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Lithium niobate ultrasonic transducer design for Enhanced Oil Recovery
Ultrasonics Sonochemistry 27 (2015) 171–177
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Ultrasonics Sonochemistry journal homepage: www.elsevier.com/locate/ultson
Lithium niobate ultrasonic transducer design for Enhanced Oil Recovery Zhenjun Wang a,⇑, Yuanming Xu a, Yuting Gu b a b
School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China The Liaoyang Campus of Shenyang University of Technology, Liaoyang 111003, China
a r t i c l e
i n f o
Article history: Received 11 January 2015 Received in revised form 20 April 2015 Accepted 15 May 2015 Available online 21 May 2015 Keywords: Lithium niobate transducer Ultrasonic oil production technology Reducing the viscosity of super heavy oil by ultrasonic wave
a b s t r a c t Due to the strong piezoelectric effect possessed by lithium niobate, a new idea that uses lithium niobate to design high-power ultrasonic transducer for Enhanced Oil Recovery technology is proposed. The purpose of this paper is to lay the foundation for the further research and development of high-power ultrasonic oil production technique. The main contents of this paper are as follows: firstly, structure design technique and application of a new high-power ultrasonic transducer are introduced; secondly, the experiment for reducing the viscosity of super heavy oil by this transducer is done, the optimum ultrasonic parameters for reducing the viscosity of super heavy oil are given. Experimental results show that heavy large molecules in super heavy oil can be cracked into light hydrocarbon substances under strong cavitation effect caused by high-intensity ultrasonic wave. Experiment proves that it is indeed feasible to design high-power ultrasonic transducer for ultrasonic oil production technology using lithium niobate. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Enhanced Oil Recovery is a generic term for techniques for improving crude oil production of the middle and later stage oil wells. Enhanced Oil Recovery is also called tertiary recovery or improved oil recovery (as opposed to primary and secondary oil recovery). Sometimes the term quaternary recovery is used to refer to more advanced, speculative, Enhanced Oil Recovery techniques [1]. Using Enhanced Oil Recovery, 30–60% or more of the reservoir’s original crude oil can be extracted [2] compared with 20– 40% using primary and secondary oil recovery [3]. Currently, most countries and some regions in the world commonly use chemical method and physical Enhanced Oil Recovery to increase oil production of oil well in the middle and later stages. Conventional method injects polymer chemical such as petroleum sulfonates into the reservoir to enhance oil displacement effect. Although some results have been obtained in field application, the disadvantages of this method are obvious – the long-term use of polymer chemicals not only directly leads to oil reservoir pollution and desertification of land, but also reduces oil recovery [4]. The most frequently used method of physical Enhanced Oil Recovery is hydrofracturing. This method uses the energy of pressure in the reservoir and increases the oil production by 2–3 times, which corresponds to 5–7 tons production increase per day in Western Siberia and reaches 23 tons/day in the Samara region ⇑ Corresponding author. http://dx.doi.org/10.1016/j.ultsonch.2015.05.017 1350-4177/Ó 2015 Elsevier B.V. All rights reserved.
[5]. Such operation requires the use of 4–8 heavy units near the well, as well as special packers and wellhead equipment. Significant loads on the equipment lead to the necessity of regular maintenance, after 4–5 operations capital repair works are required [5]. All this makes the operation very costly. Other physical Enhanced Oil Recoveries, which are currently used, are wave treatment and electromagnetic treatment. These two methods involve the use of various physical fields instead of matter to affect the oil reservoir. Such technologies are more costly and energy effective compared to hydro fracturing [5]. Ultrasonic oil production technique, according to [6–8], is one of the most promising wave methods. The effect of ultrasound on the well and the reservoir which leads to enhanced production, is based on two aspects of sonication that are relevant (1) enhancement of the flow of oil through the rocks into the pumping pool and (1) reduction of the viscosity of the oil that would make it easier to pump. As one of method of physical Enhanced Oil Recovery, ultrasonic oil production technique has many advantages such as strong adaptability, being simple to operate, low cost and no pollution to environment compared with above physical Enhanced Oil Recovery and conventional chemical methods [9]. Follow are the comparison between conventional chemical method and ultrasonic oil production technique: On one hand, in conventional chemical method, different oil reservoir conditions and the crude oil properties need different polymer chemical, and on the other hand they require periodic tests to ensure the prepared polymer chemicals are available [10]. While ultrasonic oil production technique simply creates
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sound wave directly in the oil reservoir to increase oil production. Therefore, compared with conventional chemical method, the operation of ultrasonic oil production technique is more easy and convenient. In term of equipment cost, one polymer injection device needs $230,000–$27,000, while an ultrasonic oil production device only needs $32,000–$97,000 [10]. In addition, the effective stimulation period of polymer chemical injection device on oil well is at least 4 months and up to 7 months, whereas the effective stimulation period of ultrasonic oil production technique is at least seven months and up to 15 months. So in both terms equipment investment and adaptability, ultrasonic oil production is far better than conventional chemical method. Normally, high-energy ultrasonic oil production equipment for Enhanced Oil Recovery consists of six parts: equipment special vehicle, digital control logging system, ultrasonic power source, special cables, special oil well adapter and ultrasonic transducer. Among them, high-power ultrasonic transducer is the main parts. Therefore, how to design high-power ultrasonic transducer that satisfy the practical requirement is one of the critical factors that determines Enhanced Oil Recovery technology success. In order to find a better piezoelectric material for Enhanced Oil Recovery, a new idea that uses lithium niobate to design high-power ultrasonic transducer for Enhanced Oil Recovery technology is proposed due to strong piezoelectric effect possessed by lithium niobate in this paper. Until now there are only a few studies that investigate LiNbO3 concerning its feasibility as ultrasonic transducers [17–20], The single crystal LiNbO3 shown in Fig. 1, has some excellent material properties for its use as ultrasonic transducer material. The mechanical quality factor according to manufacturer’s data is in the order of 100,000 which is a factor 50–100 higher than the available hard PZT. Also, the high Curie temperature of LiNbO3 about 1210 °C is an interesting property. It enables the possibility to design actuators for a high temperature environment. Conventional PZT has a Curie temperature about 300 °C Further, the electromechanical coupling factor is similar to PZT and the dielectric losses are significant smaller compared to PZT. The 36° y-cut LiNbO3 is the commercial available type of LiNbO3, which is closest to the optimal cut (38.9 °C) for thickness displacement [18], with the highest resulting d33. Therefore, 36° y-cut LiNbO3 can be used as the piezoelectric vibrator of the transducer that is used for Enhanced Oil Recovery. The purpose of this paper is to lay the foundation for the further research and development of high-power ultrasonic oil production technique.
