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Estimation of absolute sound pressure in a small-sized sonochemical reactor
Ultrasonics Sonochemistry 20 (2013) 468–471
Contents lists available at SciVerse ScienceDirect
Ultrasonics Sonochemistry journal homepage: www.elsevier.com/locate/ultson
Estimation of absolute sound pressure in a small-sized sonochemical reactor Shinji Sato ⇑, Yuji Wada, Daisuke Koyama, Kentaro Nakamura Precision and Intelligence Laboratory, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 226-8503, Japan
a r t i c l e
i n f o
Article history: Received 6 March 2012 Received in revised form 23 June 2012 Accepted 27 June 2012 Available online 7 July 2012 Keywords: Sonochemistry Sonochemical reactor Sonoreactor Sound pressure Refractive index
a b s t r a c t A small-sized sonochemical reactor in which the absolute value of the sound pressure amplitude can be estimated from the vibration velocity of the transducer was investigated. The sound pressure distribution in the reactor and the relationship between the vibration velocity and the sound pressure amplitude were derived through Helmholtz wave equation. The reactor consists of a bolt-clamped Langevin transducer and a rectangular cell with a tungsten reflector. A 3k/4-standing-wave-field was generated in the reactor to simplify the sound pressure distribution. The sound pressure distribution was measured from the optical refractive index change of water using a laser interferometer. The experimental and theoretical results showed a good agreement in the absolute value of the sound pressure amplitude, and it was confirmed that the sound pressure in the sonochemical reactor can be estimated from the input current of the vibrator. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent years, numbers of studies related to sonochemistry, which is chemistry assisted with ultrasonic energy, have been reported [1–5]. Sound pressure amplitude in sonochemical reactors is one of the important parameters for the quantitative evaluation of sonochemical reactions. However, it is sometimes difficult to measure the sound pressure in reactors directly because a needle hydrophone disturbs the sound pressure distribution though its diameter is around one millimeter, and is easily broken because of intense sound field. In most of the previous literatures, ‘‘strength’’ of the sound field has been evaluated from the input electric power to ultrasound transducers or maximum values of the sound pressure amplitude in the reactors. The physical methods such as thermal probes [6–8], calorimetric method [9–11], aluminum foil erosion [12–14] and measurements of ultrasonic energy supplied to medium were also conducted. Many studies on theoretical and simulated sound pressure field in sonochemical reactors have been reported [15–17]. For example, Saez et al. reported a numerical simulation of sound pressure distribution in a sonochemical reactor based on finite element method and finite difference method. They compared the simulated results with the experiments using aluminum foil erosion. In previous studies, since reactors of much larger than the wavelength of excited ultrasonic field were used, the sound pressure distributions were complicated and it was difficult to evaluate the chemical reaction in the reactor quantitatively. In this paper, a ⇑ Corresponding author. Tel.: +81 45 924 5052; fax: +81 45 924 5091. E-mail addresses: [email protected] (S. Sato), [email protected]. jp (D. Koyama), [email protected] (K. Nakamura). 1350-4177/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultsonch.2012.06.016
reactor with small dimensions comparable to the wavelength is studied to estimate the absolute value of sound pressure amplitude. Only one or two nodal positions exist in the reactor, and the sound pressure can be estimated quantitatively from the input current of ultrasound transducer being based on the simple linear relationship between the vibration velocity and the driving current of a piezoelectric transducer. 2. Theory To utilize a simple sound pressure distribution in a reactor cell, the dimensions of the cell are reduced to less than or around the wavelength of ultrasonic filed k as shown in Fig. 1. The width of the reactor is chosen smaller than k/2 to assume plane wave, while the depth is adjusted at the second lowest resonance. Consequently, one dimensional mode is excited between the output end of a longitudinal transducer with piston vibration and the bottom of the cell at the angular frequency of x. Here, the end of the transducer can be treated as a free end since the end is vibrating at the velocity of u0. On the other hand, the bottom of the cell acts as a fixed end. The depth is almost equal to 3k/4 in our experiment. According to the one-dimensional Helmholtz wave equation
(
jxquðxÞ ¼ dpðxÞ dx jxpðxÞ ¼ K duðxÞ dx
;
ð1Þ
the relationship between the vibration velocity of the ultrasound transducer u0 and the sound pressure p(x) and the particle velocity u(x) in the depth direction (x direction) can be expressed as
469
S. Sato et al. / Ultrasonics Sonochemistry 20 (2013) 468–471
Fig. 1. Experimental setup using a small-sized sonochemical reactor and a bolt-clamped Langevin transducer with a stepped horn.
