt i1 Þ2 Þ: 2ðT DtÞ i¼2
ð15Þ
3. Simulations and discussions In this section, we investigate the effectiveness and efficiency of the harvester by examining the generated power. Particularly, we study the effects of the ocean depth (d), the wave height (H), the ratio of wave length to ocean depth (L/d), the harvester location under the ocean surface (z1), and the length of the cantilever (l) on the RMS of the power harvesting. The dimensions and materials properties of the energy harvester in the simulations are provided in Table 1. The piezoelectric patches and the host cantilever are made of PZT4 (lead zirconate titanate) and steel, respectively. Fig. 2(a–c) shows the variation of the RMS of the generated power versus the ocean depth d, under wave height H = 2 m, 3 m, and 4 m and harvester location under the ocean surface z1 = 2 m, 3 m, and 4 m, respectively. The dimensions of the energy harvester are set to be: h = 0.007 m, h1 = 0.0003 m, b = 1 m, l = 1.2 m, l1 = 2l/3, a = 0.05 m, and L = 2d. From the simulations, we can find the following observations. First, the RMS of the generated power increases nonlinearly till a maximum value before a smooth reduction, when the ocean depth, d, changes from 3 m to 30 m. Hence, there is an optimal ocean depth at a given harvester location under the ocean. It is seen from Fig. 2(a–c), the optimal ocean depths corresponding to the maximum RMS at H = 4 m are 10.6 m, 14.8 m, and 20.4 m, respectively at the harvester locations under the ocean surface of 2 m, 3 m and 4 m. Second, RMS increases nonlinearly with the decrease of the harvester location under the ocean surface (z1). The observations are results of the wave force, which is expressed by the square of velocity and acceleration of water particles along the ocean depth shown in Fig. 3 and given explicitly by Eq. (3). The findings provide a guideline on choosing ocean depth and optimal locations of energy harvesters for high efficiency of an energy harvester with a certain size. In addition, when the ocean depth is very small, e.g. d < 3 m at z1 = 2 m, and d < 6 m at z1 = 3 m, and 4 m, the RMS tends to zero. This finding is owing to the fact that the harvester location is near the ocean bottom where the transverse wave motion is very small. On the other hand, when the ocean depth is very large, e.g. d > 30 m, the RMS again is found to decrease obviously since the wave at deeper ocean usually is with larger wave length (L = 2d). Such larger wave length is equivalent to a smaller angular frequency of the ocean wave motion leading to smaller amplitudes of velocity and acceleration of the water particle, as seen in Eq. (2). Third, the RMS increases nonlinearly with the increase in the wave height and results show that when wave heights are 2 m, 3 m, and 4 m, the highest RMSs are 1.5 W, 7.4 W, and 23.3 W, respectively. This finding is clearly relevant to the fact of a nonlinear increase of the wave force with the increase in the wave height as shown in Eq. (3).
Table 1 Material properties and dimensions of the piezoelectric energy harvester. Parameters
Host beam (steel)
l (m) l1 (m) a (m) b (m) h (m) h1 (m) Young’s modulus (N/m2) Mass density (kg/m3) e31 (C/m2) Cv (nF)
1.2 2 l/3 1 0.007 or 0.01 2.1e11 7800
Piezoelectric patches (PZT4)
0.05 1 0.0003 7.8e10 7500 2.8 0.75 for the piezoelectric patch with the geometry of 0.01, 0.06, 0.0003 m
46
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48 25
h=0.007m,h1=0.0003m,b=1m,l=1.2m,z1=-2m,L=2d
RMS of Power(Watt)
20 H=4m 15 10 H=3m
5
H=2m 0 0
5
10
15 Ocean depth(m)
20
25
30
(a) z1=-2m.
10
h=0.007m,h1=0.0003m,b=1m,l=1.2m,z1=-3m,L=2d
RMS of Power(Watt)
8
H=4m
6 4 H=3m 2 H=2m 0 5
10
15 20 Ocean depth(m)
25
30
(b) z1=-3m.
RMS of Power(Watt)
5
h=0.007m,h1=0.0003m,b=1m,l=1.2m,z1=-4m,L=2d H=4m
4 3 2
H=3m 1 H=2m 0 5
10
15 20 Ocean depth(m)
25
30
(c) z1=-4m.
Fig. 2. RMS of the electric power under different wave height versus the ocean depth.
2
h=0.007m,h1=0.0003m,b=1m,l=1.2m,z1=-2m,H=2m,L=2d
1.5
Amplitude of a z ( m s 2 )
RMS of Power (Watt )
1
Amplitude of u z2( m 2 s 2 )
0.5
0 0
5
10
15 Ocean depth(m)
20
25
30
Fig. 3. RMS of the electric power and the amplitudes of square of velocity and acceleration of water particles versus the ocean depth.
Fig. 4 illustrates the effect of the ratio of the wavelength to the ocean depth, L/d, on the RMS with the following geometric parameters: h = 0.007 m, h1 = 0.0003 m, l = 1.2 m, l1 = 2/3l, b = 1 m, a = 0.05 m, and H = 4 m. This figure shows a smooth decrease of the RMS with an increase in the ratio. This finding indicates that a larger harvesting energy requires an ocean wave with a smaller wave length at a certain ocean depth based on the theoretical model. It can be seen from Fig. 4 that when the ratio of L/d is 2, the highest RMS of generated power are 23.3 W, 8.4 W, 4.1 W with the conditions of d = 10.6 m, z = 2 m; d = 14.8 m, z = 3 m, and d = 20.4 m, z = 4 m. This indicates that generated RMS can be in the range of 8–24 W when the harvester location under the ocean surface is 2–3 m with ocean depths of 10–15 m.