Fig. 2. The structure diagram of lithium niobate piezoelectric vibrator.
Fig. 3. The cross sectional view of the lithium niobate transducer.
Fig. 4. The physical map of lithium niobate transducer.
uses the method of air backing. Under the action of periodicity voltage, lithium niobate wafer vibrates along thickness direction and radial direction simultaneously. Although, radial vibration frequency is low, its higher harmonics frequency is close to the fundamental frequency of the longitudinal vibration, which may affect the radiation field. Therefore, the vibrator is mounted on the base with sound absorption rubber material in order to eliminate the radial vibration as well as absorb static stress applied to the piezoelectric element. Below are the concrete transducer design.
2. Lithium niobate transducer design 2.1. Lithium niobate wafer thickness Piezoelectric vibrator is the core part of the transducer. The structure of which is shown in Fig. 2. Round electrode and ring electrode are all made of copper. The gray round electrode in the middle is connected to the signal line, the area of the gray round electrode is wafer’s effective ultrasound radiating area. The ring electrode is connected to aluminum alloy shell. The shell of this transducer is made of stainless steel. The cross sectional view of the lithium niobate transducer and its physical map are shown in Figs. 3 and 4 respectively. Piezoelectric vibrator
Fig. 1. Lithium niobate crystal.
Formula (15) can be used to determine the thickness of 36° y-cut LiNbO3. The derivation process of the Formula (15) is in the Appendix. For 36° y-cut LiNbO3, the velocity of longitudinal wave along the z axis has been measured, which is 7340 m/s. If the frequency of the ultrasound is 18 kHz, the wafer thickness is around 0.2 mm. 2.2. The diameter of lithium niobate wafer The shape of piezoelectric wafer in standard transducer is thin disks. Generally, the diameter of piezoelectric wafer should be 2 times larger than that of wafer’s effective ultrasound radiating surface so as to capture all energy beam and provide accurate sound power. For milliwatt standard ultrasound transducer, the diameter of wafer should not be more than 25 mm, while the of wafer should not be more than 40 mm for the watt standard ultrasound transducer.
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Fig. 5. The three-dimensional model of lithium niobate transducers. Fig. 7. Experimental sample.
lithium niobate transducers in series as shown in Fig. 5. Fig. 6 is a serried modal of lithium niobate transducers.
2.3. Wafer cutting Lithium niobate is a typical hard and brittle material. To achieve high efficiency and ultra-precision cutting, inner circle cutter with high rigidity and high precision spindle system must be required. The spindle radial deviation should be less than 0.02 mm, the maximum machining dimension is U60 110 mm, the minimum thickness of the cut wafer is 0.02 mm, the allowable deviation of wafer thickness is 0.2 mm, the allowable deviation of parallelism is 0.01 mm. After cut up, the wafer should be polished to ensure the flatness and roughness of the surface can reach the desired technical requirement. 2.4. The thickness and manufacture of copper electrode The harmonic frequency of early piezoelectric wafer is not high, the wafer size of which is far greater than that of the metal electrode, therefore, the effect of the electrode on wafer’s resonance frequency can be neglected. In order to obtain higher resonance frequency, the thickness of wafer must be thinner, therefore, the influence of electrode on wafer’s resonance frequency cannot be neglected. The electrode layer is, in theory, the thinner the better due to the fact that electrode thickness affects the resonance frequency of piezoelectric wafer. According to the tunneling effect in quantum mechanics, the electrical properties of electrode will not obey Ohm’s law when the electrode thickness is to nanometer scale, therefore, the conductive properties of the electrode and the quantum limit all should be taken into account. If the thicknesses of copper layer and the wafer are 2.0 105 nm and 50 nm respectively, each copper electrode has about 200 copper atom-layer. Such copper electrode can not only ensure conductivity, but also be manufactured by metal plating technique. Due to its sound field is limited, lithium niobate transducer designed (Fig. 4) cannot be used for ultrasonic oil production. In order to increase the radius of influence, one transducer should have more lithium niobate piezoelectric vibrators. As shown in Fig. 5, a new model of lithium niobate transducer for ultrasonic oil production technology is proposed in this paper. As can be seen in Fig. 5, one transducer consists of 17 lithium niobate piezoelectric vibrators, the distance between these piezoelectric vibrators located on the side of the transducer is equidistant. Due to same emission frequency and vibration phase, ultrasonic acoustic fields exited by these lithium niobate piezoelectric vibrators are coherent wave field. Thus the radius of influence is greatly increased. In order for lithium niobate transducer to be widely used for crude oil production, the best way is to connect this new modal of
3. Comparison of lithium niobate transducer with the PZT transducer used for Enhanced Oil Recovery So far, many transducers used for Enhanced Oil Recovery have been developed. In 2001, the US department of energy et al. successfully developed a vibrational transducer used for oil production, but it has lower ultrasonic frequency and lower energy density [11]; in 2007, Germany’s Hielscher company successfully created high-power cylindrical transducer [12], the most notable feature is that it can work continuously under humid and high pressure conditions, but it cannot work at high temperatures underground last for a long time; Edward H. Phillips and Los Altos proposed in their patent US3583677 that piezoelectric elastomer can be used to generate ultrasound for ultrasonic oil production, but such patents failed to be widely applied in oil production due to complex oil production environment [13,14]. In China, CSYY60H10 type high-power ultrasonic oil production equipment has been successfully developed in 2006 [15], but it cannot work at high temperatures and pressures underground last for a long time. In a word, the piezoelectric vibrators of transducers both in china and other countries are all made of PZT [16], which contains often-harmful lead. High-power ultrasonic transducer made of lithium niobate for Enhanced Oil Recovery technology is proposed in this paper due to the following two major reasons [23]: in the last years the utilizing of lead got more and more restricted, still with many exceptions for piezoelectric materials, but it is unclear how long those will be open. The second reason is an expansion of the acceptable environment-parameters, especially in for high temperature applications. Besides that, lithium niobate has good mechanical performance and chemical stability, it’s mechanical quality factor (about 105 magnitude) is much higher than common PZT. In view of the above advantages, high-power ultrasonic transducer made of lithium niobate for Enhanced Oil Recovery technology could be better than those transducers made of PZT. 4. Experiments for reducing the viscosity of super heavy oil using lithium niobate transducer When high-energy ultrasonic waves travel through a medium, rapid and successive compression and rarefaction cycles occur. The decrease in the pressure in the negative pressure cycle of the ultrasonic shock wave spontaneously generates small cavities.
Fig. 6. The three-dimensional model of serried lithium niobate transducers.