uðxÞ ¼ u0
sinðxc xÞ ; sinðxc hÞ
ð2Þ
and
pðxÞ ¼ jqcu0
cosðxc xÞ : sinðxc hÞ
ð3Þ
Here, q is the density of medium, K is a bulk modulus, and j = 1. c is the speed of sound and h is the depth of the cell. As the boundary conditions, the particle velocity at the vibrator’s surface (x = h) and the bottom of reactor (x = 0) were assumed to be u0 and 0, respectively. The theoretical sound pressure distribution in the cell is known from Eq. (3), if the vibration velocity at the end of the transducer u0 is given. 2
3. Configuration of the reactor Fig. 2 shows the configuration of the sonochemical reactor prototyped in this paper. The operation frequency is designed around 40 kHz. A rectangular cell of the reactor (10.6 10.6 25.6 mm3) was machined from a stainless steel block. Transparent observation slits made of polymethyl methacrylate (PMMA) resin were prepared to introduce a laser light for measuring the sound field in the cell. A 5 mm-thick tungsten steel plate which has a large acoustic impedance compared with water was attached so that sound wave effectively reflects at the bottom of the cell. Fig. 3
(a)
shows the configuration and the vibration distribution of the transducer. The transducer consists of a bolt-clamped Langevin transducer (BLT) with the diameter of 15 mm and a step horn. The length of the cell in x direction (25.7 mm) corresponds to 3k/4 at 43.7 kHz to have one nodal position. The horn has a 10 10 mm square radiation surface, and the theoretical transformation ratio of vibration velocity is 1:1.77. Since the vibration velocity u0 is proportional to the input current to the BLT, sound pressure can be estimated by the input current to the BLT. The force factor, which is the ratio of the input current to the vibration velocity, was measured in air before the experiment, and was 0.85 as shown in Fig. 4. This force factor was used to estimate the vibration velocity at the end surface from the measured driving current. It is the nature of piezoelectric transducer that the force factor is unchanged by the loading conditions. In the setup used in our experiment, the current to the transducer was almost in phase with the applied voltage. Consequently, the motional current can be almost equal to the total current to the transducer. However, in the case that the current to the damped admittance cannot be ignored, one should estimate the real motional current from vector calculation. 4. Sound pressure distribution In order to evaluate the proposed method, the sound pressure distribution in the reactor was measured using an optical interferometer through the refractive index modulation of water. Practically, a commercial laser Doppler velocimeter (LDV) can be utilized as the optical interferometer for sound pressure measurements without disturbing the sound field [18,19]. Principle of the measurement is illustrated in Fig. 5. Optical path length modulated by the change in refractive index due to sound pressure is detected instead of the vibration of reflector. The optical path length change due to the change in refractive index of medium Dn is equivalent to the virtual displacement of reflector Dl:
Dn l ¼ nDl:
ð4Þ
From Eq. (4) and the output from LDV mLDV, refractive index change is expressed as
n¼
n mLDV : j2pfl
ð5Þ
It should be noted that the LDV’s output is divided by j2pf to transform the velocity to the displacement.
(b)
Fig. 2. (a) Configuration of the sonochemical reactor and (b) its photograph.
