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Velocity study in an ultrasonic reactor
Ultrasonics Sonochemistry 7 (2000) 207–211 www.elsevier.nl/locate/ultsonch
Velocity study in an ultrasonic reactor M. Chouvellon a, *, A. Largillier b, T. Fournel a, P. Boldo a, Y. Gonthier c a Laboratoire Traitement du Signal et Instrumentation, UMR CNRS 5516, Universite´ Jean-Monnet, 23, rue du Dr Paul Michelon, 42023 Saint-Etienne, France b Equipe d’Analyse Nume´rique de Lyon-Saint-Etienne, UMR CNRS 5585, 23, rue du Dr Paul Michelon, 42023 Saint-Etienne, France c Laboratoire de Ge´nie des Proce´de´s, Universite´ de Savoie, Savoie Technolac, 73376 Le Bourget du Lac, France
Abstract In order to determine the parameters required to describe and to optimize sonochemical reactors, we have investigated the water flow inside such a reactor. With this aim, the experimental velocity field has been measured by tomography laser. The influence of certain parameters such as the electric power, the water height and the fluid viscosity has been evaluated. At the same time, the water movement has been studied theoretically using Nyborg’s model. We have tried to improve this model by considering a three-dimensional velocity. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Acoustic streaming; Particle image velocimetry; Sonochemistry; Velocity field
1. Introduction An ultrasonic reactor allows the degradation of some organic compounds such as pentachlorophenol which are not or poorly biodegradable by common methods in industrial flows. The experimental reactor used comprises a piezoelectric transducer stuck to the center of a stainless steel plate. This plate is placed at the bottom of a tank filled with water. The experimental reactor has been developed by the chemical engineering laboratory (Le Bourget du Lac). For this prototype, the resonance frequency of the transducer is 500 kHz and the electric power is about 100 W. The ultrasonic wave propagation induces the creation of cavitation bubbles. Their implosion generates some reactive species as hydroxyl radicals OH0 which allow the activation of chemical reactions. Many researchers have described the evolution of a single bubble in an ultrasonic field but very few have described the water movement generated by an ultrasonic wave. Nevertheless, such a movement can explain the liquid homogeneity inside the reactor. This is why our study consists in measuring the velocity field in the whole reactor using a tomography laser. The influence of certain parameters such as the electric power, the water * Corresponding author. Fax: +33-04-77-48-51-20. E-mail address: [email protected] (M. Chouvellon)
height and the fluid viscosity has been evaluated. At the same time, we have tried to study the water movement theoretically in order to obtain a model.
2. Tomography laser principle The tomography laser method consists in lighting up a thin section of the flow with a laser beam focused by a lens system. Particles (about 20 mm in diameter) are seeded into the flow as markers and particle images are periodically recorded. The particle density is close to that of water in order to ensure they follow the flow. The experimental set-up is composed of the following ( Fig. 1) $ A continuous laser of 300 mW which gives out green light at 532 nm. $ An intensified camera with a shutter control and a synchronization system. This commands the aperture of the intensifier and it allows images to be obtained with a short exposure time compared with the time between two successive images. This system is located perpendicular to the laser beam. $ A digitization board allows image recording in 512×512 pixels. The tomography laser has allowed the development of particle image velocimetry. Using this experimental set-up, some particle images were obtained (Fig. 2).
1350-4177/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S1 3 5 0 -4 1 7 7 ( 0 0 ) 0 0 06 0 - 2
208
M. Chouvellon et al. / Ultrasonics Sonochemistry 7 (2000) 207–211
Fig. 1. Laser tomography system.
Fig. 2. (a) Particle image, (b) cross correlation image, (c) velocity vectors.