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48
47
25
RMS of Power(Watt)
d=10.6m,z1=-2m
h=0.007m,h1=0.0003m,b=1m,l=1.2m,H=4m
20 15 10
d=14.8m,z1=-3m
5
d=20.4m,z1=-4m
0 2
2.5
3
3.5 4 4.5 5 Ratio of wave length to ocean depth
5.5
6
Fig. 4. RMS of the electric power versus the ratio of wave length to the ocean depth.
30
h=0.01m,h1=0.0003m,b=1m,d=10.6m,z1=-2m,H=4m,L=2d
RMS of Power(Watt)
25 20 15 10 5 0 1
1.1
1.2
1.4 1.3 Length of cantilever(m)
1.5
1.6
1.7
Fig. 5. RMS of the electric power versus the length of the cantilever.
Fig. 5 illustrates the variations of the RMS versus the cantilever length. The dimensions of the energy harvester in this simulation are set to be, h = 0.01 m, h1 = 0.0003 m, l1 = 2/3l, b = 1 m, d = 10.6 m, L = 2d, a = 0.05 m, z = 2 m, and H = 4 m. It can be found that the RMS increases with an increase in the length of the harvester. The reason for this phenomenon is obvious that longer cantilever would make better use of wave motions, and hence would increase the harvested power. It can be seen from Fig. 5 that the RMS will increase from 2.5 W to 30 W when the cantilever length changes from 1 m to 1.7 m. Hence, an increase in the harvester length would increase the efficiency significantly. It should be noted that all the RMS of the generated electric power above are calculated with the condition that the transverse velocity of water particles is much higher than the velocity of the free end of the cantilever, as indicated in the theoretical model. From our simulations, we find that even under this condition, the simple and portable piezoelectric device still generates a sufficiently large electric power to supply lamp or radio-recorder, etc. In practice, the piezoelectric energy harvester subjected to transverse ocean wave pressure could generate larger electric power when the wave height is higher and/ or the cantilever length is longer.
4. Conclusion A simple and portable ocean wave energy harvester using the transverse wave motion of water particles is developed and a corresponding mathematical model is developed to calculate the output charge and voltage from the piezoelectric patches according to the Airy linear wave theory and the classical elastic beam model. The RMS of the electric power generated from piezoelectric energy harvester is defined and the results show that the RMS increases with the increase in the length of cantilevers and the wave height and the decrease in the distance of the ocean surface to the cantilevers, and the ratio of wave length to ocean depth. For an energy harvester structure with geometric parameters of h = 0.01 m, h1 = 0.0003 m, b = 1 m, l = 1.2 m, l1 = 2/3l, d = 10.6 m, L = 2d, H = 4 m, and a = 0.05 m, the RMS can reach 30 W. It is expected that in practice the newly designed piezoelectric energy harvester could provide more efficient energy harvesting under higher wave height, larger cantilever length to satisfy the demand of normal operations of household appliances of the offshore platform.
Acknowledgement Advice and comments from Editor Mark Kachanov on a genera review of the topic are highly appreciated. This research was supported by the Canada Research Chairs Program (CRC) and the Natural Sciences and Engineering Research Council of Canada (NSERC).
48
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48
References Anton, S. R., & Sodano, H. A. (2007). A review of power harvesting using piezoelectric materials. Smart Materials and Structures, 16. R1–21. Burns, J. R. 1987. Ocean wave energy conversion using piezoelectric material members, US: 4685206. Cook-Chennault, K. A., Thambi, N., & Sastry, A. M. (2008). Powering MEMS portable devices – A review of non-regenerative and regenerative power supply systems with special emphasis on piezoelectric energy harvesting systems. Smart Materials and Structures, 17, 043001. Duan, W. H., Quek, S. T., & Wang, Q. (2005). Free vibration analysis of piezoelectric coupled thin and thick annular plate. Journal of Sound and Vibration, 281, 119–139. Falnes, J. (2007). A review of wave-energy extraction. Marine Structures, 20, 185–201. Friswell, M. I., & Adhikari, S. (2010). Sensor shape design for piezoelectric cantilever beams to harvest vibration energy. Journal of Applied Physics, 108, 014901. Khaligh, A., Zeng, P., & Zheng, C. (2010). Kinetic energy harvesting using piezoelectric and electromagnetic technologies – State of the art. IEEE Transactions on Industrial Electronics, 57, 850–860. Kim, H. S., Kim, J. H., & Kim, J. (2011). A review of piezoelectric energy harvesting based on vibration. International Journal of Precision Engineering and Manufacturing, 12, 1129–1141. Lee, C. K., & Moon, F. C. (1990). Modal sensors and actuators. Journal of Applied Mechanics, 57, 434–441. Li, S., & Lipson, H. Sep. 20–24, (2009). Vertical-stalk flapping-leaf generator for wind energy harvesting In Proceedings of ASME conference Smart Mater., Adaptive Struct. Intell. Syst. Oxnard CA. Li, S. G., Yuan, J. P., & Lipson, H. (2011). Ambient wind energy harvesting using cross-flow fluttering. Journal of Applied Physics, 109, 026104. Murray, R., & Rastegar, J. 2009. Novel two-stage piezoelectric-based ocean wave energy harvesters for moored or unmoored buoys. In Active and passive smart structures and integrated systems proceedings of SPIE 7288, pp. 1117–1129. Nader, J. R., Zhu, S. P., & Cooper, P. (2012). A finite-element study of the efficiency of arrays of oscillating water column wave energy converters. Ocean Engineering, 43, 72–81. Shenck, N. S., & Pradiso, J. A. (2001). Energy scavenging with shoes-mounted piezoelectrics. Micro IEEE, 21, 30–42. Shu, Y. C., Lien, I. C., & Wu, W. J. (2007). An improved analysis of the SSHI interface in piezoelectric energy harvesting. Smart Materials and Structures, 16, 2253–2264. Taylor, G. W. 1980. Power generation from waves near the surface of bodies of water, US: 4404490. Taylor, G. W., Burns, J. R., Kammann, S. M., Powers, W. B., & Welsh, T. R. (2001). The