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On the interaction between ultrasound waves and bubble clouds in mono- and dual-frequency sonoreactors
Ultrasonics Sonochemistry 10 (2003) 347–355 www.elsevier.com/locate/ultsonch
On the interaction between ultrasound waves and bubble clouds in mono- and dual-frequency sonoreactors q G. Servant
a,*
, J.L. Laborde a, A. Hita a, J.P. Caltagirone b, A. Gerard
c
a
c
EDF R&D, Avenue des Renardi eres––Ecuelles, 77818 Moret-sur-Loing Cedex, France b ENSCPB-MASTER, 16, Avenue Pey-Berland, 33402 Talence, France LMP––Universit e Bordeaux I, 351 cours de la Lib eration, 33405 Talence Cedex, France Received 17 March 2003; accepted 17 March 2003
Abstract Since the last decades, extensive work have been done on the numerical modeling of mono-frequency sonoreactors, we here consider the modeling of dual-frequency sonoreactors. We first present the basic features of the CAMUS code (CAvitating Medium under UltraSound), for mono-frequency excitation. Computation at low, medium and high frequency are presented. Extension of the numerical tool CAMUS is also presented: Caflisch equations are modified to take into account the dual-frequency excitation of the sound. We consider 28–56, 28–100 and 28–200 kHz sonoreactors. Fields of cavitation bubble emergence are quite different from the ones under mono-frequency. Study of spatio-temporal dynamics of cavitation bubbles in a 28–56 kHz sonoreactor is also considered. Taking into account the pressure field induced by the dual-frequency wave propagation, we compute the Bjerknes force applied on the cavitation bubble that is responsible for the bubble migration. A two phase flow approach allows to compute the bubble migration. 2003 Elsevier B.V. All rights reserved. Keywords: Ultrasound wave; Dual-frequency sonoreactor; Bubble; Cavitation
1. Introduction Since the last decade, the study of dual-frequency ultrasound waves interaction in cavitating liquids has received an increasing interest. Indeed, a bubble driven under two superposed waves may have a drastic different dynamic, compared to the one under mono-frequency excitation [1]. And this multi-mode excitation contributes in boosting sonoluminescence (see Ref. [2]), as well as other chemical processes [3,4]. Past efforts were made on the modeling of acoustic pressure field in cavitating media at low [5–12] frequency sonoreactors. Authors choose different approach, depending on the modeled phenomena: Lagrangian, Euleq This paper was originally presented at Applications of Power Ultrasound in Physical and Chemical Processing (Usound3) Paris December 2001. * Corresponding author. Tel.: +33-1-60-73-70-98; fax: +33-1-60-7371-46. E-mail address: [email protected] (G. Servant).
1350-4177/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1350-4177(03)00105-6
rian or stochastic formulation. . . For further details, we should refer to the state of the art on numerical modeling in cavitating media [13]. We mention that new methods have been presented at the 3rd Conference on ‘‘Applications of Power Ultrasound in Physical and Chemical Processing’’, Paris/France, 13–14 December 2001. Dubus et al. [14] developed a promising method for modeling ultrasound wave propagation in cavitating media. Lauterborn and Mettin [15] did an extensive study on bubble dynamics in low frequency sonoreactors. As well, Keil and Kroemer [16] improved their previous model [7,8]. Based on an Eulerian approach [17,18], they solve the coupling between 2nd order bubble dynamic (Rayleigh like equations) and nonlinear wave propagation (Navier–Stokes equations). Ref. [16] considers wave propagation in cavitating media with 20 lm initial radius in 3D complex geometries. This is a great improvement compared to the work of [19,20] that solve the same model, but considering 1D geometry. But, due to the stiffness on the equations [19] showed that some numerical problem may arise when considering small initial
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bubble radius (5 lm) and high acoustic pressure amplitude (2 atm or more). Predicting bubble dynamics in multi-frequency reactor is of a great interest to quantify the benefits of multimode excitation. Our approach is based on previous work on the modeling of pressure field as well as cavitating bubble field at low [21,22] and high [13,23] frequency. We aim to determine emergence sites of cavitation bubble under the simultaneous propagation of low and high frequency (resp. LF and HF) ultrasound waves. Numerical modeling is considered to compute both the bubble nucleation sites and spatio-temporal dynamic of cavitation bubbles in three dimensional sonoreactor.
2. Theory Major past efforts have been made on the study of cavitation at a microscale, that is to say on cavitation bubble dynamics [15]. Keller and Kolodner [24] derived equations on bubble driven under acoustic pressure. Our approach (see Refs. [13,21]), as the one of [8,11,12] is based on a macroscale study of cavitation phenomena. In order to derive a rigorous model on cavitation, we have to determine the influence of the cavitation bubbles on the sound field and vice versa. We developed two theoretical models: the first aims to describe bubbles emergence due to the acoustic driving, the second one deals with the stable cavitation bubble spatio-temporal dynamic induced by the wave propagation. One building block of the two models is the particular shape behaviour of cavitation bubbles. In Sections 2.1–2.4, we expose the theory for dualfrequency sonoreactors. Equations for mono-frequency excitation can be retrieve by setting xLF ¼ xHF ¼ x, pHF ¼ pLF ¼ p.
R ¼ R0 ½1 þ ðxLF þ xHF Þ ;
ð3Þ
xLF 1;
ð4Þ
When cavitation bubbles are driven under a time dependent sound field, they are subjected to complex shape oscillations. If we consider that bubbles remain spherical, we may consider the model of Keller and Kolodner [24]: ! ! _ R_ R 3 €þ 1 RR R_ 2 1 c 3c 2 ! R_ 1 R d ðpL ðRÞ p1 Þ; ¼ 1þ ðpL ðRÞ p1 Þ þ c qL0 qL0 c dt ð1Þ ð2Þ
xHF 1;
o2 xLF oxLF pLF p0 þ x2LF xLF ¼ þ dtotLF ; ot2 ot q0 R20
ð5Þ
o2 xHF oxHF pHF p0 þ x2HF xHF ¼ þ dtotHF ; ot2 ot q0 R20
ð6Þ
where xLF and xHF are respectively the relative radial displacement of the bubble due to the LF sound wave and the HF sound wave. pLF and pHF are respectively the acoustic pressure resulting of the LF sound wave and the HF sound wave. xLF and xHF are respectively the resonance frequency of the bubble due to the LF sound wave and the HF sound wave. Thermal, viscous and radiation losses define the damping constant dtotLF , dtotHF for a bubble of equilibrium radius R0 [25]: dtotLF ¼ ðdth þ dl þ dr ÞLF ¼
p0 þ R2r0
4l x2 R0 Im/ þ þ qR20 qxR20 c
dtotHF ¼ ðdth þ dl þ dr ÞHF ¼
2.1. Cavitation bubble dynamic
3k 2r R0 2r p1 pL ¼ p0 þ R0 R R
for air bubble in water at 20 C with a polytropic exponent of 1.4, surface tension r ¼ 0:0725 N m1 , liquid density q1 ¼ 998 kg m3 , liquid viscosity of l ¼ 0:001; static pressure, p0 ¼ 1:013 105 Pa, sound velocity in the pure liquid c ¼ 1500 m s1 . By assuming that the bubble equilibrium size radius R0 fulfils the condition 4l=ðqcÞ R0 c=x, and small amplitudes p p0 of the external sound field, and considering a two mode excitation, the linearization of the Keller–Kolodner equation gives:
p0 þ R2r0
4l x2 R0 Im/ þ þ c qR20 qxR20
! ;
ð7Þ
;
ð8Þ
LF
! HF
where / is the (complex) phase shift between the sound field and the bubble motion and x is the transducer pulsation. Because of the linearization of the Keller–Kolodner equation (see Eqs. (5) and (6)), we have to compute the low and high frequency component of the sound field. pLF and pHF are computed thanks to the following equations: pLF þ pHF ¼ p; w2LF pLF þ w2HF pHF ¼
ð9Þ o2 p ; ot2
Dt 1;
ð10Þ
Dt is the time increment considered during numerical computations. Eq. (10) assumes small perturbation hypothesis.