470
S. Sato et al. / Ultrasonics Sonochemistry 20 (2013) 468–471
(a)
(b)
Fig. 3. (a) Measurement setup and (b) the vibration distribution of the transducer.
i = 0.8455 u0
0.030
Current i [A]
Sound pressure amplitude [kPa]
0.035
0.025 0.020 0.015 0.010 0.005 0
0
0.01
0.02
0.03
50 Theoretical
40 30
Experimental
20 10 0
0.04
0
5
10
15
20
25
30
x direction [mm]
Vibrational velocity u0 [m/s]
(a)
Fig. 4. Characteristics of the transducer.
Phase [deg.]
180
Experimental
90
0
Theoretical
0
5
10
15
20
25
30
-90
-180
x direction [mm]
(b) Fig. 5. Principle of the measurement of refractive index modulation using a laser Doppler vibrometer.
Eykman’s equation [20] shows the change in refractive index as a function of pressure:
n¼
ðn0 1Þðn20 þ 1:4n0 þ 0:4Þ p; ðn20 þ 0:8n0 þ 1Þqc2
ð6Þ
Using Eqs. (5) and (6), under the assumption of plane wave propagation in the cell, the sound pressure p can be expressed as
Fig. 6. Sound pressure distribution in x direction of the cell when the vibration velocity of the transducer was 29.5 mm/s: (a) amplitude and (b) phase.
p ¼ j
n 2pfl
mLDV
ðn2 þ 0:8n þ 1Þqc2 ; ðn 1Þðn2 þ 1:4n þ 0:4Þ
ð7Þ
where n is the refractive index of media, f is the frequency, and l is the length of the cell (length of acoustic field passed by laser). In this paper, n = 1.33, f = 43.7 kHz, l = 10.6 mm, q = 1000 kg/m3, c = 1500 m/s were used. The distribution was measured by mechanically scanning the light of the LDV in the x direction.
S. Sato et al. / Ultrasonics Sonochemistry 20 (2013) 468–471
Theoretical and measured results of the sound pressure distribution are shown in Fig. 6. The phase was measured using a lock-in amplifier with the LDV’s output signal as the reference. The vibration velocity u0 was set to be 29.5 mm/s using the force factor, and the peak value of the sound pressure is estimated to be 44 kPa. These results show a good agreement on the sound pressure distribution. A 3k/4-standing-wave-field with the amplitude of 44 kPa was generated and there is a nodal point of the acoustic standing wave at x = 8.5 mm (k/4 from the bottom of the cell). The average error, which is defined as the difference between the predicted and measured values normalized by the theoretical maximum, was 13%. It is confirmed that sound pressure distribution in the sonochemical reactor can be evaluated from the vibrational velocity of the tip of the transducer or the input current to the BLT. The error might be attributed to imperfect reflectivity at the bottom and the tip of the vibrator, since the experimental results of the phase distribution change continuously at the node (x = 8.5 mm) and imply the generation of traveling wave component. The assumption of plane wave in the theoretical model may also lead these errors. The temperature rise during the experiment was enough small to ignore its effects on the results. Since the theoretical model can be applied for only linear region, we should take into account nonlinear effects in the case of higher sound pressure amplitude. 5. Conclusions The small-sized sonochemical reactor with simple sound pressure distribution was discussed. The sound pressure distribution and its absolute value in the reactor can be estimated from the input current to the transducer. The sonochemical reactor proposed in this paper shall enable quantitative evaluation in sonochemical experiments. References [1] H. Yanagawa, E. Umeki, Y. Tamura, T. Saitoh, T. Takahashi, M. Ikegami, K. Minagawa, Ultrasonic polymerization of N-isopropylacrylamide below and above critical temperature, Jpn. J. Appl. Phys. 49 (2010) 07HE07.