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Z. Wang et al. / Ultrasonics Sonochemistry 27 (2015) 171–177 Table 3 The effects of ultrasonic power on the rate of viscosity reduction of super heavy oil. Power (W) The ratio of viscosity reduction (%)
Ultrasonic frequency (kHz) Viscosity (MPa s) The ratio of viscosity reduction (%)
Table 1 The results of orthogonal experiment. Experimental number
Ultrasonic frequency (kHZ)
Ultrasonic power (W)
Time (min)
The ratio of viscosity reduction (%)
1 2 3 4 5 6 7 8 9
18 18 18 20 20 20 25 25 25
100 500 1000 100 500 1000 100 500 1000
5 15 30 15 30 5 30 5 15
45 55.4 65.8 44 48 52.2 34 38.5 42.4
5 476 54.5
10 470 62.40
15 469 63.48
20 466 64.20
Note: the initial viscosity of super heavy oil is 1, 250 MPa s (50 °C).
Theses cavities collapse in the positive pressure cycle and produce highly turbulent flow conditions and extremely high pressure and temperatures, a process that is better known as cavitation [19]. The mechanical vibration caused by cavitation effect can instantly
The ratio of viscosity reduction (%)
18 480 62
The ratio of viscosity reduction
65
60
55
50
45
0
2
4
6
8
650 44.3
850 45.6
1050 50.6
1250 51.4
20 890 28.9
25 920 26.5
break heavy large molecules in super heavy oil into light hydrocarbon substances. The viscosity of super heavy oil will never return to the pre-treatment condition when large molecules are cracked [20]. Right now, ultrasonic viscosity-reduction techniques both in China and other countries all emphasize field tests, the research on theoretical models still in its infancy, the basic theory of ultrasonic viscosity-reduction technique still has not been established. In addition, the study of cracking large molecules using ultrasound has been reported rarely [21]. In order to verify the feasibility of this new transducer for Enhanced Oil Recovery (Enhanced Oil Recovery), the experiment for reducing viscosity of super heavy oil using lithium niobate transducer has been successfully done. Experimental sample of super heavy oil, shown in Fig. 7, is taken from Binnan oil well of Shengli oilfield in china. The vertical depth of this oil well is 4364.95 m [13]. The quality of experimental sample is 250 g. Due to the temperature of the crude oil sample just out of oil field is as high as 50 °C, experimental sample must be remained at a constant temperature of 50 °C in order to accurately research the effect of ultrasonic wave on the viscosity reduction of this oil sample. The concrete steps and results are as follows: firstly, after each orthogonal experiment is completed, according to the standard rules of SY/T6316-1998, the viscosity of super heavy oil and the ratio of viscosity reduction are measured and calculated
Table 2 The relationship between viscosity, the ratio of viscosity reduction and ultrasonic treatment time (ultrasonic frequency is 18 kHz). 2 680 45.60
450 38.8
Table 4 The effects of ultrasonic frequency on the viscosity reduction of super heavy oil.
Fig. 8. Experimental sample in the water bath with constant temperature (50 °C).
Time (min) Viscosity (MPa s) The rate of viscosity reduction (%)
250 36.4
10
12
14
16
Time (min) Fig. 9. The graph of time–the ratio of viscosity reduction.
18
20
22
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Z. Wang et al. / Ultrasonics Sonochemistry 27 (2015) 171–177 Table 5 The components and mean relative molecular mass of super heavy oil before and after ultrasonic treatment. Oil samples
Saturated hydrocarbons
Aromatic hydrocarbon
Colloid
Asphaltene
Average relative molecular mass
Before processing After processing
26.51 30.25
34.64 36.04
26.48 22.35
12.35 10.38
764 598
respectively by Brookfield viscosimeter when the shear rate is 3 s1. Then the optimum ultrasound parameters are selected; secondly, the influence is analyzed of ultrasound frequency, power and treatment time on the viscosity reduction of super heavy oil respectively; finally, according to the standard rules of SY/T5119-1996, the components of super heavy oil that is processed by the ultrasound with optimum parameters are measured, and then the mean relative molecular mass of super heavy oil is measured by Knauer K-700 vapor pressure permeameter. Experimental sample in the water bath with constant temperature (50 °C) is shown in Fig. 8 [22], experimental results are shown in Table 1. As can be seen in Table 1, the optimum values of the ultrasonic power, frequency and treatment time are 1000 W, 18 kHz and 30 min respectively. But what is the influence of ultrasound frequency, power and treatment time on the viscosity reduction of super heavy oil respectively? The results are as follows: As can be seen in Table 2 and Fig. 9, under constant ultrasonic frequency, the ratio of viscosity reduction of super heavy oil increases with the increase of ultrasonic treatment time. The longer the duration of cavitation effect is, the easier it is to reduce the viscosity of super heavy oil under constant ultrasonic frequency. As can be seen form Table 3, with the increase of ultrasonic power, the ratio of viscosity reduction of super heavy oil increases accordingly. The greater the ultrasonic power is, the more obvious the cavitation effect is and the easier it is to reduce the viscosity of super heavy oil. As can be seen in Table 4, the ratio of viscosity reduction of super heavy oil decreases with the increase of ultrasonic frequency. The cavitation effect caused by ultrasonic wave needs enough time, the higher the ultrasonic frequency is, the shorter the period of ultrasonic wave is and the shorter the time needed for cavitation effect is. Lack of enough time is not conducive to generate cavitation effect so that the viscosity of super heavy oil cannot be effectively reduced. Therefore, ultrasonic frequency cannot be too high. As can be seen in Table 5, heavy large molecules (Resin and Asphaltene) in super heavy oil can be cracked into light hydrocarbon substances (saturated hydrocarbons, aromatic hydrocarbons) under ultrasonic treatment. Compound molecular bond such as C-C chemical bond can be cracked under strong cavitation effect caused by high-intensity ultrasonic wave. The viscosity of super heavy oil is reduced when the large molecules are cracked. The front end of high-power transducers used for Enhanced Oil Recovery are normally installed with spring-piston pressure auto-balancing device, which can make ultrasonic transducer work
Fig. 10. Wafer modal.