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We retain Eqs. (5) and (6) to model the bubble dynamic for the two studies, that is to say for the modeling of the dual-frequency wave propagation in bubbly liquid and of the cavitation bubble migration induced by the dual-frequency ultrasound wave propagation. 2.2. Bubble radii distribution We consider two kind of cavitation bubbles. The fragmentary transient cavitation bubble have a fragmentary collapse. They have a short life time of one acoustic cycle [26]. These bubbles are considered when studying the emergence of cavitation bubbles. The repetitive transient cavitation bubbles have a nonfragmentary collapse, their life time is several acoustic cycles [26]. These bubbles are considered when studying the Ôspatio-temporal dynamic of cavitation bubblesÕ. 2.2.1. Repetitive transient cavitation bubbles In the literature, we may find measurements of the radii distribution of Ôrepetitive transient cavitation bubblesÕ (see Refs. [22,27,28]). All measurements agree with a micron size (see 1). Ref. [27] reports that at low frequency, experimental average diameter is very low and smaller than the resonance diameter predicted by the linear theory. Experiments of [28] revealed as well, a large discrepancy from the common, but obviously wrong assumption of a bubble size distribution near the linear resonance radii. Explanation of a such remarkable result (see Ref. [28]) is based on consideration of shape stability. It shows that surface-stable bubble sizes, that is to say size of Ôlong lifeÕ bubbles (repetitive transient cavitation bubbles in our case) are of micron size. When considering the spatio-temporal dynamic of repetitive transient cavitation bubble, we assume a 5 lm initial radius. 2.2.2. Fragmentary transient cavitation bubbles Although there are some recent publications [27,28] on bubble volume fraction and radii distribution of repetitive transient cavitation bubbles in sonochemical reactors, we did not find (up to now) any data on fragmentary transient bubbles radii distribution. Even though, we assume that, as the repetitive transient cavitation bubbles, their radii distribution follow a truncated Gaussian, the one of [25], that is to say that the bubble volume fraction is given by: Z 1 4 bðr; tÞ ¼ p R3 ðr; R0 ; tÞf ðx; y; z; R0 Þ dR0 ; ð11Þ 3 0 where Rðx; y; z; R0 ; tÞ is the instantaneous bubble radius at time t at position ðx; y; zÞ in the sonoreactor, having an equilibrium radius R0 . f ðr; R0 Þ dR0 is the number of bubbles per unit volume with equilibrium radius R0 , where f is a truncated Gaussian (see Fig. 1):
Fig. 1. Measurement of repetitive transient cavitation bubble radii by use of Phase Doppler equipment in a 28 kHz sonoreactor. For further details on the measuring technique, see Refs. [22,29].
2
f ðR0 Þ ¼ C exp½ðR0 R3 Þ =f2 ;
R1 < R 0 < R2
¼ 0; otherwise:
ð12Þ
We assume a bubble radii ranging from R1 ¼ 5 lm and R2 ¼ 3 mm. The deviation f is set to 2 mm. C is a parameter chosen to match the variable gas volume void fraction b; R3 ¼ ðR1 þ R2 Þ=2. By assessing the presence of large bubbles (millimeter size) we take into account coalescence of some emerged bubbles. 2.3. Emergence of cavitation bubbles 2.3.1. Governing equations When considering the modeling of the wave propagation in bubbly liquids, a model widely used is the one of [30] or [31]. Van Wijngaarden [30] heuristically derived bubbly liquids motion equations that were confirmed by [31] using a a more rigorous mathematical approach. The validity domain of this model is for weakly compressible fluid and assuming a small volume fraction of gas within the liquid. We retain the Caflisch model [31]: oq oðxLF þ xHF Þ þ divðquÞ ¼ 3bq0 ; ot ot
ð13Þ
ou þ q0 uru ¼ rp þ lr2 u: ð14Þ ot The set of Eqs. (5)–(10), (13), (14) allow to compute the LF, HF sound waves propagation and their interaction with cavitation bubbles. See Ref. [22] for a detailed algorithm of the Caflisch model, in the case of dual-frequency excitation. q0
2.3.2. Dependency of the bubble volume fraction and the pressure amplitude In the present work we take into account the generation of bubbles due to the acoustic driving. We assume a linear dependency between the bubble volume fraction and the acoustic driving.
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2.4. Spatio-temporal dynamics of repetitive transient bubble clouds
• The expression of the viscous drag force FD is the one derived analytically by [34] considering a time dependent radius:
In the following paragraph, we expose the retained method to model the motion of the repetitive transient cavitation bubbles.
FD ¼ 12pl1 RðtÞur : ð18Þ • The force FUS applied on the bubble by the undisturbed surrounding fluid flow. Because of the dualfrequency of the sound field, we assume that FUS ¼ FUSLF þ FUSHF , where FUSLF and FUSHF are respectively the Bjerknes force resulting on the interaction between cavitation bubbles and the low (resp. high) frequency component of the pressure field.