Each of them is then divided into several subimages. The calculation of the correlation allows a local measurement of a mean displacement of particles in a subimage. When there is a single exposure per frame to the light diffused by particles, two consecutive subimages s1 and s2 give two successive positions of the particles. From an ideal point of view, their displacement is a translation and the second subimage s2 is a shifted version of the subimage s1. The cross-correlation method [1] consists in associating the best superposition with the local displacement by relative shifting between subimages. It corresponds to the maximum of the correlation of the two successive subimage. However, this method allows only two components of the velocity to be obtained with a single camera.
two successive images and the exposure time are respectively 40 and 5 ms. A series of images was used to obtain the following mean velocity field ( Fig. 3). We notice two main vortices on each side of the central axis. The velocity is maximum in a column above the transducer. The vorticity calculated from the last velocity field indicates the location of vortex motions. The turbulence rate will allow us to quantify the efficiency of the chemical species mixing. In order to evaluate the influence of certain factors on the flow, we have varied some experimental parameters.
4. Influence of experimental factors 3. Velocity field The mean velocity inside the ultrasonic reactor has been measured by this method for an electric power of 100 W and for a 10 cm water height. The time between
The flow depends on many parameters such as the electric power, the water height and the fluid viscosity. For the study of each of these parameters, the same procedure has been used. With this aim, several images have been taken at different times at the reactor center.
M. Chouvellon et al. / Ultrasonics Sonochemistry 7 (2000) 207–211
209
Fig. 3. Mean velocity and vorticity field measured by particle image velocimetry (r, radial coordinate; z, vertical coordinate). Velocity scale4 cm s−1; non-dimensional vorticity scale: vorticity =vorticity/(U /water height). ad max
The space and time average of the velocity modulus is then calculated and it is represented in the different figures. The velocity field is about of 6 cm×6 cm and the water height is 10 cm. 4.1. Electric power A luminol experiment [2] has shown that the chemical activity increases quasi-linearly with the electric power. Hence, we are interested in studying the influence of this parameter on the flow. Contrary to the chemical activity, the velocity increases in a non-linear way with the electric power: it increases more from 50 to 100 W than above 100 W (Fig. 4)
can be used with a good mixing at a given electric power. The greater the water height, the lower is the averaged velocity. This velocity decreases in a regular way. The decrease of the velocity (Fig. 5) can be explained by the decrease in the electric power per water volume unit as the water height increases. 4.3. Fluid viscosity
The study of this parameter allows information to be obtained on the maximum of the water volume which
The last parameter studied is the liquid viscosity. In this experiment, the viscosity is increased by increasing the percentage of glycerol. The liquid height and the electric power do not vary during this experiment ( Fig. 6). Contrary to classical flow, the average velocity increases with the liquid viscosity until it reaches a threshold for m/m =20. However, it seems that the eau viscosity impedes chemical activity by decreasing the fluctuations in bubble diameter. In order to understand better the physical phenomena
Fig. 4. Velocity modulus as a function of electric power (water height: 0.1 m).
Fig. 5. Velocity modulus as a function of water height (electric power: 100 W ).
4.2. Water height
210
M. Chouvellon et al. / Ultrasonics Sonochemistry 7 (2000) 207–211
ity equation: r
C
∂u ∂t
D
A
+(u · V )u =−Vp+ m∞+
4 3
B
m VVu−mV×Vu, (1)
∂r ∂t
+V · (ru)=0.
(2)
Then, he breaks down the pressure, the fluid density and the velocity into several terms: Fig. 6. Velocity modulus as a function of viscosity (electric power: 100 W; water height: 0.1 m).
responsible for this flow, we have tried to study the water movement theoretically.
5. Acoustic streaming 5.1. Nyborg’s model The water flow cannot be explained only in terms of thermal convection, so some authors have considered acoustic streaming [3–5]. Nyborg [6 ] has formulated a model from both the dynamic equation and the continu-
Fig. 7. Profiles of the experimental and the theoretical velocities.