energy harvesting eel: A small subsurface ocean/river power generator. IEEE Journal of Oceanic Engineering, 26, 539–547. Vatansever, D., Hadimani, R. L., Shah, T., & Siores, E. (2011). An investigation of energy harvesting from renewable sources with PVDF and PZT. Smart Materials and Structures, 20, 055019. Wang, Q., & Quek, S. T. (2000). On dispersion relations in piezoelectric coupled beams. AIAA Journal, 38, 2357–2361. Wang, Q., & Quek, S. T. (2002). A model for the analysis of beam embedded with piezoelectric layers. Journal of Intelligent Systems and Structures, 13, 61–70. Wang, Q., Quek, S. T., Sun, C. T., & Liu, X. (2001). Analysis of piezoelectric coupled circular plate. Smart Materials and Structures, 10, 229–239. Wang, Q., & Wang, C. M. (2000). Optimal placement and size of piezoelectric patches on beams from the controllability perspective. Smart Materials and Structures, 9, 558–567. Wang, Q., & Wu, N. (2012). Optimal design of piezoelectric coupled beam for power harvesting. Smart Materials and Structures, 21, 085013. Wickenheiser, A. M., Reissman, T., Wu, W. J., & Garcia, E. (2010). Modeling the effects of electromechanical coupling on energy storage through piezoelectric energy harvesting. IEEE/ASME Transactions on Mechatronics, 15, 400–411. Wu, N., Wang, Q., & Xie, X. D. (2013). Wind energy harvesting with a piezoelectric harvester. Smart Materials and Structures, 22, 095023. Xie, X. D., Wang, Q., & Wu, N. (2014). A ring piezoelectric energy harvester excited by magnetic forces. International Journal of Engineering Science, 77, 71–78. Xie, X. D., Wu, N., Yuen, K. V., & Wang, Q. (2013). Energy harvesting from high-rise buildings by a piezoelectric coupled cantilever with a proof mass. International Journal of Engineering Science, 72, 98–106. Yang, Y., Yeo, J., & Priya, S. (2012). Harvesting energy from counter balancing movement in bicycle riding. Sensors, 12, 10248–10258. Zhang, Y. L., & Lin, Z. (2011). Advances in technology of ocean wave energy converters using piezoelectric materials. Journal of Water Resources and Electric Power, 5. Zhu, Y. G. (1991). Wave mechanics for ocean engineering. Tianjin: Tianjin University Press. Zurkinden, A. S., Campanile, F., & Martinelli, L. Oct 4–6 (2007). Wave energy converter through piezoelectric polymers, In Proceedings of the COMSOL users conference, Boston Marriott Newton.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Energy harvesting from transverse ocean waves by a piezoelectric plate X.D. Xie a,b, Q. Wang b,⇑, N. Wu b a b
School of Mechanical and Automotive Engineering, Hubei University of Arts and Science, Xiangyang, Hubei 441053, China Department of Mechanical and Manufacturing Engineering, University of Manitoba, Winnipeg, Manitoba R3T 5V6, Canada
a r t i c l e
i n f o
Article history: Received 19 February 2014 Received in revised form 24 March 2014 Accepted 9 April 2014
Keywords: Energy harvesting Ocean waves Transverse wave motion Piezoelectric harvester Root mean square (RMS)
a b s t r a c t An ocean wave energy harvester from the transverse wave motion of water particles is developed by the piezoelectric effects. The harvester is made of two horizontal cantilever plates attached by piezoelectric patches and fixed on a vertical rectangular column. To describe the energy harvesting process, a mathematical model is developed to calculate the output charge and voltage from the piezoelectric patches according to the Airy linear wave theory and the elastic beam model. The influences on the root mean square (RMS) of the generated power from the piezoelectric patches, such as the ocean depth, the harvester location under the ocean surface, the length of the cantilevers, the wave height, and the ratio of wave length to ocean depth, are discussed. Results show that the RMS increases with the increase in the length of cantilevers and the wave height, and decrease in the distance of the ocean surface to the cantilevers and the ratio of the wave length to ocean depth. As a result, an optimum ocean depth is obtained to achieve a maximum RMS at different harvester locations under the ocean surface. A value of the power up to 30 W can be realized for a practical transverse wave with the values of the ocean depth, wave length, wave height and harvester location under the ocean surface to be 10.6 m, 21.2 m, 4 m, and 2 m, respectively. This research develops a novel technique leading to efficient and practical energy harvesting from transverse waves by piezoelectric energy harvesters that could be easily fixed on an offshore platform. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction Energy crisis and environmental problems such as global warming and atmospheric pollution have prompted people to study technologies of exploiting new energy. In the past decade, applications of piezoelectric materials in energy generation and harvesting have received much attention owing to their unique characteristics (Anton & Sodano, 2007; Cook-Chennault, Thambi, & Sastry, 2008; Kim, Kim, & Kim, 2011). A piezoelectric material is a type of smart materials exhibiting a direct piezoelectric effect of the internal generation of electrical charge resulting from an applied mechanical force and a reverse piezoelectric effect of the internal generation of a mechanical strain resulting from an applied electrical field http://en.wikipedia.org/wiki/Piezoelectricity, (Duan, Quek, & Wang, 2005; Wang & Quek, 2000, 2002; Wang, Quek, Sun, & Liu, 2001). Since early 2000, an amount of energy generators and harvesters by using piezoelectric effects, such as piezoelectric coupled cantilevers, shells, cymbals and stacks, with various designs of electrical circuits have been developed ⇑ Corresponding author. Tel.: +1 204 474 6443. E-mail address: [email protected] (Q. Wang). http://dx.doi.org/10.1016/j.ijengsci.2014.04.003 0020-7225/Ó 2014 Elsevier Ltd. All rights reserved.