2.4.1. Governing equations We consider the medium as a bubbly liquid. The model is based on the Eulerian two-fluid approach (see Ref. [32]). The advantage of this approach, is to compute the motion of a large number of bubbles with only one set of governing equations, instead of having one equation of motion for each bubble. Separate Eulerian conservation equations are formulated for both phases coupled through interfacial transfer terms. Each phase is governed by the Navier–Stokes equations. Interactions between the liquid and bubbles (such as added mass force, drag force. . .) appear in the equations via coupling terms Ik . Ik is obtained from the analysis of the balance of forces acting on an isolated particle. These equations can be derived in conservative, transient form as: oðaqÞk þ rðaquÞk ¼ 0; ot
FUSLF ¼ V ðtÞrpLF ;
ð19Þ
FUSHF ¼ V ðtÞrpHF :
ð20Þ
2.4.3. Forces time averaging We consider that bubbles encounter only one sound field amplitude during one of their radial oscillation period (which is assumed equal to the sound field oscillation period T ¼ 1=f for all bubbles). Then the time averaging over a T period is written as follow: hFD iT ¼ 12pl1 hRðtÞiT ur ;
ð21Þ
1 dur ; hFAM iT ¼ q1 hV ðtÞiT 2 dt
ð22Þ
ð16Þ
hFUSLF iT ¼ hV ðtÞrpLF iT ;
ð23Þ
where p1 is the mean pressure of the continuous phase and Ik is the part of the interfacial momentum transfer rate between phases. u, q and l, are respectively velocity, density and dynamic viscosity. k ¼ 1 or 2, 1 refers to the fluid phase and 2 to the bubble phase, a is the volume fraction of the considered phase. The CFD code ESTET-ASTRID solves these equations (see Refs. [32,33]). ESTET-ASTRID is part of the CAMUS code.
hFUSHF iT ¼ hV ðtÞrpHF iT :
ð24Þ
ð15Þ
oðaquÞk ~p1 þ lk r2 ðauÞ þ Ik ; þ r:ðaqu uÞk ¼ r k ot
2.4.2. Forces exerted on the repetitive transient cavitation bubbles Because repetitive transient cavitation bubbles migrate, they induce disturbances within the liquid, the interaction between cavitation bubbles and the acoustic pressure field is rendered through the stresses applied by the surrounding liquid on the cavitation bubbles. We assume that the considered bubbles have the same equilibrium radius and follow linear oscillations. The forces they experienced are the following:
fV ¼ hFD iT þ hFMA iT þ FUS iT :
ð25Þ
And the interfacial momentum transfer (see Eq. (16)) has the following expression: I2;i ¼ I1;i ¼ a2 f:
ð26Þ
3. Results: mono-frequency sonoreactors
• The added mass force FAM , 1 dur ; FAM ¼ q1 V ðtÞ 2 dt
But we take into account the change of the mechanical properties of the sound wave due to the emergence of unstable bubbles in the computation of the repetitive transient bubble motion. Indeed, the acoustic force FUS is calculated upon the pressure field and bubble field computed taking into account the emergence of the fragmentary transient cavitation bubbles. This force is implemented in the CAMUS code as an external force (applied on the stable cavitation bubbles). We then deduce the resultant force applied on each point included in the dispersed phase:
ð17Þ
where ur is the local instantaneous relative velocity on each point included in the dispersed phase and given by ur ¼ u2 u1 .
CAMUS code is able to model a wide range of phenomena [22] such as bubble emergence, spatio-temporal dynamic of bubble clouds [13,21], acoustic streaming [23] in the case of low (20–30 kHz) and high (500 kHz) frequencies.
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3.1. Low frequency sonoreactor: f ¼ 28 kHz We consider a cylindrical a cylindrical reactor: 12 cm diameter and a water height of 9.4 cm. The piezoelectric transducer is 6 cm diameter, and placed at the bottom of the reactor. Looking at Fig. 2(a) and (b), we find good agreement between the test of aluminium foil degradation and computed bubble volume fraction. Comparing the pressure field taking into account the cavitation bubble emergence and the one of a pure liquid (see Fig. 3), we notice a slight damping. Indeed, the power input of the ultrasound emitter is not big, only 50 W. Thus few bubble emerge (compared to sonotrode emitter where the power input is up to 1000 W). Comparing pressure measurements (Fig. 4) and computed bubble volume fraction (Fig. 2(b)), we see that bubbles emerge at the pressure antinodes.
Fig. 4. Axial cut of the measured acoustic pressure field, through the use of an hydrophone. Sonoreactor diameter is 12 cm, transducer diameter is 6 cm, f ¼ 28 kHz.
3.2. Medium frequency sonoreactor: f ¼ 200 kHz The CAMUS code also predicts bubble emergence field at medium frequency. Considering a 200 kHz sonoreactor (see Fig. 5(a)) we have a good agreement between experiments and modeling (Figs. 5(b) and 6). For higher power input, an acoustic fountain occur at
Fig. 2. Semi-axial cut of an aluminium foil immersed in a 28 kHz sonoreactor, bubble jet pittings degrade the aluminium foil (a). Semiaxial cut of computed bubble volume fraction (f ¼ 28 kHz) (b). Sonoreactor diameter is 12 cm, transducer diameter is 6 cm.
Fig. 3. Semi-axial cut of the acoustic pressure field, taking into account the cavitation bubble emergence and in a pure liquid. Sonoreactor diameter is 12 cm, transducer diameter is 6 cm, f ¼ 28 kHz.
Fig. 5. 200 kHz sonoreactor geometry (a); Sarvazyan method: Chromolux foil immersed in methylene blue dye, darker area reveal higher bubble volume fraction (b).
Fig. 6. Cavitation bubble fields in a 200 kHz sonoreactor.
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excitation. As seen on Fig. 8(a) and (b), bubble clouds shape consists of concentric circles.
4. Results: dual-frequency sonoreactors 4.1. Dual-frequency reactor geometry Considered sonoreactors have a cylindrical shape. The reactors diameter is 8 cm, the water level is 9.4 cm. LF and HF sound waves are respectively generated at the bottom and the top of the sonoreactors, through the use of circular transducers (Fig. 9). This reactor shape allows to have interaction of low and high frequency ultrasound waves, throughout the reactor. 4.2. 28–56 kHz sonoreactor Fig. 7. Cavitation bubble fields in a 200 kHz sonoreactor, taking into account of the acoustic fountain.
the water–air interface. Taking into account of this phenomena in the modeled reactor, we get the Ôself focusingÕ effect. That is to say, a higher bubble volume fraction nearby the acoustic fountain (Fig. 7). This is in good agreement with the experiments of [29]. 3.3. High frequency sonoreactor: f ¼ 540 kHz As well, the CAMUS code predict the particular shape of cavitation bubble field in the case of a 540 kHz
Fig. 10 shows Cavitation bubble cartography. We notice a cavitation zone with a higher bubble volume fraction, compared to mono-frequency reactors (see Fig. 2). Fig. 12 gives the decomposition of the pressure field (Fig. 11). Low and high frequency component of the acoustic pressure field are accurately computed, even though we assume small perturbation hypothesis (see Eq. (10)). We notice a drastic change in bubble cartography, compared to mono-frequency reactor (see Fig. 2). 4.3. 28–100 kHz sonoreactor Fig. 13 gives the cavitation bubble cartography in a 28–100 kHz sonoreactor. Computation of the bubble emergence field is achieved by taking into account (throughout direct simulation) of the dual-frequency wave propagation (see Fig. 14). Computation of the bubble–sound wave interaction is done through the decomposition (at each time step) of the dual-frequency pressure field (see Fig. 15).