471
[2] S. Hatanaka, S. Hayashi, P. Choi, Sonoluminescence of alkali-metal atoms in sulfuric acid: comparison with that in water, Jpn. J. Appl. Phys. 49 (2010) 07HE01. [3] T. Uchida, T. Kikuchi, T.Aoki, N. Kawashima, S. Takeuchi, Effects of ultrasound exposure time on nanometer-sized diamond particles dispersion, Jpn. J. Appl. Phys. 48 (2009) 07GH03. [4] M. Cui, M. Jang, S. Lee, B. Kweon, S. Yun, J. Khim, Arsenite oxidation and treatment by ultrasound/iron in aqueous solutions, Jpn. J. Appl. Phys. 50 (2011) 07HE08. [5] H. Harada, Overall Water Splitting by Sono-Catalysis Using Homogeneous Catalysts of Fe(III) Compounds: Aim of Construction of a Photosynthesis-Like System under Sonication, Jpn. J. Appl. Phys. 41 (2002) 3213–3216. [6] C.J. Martin, A.N.R. Law, The use of thermistor probes to measure energy distribution in ultrasound fields, Ultrasonics 18 (1980) 127–133. [7] C.J. Martin, A.N.R. Law, Design of thermistor probes for measurement of ultrasound intensity distributions, Ultrasonics 21 (1983) 85–90. [8] M. Romdhane, C. Gourdon, G. Casamatta, Local investigation of some ultrasonic devices by means of a thermal sensor, Ultrasonics 33 (1995) 221–227. [9] M.A. Margulis, I.M. Margulis, Calorimetric method for measurement of acoustic power absorbed in a volume of a liquid, Ultrason. Sonochem. 10 (2003) 343– 345. [10] T. Kimura, T. Sakamoto, J.M. Leveque, H. Sohmiya, M. Fujita, S. Ikeda, T. Ando, Standardization of ultrasonic power for sonochemical reaction, Ultrason. Sonochem. 3 (1996) 157–161. [11] R.F. Contamine, A.M. Wilhelm, J. Berlian, H. Delmas, Power measurement in sonochemistry, Ultrason. Sonochem. 2 (1995) 43–47. [12] B. Pugin, Qualitative characterization of ultrasound reactors for heterogeneous sonochemistry, Ultrasonics 25 (1987) 49–55. [13] M.M. Chivate, A.B. Pandit, Quantification of cavitation intensity in fluid bulk, Ultrason. Sonochem. 2 (1995) 19–25. [14] C. Wu, N. Nakagawa, M. Fujihara, Study on acoustic cavitations in the ultrasonic radiation field, JSME Int J., Ser. C 49 (3) (2006) 758–763. [15] S. Dahnke, K.M. Swamy, F.J. Keil, A comparative study on the modeling of sound pressure field distributions in a sonoreactor with experimental investigation, Ultrason. Sonochem. 6 (1999) 221–226. [16] S.W. DaKhnke, F.J. Keil, Modeling of linear pressure fields in sonochemical reactors considering an inhomogeneous density distribution of cavitation bubbles, Chem. Eng. Sci. 54 (1999) 2865–2872. [17] V. Saez, A. Frıas-Ferrer, J. Iniesta, J. Gonzalez-Garcıa, A. Aldaz, E. Riera, Chacterization of a 20 kHz sonoreactor. Part I: analysis of mechanical effects by classical and numerical methods, Ultrason. Sonochem. 12 (2005) 59–65. [18] K. Nakamura, M. Hirayama, S.Ueha, Measurements of air-borne ultrasound by detecting the modulation in optical refractive index of air, Proc. IEEE Ultrason. Symp. (2002) 609–612. [19] K. Aoki, Y. Uno and K. Nakamura, Noninvasive measurements of ultrasonic fields by using the optical path modulation of the transmitted laser beam, Proc. Annual Meeting of Acoustical Society of Japan (Spring, 1997) 1099. [20] O. Nomoto, K. Negishi, Diffraction of light by ultrasonic waves of finite amplitude, Acustica 15 (1965) 223–235.