42
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48
(Cook-Chennault et al., 2008; Kim et al., 2011; Shu, Lien, & Wu, 2007; Wickenheiser, Reissman, Wu, & Garcia, 2010; Xie, Wang, & Wu, 2014; Xie, Wu, Yuen, & Wang, 2013). In addition, many research works were conducted on optimizing designs of piezoelectric coupled structures (Shenck & Pradiso, 2001; Wang & Wang, 2000; Yang, Yeo, & Priya, 2012) for more effective energy harvesting. These devices aimed to achieve practical portable microelectromechanical systems (MEMS) Cook-Chennault et al., 2008 via collecting energy of human actives, such as men’s working (Wang & Wu, 2012) and the bikes’ motions (Friswell & Adhikari, 2010). Recently, many research studies were conducted on piezoelectric energy harvesting from ambient vibrations (Khaligh, Zeng, & Zheng, 2010; Li, Yuan, & Lipson, 2011; Nader, Zhu, & Cooper, 2012; Vatansever, Hadimani, Shah, & Siores, 2011; Wu, Wang, & Xie, 2013) by natural energies such as solar energy, wind energy, and ocean-wave energy. Vatansever et al. (2011) evaluated energy harvesting from ceramic based piezoelectric fiber composite structures subjected to different wind loadings. Li et al. (2011) introduced a bio-inspired piezo-leaf architecture, which was in dangling cross-flow stalk that converted wind energy into electrical energy, by wind-induced flutter motion. Wu et al. (2013) developed an effective and compact wind energy cantilever harvester subjected to a cross wind. Sufficient electrical energy output as high as 2 watts was realized by tuning the resonant frequency of the harvester with a proof mass on the tip of the cantilever. As is indicated, wind energy has a larger energy density than the solar energy, and ocean-wave energy is more persistent and spatially concentrated than the wind energy (Falnes, 2007). Generally, the density of the ocean wave-energy is 4–30 times of that of wind energy (Zhang & Lin, 2011). Therefore, it is more efficient and effective to develop new technologies in harvesting the ocean wave-energy with piezoelectric materials. In view of considerable large powers from ocean wave motions, which can easily exceed 50 kW per meter of wave front (Murray & Rastegar, 2009), harvesting energy from ocean waves to electrical energy by piezoelectric effects has long been pursued as an alternative or self-contained power source. Zurkinden, Campanile, and Martinelli (2007) designed an ocean wave energy converter consisting of a flexible foam substrate attached by piezoelectric layers and simulated the efficiency of converting wave energy to electric energy by considering influences from different aspects such as the free surface wave, the fluid–structure–interaction, the mechanical energy input to the piezoelectric material, and the electric power output using an equivalent open circuit model. Taylor, Burns, Kammann, Powers, and Welsh (2001) developed a flexible polyvinylidene fluoride (PVDF) ‘‘eel’’ device to convert the mechanical energy in ocean or river water flows into electricity using the regular traveling vortex behind the bluff body to strain the piezoelectric elements. Using a similar principle, Li and Lipson (2009) explored a ‘‘piezo-leaf’’ energy-harvesting system where the PVDF strip of the ‘‘eel’’ system was replaced by a PVDF cantilever with a large triangular plastic ‘‘leaf’’ attached to the free end of the cantilever to improve the power generation. Burns (1987) provided a piezo device consisted of a buoy floating on the ocean surface, a few anchor chains fixed on the ocean-bed, and an array of piezoelectric micro thin films between the buoy and chains, and showed that the device can generate electric power when the piezoelectric films bear tension and compression alternatively duo to the up and down motion of the buoy. Taylor (1980) developed a piezoelectric device made of a buoy, the supporting structure, and piezoelectric layered sheets floating on the surface and anchor chains. The dimensions of each piezoelectric layered sheet were designed according to the wave length to improve the efficiency of wave energy conversion. Murray and Rastegar (2009) presented a two-stage electrical energy generators with two decoupled systems using the mechanism of strumming a guitar, in which low-frequency heaving-buoy can successively excite an array of vibratory elements (secondary system) into resonance and wave energy can be efficiently harvested using piezoelectric elements from the resonant secondary system. Previous studies on the source of the ocean wave energy harvesting with piezoelectric materials can be classified into two main categories: (1) vibrations caused by small longitudinal wave motions near the seabed or the vortex caused by the bluff body fixed on the seabed, and (2) vibrations of heaving-based buoy on the sea surface. In all these cases, only limited electric energy could be generated since the wave motions were not used directly and efficiently. Meanwhile, there has been very little amount of work dedicated to the theoretical framework of the energy harvesting from the transverse ocean wave motions using piezoelectric coupled structures. Therefore, a comprehensive model is indispensable in designs of the piezoelectric harvesters. In the monochromatic linear plane waves in deep water, particles near the surface move in circular paths, making water waves a combination of longitudinal and transverse wave motions. As the depth below the free surface increases, the radius of the circular motion decreases. At a depth equal to half the wavelength, the orbital movement has decayed to less than 5% of its value at the surface (http://en.wikipedia.org/wiki/Ocean_wave). In other words, in deep oceans, the energy from the longitudinal wave motion near the surface decays exponentially, and only transverse wave motions could be used efficiently by a harvester under the ocean. Therefore, it is anticipated that a more efficient energy harvesting technique can be developed to fully utilize the transverse wave motion in contrast to the aforementioned techniques that benefit only from the vertex, longitudinal wave vibration near the ocean-bed or the heaving buoy. In order to achieve this goal, a mathematical model is developed to investigate and calculate the energy harvesting from a simple and portable piezoelectric cantilever that can be easily fixed on a fixed structure under ocean or floating structures such as an offshore platform (see Fig. 1(b)). The collected electrical energy is realized by the electromechanical coupling effect of the piezoelectric patches from the transverse wave motion near the surface of ocean according to the linear Airy wave theory and the Euler–Bernoulli beam theory. The energy harvesting under different conditions, such as the cantilever location under ocean surface, the ocean depth, the wave height, the length of the cantilevers and the ratio of the wave length to the ocean depth, is calculated and discussed. It is expected that the device directly makes full use of the energy from the transverse wave motion near the surface of ocean to generate higher electric power that is sufficient for small power electric appliances of offshore platform.