Fig. 8. Chromolux foil colored by methylene blue dye facing a 540 kHz piezoelectric transducer (Sarvazyan method) (a); computed bubble volume fraction (f ¼ 540 kHz) (b). Sonoreactor diameter is 10 cm, water height is 1 cm.
Fig. 9. Geometry of dual-frequency reactor.
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Fig. 10. 3D cavitation bubble fields in a 28–56 kHz sonoreactor.
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Fig. 13. 3D cavitation bubble fields in a 28–100 kHz sonoreactor.
Fig. 11. Axial cut of the acoustic pressure field in a 28–56 kHz sonoreactor.
Fig. 14. Axial cut of the acoustic pressure field in a 28–100 kHz sonoreactor.
Fig. 12. Axial cut of the low and high frequency component of the acoustic pressure field in a 28–56 kHz sonoreactor.
Fig. 15. Axial cut of the acoustic pressure field in a 28–100 kHz sonoreactor.
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Fig. 16. Axial cut of the cavitation bubble fields in a 28–200 kHz sonoreactor. Fig. 19. Velocity field of cavitation bubbles in a 28–56 kHz reactor. Bubbles have a 5 lm initial radius. Absolute magnitude of the velocities is 50 cm/s1 .
4.4. 28–200 kHz sonoreactor The use of a dual-frequency excitation allows to get a more intense cavitation bubble field (see Fig. 16) than in the case of mono-frequency excitation. Fig. 18 shows the decomposition of the dual-frequency acoustic pressure field (see Fig. 17). 4.5. Spatio-temporal dynamic of cavitation bubble in a 28–56 kHz sonoreactor
Fig. 17. Axial cut of the acoustic pressure field in a 28–200 kHz sonoreactor.
Comparing Figs. 11 and 19, we notice that cavitation bubbles go to pressure antinodes which result in the interaction of the low and high frequency ultrasound waves propagation. Before reaching the pressure antinodes of the highest amplitude, bubbles reach pressure antinodes of smaller amplitude (see Fig. 11).
5. Conclusion
Fig. 18. Axial cut of the low and high frequency component of the acoustic pressure field in a 28–200 kHz sonoreactor.
Acoustic cavitation covers a wide range of different phenomena. It has been shown that numerical modeling may be useful to predict encountered phenomena. The use of the CAMUS is able to predict the self-focusing phenomena at medium frequency. As well, the code accurately predicts the cavitation bubble emergence field at low and high frequency (28–540 kHz). Extension of the CAMUS code have been presented, to take into account dual-frequency sound wave propagation. Computations in 28–56, 28–100 and 28–200 kHz sonoreactors show that it is of a great interest to use dual-frequency sound waves: cavitation bubble volume fraction field are higher than in mono-frequency sonoreactors. As well, cavitation bubbles have a specific spatio-temporal dynamic. Future work will have to be
G. Servant et al. / Ultrasonics Sonochemistry 10 (2003) 347–355
undertaken to optimize the dual-frequency sonoreactor geometry. Acknowledgements Authors are grateful to Dr. Breitbach (Dortmund University, Germany) for his useful advices on the ÔSarvazyan methodÕ, as well as Prof. Simonin, IMFT/ Toulouse, France) for constructive discussion. References [1] A.H. Nayfeh, D.T. Mook, J. Phys., Colloque C8 40 (Suppl. 11) (1979) 3. [2] J. Holzfuss, M. R€ uggeberg, R. Mettin, Phys. Rev. Lett. 81 (1998) 1961. [3] K.M. Swamy, K.L. Narayana, Intensification of leaching process by dual-frequency ultrasound, Ultrason. Sonochem. 8 (2001) 341– 346. [4] S. Manickam, Aniruddha B. Pandit, Ultrasound enhanced degradation of Rhodamine B: optimization with power density, Ultrason. Sonochem. 8 (2001) 233–240. [5] O. Dahlem, Contribution a 1Õetude des problemes de montee en echelle des reacteurs sonochimiques, et investigation de la cavitation hydrodynamique comme technique alternative a la cavitation Acoustique, these de troisieme cycle, Universite libre de Bruxelles, 1999. [6] S. D€ ahnke, F.J. Keil, Modeling of three-dimensional linear pressure fields in sonochemical reactors with homogeneous and inhomogeneous density distribution of cavitation bubbles, Ind. Eng. Chem. Res. 37 (1998) 848–864. [7] S. D€ ahnke, K.M. Swamy, F.J. Keil, Modeling of three-dimensional pressure fields in sonochemical rectors with an inhomogeneous density distribution of cavitation bubbles, Comparison of theoretical and experimental results, Ultrason. Sonochem. 6 (1999) 31–41. [8] S. D€ ahnke, 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. [9] C. Horst, U. Hoffmann, U. Kunz, Modelling of sound fields in ultrasound reactors as a first step in scale up of heterogeneous reactions, European Congress on Chemical Engineering, ECCE 1, Florenz, 4–7 Mai 1997. [10] J.M. L€ oning, C. Horst, U. Hoffmann, Sonochemical reactors for heterogeneous solid–liquid reactions, A chemical engineering approach, in: Proceedings 2nd Conference: Applications of Power Ultrasound in Physical and Chemical Processing, Toulouse, 6–7 May 1999. [11] C. Horst, U. Kunz, A. Rosenpl€anter, U. Hoffmann, Activated solid–fluid reactions in ultrasound reactors, Chem. Eng. Sci. 54 (1999) 2849–2858. [12] U. Parlitz, R. Mettin, S. Luther, I. Akhatov, M. Voss, W. Lauterborn, Spatio-temporal dynamics of Acoustic cavitation bubble clouds, Philos. Trans. R. Soc. Lond. A 357 (1999) 313–334. [13] G. Servant, J.L. Laborde, A. Hita, J.P. Caltagirone, A. Gerard, Ultrason. Sonochem. 8 (2001) 63–174. [14] B. Dubus, O. Ledez, C. Granger, P. Mosbah, C. Campos/Pozuelo, in: Proceeding of the 3rd Conference on Applications of Power Ultrasound in Physical and Chemical Processing, Paris, France, 13–14 December 2001.