Fig. 11. Lumped equivalent circuit of wafer.
safely at high temperatures and pressures underground last for a long time. This device has been very mature, it can made transducers work safely under external pressure of 30 MPa [15]. Spring-piston pressure auto-balancing device can also be installed on the front end of lithium niobate transducer designed in this paper. Crude oil viscosity lowering and production increasing field test by high-power transducer made of PZT has been carried out in 90– 1 well in west oilfield in China [15]. Oil pressure and temperature are 9.1 MPa and 50 °C respectively. The performance parameters of this transducer are: output frequency is 3–35 kHz, service temperature of about 120 °C. Test result showed that oil production has remained unchanged at 15 m3 per day, the daily increased oil is 7.8 m3 and the cumulative increased oil is 600 m3 after ultrasonic transducer (vibrates at 22 kHz) processing 15 h. Vibrational frequency of each lithium niobate piezoelectric wafer designed in this paper (25 mm in diameter and 0.2 mm thick) can reach about 18.5 MHz according to Formula (15), besides, lithium niobate piezoelectric wafer can work safely at high temperatures of 1210 °C last for a long time, The spring-piston pressure auto-balancing device installed on the front end of lithium niobate transducer designed in this paper can be sufficient to overcome 9.1 MPa oil pressure. In a word, by meeting the above parameter requirement, lithium niobate ultrasonic transducer described in this paper is feasible for Enhanced Oil Recovery. 5. Conclusion 1. As can be clearly seen in Table 2, Table 3 and Table 4, the most important factors for reducing the viscosity of super heavy oil are ultrasonic power, followed by ultrasonic frequency and time. 2. As can be seen in Table 5, compared with no ultrasonic treatment, mean relative molecular mass of the light components such as Saturated hydrocarbon and Aromatic hydrocarbon in super heavy oil that is processed by the ultrasound with optimum parameters are increased by 14.1% and 4% respectively, while heavy components such as Colloid and Asphaltene are decreased by 15.6% and 16% respectively, in addition, the mean relative molecular mass of super heavy oil is decreased by 21.73%. Experimental results prove that heavy large molecules in super heavy oil can be cracked into light hydrocarbon substances under strong cavitation effect caused by high-intensity ultrasonic wave. 3. Research results prove that it is indeed feasible to design high-power ultrasonic transducer for ultrasonic oil production technology using lithium niobate.
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Z. Wang et al. / Ultrasonics Sonochemistry 27 (2015) 171–177
Acknowledgement
q
The authors would like to thank financial support from ‘‘863’’project fund in China and deep instruction from Prof. Shushan Zhao in Harbin Institute of Technology in China.
Under the action of alternating electric field E, as shown in Fig. 10, particles in wafer do stretching vibration along the z axis. As can be seen in Fig. 2, the following boundary conditions are obtained: the strain S3 –0; and S1 ¼ S2 ¼ S4 ¼ S5 ¼ S6 ¼ 0, electric displacement: D3 –0; D1 ¼ D2 ¼ 0. According to piezoelectric Formula (1) and the above boundary conditions, the following simplified Formula (2) can be acquired.
T m ¼ cDmn Sh hjn Dj ðm; n ¼ 1; 2; ; 6Þ
ð1Þ
Ei ¼ hin Sn þ bSij Dj ði; j ¼ 1; 2; 3Þ (
T 3 ¼ cD33 S3 h33 D3
ð2Þ
E3 ¼ h33 S3 þ bS33 D3 where
cDmn
the condition of constant strain, h represents the thickness of wafer. According to Newton’s second law, vibration equation can be written as follows
@ 2 g @T q 2¼ 3 @z @t
ð3Þ
where g is the displacement along direction of z axis, q is wafer density. The second equation in Formula (2) can be written as
E3 bS33
þ
h33 bS33
@D3 @z
S3
ð4Þ
Take the partial derivative of Formula (4) with respect to z. Due 3 ¼ 0 can be to the fact that there is no free charge in wafer, @D @z obtained. It is known that S3 ¼ @@zg ; so can be obtain after integrating.
@E3 @z
2
¼ h33 @@z2g, Formula (5)
where
¼ 0 and S3 ¼ @@zg, vibration equation can be written
where B is integral constant, according to the applied voltage Rt E dt; integral constant can be calculated as 0 3
qffiffiffiffiffiffiffiffiffiffiffiffi cD33 =q is longitudinal wave velocity in wafer, Formula
@2g 2 þk g¼0 @z2
ð11Þ
where k ¼ x=t is wave number, the solution of Formula (11) is g2 sin kz g ¼ g1 sin kðtzÞ sin kt Stress distribution in wafer is determined by the first equation in Formula (2). According to the equilibrium conditions, the following equations can be obtained.
(
F 1 ¼ cD33 ð@@zgÞz¼0 A þ h33 D3 A F 2 ¼ cD33 ð@@zgÞz¼t A þ h33 D3 A
ð12Þ
where F 1 and F 2 are the forces applied on the bottom surface (z ¼ 0) and the top surface (z ¼ t) respectively. On the basis of Formula (4), the following Formula (13) can be obtained.
h33 D3 A ¼ rV
r2 ðg þ g 2 Þ jxC 0 1
ð13Þ
Substituting Formula (8) and Formula (10) into Formula (9), mechanical vibration Formula (11) can be acquired.
8 < F 1 ¼ ð qtA r2 Þðg1 þ g2 Þ þ jqtA tan 1 ktg1 þ rV j sin kt 2 jx C 0 : F 2 ¼ ð qtA r2 Þðg þ g Þ þ jqtA tan 1 ktg þ rV 1 2 2 j sin kt 2 jx C 0
ð14Þ
The lumped equivalent circuit of wafer is as follows As can be seen in Fig. 11, piezoelectric wafer has strict odd-harmonic structure, therefore, the resonance frequency under first harmonic can be written as 1
ð5Þ
t¼
ð10Þ
(7) can be written as the following Formula (11) under sinusoidal voltage.
fP ¼
@g E3 ¼ h33 þB @z
ð9Þ
as
is elastic stiffness tensor under the condition of constant
electric displacement, bSij is dielectric impermeability tensor under
D3 ¼
Based on
@2g @2g ¼ t2 2 2 @z @t
Appendix A.