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48
43
Fig. 1. Set-up of a piezoelectric energy harvester.
2. Energy harvesting from ocean waves by a piezoelectric harvester A mathematical model is developed to describe the principle of a piezoelectric energy harvester by exploring the transverse motions of water particles in ocean waves. Fig. 1 illustrates schematically a piezoelectric energy harvester, consisting of two cantilever substrates attached by piezoelectric patches and fixed on a vertical rectangular column, subjected to a transverse wave pressure from water particles. From this configuration, it is clearly seen that the design only uses the transverse motions for energy harvesting. Piezoelectric patches with equal length, denoted by a, are mounted one by one on both faces of the two cantilevers with a length of l (see Fig. 1). The distance from the bottom of the cantilever to the end of the mth piezoelectric patch is l1; and h1 and h are the thicknesses of the piezoelectric patch and the cantilever, respectively. The width of the cantilever and the piezoelectric patches is b. According to the Airy linear wave theory, the transverse wave pressure applied to the harvester at height z, counting from the bottom of the ocean, can be modeled as (Zhu, 1991):
fV ¼ (
1 @wðx; tÞ @wðx; tÞ @ 2 wðx; tÞ uz C D qw b uz þ C M qw bhaz C m qw bh ; 2 @t @t @t 2 sinh kðzþdÞ sinðkx sinh kd 02 H sinh kðzþdÞ cosðkx 2 sinh kd
uz ¼ x0 H2 az ¼ x
x0 tÞ ; x0 tÞ
ð1Þ
ð2Þ
where uz and az are the transverse velocity and acceleration of water particles in the ocean, respectively; x is the position in the longitudinal direction of ocean waves; z is the position in the transverse direction from the ocean surface; CD, CM, and Cm are coefficients of the drag force and inertia forces of the cantilever and the added mass, respectively; qw is the density of the ocean water; w(x, t) is the transverse displacement function of the cantilever (0 6 x 6 l); H, k and x0 are the height, wave number and angular frequency of the ocean wave motion, respectively; d is the ocean depth. In practice, it is reasonable to assume uz Ew(x, t)/Et at z = z1, so Eq. (1) can be simplified as follows:
fV ¼
1 @ 2 wðx; tÞ C D qw b uz juz j þ C M qw bhaz C m qw bh 2 @t2
¼ K 1 ðzÞ sinðkx x0 tÞj sinðkx x0 tÞj þ K 2 ðzÞ cosðkx x0 tÞ K 3
@ 2 wðx; tÞ ; @t2
ð3Þ
44
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48 2
kðzþdÞ kðzþdÞ where K 1 ðzÞ ¼ 18 C D qw bx02 H2 sinh , K 2 ðzÞ ¼ 12 C M qw bhx02 H sinh , and K3 = Cmqwbh. It is worth noting that the transverse sinh kd sinh2 kd
pressure, fv, is a complex function of the space coordinates (x, z) and time coordinate (t), which includes a drag force (the first item on the right side of Eq. (3)) and an inertial force (the last two items on the right side of Eq. (3)) that are induced by the water particles flowing around a body. Therefore, the mathematic solution of the piezoelectric coupled cantilever loaded by the transverse wave pressure cannot be obtained from a standard procedure and requires the following investigation. According to the Euler–Bernoulli beam theory, the governing equation of the piezoelectric energy harvester subjected to the transverse wave pressure is expressed as:
EI
@4w @2w þ ðK þ q AÞ ¼ f ðx; tÞ ¼ K 1 ðzÞ sinðkx x0 tÞj sinðkx x0 tÞj þ K 2 ðzÞ cosðkx x0 tÞ; 3 1 @x4 @t2
ð4Þ
where q1 is the material density of the cantilever substrate; EI and A are the flexural rigidity and cross section area of the cantilever substrate, respectively. It is assumed that the piezoelectric patches are mounted tightly on the surface of the cantilever. Due to the bending motion of the cantilever, the two faces of the beam exhibit positive and negative axial strains alternatively. As a result, the generated charge Q ig and voltage V ig on the surface of the ith piezoelectric patch is hence provided as (Lee & Moon, 1990):
Q ig ðtÞ
! e31 bðh þ h1 Þ @wðx; tÞ @wðx; tÞ ; ¼ 2 @x x¼ia @x x¼ði1Þa
V ig ðtÞ ¼
! e31 ðh þ h1 Þ @wðx; tÞ @wðx; tÞ ; @x x¼ia @x x¼ði1Þa 2C 0V
ð5Þ
ð6Þ
where e31 is the piezoelectric constant; CV is the electrical capacity of the piezoelectric patches; and C 0v is the electrical capacity per unit width of the piezoelectric patches (e.g. C 0V = CV/b). To obtain the displacement field, w(x, t), in Eq. (4), the vibration governing equation should be solved. First, the free vibration solution of the piezoelectric energy harvester in Eq. (4) can be obtained:
WðxÞ ¼ C 1 cosh sx þ C 2 sinh sx þ C 3 cos sx þ C 4 sin sx 0 6 x 6 l;
ð7Þ
where W(x) is the modal function of the cantilever of a particular mode; C1, C2,. . . , C4 are the corresponding unknown coefficients and s are given as
s¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 ðK 3 þ q AÞx 1 ; EI
ð8Þ
and x is the angular frequency of the structure. The boundary conditions of the piezoelectric energy harvester are given as:
WðxÞ ¼ 0;
dWðxÞ dx
2 WðxÞ EI d dx 2
3 WðxÞ EI d dx 3
¼ 0;
¼ 0;
if x ¼ 0;
¼ 0; if x ¼ l:
ð9Þ
Substituting Eq. (7) into these boundary conditions of the piezoelectric energy harvester leads to four linear equations, from which we can easily obtain the solution of the nth normal mode of the cantilever, Wn(x), corresponding to the nth natural frequency, xn, of the piezoelectric energy harvester. The forced vibration of the piezoelectric energy harvester subjected to the transverse wave pressure can thus be provided as follows:
wðx; tÞ ¼
1 X W n ðxÞ qn ðtÞ;
ð10Þ
n¼1
where qn(t) is the generalized coordinate, which is determined by the distributed transverse wave pressure applied to the harvester given in Eq. (3).