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On the interaction between ultrasound waves and bubble clouds in mono- and dual-frequency sonoreactors q G. Servant
a,*
, J.L. Laborde a, A. Hita a, J.P. Caltagirone b, A. Gerard
c
a
c
EDF R&D, Avenue des Renardi eres––Ecuelles, 77818 Moret-sur-Loing Cedex, France b ENSCPB-MASTER, 16, Avenue Pey-Berland, 33402 Talence, France LMP––Universit e Bordeaux I, 351 cours de la Lib eration, 33405 Talence Cedex, France Received 17 March 2003; accepted 17 March 2003
Abstract Since the last decades, extensive work have been done on the numerical modeling of mono-frequency sonoreactors, we here consider the modeling of dual-frequency sonoreactors. We first present the basic features of the CAMUS code (CAvitating Medium under UltraSound), for mono-frequency excitation. Computation at low, medium and high frequency are presented. Extension of the numerical tool CAMUS is also presented: Caflisch equations are modified to take into account the dual-frequency excitation of the sound. We consider 28–56, 28–100 and 28–200 kHz sonoreactors. Fields of cavitation bubble emergence are quite different from the ones under mono-frequency. Study of spatio-temporal dynamics of cavitation bubbles in a 28–56 kHz sonoreactor is also considered. Taking into account the pressure field induced by the dual-frequency wave propagation, we compute the Bjerknes force applied on the cavitation bubble that is responsible for the bubble migration. A two phase flow approach allows to compute the bubble migration. 2003 Elsevier B.V. All rights reserved. Keywords: Ultrasound wave; Dual-frequency sonoreactor; Bubble; Cavitation
1. Introduction Since the last decade, the study of dual-frequency ultrasound waves interaction in cavitating liquids has received an increasing interest. Indeed, a bubble driven under two superposed waves may have a drastic different dynamic, compared to the one under mono-frequency excitation [1]. And this multi-mode excitation contributes in boosting sonoluminescence (see Ref. [2]), as well as other chemical processes [3,4]. Past efforts were made on the modeling of acoustic pressure field in cavitating media at low [5–12] frequency sonoreactors. Authors choose different approach, depending on the modeled phenomena: Lagrangian, Euleq This paper was originally presented at Applications of Power Ultrasound in Physical and Chemical Processing (Usound3) Paris December 2001. * Corresponding author. Tel.: +33-1-60-73-70-98; fax: +33-1-60-7371-46. E-mail address: [email protected] (G. Servant).
1350-4177/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1350-4177(03)00105-6
rian or stochastic formulation. . . For further details, we should refer to the state of the art on numerical modeling in cavitating media [13]. We mention that new methods have been presented at the 3rd Conference on ‘‘Applications of Power Ultrasound in Physical and Chemical Processing’’, Paris/France, 13–14 December 2001. Dubus et al. [14] developed a promising method for modeling ultrasound wave propagation in cavitating media. Lauterborn and Mettin [15] did an extensive study on bubble dynamics in low frequency sonoreactors. As well, Keil and Kroemer [16] improved their previous model [7,8]. Based on an Eulerian approach [17,18], they solve the coupling between 2nd order bubble dynamic (Rayleigh like equations) and nonlinear wave propagation (Navier–Stokes equations). Ref. [16] considers wave propagation in cavitating media with 20 lm initial radius in 3D complex geometries. This is a great improvement compared to the work of [19,20] that solve the same model, but considering 1D geometry. But, due to the stiffness on the equations [19] showed that some numerical problem may arise when considering small initial
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bubble radius (5 lm) and high acoustic pressure amplitude (2 atm or more). Predicting bubble dynamics in multi-frequency reactor is of a great interest to quantify the benefits of multimode excitation. Our approach is based on previous work on the modeling of pressure field as well as cavitating bubble field at low [21,22] and high [13,23] frequency. We aim to determine emergence sites of cavitation bubble under the simultaneous propagation of low and high frequency (resp. LF and HF) ultrasound waves. Numerical modeling is considered to compute both the bubble nucleation sites and spatio-temporal dynamic of cavitation bubbles in three dimensional sonoreactor.
2. Theory Major past efforts have been made on the study of cavitation at a microscale, that is to say on cavitation bubble dynamics [15]. Keller and Kolodner [24] derived equations on bubble driven under acoustic pressure. Our approach (see Refs. [13,21]), as the one of [8,11,12] is based on a macroscale study of cavitation phenomena. In order to derive a rigorous model on cavitation, we have to determine the influence of the cavitation bubbles on the sound field and vice versa. We developed two theoretical models: the first aims to describe bubbles emergence due to the acoustic driving, the second one deals with the stable cavitation bubble spatio-temporal dynamic induced by the wave propagation. One building block of the two models is the particular shape behaviour of cavitation bubbles. In Sections 2.1–2.4, we expose the theory for dualfrequency sonoreactors. Equations for mono-frequency excitation can be retrieve by setting xLF ¼ xHF ¼ x, pHF ¼ pLF ¼ p.
R ¼ R0 ½1 þ ðxLF þ xHF Þ ;
ð3Þ
xLF 1;
ð4Þ
When cavitation bubbles are driven under a time dependent sound field, they are subjected to complex shape oscillations. If we consider that bubbles remain spherical, we may consider the model of Keller and Kolodner [24]: ! ! _ R_ R 3 €þ 1 RR R_ 2 1 c 3c 2 ! R_ 1 R d ðpL ðRÞ p1 Þ; ¼ 1þ ðpL ðRÞ p1 Þ þ c qL0 qL0 c dt ð1Þ ð2Þ
xHF 1;
o2 xLF oxLF pLF p0 þ x2LF xLF ¼ þ dtotLF ; ot2 ot q0 R20
ð5Þ
o2 xHF oxHF pHF p0 þ x2HF xHF ¼ þ dtotHF ; ot2 ot q0 R20
ð6Þ
where xLF and xHF are respectively the relative radial displacement of the bubble due to the LF sound wave and the HF sound wave. pLF and pHF are respectively the acoustic pressure resulting of the LF sound wave and the HF sound wave. xLF and xHF are respectively the resonance frequency of the bubble due to the LF sound wave and the HF sound wave. Thermal, viscous and radiation losses define the damping constant dtotLF , dtotHF for a bubble of equilibrium radius R0 [25]: dtotLF ¼ ðdth þ dl þ dr ÞLF ¼
p0 þ R2r0
4l x2 R0 Im/ þ þ qR20 qxR20 c
dtotHF ¼ ðdth þ dl þ dr ÞHF ¼
2.1. Cavitation bubble dynamic
3k 2r R0 2r p1 pL ¼ p0 þ R0 R R
for air bubble in water at 20 C with a polytropic exponent of 1.4, surface tension r ¼ 0:0725 N m1 , liquid density q1 ¼ 998 kg m3 , liquid viscosity of l ¼ 0:001; static pressure, p0 ¼ 1:013 105 Pa, sound velocity in the pure liquid c ¼ 1500 m s1 . By assuming that the bubble equilibrium size radius R0 fulfils the condition 4l=ðqcÞ R0 c=x, and small amplitudes p p0 of the external sound field, and considering a two mode excitation, the linearization of the Keller–Kolodner equation gives:
p0 þ R2r0
4l x2 R0 Im/ þ þ c qR20 qxR20
! ;
ð7Þ
;
ð8Þ
LF
! HF
where / is the (complex) phase shift between the sound field and the bubble motion and x is the transducer pulsation. Because of the linearization of the Keller–Kolodner equation (see Eqs. (5) and (6)), we have to compute the low and high frequency component of the sound field. pLF and pHF are computed thanks to the following equations: pLF þ pHF ¼ p; w2LF pLF þ w2HF pHF ¼
ð9Þ o2 p ; ot2
Dt 1;
ð10Þ
Dt is the time increment considered during numerical computations. Eq. (10) assumes small perturbation hypothesis.