Contents lists available at SciVerse ScienceDirect
Ultrasonics Sonochemistry journal homepage: www.elsevier.com/locate/ultson
Estimation of absolute sound pressure in a small-sized sonochemical reactor Shinji Sato ⇑, Yuji Wada, Daisuke Koyama, Kentaro Nakamura Precision and Intelligence Laboratory, Tokyo Institute of Technology, 4259 Nagatsuta, Midori-ku, Yokohama 226-8503, Japan
a r t i c l e
i n f o
Article history: Received 6 March 2012 Received in revised form 23 June 2012 Accepted 27 June 2012 Available online 7 July 2012 Keywords: Sonochemistry Sonochemical reactor Sonoreactor Sound pressure Refractive index
a b s t r a c t A small-sized sonochemical reactor in which the absolute value of the sound pressure amplitude can be estimated from the vibration velocity of the transducer was investigated. The sound pressure distribution in the reactor and the relationship between the vibration velocity and the sound pressure amplitude were derived through Helmholtz wave equation. The reactor consists of a bolt-clamped Langevin transducer and a rectangular cell with a tungsten reflector. A 3k/4-standing-wave-field was generated in the reactor to simplify the sound pressure distribution. The sound pressure distribution was measured from the optical refractive index change of water using a laser interferometer. The experimental and theoretical results showed a good agreement in the absolute value of the sound pressure amplitude, and it was confirmed that the sound pressure in the sonochemical reactor can be estimated from the input current of the vibrator. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In recent years, numbers of studies related to sonochemistry, which is chemistry assisted with ultrasonic energy, have been reported [1–5]. Sound pressure amplitude in sonochemical reactors is one of the important parameters for the quantitative evaluation of sonochemical reactions. However, it is sometimes difficult to measure the sound pressure in reactors directly because a needle hydrophone disturbs the sound pressure distribution though its diameter is around one millimeter, and is easily broken because of intense sound field. In most of the previous literatures, ‘‘strength’’ of the sound field has been evaluated from the input electric power to ultrasound transducers or maximum values of the sound pressure amplitude in the reactors. The physical methods such as thermal probes [6–8], calorimetric method [9–11], aluminum foil erosion [12–14] and measurements of ultrasonic energy supplied to medium were also conducted. Many studies on theoretical and simulated sound pressure field in sonochemical reactors have been reported [15–17]. For example, Saez et al. reported a numerical simulation of sound pressure distribution in a sonochemical reactor based on finite element method and finite difference method. They compared the simulated results with the experiments using aluminum foil erosion. In previous studies, since reactors of much larger than the wavelength of excited ultrasonic field were used, the sound pressure distributions were complicated and it was difficult to evaluate the chemical reaction in the reactor quantitatively. In this paper, a ⇑ Corresponding author. Tel.: +81 45 924 5052; fax: +81 45 924 5091. E-mail addresses: [email protected] (S. Sato), [email protected]. jp (D. Koyama), [email protected] (K. Nakamura). 1350-4177/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultsonch.2012.06.016
reactor with small dimensions comparable to the wavelength is studied to estimate the absolute value of sound pressure amplitude. Only one or two nodal positions exist in the reactor, and the sound pressure can be estimated quantitatively from the input current of ultrasound transducer being based on the simple linear relationship between the vibration velocity and the driving current of a piezoelectric transducer. 2. Theory To utilize a simple sound pressure distribution in a reactor cell, the dimensions of the cell are reduced to less than or around the wavelength of ultrasonic filed k as shown in Fig. 1. The width of the reactor is chosen smaller than k/2 to assume plane wave, while the depth is adjusted at the second lowest resonance. Consequently, one dimensional mode is excited between the output end of a longitudinal transducer with piston vibration and the bottom of the cell at the angular frequency of x. Here, the end of the transducer can be treated as a free end since the end is vibrating at the velocity of u0. On the other hand, the bottom of the cell acts as a fixed end. The depth is almost equal to 3k/4 in our experiment. According to the one-dimensional Helmholtz wave equation
(
jxquðxÞ ¼ dpðxÞ dx jxpðxÞ ¼ K duðxÞ dx
;
ð1Þ
the relationship between the vibration velocity of the ultrasound transducer u0 and the sound pressure p(x) and the particle velocity u(x) in the depth direction (x direction) can be expressed as
469
S. Sato et al. / Ultrasonics Sonochemistry 20 (2013) 468–471
Fig. 1. Experimental setup using a small-sized sonochemical reactor and a bolt-clamped Langevin transducer with a stepped horn.