qn ¼
1 ðK 3 þ q1 AÞBn xn
Z
t
Q n ðsÞ sin xn ðt sÞds;
ð11Þ
0
Rl Rl where Q n ðsÞ ¼ 0 f ðx; sÞW n ðxÞdx and Bn ¼ 0 W 2n ðxÞdx. Finally, by substituting Eqs. (10) and (11) into Eqs. (5) and (6), the generated charge Q ig ðtÞ and voltage V ig ðtÞ from the ith piezoelectric patch bonded on the piezoelectric energy harvester subjected to transverse wave pressure at time t can be solved:
Q ig ðtÞ
! 1 e31 bðh þ h1 Þ X dW n ðxÞ dW n ðxÞ qn ðtÞ; ¼ 2 dx x¼ia dx x¼ði1Þa n¼1
ð12Þ
45
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48
V ig ðtÞ ¼
! 1 e31 ðh þ h1 Þ X dW n ðxÞ dW n ðxÞ qn ðtÞ; dx x¼ia dx x¼ði1Þa 2C 0V n¼1
ð13Þ
where 1 6 i 6 m, and m is the number of the piezoelectric patches mounted on one face of one of the cantilevers. Finally, the expression of RMS of the electric power is provided. When the transverse wave oscillates for a period of T, the RMS of the generated electric power from time 0 to T can be given as:
prms e
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 T ¼ ½pe ðtÞ2 dt ; T 0
ð14Þ
where pe(t) is the total generated electric power of all the piezoelectric patches mounted on the cantilevers at time t P dQ ig ðtÞ i (0 < t < T) and it is given by pe ðtÞ ¼ 4 m i¼1 dt V g ðtÞ. To compute the RMS of the generated electric power, the duration can be partitioned into j time steps with a sufficiently short time interval Dt. As a result, the expression in Eq. (14) can be rewritten in a discrete form for computation:
prms e
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u j u X Dt ¼ 4t ð½Pe ðt i Þ2 þ ½Pe ðt i1 Þ2 Þ: 2ðT DtÞ i¼2
ð15Þ
3. Simulations and discussions In this section, we investigate the effectiveness and efficiency of the harvester by examining the generated power. Particularly, we study the effects of the ocean depth (d), the wave height (H), the ratio of wave length to ocean depth (L/d), the harvester location under the ocean surface (z1), and the length of the cantilever (l) on the RMS of the power harvesting. The dimensions and materials properties of the energy harvester in the simulations are provided in Table 1. The piezoelectric patches and the host cantilever are made of PZT4 (lead zirconate titanate) and steel, respectively. Fig. 2(a–c) shows the variation of the RMS of the generated power versus the ocean depth d, under wave height H = 2 m, 3 m, and 4 m and harvester location under the ocean surface z1 = 2 m, 3 m, and 4 m, respectively. The dimensions of the energy harvester are set to be: h = 0.007 m, h1 = 0.0003 m, b = 1 m, l = 1.2 m, l1 = 2l/3, a = 0.05 m, and L = 2d. From the simulations, we can find the following observations. First, the RMS of the generated power increases nonlinearly till a maximum value before a smooth reduction, when the ocean depth, d, changes from 3 m to 30 m. Hence, there is an optimal ocean depth at a given harvester location under the ocean. It is seen from Fig. 2(a–c), the optimal ocean depths corresponding to the maximum RMS at H = 4 m are 10.6 m, 14.8 m, and 20.4 m, respectively at the harvester locations under the ocean surface of 2 m, 3 m and 4 m. Second, RMS increases nonlinearly with the decrease of the harvester location under the ocean surface (z1). The observations are results of the wave force, which is expressed by the square of velocity and acceleration of water particles along the ocean depth shown in Fig. 3 and given explicitly by Eq. (3). The findings provide a guideline on choosing ocean depth and optimal locations of energy harvesters for high efficiency of an energy harvester with a certain size. In addition, when the ocean depth is very small, e.g. d < 3 m at z1 = 2 m, and d < 6 m at z1 = 3 m, and 4 m, the RMS tends to zero. This finding is owing to the fact that the harvester location is near the ocean bottom where the transverse wave motion is very small. On the other hand, when the ocean depth is very large, e.g. d > 30 m, the RMS again is found to decrease obviously since the wave at deeper ocean usually is with larger wave length (L = 2d). Such larger wave length is equivalent to a smaller angular frequency of the ocean wave motion leading to smaller amplitudes of velocity and acceleration of the water particle, as seen in Eq. (2). Third, the RMS increases nonlinearly with the increase in the wave height and results show that when wave heights are 2 m, 3 m, and 4 m, the highest RMSs are 1.5 W, 7.4 W, and 23.3 W, respectively. This finding is clearly relevant to the fact of a nonlinear increase of the wave force with the increase in the wave height as shown in Eq. (3).