p=p +p +p , 0 1 2
r=r +r +r , 0 1 2
u=u +u , 1 2
where p and r correspond respectively to the equilib0 0 rium pressure and fluid density. The other terms such as p , r and u are the first-order approximations to 1 1 1 solutions of the equations, which vary sinusoidally with time. The second terms p , r and u are the second2 2 2 order approximations to the solutions and they yield correction terms to be added to the first terms. These corrections include time-independent quantities which interest us here. In this method, the first-order approximation u is determined then the second-order approxi1 mation is deduced. The latter calculation has recently been published [7]. Nyborg calculated the theoretical velocities in a simple case: he supposed that a plane wave propagates in unbounded medium so u =A e−az cos(vt−kx) where 1 a is the wave attenuation coefficient. Then, he could easily obtain an expression for the velocity. We have compared the Nyborg model with the measured velocity in Fig. 7: The Nyborg model is already in good agreement with the experiment. Indeed, the velocity is zero almost at the same place inside the reactor. Moreover, the velocity magnitudes are nearly the same. However, the reactor delimits a bounded volume, which is why we have tried to find a solution for u and so u closer to reality. 1 2
Fig. 8. (a) High view of the networking; (b) side view of the networking (the circles represent the nodes).
M. Chouvellon et al. / Ultrasonics Sonochemistry 7 (2000) 207–211
211
5.2. Improvement of Nyborg’s model
6. Conclusion
This model can be improved by considering a threedimensional velocity which depends on the three components in a bounded space. With this aim, we have used cylindrical coordinates. Moreover, the ultrasonic wave is supposed to be reflected at the free surface. The limit conditions for u are based on the measurement of the 1 amplitude of vibration of the steel plate by an optical method [8]. An analytic expression for u has been 1 obtained:
This study allows us to determine the influence of several parameters of the flow inside an ultrasonic reactor. In this way, an experiment has shown that the fluid viscosity increases the velocity until it reaches a threshold. Hence, in order to understand the physical phenomena better, we have tried to give theoretical values for the velocities based on the non-linear terms of both the dynamic equation and the continuity equation. For the moment, some mathematical problems prevent us from finding a solution.
C A BD w(r, z) cos h
u =Re ejvt w(r, z) sin h 1 v (r, z) 3
,
where w is a sum of Bessel functions of order 1 and v 3 is a sum of Bessel functions of order 0. These two terms depend on the same parameters: the viscosity m, the fluid density r and the pulsation v of the acoustic wave. In order to obtain u , partial differential equations 2 are solved with finite elements [9]. With this aim, the domain is divided into several squared elements (Fig. 8). The time-independent solution u is then the sum of 2 basis functions Q : i,j ∑ u Q (r, z), i,j i,j iµdomain where Q (r, z)=d , i and j are the nodes coordinates i,j i,j and u is a vector. This approximation is obtained by i,j resolution of a linear problem. For the moment, we do not have an expression for the theoretical velocities, as there are still mathematical problems with the boundary conditions. u (r, z)= 2
References [1] R.D. Keane, R.J. Adrian, Theory of cross-correlation analysis of PIV images, Appl. Sci. Res. 49 (1992) 191–215. [2] V. Renaudin, Etude et caracte´risation de la zone re´actionnelle et de la re´partition e´nerge´tique dans un re´acteur a` ultrasons, Ph.D. Thesis, Universite´ de Savoie, Chambe´ry, 1994. [3] L. Rayleigh, On the circulation of air observed in Kundt’s tubes and some allied acoustical problems, Philos. Trans. R. Soc. London 11 (1883) 1–21. [4] H. Mitome, Acoustic streaming and nonlinear acoustics research activities in Japan, Adv. Nonlinear Acoust. 11 (1993) 43–54. [5] C. Eckart, Vortices and streams caused by sound waves, Phys. Rev. 73 (1) (1948) 68–76. [6 ] W.L. Nyborg, Acoustic streaming, in: Physical Acoustics Vol. 2, Academic Press, New York, 1965, pp. 265–331. [7] M.F. Hamilton, D.T. Blackstock, in: Nonlinear Acoustics, Academic Press, New York, 1997, pp. 207–231. [8] M. Chouvellon, Transmitter vibrations and flow displacements measurements in a sonochemical reactor, Applications of Power Ultrasound in Physical and Chemical Processing (1997) 199–204. [9] C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, New York, 1987.