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We retain Eqs. (5) and (6) to model the bubble dynamic for the two studies, that is to say for the modeling of the dual-frequency wave propagation in bubbly liquid and of the cavitation bubble migration induced by the dual-frequency ultrasound wave propagation. 2.2. Bubble radii distribution We consider two kind of cavitation bubbles. The fragmentary transient cavitation bubble have a fragmentary collapse. They have a short life time of one acoustic cycle [26]. These bubbles are considered when studying the emergence of cavitation bubbles. The repetitive transient cavitation bubbles have a nonfragmentary collapse, their life time is several acoustic cycles [26]. These bubbles are considered when studying the Ôspatio-temporal dynamic of cavitation bubblesÕ. 2.2.1. Repetitive transient cavitation bubbles In the literature, we may find measurements of the radii distribution of Ôrepetitive transient cavitation bubblesÕ (see Refs. [22,27,28]). All measurements agree with a micron size (see 1). Ref. [27] reports that at low frequency, experimental average diameter is very low and smaller than the resonance diameter predicted by the linear theory. Experiments of [28] revealed as well, a large discrepancy from the common, but obviously wrong assumption of a bubble size distribution near the linear resonance radii. Explanation of a such remarkable result (see Ref. [28]) is based on consideration of shape stability. It shows that surface-stable bubble sizes, that is to say size of Ôlong lifeÕ bubbles (repetitive transient cavitation bubbles in our case) are of micron size. When considering the spatio-temporal dynamic of repetitive transient cavitation bubble, we assume a 5 lm initial radius. 2.2.2. Fragmentary transient cavitation bubbles Although there are some recent publications [27,28] on bubble volume fraction and radii distribution of repetitive transient cavitation bubbles in sonochemical reactors, we did not find (up to now) any data on fragmentary transient bubbles radii distribution. Even though, we assume that, as the repetitive transient cavitation bubbles, their radii distribution follow a truncated Gaussian, the one of [25], that is to say that the bubble volume fraction is given by: Z 1 4 bðr; tÞ ¼ p R3 ðr; R0 ; tÞf ðx; y; z; R0 Þ dR0 ; ð11Þ 3 0 where Rðx; y; z; R0 ; tÞ is the instantaneous bubble radius at time t at position ðx; y; zÞ in the sonoreactor, having an equilibrium radius R0 . f ðr; R0 Þ dR0 is the number of bubbles per unit volume with equilibrium radius R0 , where f is a truncated Gaussian (see Fig. 1):
Fig. 1. Measurement of repetitive transient cavitation bubble radii by use of Phase Doppler equipment in a 28 kHz sonoreactor. For further details on the measuring technique, see Refs. [22,29].
2
f ðR0 Þ ¼ C exp½ðR0 R3 Þ =f2 ;
R1 < R 0 < R2
¼ 0; otherwise:
ð12Þ
We assume a bubble radii ranging from R1 ¼ 5 lm and R2 ¼ 3 mm. The deviation f is set to 2 mm. C is a parameter chosen to match the variable gas volume void fraction b; R3 ¼ ðR1 þ R2 Þ=2. By assessing the presence of large bubbles (millimeter size) we take into account coalescence of some emerged bubbles. 2.3. Emergence of cavitation bubbles 2.3.1. Governing equations When considering the modeling of the wave propagation in bubbly liquids, a model widely used is the one of [30] or [31]. Van Wijngaarden [30] heuristically derived bubbly liquids motion equations that were confirmed by [31] using a a more rigorous mathematical approach. The validity domain of this model is for weakly compressible fluid and assuming a small volume fraction of gas within the liquid. We retain the Caflisch model [31]: oq oðxLF þ xHF Þ þ divðquÞ ¼ 3bq0 ; ot ot
ð13Þ
ou þ q0 uru ¼ rp þ lr2 u: ð14Þ ot The set of Eqs. (5)–(10), (13), (14) allow to compute the LF, HF sound waves propagation and their interaction with cavitation bubbles. See Ref. [22] for a detailed algorithm of the Caflisch model, in the case of dual-frequency excitation. q0
2.3.2. Dependency of the bubble volume fraction and the pressure amplitude In the present work we take into account the generation of bubbles due to the acoustic driving. We assume a linear dependency between the bubble volume fraction and the acoustic driving.
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2.4. Spatio-temporal dynamics of repetitive transient bubble clouds
• The expression of the viscous drag force FD is the one derived analytically by [34] considering a time dependent radius:
In the following paragraph, we expose the retained method to model the motion of the repetitive transient cavitation bubbles.
FD ¼ 12pl1 RðtÞur : ð18Þ • The force FUS applied on the bubble by the undisturbed surrounding fluid flow. Because of the dualfrequency of the sound field, we assume that FUS ¼ FUSLF þ FUSHF , where FUSLF and FUSHF are respectively the Bjerknes force resulting on the interaction between cavitation bubbles and the low (resp. high) frequency component of the pressure field.