Table 1 Material properties and dimensions of the piezoelectric energy harvester. Parameters
Host beam (steel)
l (m) l1 (m) a (m) b (m) h (m) h1 (m) Young’s modulus (N/m2) Mass density (kg/m3) e31 (C/m2) Cv (nF)
1.2 2 l/3 1 0.007 or 0.01 2.1e11 7800
Piezoelectric patches (PZT4)
0.05 1 0.0003 7.8e10 7500 2.8 0.75 for the piezoelectric patch with the geometry of 0.01, 0.06, 0.0003 m
46
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48 25
h=0.007m,h1=0.0003m,b=1m,l=1.2m,z1=-2m,L=2d
RMS of Power(Watt)
20 H=4m 15 10 H=3m
5
H=2m 0 0
5
10
15 Ocean depth(m)
20
25
30
(a) z1=-2m.
10
h=0.007m,h1=0.0003m,b=1m,l=1.2m,z1=-3m,L=2d
RMS of Power(Watt)
8
H=4m
6 4 H=3m 2 H=2m 0 5
10
15 20 Ocean depth(m)
25
30
(b) z1=-3m.
RMS of Power(Watt)
5
h=0.007m,h1=0.0003m,b=1m,l=1.2m,z1=-4m,L=2d H=4m
4 3 2
H=3m 1 H=2m 0 5
10
15 20 Ocean depth(m)
25
30
(c) z1=-4m.
Fig. 2. RMS of the electric power under different wave height versus the ocean depth.
2
h=0.007m,h1=0.0003m,b=1m,l=1.2m,z1=-2m,H=2m,L=2d
1.5
Amplitude of a z ( m s 2 )
RMS of Power (Watt )
1
Amplitude of u z2( m 2 s 2 )
0.5
0 0
5
10
15 Ocean depth(m)
20
25
30
Fig. 3. RMS of the electric power and the amplitudes of square of velocity and acceleration of water particles versus the ocean depth.
Fig. 4 illustrates the effect of the ratio of the wavelength to the ocean depth, L/d, on the RMS with the following geometric parameters: h = 0.007 m, h1 = 0.0003 m, l = 1.2 m, l1 = 2/3l, b = 1 m, a = 0.05 m, and H = 4 m. This figure shows a smooth decrease of the RMS with an increase in the ratio. This finding indicates that a larger harvesting energy requires an ocean wave with a smaller wave length at a certain ocean depth based on the theoretical model. It can be seen from Fig. 4 that when the ratio of L/d is 2, the highest RMS of generated power are 23.3 W, 8.4 W, 4.1 W with the conditions of d = 10.6 m, z = 2 m; d = 14.8 m, z = 3 m, and d = 20.4 m, z = 4 m. This indicates that generated RMS can be in the range of 8–24 W when the harvester location under the ocean surface is 2–3 m with ocean depths of 10–15 m.
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48
47
25
RMS of Power(Watt)
d=10.6m,z1=-2m
h=0.007m,h1=0.0003m,b=1m,l=1.2m,H=4m
20 15 10
d=14.8m,z1=-3m
5
d=20.4m,z1=-4m
0 2
2.5
3
3.5 4 4.5 5 Ratio of wave length to ocean depth
5.5
6
Fig. 4. RMS of the electric power versus the ratio of wave length to the ocean depth.
30
h=0.01m,h1=0.0003m,b=1m,d=10.6m,z1=-2m,H=4m,L=2d
RMS of Power(Watt)
25 20 15 10 5 0 1
1.1
1.2
1.4 1.3 Length of cantilever(m)
1.5
1.6
1.7
Fig. 5. RMS of the electric power versus the length of the cantilever.
Fig. 5 illustrates the variations of the RMS versus the cantilever length. The dimensions of the energy harvester in this simulation are set to be, h = 0.01 m, h1 = 0.0003 m, l1 = 2/3l, b = 1 m, d = 10.6 m, L = 2d, a = 0.05 m, z = 2 m, and H = 4 m. It can be found that the RMS increases with an increase in the length of the harvester. The reason for this phenomenon is obvious that longer cantilever would make better use of wave motions, and hence would increase the harvested power. It can be seen from Fig. 5 that the RMS will increase from 2.5 W to 30 W when the cantilever length changes from 1 m to 1.7 m. Hence, an increase in the harvester length would increase the efficiency significantly. It should be noted that all the RMS of the generated electric power above are calculated with the condition that the transverse velocity of water particles is much higher than the velocity of the free end of the cantilever, as indicated in the theoretical model. From our simulations, we find that even under this condition, the simple and portable piezoelectric device still generates a sufficiently large electric power to supply lamp or radio-recorder, etc. In practice, the piezoelectric energy harvester subjected to transverse ocean wave pressure could generate larger electric power when the wave height is higher and/ or the cantilever length is longer.
4. Conclusion A simple and portable ocean wave energy harvester using the transverse wave motion of water particles is developed and a corresponding mathematical model is developed to calculate the output charge and voltage from the piezoelectric patches according to the Airy linear wave theory and the classical elastic beam model. The RMS of the electric power generated from piezoelectric energy harvester is defined and the results show that the RMS increases with the increase in the length of cantilevers and the wave height and the decrease in the distance of the ocean surface to the cantilevers, and the ratio of wave length to ocean depth. For an energy harvester structure with geometric parameters of h = 0.01 m, h1 = 0.0003 m, b = 1 m, l = 1.2 m, l1 = 2/3l, d = 10.6 m, L = 2d, H = 4 m, and a = 0.05 m, the RMS can reach 30 W. It is expected that in practice the newly designed piezoelectric energy harvester could provide more efficient energy harvesting under higher wave height, larger cantilever length to satisfy the demand of normal operations of household appliances of the offshore platform.