uðxÞ ¼ u0
sinðxc xÞ ; sinðxc hÞ
ð2Þ
and
pðxÞ ¼ jqcu0
cosðxc xÞ : sinðxc hÞ
ð3Þ
Here, q is the density of medium, K is a bulk modulus, and j = 1. c is the speed of sound and h is the depth of the cell. As the boundary conditions, the particle velocity at the vibrator’s surface (x = h) and the bottom of reactor (x = 0) were assumed to be u0 and 0, respectively. The theoretical sound pressure distribution in the cell is known from Eq. (3), if the vibration velocity at the end of the transducer u0 is given. 2
3. Configuration of the reactor Fig. 2 shows the configuration of the sonochemical reactor prototyped in this paper. The operation frequency is designed around 40 kHz. A rectangular cell of the reactor (10.6 10.6 25.6 mm3) was machined from a stainless steel block. Transparent observation slits made of polymethyl methacrylate (PMMA) resin were prepared to introduce a laser light for measuring the sound field in the cell. A 5 mm-thick tungsten steel plate which has a large acoustic impedance compared with water was attached so that sound wave effectively reflects at the bottom of the cell. Fig. 3
(a)
shows the configuration and the vibration distribution of the transducer. The transducer consists of a bolt-clamped Langevin transducer (BLT) with the diameter of 15 mm and a step horn. The length of the cell in x direction (25.7 mm) corresponds to 3k/4 at 43.7 kHz to have one nodal position. The horn has a 10 10 mm square radiation surface, and the theoretical transformation ratio of vibration velocity is 1:1.77. Since the vibration velocity u0 is proportional to the input current to the BLT, sound pressure can be estimated by the input current to the BLT. The force factor, which is the ratio of the input current to the vibration velocity, was measured in air before the experiment, and was 0.85 as shown in Fig. 4. This force factor was used to estimate the vibration velocity at the end surface from the measured driving current. It is the nature of piezoelectric transducer that the force factor is unchanged by the loading conditions. In the setup used in our experiment, the current to the transducer was almost in phase with the applied voltage. Consequently, the motional current can be almost equal to the total current to the transducer. However, in the case that the current to the damped admittance cannot be ignored, one should estimate the real motional current from vector calculation. 4. Sound pressure distribution In order to evaluate the proposed method, the sound pressure distribution in the reactor was measured using an optical interferometer through the refractive index modulation of water. Practically, a commercial laser Doppler velocimeter (LDV) can be utilized as the optical interferometer for sound pressure measurements without disturbing the sound field [18,19]. Principle of the measurement is illustrated in Fig. 5. Optical path length modulated by the change in refractive index due to sound pressure is detected instead of the vibration of reflector. The optical path length change due to the change in refractive index of medium Dn is equivalent to the virtual displacement of reflector Dl:
Dn l ¼ nDl:
ð4Þ
From Eq. (4) and the output from LDV mLDV, refractive index change is expressed as
n¼
n mLDV : j2pfl
ð5Þ
It should be noted that the LDV’s output is divided by j2pf to transform the velocity to the displacement.
(b)
Fig. 2. (a) Configuration of the sonochemical reactor and (b) its photograph.