2.4.1. Governing equations We consider the medium as a bubbly liquid. The model is based on the Eulerian two-fluid approach (see Ref. [32]). The advantage of this approach, is to compute the motion of a large number of bubbles with only one set of governing equations, instead of having one equation of motion for each bubble. Separate Eulerian conservation equations are formulated for both phases coupled through interfacial transfer terms. Each phase is governed by the Navier–Stokes equations. Interactions between the liquid and bubbles (such as added mass force, drag force. . .) appear in the equations via coupling terms Ik . Ik is obtained from the analysis of the balance of forces acting on an isolated particle. These equations can be derived in conservative, transient form as: oðaqÞk þ rðaquÞk ¼ 0; ot
FUSLF ¼ V ðtÞrpLF ;
ð19Þ
FUSHF ¼ V ðtÞrpHF :
ð20Þ
2.4.3. Forces time averaging We consider that bubbles encounter only one sound field amplitude during one of their radial oscillation period (which is assumed equal to the sound field oscillation period T ¼ 1=f for all bubbles). Then the time averaging over a T period is written as follow: hFD iT ¼ 12pl1 hRðtÞiT ur ;
ð21Þ
1 dur ; hFAM iT ¼ q1 hV ðtÞiT 2 dt
ð22Þ
ð16Þ
hFUSLF iT ¼ hV ðtÞrpLF iT ;
ð23Þ
where p1 is the mean pressure of the continuous phase and Ik is the part of the interfacial momentum transfer rate between phases. u, q and l, are respectively velocity, density and dynamic viscosity. k ¼ 1 or 2, 1 refers to the fluid phase and 2 to the bubble phase, a is the volume fraction of the considered phase. The CFD code ESTET-ASTRID solves these equations (see Refs. [32,33]). ESTET-ASTRID is part of the CAMUS code.
hFUSHF iT ¼ hV ðtÞrpHF iT :
ð24Þ
ð15Þ
oðaquÞk ~p1 þ lk r2 ðauÞ þ Ik ; þ r:ðaqu uÞk ¼ r k ot
2.4.2. Forces exerted on the repetitive transient cavitation bubbles Because repetitive transient cavitation bubbles migrate, they induce disturbances within the liquid, the interaction between cavitation bubbles and the acoustic pressure field is rendered through the stresses applied by the surrounding liquid on the cavitation bubbles. We assume that the considered bubbles have the same equilibrium radius and follow linear oscillations. The forces they experienced are the following:
fV ¼ hFD iT þ hFMA iT þ FUS iT :
ð25Þ
And the interfacial momentum transfer (see Eq. (16)) has the following expression: I2;i ¼ I1;i ¼ a2 f:
ð26Þ
3. Results: mono-frequency sonoreactors
• The added mass force FAM , 1 dur ; FAM ¼ q1 V ðtÞ 2 dt
But we take into account the change of the mechanical properties of the sound wave due to the emergence of unstable bubbles in the computation of the repetitive transient bubble motion. Indeed, the acoustic force FUS is calculated upon the pressure field and bubble field computed taking into account the emergence of the fragmentary transient cavitation bubbles. This force is implemented in the CAMUS code as an external force (applied on the stable cavitation bubbles). We then deduce the resultant force applied on each point included in the dispersed phase:
ð17Þ
where ur is the local instantaneous relative velocity on each point included in the dispersed phase and given by ur ¼ u2 u1 .
CAMUS code is able to model a wide range of phenomena [22] such as bubble emergence, spatio-temporal dynamic of bubble clouds [13,21], acoustic streaming [23] in the case of low (20–30 kHz) and high (500 kHz) frequencies.
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3.1. Low frequency sonoreactor: f ¼ 28 kHz We consider a cylindrical a cylindrical reactor: 12 cm diameter and a water height of 9.4 cm. The piezoelectric transducer is 6 cm diameter, and placed at the bottom of the reactor. Looking at Fig. 2(a) and (b), we find good agreement between the test of aluminium foil degradation and computed bubble volume fraction. Comparing the pressure field taking into account the cavitation bubble emergence and the one of a pure liquid (see Fig. 3), we notice a slight damping. Indeed, the power input of the ultrasound emitter is not big, only 50 W. Thus few bubble emerge (compared to sonotrode emitter where the power input is up to 1000 W). Comparing pressure measurements (Fig. 4) and computed bubble volume fraction (Fig. 2(b)), we see that bubbles emerge at the pressure antinodes.
Fig. 4. Axial cut of the measured acoustic pressure field, through the use of an hydrophone. Sonoreactor diameter is 12 cm, transducer diameter is 6 cm, f ¼ 28 kHz.
3.2. Medium frequency sonoreactor: f ¼ 200 kHz The CAMUS code also predicts bubble emergence field at medium frequency. Considering a 200 kHz sonoreactor (see Fig. 5(a)) we have a good agreement between experiments and modeling (Figs. 5(b) and 6). For higher power input, an acoustic fountain occur at
Fig. 2. Semi-axial cut of an aluminium foil immersed in a 28 kHz sonoreactor, bubble jet pittings degrade the aluminium foil (a). Semiaxial cut of computed bubble volume fraction (f ¼ 28 kHz) (b). Sonoreactor diameter is 12 cm, transducer diameter is 6 cm.
Fig. 3. Semi-axial cut of the acoustic pressure field, taking into account the cavitation bubble emergence and in a pure liquid. Sonoreactor diameter is 12 cm, transducer diameter is 6 cm, f ¼ 28 kHz.
Fig. 5. 200 kHz sonoreactor geometry (a); Sarvazyan method: Chromolux foil immersed in methylene blue dye, darker area reveal higher bubble volume fraction (b).
Fig. 6. Cavitation bubble fields in a 200 kHz sonoreactor.
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excitation. As seen on Fig. 8(a) and (b), bubble clouds shape consists of concentric circles.
4. Results: dual-frequency sonoreactors 4.1. Dual-frequency reactor geometry Considered sonoreactors have a cylindrical shape. The reactors diameter is 8 cm, the water level is 9.4 cm. LF and HF sound waves are respectively generated at the bottom and the top of the sonoreactors, through the use of circular transducers (Fig. 9). This reactor shape allows to have interaction of low and high frequency ultrasound waves, throughout the reactor. 4.2. 28–56 kHz sonoreactor Fig. 7. Cavitation bubble fields in a 200 kHz sonoreactor, taking into account of the acoustic fountain.
the water–air interface. Taking into account of this phenomena in the modeled reactor, we get the Ôself focusingÕ effect. That is to say, a higher bubble volume fraction nearby the acoustic fountain (Fig. 7). This is in good agreement with the experiments of [29]. 3.3. High frequency sonoreactor: f ¼ 540 kHz As well, the CAMUS code predict the particular shape of cavitation bubble field in the case of a 540 kHz
Fig. 10 shows Cavitation bubble cartography. We notice a cavitation zone with a higher bubble volume fraction, compared to mono-frequency reactors (see Fig. 2). Fig. 12 gives the decomposition of the pressure field (Fig. 11). Low and high frequency component of the acoustic pressure field are accurately computed, even though we assume small perturbation hypothesis (see Eq. (10)). We notice a drastic change in bubble cartography, compared to mono-frequency reactor (see Fig. 2). 4.3. 28–100 kHz sonoreactor Fig. 13 gives the cavitation bubble cartography in a 28–100 kHz sonoreactor. Computation of the bubble emergence field is achieved by taking into account (throughout direct simulation) of the dual-frequency wave propagation (see Fig. 14). Computation of the bubble–sound wave interaction is done through the decomposition (at each time step) of the dual-frequency pressure field (see Fig. 15).