(
@2g @S3 @D3 ¼ cD33 h33 @z @z @t 2
t 2h
ð15Þ
References
V¼
ðg1 þ g2 Þ where g1 and g2 are the longitudinal displaceB ¼ Vt h33 t ment of the bottom surface (z ¼ 0) and the top surface (z ¼ t) respectively, then E3 can be written as
E3 ¼ h33
@ g V h33 þ ðg1 þ g2 Þ @t t t
ð6Þ
Substituting Formula (6) into Formula (2), electric displacement D3 can be written as
D3 ¼
h33 bS33 t
ðg1 þ g2 Þ þ
V
ð7Þ
bS33 t
If the wafer is excited by sinusoidal voltage, frequency of which is x; the current can be obtained
I ¼ jx0 C 0 V rðg1 þ g2 Þ
ð8Þ
where C 0 ¼ A=tbS33 is the cut-off capacitance, r ¼ Ah33 =tbS33 is elec2
tromechanical conversion coefficient, A ¼ pðd=2Þ is area. Substituting the first equation in Formula (2) into Formula (3), Formula (9) can be obtained
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Contents lists available at ScienceDirect
Ultrasonics Sonochemistry journal homepage: www.elsevier.com/locate/ultson
Lithium niobate ultrasonic transducer design for Enhanced Oil Recovery Zhenjun Wang a,⇑, Yuanming Xu a, Yuting Gu b a b
School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China The Liaoyang Campus of Shenyang University of Technology, Liaoyang 111003, China
a r t i c l e
i n f o
Article history: Received 11 January 2015 Received in revised form 20 April 2015 Accepted 15 May 2015 Available online 21 May 2015 Keywords: Lithium niobate transducer Ultrasonic oil production technology Reducing the viscosity of super heavy oil by ultrasonic wave
a b s t r a c t Due to the strong piezoelectric effect possessed by lithium niobate, a new idea that uses lithium niobate to design high-power ultrasonic transducer for Enhanced Oil Recovery technology is proposed. The purpose of this paper is to lay the foundation for the further research and development of high-power ultrasonic oil production technique. The main contents of this paper are as follows: firstly, structure design technique and application of a new high-power ultrasonic transducer are introduced; secondly, the experiment for reducing the viscosity of super heavy oil by this transducer is done, the optimum ultrasonic parameters for reducing the viscosity of super heavy oil are given. Experimental results show that heavy large molecules in super heavy oil can be cracked into light hydrocarbon substances under strong cavitation effect caused by high-intensity ultrasonic wave. Experiment proves that it is indeed feasible to design high-power ultrasonic transducer for ultrasonic oil production technology using lithium niobate. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Enhanced Oil Recovery is a generic term for techniques for improving crude oil production of the middle and later stage oil wells. Enhanced Oil Recovery is also called tertiary recovery or improved oil recovery (as opposed to primary and secondary oil recovery). Sometimes the term quaternary recovery is used to refer to more advanced, speculative, Enhanced Oil Recovery techniques [1]. Using Enhanced Oil Recovery, 30–60% or more of the reservoir’s original crude oil can be extracted [2] compared with 20– 40% using primary and secondary oil recovery [3]. Currently, most countries and some regions in the world commonly use chemical method and physical Enhanced Oil Recovery to increase oil production of oil well in the middle and later stages. Conventional method injects polymer chemical such as petroleum sulfonates into the reservoir to enhance oil displacement effect. Although some results have been obtained in field application, the disadvantages of this method are obvious – the long-term use of polymer chemicals not only directly leads to oil reservoir pollution and desertification of land, but also reduces oil recovery [4]. The most frequently used method of physical Enhanced Oil Recovery is hydrofracturing. This method uses the energy of pressure in the reservoir and increases the oil production by 2–3 times, which corresponds to 5–7 tons production increase per day in Western Siberia and reaches 23 tons/day in the Samara region ⇑ Corresponding author. http://dx.doi.org/10.1016/j.ultsonch.2015.05.017 1350-4177/Ó 2015 Elsevier B.V. All rights reserved.
[5]. Such operation requires the use of 4–8 heavy units near the well, as well as special packers and wellhead equipment. Significant loads on the equipment lead to the necessity of regular maintenance, after 4–5 operations capital repair works are required [5]. All this makes the operation very costly. Other physical Enhanced Oil Recoveries, which are currently used, are wave treatment and electromagnetic treatment. These two methods involve the use of various physical fields instead of matter to affect the oil reservoir. Such technologies are more costly and energy effective compared to hydro fracturing [5]. Ultrasonic oil production technique, according to [6–8], is one of the most promising wave methods. The effect of ultrasound on the well and the reservoir which leads to enhanced production, is based on two aspects of sonication that are relevant (1) enhancement of the flow of oil through the rocks into the pumping pool and (1) reduction of the viscosity of the oil that would make it easier to pump. As one of method of physical Enhanced Oil Recovery, ultrasonic oil production technique has many advantages such as strong adaptability, being simple to operate, low cost and no pollution to environment compared with above physical Enhanced Oil Recovery and conventional chemical methods [9]. Follow are the comparison between conventional chemical method and ultrasonic oil production technique: On one hand, in conventional chemical method, different oil reservoir conditions and the crude oil properties need different polymer chemical, and on the other hand they require periodic tests to ensure the prepared polymer chemicals are available [10]. While ultrasonic oil production technique simply creates
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sound wave directly in the oil reservoir to increase oil production. Therefore, compared with conventional chemical method, the operation of ultrasonic oil production technique is more easy and convenient. In term of equipment cost, one polymer injection device needs $230,000–$27,000, while an ultrasonic oil production device only needs $32,000–$97,000 [10]. In addition, the effective stimulation period of polymer chemical injection device on oil well is at least 4 months and up to 7 months, whereas the effective stimulation period of ultrasonic oil production technique is at least seven months and up to 15 months. So in both terms equipment investment and adaptability, ultrasonic oil production is far better than conventional chemical method. Normally, high-energy ultrasonic oil production equipment for Enhanced Oil Recovery consists of six parts: equipment special vehicle, digital control logging system, ultrasonic power source, special cables, special oil well adapter and ultrasonic transducer. Among them, high-power ultrasonic transducer is the main parts. Therefore, how to design high-power ultrasonic transducer that satisfy the practical requirement is one of the critical factors that determines Enhanced Oil Recovery technology success. In order to find a better piezoelectric material for Enhanced Oil Recovery, a new idea that uses lithium niobate to design high-power ultrasonic transducer for Enhanced Oil Recovery technology is proposed due to strong piezoelectric effect possessed by lithium niobate in this paper. Until now there are only a few studies that investigate LiNbO3 concerning its feasibility as ultrasonic transducers [17–20], The single crystal LiNbO3 shown in Fig. 1, has some excellent material properties for its use as ultrasonic transducer material. The mechanical quality factor according to manufacturer’s data is in the order of 100,000 which is a factor 50–100 higher than the available hard PZT. Also, the high Curie temperature of LiNbO3 about 1210 °C is an interesting property. It enables the possibility to design actuators for a high temperature environment. Conventional PZT has a Curie temperature about 300 °C Further, the electromechanical coupling factor is similar to PZT and the dielectric losses are significant smaller compared to PZT. The 36° y-cut LiNbO3 is the commercial available type of LiNbO3, which is closest to the optimal cut (38.9 °C) for thickness displacement [18], with the highest resulting d33. Therefore, 36° y-cut LiNbO3 can be used as the piezoelectric vibrator of the transducer that is used for Enhanced Oil Recovery. The purpose of this paper is to lay the foundation for the further research and development of high-power ultrasonic oil production technique.