Velocity study in an ultrasonic reactor M. Chouvellon a, *, A. Largillier b, T. Fournel a, P. Boldo a, Y. Gonthier c a Laboratoire Traitement du Signal et Instrumentation, UMR CNRS 5516, Universite´ Jean-Monnet, 23, rue du Dr Paul Michelon, 42023 Saint-Etienne, France b Equipe d’Analyse Nume´rique de Lyon-Saint-Etienne, UMR CNRS 5585, 23, rue du Dr Paul Michelon, 42023 Saint-Etienne, France c Laboratoire de Ge´nie des Proce´de´s, Universite´ de Savoie, Savoie Technolac, 73376 Le Bourget du Lac, France
Abstract In order to determine the parameters required to describe and to optimize sonochemical reactors, we have investigated the water flow inside such a reactor. With this aim, the experimental velocity field has been measured by tomography laser. The influence of certain parameters such as the electric power, the water height and the fluid viscosity has been evaluated. At the same time, the water movement has been studied theoretically using Nyborg’s model. We have tried to improve this model by considering a three-dimensional velocity. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Acoustic streaming; Particle image velocimetry; Sonochemistry; Velocity field
1. Introduction An ultrasonic reactor allows the degradation of some organic compounds such as pentachlorophenol which are not or poorly biodegradable by common methods in industrial flows. The experimental reactor used comprises a piezoelectric transducer stuck to the center of a stainless steel plate. This plate is placed at the bottom of a tank filled with water. The experimental reactor has been developed by the chemical engineering laboratory (Le Bourget du Lac). For this prototype, the resonance frequency of the transducer is 500 kHz and the electric power is about 100 W. The ultrasonic wave propagation induces the creation of cavitation bubbles. Their implosion generates some reactive species as hydroxyl radicals OH0 which allow the activation of chemical reactions. Many researchers have described the evolution of a single bubble in an ultrasonic field but very few have described the water movement generated by an ultrasonic wave. Nevertheless, such a movement can explain the liquid homogeneity inside the reactor. This is why our study consists in measuring the velocity field in the whole reactor using a tomography laser. The influence of certain parameters such as the electric power, the water * Corresponding author. Fax: +33-04-77-48-51-20. E-mail address: [email protected] (M. Chouvellon)
height and the fluid viscosity has been evaluated. At the same time, we have tried to study the water movement theoretically in order to obtain a model.
2. Tomography laser principle The tomography laser method consists in lighting up a thin section of the flow with a laser beam focused by a lens system. Particles (about 20 mm in diameter) are seeded into the flow as markers and particle images are periodically recorded. The particle density is close to that of water in order to ensure they follow the flow. The experimental set-up is composed of the following ( Fig. 1) $ A continuous laser of 300 mW which gives out green light at 532 nm. $ An intensified camera with a shutter control and a synchronization system. This commands the aperture of the intensifier and it allows images to be obtained with a short exposure time compared with the time between two successive images. This system is located perpendicular to the laser beam. $ A digitization board allows image recording in 512×512 pixels. The tomography laser has allowed the development of particle image velocimetry. Using this experimental set-up, some particle images were obtained (Fig. 2).
1350-4177/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S1 3 5 0 -4 1 7 7 ( 0 0 ) 0 0 06 0 - 2
208
M. Chouvellon et al. / Ultrasonics Sonochemistry 7 (2000) 207–211
Fig. 1. Laser tomography system.
Fig. 2. (a) Particle image, (b) cross correlation image, (c) velocity vectors.