470
S. Sato et al. / Ultrasonics Sonochemistry 20 (2013) 468–471
(a)
(b)
Fig. 3. (a) Measurement setup and (b) the vibration distribution of the transducer.
i = 0.8455 u0
0.030
Current i [A]
Sound pressure amplitude [kPa]
0.035
0.025 0.020 0.015 0.010 0.005 0
0
0.01
0.02
0.03
50 Theoretical
40 30
Experimental
20 10 0
0.04
0
5
10
15
20
25
30
x direction [mm]
Vibrational velocity u0 [m/s]
(a)
Fig. 4. Characteristics of the transducer.
Phase [deg.]
180
Experimental
90
0
Theoretical
0
5
10
15
20
25
30
-90
-180
x direction [mm]
(b) Fig. 5. Principle of the measurement of refractive index modulation using a laser Doppler vibrometer.
Eykman’s equation [20] shows the change in refractive index as a function of pressure:
n¼
ðn0 1Þðn20 þ 1:4n0 þ 0:4Þ p; ðn20 þ 0:8n0 þ 1Þqc2
ð6Þ
Using Eqs. (5) and (6), under the assumption of plane wave propagation in the cell, the sound pressure p can be expressed as
Fig. 6. Sound pressure distribution in x direction of the cell when the vibration velocity of the transducer was 29.5 mm/s: (a) amplitude and (b) phase.
p ¼ j
n 2pfl
mLDV
ðn2 þ 0:8n þ 1Þqc2 ; ðn 1Þðn2 þ 1:4n þ 0:4Þ
ð7Þ
where n is the refractive index of media, f is the frequency, and l is the length of the cell (length of acoustic field passed by laser). In this paper, n = 1.33, f = 43.7 kHz, l = 10.6 mm, q = 1000 kg/m3, c = 1500 m/s were used. The distribution was measured by mechanically scanning the light of the LDV in the x direction.
S. Sato et al. / Ultrasonics Sonochemistry 20 (2013) 468–471
Theoretical and measured results of the sound pressure distribution are shown in Fig. 6. The phase was measured using a lock-in amplifier with the LDV’s output signal as the reference. The vibration velocity u0 was set to be 29.5 mm/s using the force factor, and the peak value of the sound pressure is estimated to be 44 kPa. These results show a good agreement on the sound pressure distribution. A 3k/4-standing-wave-field with the amplitude of 44 kPa was generated and there is a nodal point of the acoustic standing wave at x = 8.5 mm (k/4 from the bottom of the cell). The average error, which is defined as the difference between the predicted and measured values normalized by the theoretical maximum, was 13%. It is confirmed that sound pressure distribution in the sonochemical reactor can be evaluated from the vibrational velocity of the tip of the transducer or the input current to the BLT. The error might be attributed to imperfect reflectivity at the bottom and the tip of the vibrator, since the experimental results of the phase distribution change continuously at the node (x = 8.5 mm) and imply the generation of traveling wave component. The assumption of plane wave in the theoretical model may also lead these errors. The temperature rise during the experiment was enough small to ignore its effects on the results. Since the theoretical model can be applied for only linear region, we should take into account nonlinear effects in the case of higher sound pressure amplitude. 5. Conclusions The small-sized sonochemical reactor with simple sound pressure distribution was discussed. The sound pressure distribution and its absolute value in the reactor can be estimated from the input current to the transducer. The sonochemical reactor proposed in this paper shall enable quantitative evaluation in sonochemical experiments. References [1] H. Yanagawa, E. Umeki, Y. Tamura, T. Saitoh, T. Takahashi, M. Ikegami, K. Minagawa, Ultrasonic polymerization of N-isopropylacrylamide below and above critical temperature, Jpn. J. Appl. Phys. 49 (2010) 07HE07.