Fig. 2. The structure diagram of lithium niobate piezoelectric vibrator.
Fig. 3. The cross sectional view of the lithium niobate transducer.
Fig. 4. The physical map of lithium niobate transducer.
uses the method of air backing. Under the action of periodicity voltage, lithium niobate wafer vibrates along thickness direction and radial direction simultaneously. Although, radial vibration frequency is low, its higher harmonics frequency is close to the fundamental frequency of the longitudinal vibration, which may affect the radiation field. Therefore, the vibrator is mounted on the base with sound absorption rubber material in order to eliminate the radial vibration as well as absorb static stress applied to the piezoelectric element. Below are the concrete transducer design.
2. Lithium niobate transducer design 2.1. Lithium niobate wafer thickness Piezoelectric vibrator is the core part of the transducer. The structure of which is shown in Fig. 2. Round electrode and ring electrode are all made of copper. The gray round electrode in the middle is connected to the signal line, the area of the gray round electrode is wafer’s effective ultrasound radiating area. The ring electrode is connected to aluminum alloy shell. The shell of this transducer is made of stainless steel. The cross sectional view of the lithium niobate transducer and its physical map are shown in Figs. 3 and 4 respectively. Piezoelectric vibrator
Fig. 1. Lithium niobate crystal.
Formula (15) can be used to determine the thickness of 36° y-cut LiNbO3. The derivation process of the Formula (15) is in the Appendix. For 36° y-cut LiNbO3, the velocity of longitudinal wave along the z axis has been measured, which is 7340 m/s. If the frequency of the ultrasound is 18 kHz, the wafer thickness is around 0.2 mm. 2.2. The diameter of lithium niobate wafer The shape of piezoelectric wafer in standard transducer is thin disks. Generally, the diameter of piezoelectric wafer should be 2 times larger than that of wafer’s effective ultrasound radiating surface so as to capture all energy beam and provide accurate sound power. For milliwatt standard ultrasound transducer, the diameter of wafer should not be more than 25 mm, while the of wafer should not be more than 40 mm for the watt standard ultrasound transducer.
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Fig. 5. The three-dimensional model of lithium niobate transducers. Fig. 7. Experimental sample.
lithium niobate transducers in series as shown in Fig. 5. Fig. 6 is a serried modal of lithium niobate transducers.
2.3. Wafer cutting Lithium niobate is a typical hard and brittle material. To achieve high efficiency and ultra-precision cutting, inner circle cutter with high rigidity and high precision spindle system must be required. The spindle radial deviation should be less than 0.02 mm, the maximum machining dimension is U60 110 mm, the minimum thickness of the cut wafer is 0.02 mm, the allowable deviation of wafer thickness is 0.2 mm, the allowable deviation of parallelism is 0.01 mm. After cut up, the wafer should be polished to ensure the flatness and roughness of the surface can reach the desired technical requirement. 2.4. The thickness and manufacture of copper electrode The harmonic frequency of early piezoelectric wafer is not high, the wafer size of which is far greater than that of the metal electrode, therefore, the effect of the electrode on wafer’s resonance frequency can be neglected. In order to obtain higher resonance frequency, the thickness of wafer must be thinner, therefore, the influence of electrode on wafer’s resonance frequency cannot be neglected. The electrode layer is, in theory, the thinner the better due to the fact that electrode thickness affects the resonance frequency of piezoelectric wafer. According to the tunneling effect in quantum mechanics, the electrical properties of electrode will not obey Ohm’s law when the electrode thickness is to nanometer scale, therefore, the conductive properties of the electrode and the quantum limit all should be taken into account. If the thicknesses of copper layer and the wafer are 2.0 105 nm and 50 nm respectively, each copper electrode has about 200 copper atom-layer. Such copper electrode can not only ensure conductivity, but also be manufactured by metal plating technique. Due to its sound field is limited, lithium niobate transducer designed (Fig. 4) cannot be used for ultrasonic oil production. In order to increase the radius of influence, one transducer should have more lithium niobate piezoelectric vibrators. As shown in Fig. 5, a new model of lithium niobate transducer for ultrasonic oil production technology is proposed in this paper. As can be seen in Fig. 5, one transducer consists of 17 lithium niobate piezoelectric vibrators, the distance between these piezoelectric vibrators located on the side of the transducer is equidistant. Due to same emission frequency and vibration phase, ultrasonic acoustic fields exited by these lithium niobate piezoelectric vibrators are coherent wave field. Thus the radius of influence is greatly increased. In order for lithium niobate transducer to be widely used for crude oil production, the best way is to connect this new modal of
3. Comparison of lithium niobate transducer with the PZT transducer used for Enhanced Oil Recovery So far, many transducers used for Enhanced Oil Recovery have been developed. In 2001, the US department of energy et al. successfully developed a vibrational transducer used for oil production, but it has lower ultrasonic frequency and lower energy density [11]; in 2007, Germany’s Hielscher company successfully created high-power cylindrical transducer [12], the most notable feature is that it can work continuously under humid and high pressure conditions, but it cannot work at high temperatures underground last for a long time; Edward H. Phillips and Los Altos proposed in their patent US3583677 that piezoelectric elastomer can be used to generate ultrasound for ultrasonic oil production, but such patents failed to be widely applied in oil production due to complex oil production environment [13,14]. In China, CSYY60H10 type high-power ultrasonic oil production equipment has been successfully developed in 2006 [15], but it cannot work at high temperatures and pressures underground last for a long time. In a word, the piezoelectric vibrators of transducers both in china and other countries are all made of PZT [16], which contains often-harmful lead. High-power ultrasonic transducer made of lithium niobate for Enhanced Oil Recovery technology is proposed in this paper due to the following two major reasons [23]: in the last years the utilizing of lead got more and more restricted, still with many exceptions for piezoelectric materials, but it is unclear how long those will be open. The second reason is an expansion of the acceptable environment-parameters, especially in for high temperature applications. Besides that, lithium niobate has good mechanical performance and chemical stability, it’s mechanical quality factor (about 105 magnitude) is much higher than common PZT. In view of the above advantages, high-power ultrasonic transducer made of lithium niobate for Enhanced Oil Recovery technology could be better than those transducers made of PZT. 4. Experiments for reducing the viscosity of super heavy oil using lithium niobate transducer When high-energy ultrasonic waves travel through a medium, rapid and successive compression and rarefaction cycles occur. The decrease in the pressure in the negative pressure cycle of the ultrasonic shock wave spontaneously generates small cavities.
Fig. 6. The three-dimensional model of serried lithium niobate transducers.