Each of them is then divided into several subimages. The calculation of the correlation allows a local measurement of a mean displacement of particles in a subimage. When there is a single exposure per frame to the light diffused by particles, two consecutive subimages s1 and s2 give two successive positions of the particles. From an ideal point of view, their displacement is a translation and the second subimage s2 is a shifted version of the subimage s1. The cross-correlation method [1] consists in associating the best superposition with the local displacement by relative shifting between subimages. It corresponds to the maximum of the correlation of the two successive subimage. However, this method allows only two components of the velocity to be obtained with a single camera.
two successive images and the exposure time are respectively 40 and 5 ms. A series of images was used to obtain the following mean velocity field ( Fig. 3). We notice two main vortices on each side of the central axis. The velocity is maximum in a column above the transducer. The vorticity calculated from the last velocity field indicates the location of vortex motions. The turbulence rate will allow us to quantify the efficiency of the chemical species mixing. In order to evaluate the influence of certain factors on the flow, we have varied some experimental parameters.
4. Influence of experimental factors 3. Velocity field The mean velocity inside the ultrasonic reactor has been measured by this method for an electric power of 100 W and for a 10 cm water height. The time between
The flow depends on many parameters such as the electric power, the water height and the fluid viscosity. For the study of each of these parameters, the same procedure has been used. With this aim, several images have been taken at different times at the reactor center.
M. Chouvellon et al. / Ultrasonics Sonochemistry 7 (2000) 207–211
209
Fig. 3. Mean velocity and vorticity field measured by particle image velocimetry (r, radial coordinate; z, vertical coordinate). Velocity scale4 cm s−1; non-dimensional vorticity scale: vorticity =vorticity/(U /water height). ad max
The space and time average of the velocity modulus is then calculated and it is represented in the different figures. The velocity field is about of 6 cm×6 cm and the water height is 10 cm. 4.1. Electric power A luminol experiment [2] has shown that the chemical activity increases quasi-linearly with the electric power. Hence, we are interested in studying the influence of this parameter on the flow. Contrary to the chemical activity, the velocity increases in a non-linear way with the electric power: it increases more from 50 to 100 W than above 100 W (Fig. 4)
can be used with a good mixing at a given electric power. The greater the water height, the lower is the averaged velocity. This velocity decreases in a regular way. The decrease of the velocity (Fig. 5) can be explained by the decrease in the electric power per water volume unit as the water height increases. 4.3. Fluid viscosity
The study of this parameter allows information to be obtained on the maximum of the water volume which
The last parameter studied is the liquid viscosity. In this experiment, the viscosity is increased by increasing the percentage of glycerol. The liquid height and the electric power do not vary during this experiment ( Fig. 6). Contrary to classical flow, the average velocity increases with the liquid viscosity until it reaches a threshold for m/m =20. However, it seems that the eau viscosity impedes chemical activity by decreasing the fluctuations in bubble diameter. In order to understand better the physical phenomena
Fig. 4. Velocity modulus as a function of electric power (water height: 0.1 m).
Fig. 5. Velocity modulus as a function of water height (electric power: 100 W ).
4.2. Water height
210
M. Chouvellon et al. / Ultrasonics Sonochemistry 7 (2000) 207–211
ity equation: r
C
∂u ∂t
D
A
+(u · V )u =−Vp+ m∞+
4 3
B
m VVu−mV×Vu, (1)
∂r ∂t
+V · (ru)=0.
(2)
Then, he breaks down the pressure, the fluid density and the velocity into several terms: Fig. 6. Velocity modulus as a function of viscosity (electric power: 100 W; water height: 0.1 m).
responsible for this flow, we have tried to study the water movement theoretically.
5. Acoustic streaming 5.1. Nyborg’s model The water flow cannot be explained only in terms of thermal convection, so some authors have considered acoustic streaming [3–5]. Nyborg [6 ] has formulated a model from both the dynamic equation and the continu-
Fig. 7. Profiles of the experimental and the theoretical velocities.