471
[2] S. Hatanaka, S. Hayashi, P. Choi, Sonoluminescence of alkali-metal atoms in sulfuric acid: comparison with that in water, Jpn. J. Appl. Phys. 49 (2010) 07HE01. [3] T. Uchida, T. Kikuchi, T.Aoki, N. Kawashima, S. Takeuchi, Effects of ultrasound exposure time on nanometer-sized diamond particles dispersion, Jpn. J. Appl. Phys. 48 (2009) 07GH03. [4] M. Cui, M. Jang, S. Lee, B. Kweon, S. Yun, J. Khim, Arsenite oxidation and treatment by ultrasound/iron in aqueous solutions, Jpn. J. Appl. Phys. 50 (2011) 07HE08. [5] H. Harada, Overall Water Splitting by Sono-Catalysis Using Homogeneous Catalysts of Fe(III) Compounds: Aim of Construction of a Photosynthesis-Like System under Sonication, Jpn. J. Appl. Phys. 41 (2002) 3213–3216. [6] C.J. Martin, A.N.R. Law, The use of thermistor probes to measure energy distribution in ultrasound fields, Ultrasonics 18 (1980) 127–133. [7] C.J. Martin, A.N.R. Law, Design of thermistor probes for measurement of ultrasound intensity distributions, Ultrasonics 21 (1983) 85–90. [8] M. Romdhane, C. Gourdon, G. Casamatta, Local investigation of some ultrasonic devices by means of a thermal sensor, Ultrasonics 33 (1995) 221–227. [9] M.A. Margulis, I.M. Margulis, Calorimetric method for measurement of acoustic power absorbed in a volume of a liquid, Ultrason. Sonochem. 10 (2003) 343– 345. [10] T. Kimura, T. Sakamoto, J.M. Leveque, H. Sohmiya, M. Fujita, S. Ikeda, T. Ando, Standardization of ultrasonic power for sonochemical reaction, Ultrason. Sonochem. 3 (1996) 157–161. [11] R.F. Contamine, A.M. Wilhelm, J. Berlian, H. Delmas, Power measurement in sonochemistry, Ultrason. Sonochem. 2 (1995) 43–47. [12] B. Pugin, Qualitative characterization of ultrasound reactors for heterogeneous sonochemistry, Ultrasonics 25 (1987) 49–55. [13] M.M. Chivate, A.B. Pandit, Quantification of cavitation intensity in fluid bulk, Ultrason. Sonochem. 2 (1995) 19–25. [14] C. Wu, N. Nakagawa, M. Fujihara, Study on acoustic cavitations in the ultrasonic radiation field, JSME Int J., Ser. C 49 (3) (2006) 758–763. [15] S. Dahnke, K.M. Swamy, F.J. Keil, A comparative study on the modeling of sound pressure field distributions in a sonoreactor with experimental investigation, Ultrason. Sonochem. 6 (1999) 221–226. [16] S.W. DaKhnke, F.J. Keil, Modeling of linear pressure fields in sonochemical reactors considering an inhomogeneous density distribution of cavitation bubbles, Chem. Eng. Sci. 54 (1999) 2865–2872. [17] V. Saez, A. Frıas-Ferrer, J. Iniesta, J. Gonzalez-Garcıa, A. Aldaz, E. Riera, Chacterization of a 20 kHz sonoreactor. Part I: analysis of mechanical effects by classical and numerical methods, Ultrason. Sonochem. 12 (2005) 59–65. [18] K. Nakamura, M. Hirayama, S.Ueha, Measurements of air-borne ultrasound by detecting the modulation in optical refractive index of air, Proc. IEEE Ultrason. Symp. (2002) 609–612. [19] K. Aoki, Y. Uno and K. Nakamura, Noninvasive measurements of ultrasonic fields by using the optical path modulation of the transmitted laser beam, Proc. Annual Meeting of Acoustical Society of Japan (Spring, 1997) 1099. [20] O. Nomoto, K. Negishi, Diffraction of light by ultrasonic waves of finite amplitude, Acustica 15 (1965) 223–235.