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Z. Wang et al. / Ultrasonics Sonochemistry 27 (2015) 171–177 Table 3 The effects of ultrasonic power on the rate of viscosity reduction of super heavy oil. Power (W) The ratio of viscosity reduction (%)
Ultrasonic frequency (kHz) Viscosity (MPa s) The ratio of viscosity reduction (%)
Table 1 The results of orthogonal experiment. Experimental number
Ultrasonic frequency (kHZ)
Ultrasonic power (W)
Time (min)
The ratio of viscosity reduction (%)
1 2 3 4 5 6 7 8 9
18 18 18 20 20 20 25 25 25
100 500 1000 100 500 1000 100 500 1000
5 15 30 15 30 5 30 5 15
45 55.4 65.8 44 48 52.2 34 38.5 42.4
5 476 54.5
10 470 62.40
15 469 63.48
20 466 64.20
Note: the initial viscosity of super heavy oil is 1, 250 MPa s (50 °C).
Theses cavities collapse in the positive pressure cycle and produce highly turbulent flow conditions and extremely high pressure and temperatures, a process that is better known as cavitation [19]. The mechanical vibration caused by cavitation effect can instantly
The ratio of viscosity reduction (%)
18 480 62
The ratio of viscosity reduction
65
60
55
50
45
0
2
4
6
8
650 44.3
850 45.6
1050 50.6
1250 51.4
20 890 28.9
25 920 26.5
break heavy large molecules in super heavy oil into light hydrocarbon substances. The viscosity of super heavy oil will never return to the pre-treatment condition when large molecules are cracked [20]. Right now, ultrasonic viscosity-reduction techniques both in China and other countries all emphasize field tests, the research on theoretical models still in its infancy, the basic theory of ultrasonic viscosity-reduction technique still has not been established. In addition, the study of cracking large molecules using ultrasound has been reported rarely [21]. In order to verify the feasibility of this new transducer for Enhanced Oil Recovery (Enhanced Oil Recovery), the experiment for reducing viscosity of super heavy oil using lithium niobate transducer has been successfully done. Experimental sample of super heavy oil, shown in Fig. 7, is taken from Binnan oil well of Shengli oilfield in china. The vertical depth of this oil well is 4364.95 m [13]. The quality of experimental sample is 250 g. Due to the temperature of the crude oil sample just out of oil field is as high as 50 °C, experimental sample must be remained at a constant temperature of 50 °C in order to accurately research the effect of ultrasonic wave on the viscosity reduction of this oil sample. The concrete steps and results are as follows: firstly, after each orthogonal experiment is completed, according to the standard rules of SY/T6316-1998, the viscosity of super heavy oil and the ratio of viscosity reduction are measured and calculated
Table 2 The relationship between viscosity, the ratio of viscosity reduction and ultrasonic treatment time (ultrasonic frequency is 18 kHz). 2 680 45.60
450 38.8
Table 4 The effects of ultrasonic frequency on the viscosity reduction of super heavy oil.
Fig. 8. Experimental sample in the water bath with constant temperature (50 °C).
Time (min) Viscosity (MPa s) The rate of viscosity reduction (%)
250 36.4
10
12
14
16
Time (min) Fig. 9. The graph of time–the ratio of viscosity reduction.
18
20
22
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Z. Wang et al. / Ultrasonics Sonochemistry 27 (2015) 171–177 Table 5 The components and mean relative molecular mass of super heavy oil before and after ultrasonic treatment. Oil samples
Saturated hydrocarbons
Aromatic hydrocarbon
Colloid
Asphaltene
Average relative molecular mass
Before processing After processing
26.51 30.25
34.64 36.04
26.48 22.35
12.35 10.38
764 598
respectively by Brookfield viscosimeter when the shear rate is 3 s1. Then the optimum ultrasound parameters are selected; secondly, the influence is analyzed of ultrasound frequency, power and treatment time on the viscosity reduction of super heavy oil respectively; finally, according to the standard rules of SY/T5119-1996, the components of super heavy oil that is processed by the ultrasound with optimum parameters are measured, and then the mean relative molecular mass of super heavy oil is measured by Knauer K-700 vapor pressure permeameter. Experimental sample in the water bath with constant temperature (50 °C) is shown in Fig. 8 [22], experimental results are shown in Table 1. As can be seen in Table 1, the optimum values of the ultrasonic power, frequency and treatment time are 1000 W, 18 kHz and 30 min respectively. But what is the influence of ultrasound frequency, power and treatment time on the viscosity reduction of super heavy oil respectively? The results are as follows: As can be seen in Table 2 and Fig. 9, under constant ultrasonic frequency, the ratio of viscosity reduction of super heavy oil increases with the increase of ultrasonic treatment time. The longer the duration of cavitation effect is, the easier it is to reduce the viscosity of super heavy oil under constant ultrasonic frequency. As can be seen form Table 3, with the increase of ultrasonic power, the ratio of viscosity reduction of super heavy oil increases accordingly. The greater the ultrasonic power is, the more obvious the cavitation effect is and the easier it is to reduce the viscosity of super heavy oil. As can be seen in Table 4, the ratio of viscosity reduction of super heavy oil decreases with the increase of ultrasonic frequency. The cavitation effect caused by ultrasonic wave needs enough time, the higher the ultrasonic frequency is, the shorter the period of ultrasonic wave is and the shorter the time needed for cavitation effect is. Lack of enough time is not conducive to generate cavitation effect so that the viscosity of super heavy oil cannot be effectively reduced. Therefore, ultrasonic frequency cannot be too high. As can be seen in Table 5, heavy large molecules (Resin and Asphaltene) in super heavy oil can be cracked into light hydrocarbon substances (saturated hydrocarbons, aromatic hydrocarbons) under ultrasonic treatment. Compound molecular bond such as C-C chemical bond can be cracked under strong cavitation effect caused by high-intensity ultrasonic wave. The viscosity of super heavy oil is reduced when the large molecules are cracked. The front end of high-power transducers used for Enhanced Oil Recovery are normally installed with spring-piston pressure auto-balancing device, which can make ultrasonic transducer work
Fig. 10. Wafer modal.