p=p +p +p , 0 1 2
r=r +r +r , 0 1 2
u=u +u , 1 2
where p and r correspond respectively to the equilib0 0 rium pressure and fluid density. The other terms such as p , r and u are the first-order approximations to 1 1 1 solutions of the equations, which vary sinusoidally with time. The second terms p , r and u are the second2 2 2 order approximations to the solutions and they yield correction terms to be added to the first terms. These corrections include time-independent quantities which interest us here. In this method, the first-order approximation u is determined then the second-order approxi1 mation is deduced. The latter calculation has recently been published [7]. Nyborg calculated the theoretical velocities in a simple case: he supposed that a plane wave propagates in unbounded medium so u =A e−az cos(vt−kx) where 1 a is the wave attenuation coefficient. Then, he could easily obtain an expression for the velocity. We have compared the Nyborg model with the measured velocity in Fig. 7: The Nyborg model is already in good agreement with the experiment. Indeed, the velocity is zero almost at the same place inside the reactor. Moreover, the velocity magnitudes are nearly the same. However, the reactor delimits a bounded volume, which is why we have tried to find a solution for u and so u closer to reality. 1 2
Fig. 8. (a) High view of the networking; (b) side view of the networking (the circles represent the nodes).
M. Chouvellon et al. / Ultrasonics Sonochemistry 7 (2000) 207–211
211
5.2. Improvement of Nyborg’s model
6. Conclusion
This model can be improved by considering a threedimensional velocity which depends on the three components in a bounded space. With this aim, we have used cylindrical coordinates. Moreover, the ultrasonic wave is supposed to be reflected at the free surface. The limit conditions for u are based on the measurement of the 1 amplitude of vibration of the steel plate by an optical method [8]. An analytic expression for u has been 1 obtained:
This study allows us to determine the influence of several parameters of the flow inside an ultrasonic reactor. In this way, an experiment has shown that the fluid viscosity increases the velocity until it reaches a threshold. Hence, in order to understand the physical phenomena better, we have tried to give theoretical values for the velocities based on the non-linear terms of both the dynamic equation and the continuity equation. For the moment, some mathematical problems prevent us from finding a solution.
C A BD w(r, z) cos h
u =Re ejvt w(r, z) sin h 1 v (r, z) 3
,
where w is a sum of Bessel functions of order 1 and v 3 is a sum of Bessel functions of order 0. These two terms depend on the same parameters: the viscosity m, the fluid density r and the pulsation v of the acoustic wave. In order to obtain u , partial differential equations 2 are solved with finite elements [9]. With this aim, the domain is divided into several squared elements (Fig. 8). The time-independent solution u is then the sum of 2 basis functions Q : i,j ∑ u Q (r, z), i,j i,j iµdomain where Q (r, z)=d , i and j are the nodes coordinates i,j i,j and u is a vector. This approximation is obtained by i,j resolution of a linear problem. For the moment, we do not have an expression for the theoretical velocities, as there are still mathematical problems with the boundary conditions. u (r, z)= 2
References [1] R.D. Keane, R.J. Adrian, Theory of cross-correlation analysis of PIV images, Appl. Sci. Res. 49 (1992) 191–215. [2] V. Renaudin, Etude et caracte´risation de la zone re´actionnelle et de la re´partition e´nerge´tique dans un re´acteur a` ultrasons, Ph.D. Thesis, Universite´ de Savoie, Chambe´ry, 1994. [3] L. Rayleigh, On the circulation of air observed in Kundt’s tubes and some allied acoustical problems, Philos. Trans. R. Soc. London 11 (1883) 1–21. [4] H. Mitome, Acoustic streaming and nonlinear acoustics research activities in Japan, Adv. Nonlinear Acoust. 11 (1993) 43–54. [5] C. Eckart, Vortices and streams caused by sound waves, Phys. Rev. 73 (1) (1948) 68–76. [6 ] W.L. Nyborg, Acoustic streaming, in: Physical Acoustics Vol. 2, Academic Press, New York, 1965, pp. 265–331. [7] M.F. Hamilton, D.T. Blackstock, in: Nonlinear Acoustics, Academic Press, New York, 1997, pp. 207–231. [8] M. Chouvellon, Transmitter vibrations and flow displacements measurements in a sonochemical reactor, Applications of Power Ultrasound in Physical and Chemical Processing (1997) 199–204. [9] C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press, New York, 1987.