Acknowledgement Advice and comments from Editor Mark Kachanov on a genera review of the topic are highly appreciated. This research was supported by the Canada Research Chairs Program (CRC) and the Natural Sciences and Engineering Research Council of Canada (NSERC).
48
X.D. Xie et al. / International Journal of Engineering Science 81 (2014) 41–48
References Anton, S. R., & Sodano, H. A. (2007). A review of power harvesting using piezoelectric materials. Smart Materials and Structures, 16. R1–21. Burns, J. R. 1987. Ocean wave energy conversion using piezoelectric material members, US: 4685206. Cook-Chennault, K. A., Thambi, N., & Sastry, A. M. (2008). Powering MEMS portable devices – A review of non-regenerative and regenerative power supply systems with special emphasis on piezoelectric energy harvesting systems. Smart Materials and Structures, 17, 043001. Duan, W. H., Quek, S. T., & Wang, Q. (2005). Free vibration analysis of piezoelectric coupled thin and thick annular plate. Journal of Sound and Vibration, 281, 119–139. Falnes, J. (2007). A review of wave-energy extraction. Marine Structures, 20, 185–201. Friswell, M. I., & Adhikari, S. (2010). Sensor shape design for piezoelectric cantilever beams to harvest vibration energy. Journal of Applied Physics, 108, 014901. Khaligh, A., Zeng, P., & Zheng, C. (2010). Kinetic energy harvesting using piezoelectric and electromagnetic technologies – State of the art. IEEE Transactions on Industrial Electronics, 57, 850–860. Kim, H. S., Kim, J. H., & Kim, J. (2011). A review of piezoelectric energy harvesting based on vibration. International Journal of Precision Engineering and Manufacturing, 12, 1129–1141. Lee, C. K., & Moon, F. C. (1990). Modal sensors and actuators. Journal of Applied Mechanics, 57, 434–441. Li, S., & Lipson, H. Sep. 20–24, (2009). Vertical-stalk flapping-leaf generator for wind energy harvesting In Proceedings of ASME conference Smart Mater., Adaptive Struct. Intell. Syst. Oxnard CA. Li, S. G., Yuan, J. P., & Lipson, H. (2011). Ambient wind energy harvesting using cross-flow fluttering. Journal of Applied Physics, 109, 026104. Murray, R., & Rastegar, J. 2009. Novel two-stage piezoelectric-based ocean wave energy harvesters for moored or unmoored buoys. In Active and passive smart structures and integrated systems proceedings of SPIE 7288, pp. 1117–1129. Nader, J. R., Zhu, S. P., & Cooper, P. (2012). A finite-element study of the efficiency of arrays of oscillating water column wave energy converters. Ocean Engineering, 43, 72–81. Shenck, N. S., & Pradiso, J. A. (2001). Energy scavenging with shoes-mounted piezoelectrics. Micro IEEE, 21, 30–42. Shu, Y. C., Lien, I. C., & Wu, W. J. (2007). An improved analysis of the SSHI interface in piezoelectric energy harvesting. Smart Materials and Structures, 16, 2253–2264. Taylor, G. W. 1980. Power generation from waves near the surface of bodies of water, US: 4404490. Taylor, G. W., Burns, J. R., Kammann, S. M., Powers, W. B., & Welsh, T. R. (2001). The energy harvesting eel: A small subsurface ocean/river power generator. IEEE Journal of Oceanic Engineering, 26, 539–547. Vatansever, D., Hadimani, R. L., Shah, T., & Siores, E. (2011). An investigation of energy harvesting from renewable sources with PVDF and PZT. Smart Materials and Structures, 20, 055019. Wang, Q., & Quek, S. T. (2000). On dispersion relations in piezoelectric coupled beams. AIAA Journal, 38, 2357–2361. Wang, Q., & Quek, S. T. (2002). A model for the analysis of beam embedded with piezoelectric layers. Journal of Intelligent Systems and Structures, 13, 61–70. Wang, Q., Quek, S. T., Sun, C. T., & Liu, X. (2001). Analysis of piezoelectric coupled circular plate. Smart Materials and Structures, 10, 229–239. Wang, Q., & Wang, C. M. (2000). Optimal placement and size of piezoelectric patches on beams from the controllability perspective. Smart Materials and Structures, 9, 558–567. Wang, Q., & Wu, N. (2012). Optimal design of piezoelectric coupled beam for power harvesting. Smart Materials and Structures, 21, 085013. Wickenheiser, A. M., Reissman, T., Wu, W. J., & Garcia, E. (2010). Modeling the effects of electromechanical coupling on energy storage through piezoelectric energy harvesting. IEEE/ASME Transactions on Mechatronics, 15, 400–411. Wu, N., Wang, Q., & Xie, X. D. (2013). Wind energy harvesting with a piezoelectric harvester. Smart Materials and Structures, 22, 095023. Xie, X. D., Wang, Q., & Wu, N. (2014). A ring piezoelectric energy harvester excited by magnetic forces. International Journal of Engineering Science, 77, 71–78. Xie, X. D., Wu, N., Yuen, K. V., & Wang, Q. (2013). Energy harvesting from high-rise buildings by a piezoelectric coupled cantilever with a proof mass. International Journal of Engineering Science, 72, 98–106. Yang, Y., Yeo, J., & Priya, S. (2012). Harvesting energy from counter balancing movement in bicycle riding. Sensors, 12, 10248–10258. Zhang, Y. L., & Lin, Z. (2011). Advances in technology of ocean wave energy converters using piezoelectric materials. Journal of Water Resources and Electric Power, 5. Zhu, Y. G. (1991). Wave mechanics for ocean engineering. Tianjin: Tianjin University Press. Zurkinden, A. S., Campanile, F., & Martinelli, L. Oct 4–6 (2007). Wave energy converter through piezoelectric polymers, In Proceedings of the COMSOL users conference, Boston Marriott Newton.