Fig. 11. Lumped equivalent circuit of wafer.
safely at high temperatures and pressures underground last for a long time. This device has been very mature, it can made transducers work safely under external pressure of 30 MPa [15]. Spring-piston pressure auto-balancing device can also be installed on the front end of lithium niobate transducer designed in this paper. Crude oil viscosity lowering and production increasing field test by high-power transducer made of PZT has been carried out in 90– 1 well in west oilfield in China [15]. Oil pressure and temperature are 9.1 MPa and 50 °C respectively. The performance parameters of this transducer are: output frequency is 3–35 kHz, service temperature of about 120 °C. Test result showed that oil production has remained unchanged at 15 m3 per day, the daily increased oil is 7.8 m3 and the cumulative increased oil is 600 m3 after ultrasonic transducer (vibrates at 22 kHz) processing 15 h. Vibrational frequency of each lithium niobate piezoelectric wafer designed in this paper (25 mm in diameter and 0.2 mm thick) can reach about 18.5 MHz according to Formula (15), besides, lithium niobate piezoelectric wafer can work safely at high temperatures of 1210 °C last for a long time, The spring-piston pressure auto-balancing device installed on the front end of lithium niobate transducer designed in this paper can be sufficient to overcome 9.1 MPa oil pressure. In a word, by meeting the above parameter requirement, lithium niobate ultrasonic transducer described in this paper is feasible for Enhanced Oil Recovery. 5. Conclusion 1. As can be clearly seen in Table 2, Table 3 and Table 4, the most important factors for reducing the viscosity of super heavy oil are ultrasonic power, followed by ultrasonic frequency and time. 2. As can be seen in Table 5, compared with no ultrasonic treatment, mean relative molecular mass of the light components such as Saturated hydrocarbon and Aromatic hydrocarbon in super heavy oil that is processed by the ultrasound with optimum parameters are increased by 14.1% and 4% respectively, while heavy components such as Colloid and Asphaltene are decreased by 15.6% and 16% respectively, in addition, the mean relative molecular mass of super heavy oil is decreased by 21.73%. Experimental results prove that heavy large molecules in super heavy oil can be cracked into light hydrocarbon substances under strong cavitation effect caused by high-intensity ultrasonic wave. 3. Research results prove that it is indeed feasible to design high-power ultrasonic transducer for ultrasonic oil production technology using lithium niobate.
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Acknowledgement
q
The authors would like to thank financial support from ‘‘863’’project fund in China and deep instruction from Prof. Shushan Zhao in Harbin Institute of Technology in China.
Under the action of alternating electric field E, as shown in Fig. 10, particles in wafer do stretching vibration along the z axis. As can be seen in Fig. 2, the following boundary conditions are obtained: the strain S3 –0; and S1 ¼ S2 ¼ S4 ¼ S5 ¼ S6 ¼ 0, electric displacement: D3 –0; D1 ¼ D2 ¼ 0. According to piezoelectric Formula (1) and the above boundary conditions, the following simplified Formula (2) can be acquired.
T m ¼ cDmn Sh hjn Dj ðm; n ¼ 1; 2; ; 6Þ
ð1Þ
Ei ¼ hin Sn þ bSij Dj ði; j ¼ 1; 2; 3Þ (
T 3 ¼ cD33 S3 h33 D3
ð2Þ
E3 ¼ h33 S3 þ bS33 D3 where
cDmn
the condition of constant strain, h represents the thickness of wafer. According to Newton’s second law, vibration equation can be written as follows
@ 2 g @T q 2¼ 3 @z @t
ð3Þ
where g is the displacement along direction of z axis, q is wafer density. The second equation in Formula (2) can be written as
E3 bS33
þ
h33 bS33
@D3 @z
S3
ð4Þ
Take the partial derivative of Formula (4) with respect to z. Due 3 ¼ 0 can be to the fact that there is no free charge in wafer, @D @z obtained. It is known that S3 ¼ @@zg ; so can be obtain after integrating.
@E3 @z
2
¼ h33 @@z2g, Formula (5)
where
¼ 0 and S3 ¼ @@zg, vibration equation can be written
where B is integral constant, according to the applied voltage Rt E dt; integral constant can be calculated as 0 3
qffiffiffiffiffiffiffiffiffiffiffiffi cD33 =q is longitudinal wave velocity in wafer, Formula
@2g 2 þk g¼0 @z2
ð11Þ
where k ¼ x=t is wave number, the solution of Formula (11) is g2 sin kz g ¼ g1 sin kðtzÞ sin kt Stress distribution in wafer is determined by the first equation in Formula (2). According to the equilibrium conditions, the following equations can be obtained.
(
F 1 ¼ cD33 ð@@zgÞz¼0 A þ h33 D3 A F 2 ¼ cD33 ð@@zgÞz¼t A þ h33 D3 A
ð12Þ
where F 1 and F 2 are the forces applied on the bottom surface (z ¼ 0) and the top surface (z ¼ t) respectively. On the basis of Formula (4), the following Formula (13) can be obtained.
h33 D3 A ¼ rV
r2 ðg þ g 2 Þ jxC 0 1
ð13Þ
Substituting Formula (8) and Formula (10) into Formula (9), mechanical vibration Formula (11) can be acquired.
8 < F 1 ¼ ð qtA r2 Þðg1 þ g2 Þ þ jqtA tan 1 ktg1 þ rV j sin kt 2 jx C 0 : F 2 ¼ ð qtA r2 Þðg þ g Þ þ jqtA tan 1 ktg þ rV 1 2 2 j sin kt 2 jx C 0
ð14Þ
The lumped equivalent circuit of wafer is as follows As can be seen in Fig. 11, piezoelectric wafer has strict odd-harmonic structure, therefore, the resonance frequency under first harmonic can be written as 1
ð5Þ
t¼
ð10Þ
(7) can be written as the following Formula (11) under sinusoidal voltage.
fP ¼
@g E3 ¼ h33 þB @z
ð9Þ
as
is elastic stiffness tensor under the condition of constant
electric displacement, bSij is dielectric impermeability tensor under
D3 ¼
Based on
@2g @2g ¼ t2 2 2 @z @t
Appendix A.
(
@2g @S3 @D3 ¼ cD33 h33 @z @z @t 2
t 2h
ð15Þ
References
V¼
ðg1 þ g2 Þ where g1 and g2 are the longitudinal displaceB ¼ Vt h33 t ment of the bottom surface (z ¼ 0) and the top surface (z ¼ t) respectively, then E3 can be written as
E3 ¼ h33
@ g V h33 þ ðg1 þ g2 Þ @t t t
ð6Þ
Substituting Formula (6) into Formula (2), electric displacement D3 can be written as
D3 ¼
h33 bS33 t
ðg1 þ g2 Þ þ
V
ð7Þ
bS33 t
If the wafer is excited by sinusoidal voltage, frequency of which is x; the current can be obtained
I ¼ jx0 C 0 V rðg1 þ g2 Þ
ð8Þ
where C 0 ¼ A=tbS33 is the cut-off capacitance, r ¼ Ah33 =tbS33 is elec2
tromechanical conversion coefficient, A ¼ pðd=2Þ is area. Substituting the first equation in Formula (2) into Formula (3), Formula (9) can be obtained
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