Fig. 8. Chromolux foil colored by methylene blue dye facing a 540 kHz piezoelectric transducer (Sarvazyan method) (a); computed bubble volume fraction (f ¼ 540 kHz) (b). Sonoreactor diameter is 10 cm, water height is 1 cm.
Fig. 9. Geometry of dual-frequency reactor.
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Fig. 10. 3D cavitation bubble fields in a 28–56 kHz sonoreactor.
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Fig. 13. 3D cavitation bubble fields in a 28–100 kHz sonoreactor.
Fig. 11. Axial cut of the acoustic pressure field in a 28–56 kHz sonoreactor.
Fig. 14. Axial cut of the acoustic pressure field in a 28–100 kHz sonoreactor.
Fig. 12. Axial cut of the low and high frequency component of the acoustic pressure field in a 28–56 kHz sonoreactor.
Fig. 15. Axial cut of the acoustic pressure field in a 28–100 kHz sonoreactor.
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Fig. 16. Axial cut of the cavitation bubble fields in a 28–200 kHz sonoreactor. Fig. 19. Velocity field of cavitation bubbles in a 28–56 kHz reactor. Bubbles have a 5 lm initial radius. Absolute magnitude of the velocities is 50 cm/s1 .
4.4. 28–200 kHz sonoreactor The use of a dual-frequency excitation allows to get a more intense cavitation bubble field (see Fig. 16) than in the case of mono-frequency excitation. Fig. 18 shows the decomposition of the dual-frequency acoustic pressure field (see Fig. 17). 4.5. Spatio-temporal dynamic of cavitation bubble in a 28–56 kHz sonoreactor
Fig. 17. Axial cut of the acoustic pressure field in a 28–200 kHz sonoreactor.
Comparing Figs. 11 and 19, we notice that cavitation bubbles go to pressure antinodes which result in the interaction of the low and high frequency ultrasound waves propagation. Before reaching the pressure antinodes of the highest amplitude, bubbles reach pressure antinodes of smaller amplitude (see Fig. 11).
5. Conclusion
Fig. 18. Axial cut of the low and high frequency component of the acoustic pressure field in a 28–200 kHz sonoreactor.
Acoustic cavitation covers a wide range of different phenomena. It has been shown that numerical modeling may be useful to predict encountered phenomena. The use of the CAMUS is able to predict the self-focusing phenomena at medium frequency. As well, the code accurately predicts the cavitation bubble emergence field at low and high frequency (28–540 kHz). Extension of the CAMUS code have been presented, to take into account dual-frequency sound wave propagation. Computations in 28–56, 28–100 and 28–200 kHz sonoreactors show that it is of a great interest to use dual-frequency sound waves: cavitation bubble volume fraction field are higher than in mono-frequency sonoreactors. As well, cavitation bubbles have a specific spatio-temporal dynamic. Future work will have to be
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undertaken to optimize the dual-frequency sonoreactor geometry. Acknowledgements Authors are grateful to Dr. Breitbach (Dortmund University, Germany) for his useful advices on the ÔSarvazyan methodÕ, as well as Prof. Simonin, IMFT/ Toulouse, France) for constructive discussion. References [1] A.H. Nayfeh, D.T. Mook, J. Phys., Colloque C8 40 (Suppl. 11) (1979) 3. [2] J. Holzfuss, M. R€ uggeberg, R. Mettin, Phys. Rev. Lett. 81 (1998) 1961. [3] K.M. Swamy, K.L. Narayana, Intensification of leaching process by dual-frequency ultrasound, Ultrason. Sonochem. 8 (2001) 341– 346. [4] S. Manickam, Aniruddha B. Pandit, Ultrasound enhanced degradation of Rhodamine B: optimization with power density, Ultrason. Sonochem. 8 (2001) 233–240. [5] O. Dahlem, Contribution a 1Õetude des problemes de montee en echelle des reacteurs sonochimiques, et investigation de la cavitation hydrodynamique comme technique alternative a la cavitation Acoustique, these de troisieme cycle, Universite libre de Bruxelles, 1999. [6] S. D€ ahnke, F.J. Keil, Modeling of three-dimensional linear pressure fields in sonochemical reactors with homogeneous and inhomogeneous density distribution of cavitation bubbles, Ind. Eng. Chem. Res. 37 (1998) 848–864. [7] S. D€ ahnke, K.M. Swamy, F.J. Keil, Modeling of three-dimensional pressure fields in sonochemical rectors with an inhomogeneous density distribution of cavitation bubbles, Comparison of theoretical and experimental results, Ultrason. Sonochem. 6 (1999) 31–41. [8] S. D€ ahnke, 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. [9] C. Horst, U. Hoffmann, U. Kunz, Modelling of sound fields in ultrasound reactors as a first step in scale up of heterogeneous reactions, European Congress on Chemical Engineering, ECCE 1, Florenz, 4–7 Mai 1997. [10] J.M. L€ oning, C. Horst, U. Hoffmann, Sonochemical reactors for heterogeneous solid–liquid reactions, A chemical engineering approach, in: Proceedings 2nd Conference: Applications of Power Ultrasound in Physical and Chemical Processing, Toulouse, 6–7 May 1999. [11] C. Horst, U. Kunz, A. Rosenpl€anter, U. Hoffmann, Activated solid–fluid reactions in ultrasound reactors, Chem. Eng. Sci. 54 (1999) 2849–2858. [12] U. Parlitz, R. Mettin, S. Luther, I. Akhatov, M. Voss, W. Lauterborn, Spatio-temporal dynamics of Acoustic cavitation bubble clouds, Philos. Trans. R. Soc. Lond. A 357 (1999) 313–334. [13] G. Servant, J.L. Laborde, A. Hita, J.P. Caltagirone, A. Gerard, Ultrason. Sonochem. 8 (2001) 63–174. [14] B. Dubus, O. Ledez, C. Granger, P. Mosbah, C. Campos/Pozuelo, in: Proceeding of the 3rd Conference on Applications of Power Ultrasound in Physical and Chemical Processing, Paris, France, 13–14 December 2001.
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