Chinese Journal of Engineering

Volume 2013 (2013), Article ID 362682, 14 pages

http://dx.doi.org/10.1155/2013/362682

## Numerical Simulation and Investigation of System Parameters of Sonochemical Process

Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati 781 039, Assam, India

Received 25 July 2013; Accepted 19 August 2013

Academic Editors: B.-Y. Cao, S. Tang, and G. Xiao

Copyright © 2013 Sankar Chakma and Vijayanand S. Moholkar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents the effects of various parameters that significantly affect the cavitation. In this study, three types of liquid mediums with different physicochemical properties were considered as the cavitation medium. The effects of various operating parameters such as temperature, pressure, initial bubble radius, dissolved gas content and so forth, were investigated in detail. The simulation results of cavitation bubble dynamics model showed a very interesting link among these parameters for production of oxidizing species. The formation of ^{•}OH radical and H_{2}O_{2} is considered as the results of main effects of sonochemical process. Simulation results of radial motion of cavitation bubble dynamics revealed that bubble with small initial radius gives higher sonochemical effects. This is due to the bubble with small radius can undergo many acoustic cycles before reaching its critical radius when it collapses and produces higher temperature and pressure inside the bubble. On the other hand, due to the low surface tension and high vapor pressure, organic solvents are not suitable for sonochemical reactions.

#### 1. Introduction

The sonochemistry concept is a well-established technique, and now it is considered as one of the most popular advanced oxidation processes. In the last few decades, sonochemical process became one of the most popular techniques for synthesis of catalysts as well as different types of materials [4–7], degradation of pollutants [8–12], synthesis of biodiesel [13, 14], and so forth. Numerous literatures on sonochemistry have already been published [15–18] which reported the various beneficial effects of sonochemistry. The principal phenomenon behind all of these effects is the cavitation. Cavitation is nothing but nucleation, growth, and implosive collapse of a bubble. Cavitation occurs through the formation of bubbles or cavities present in liquid medium. Each cavitating bubble contains gas or vapor or a mixture of gas-vapor. When bubble contains only gas, the expansion of bubble is mainly by diffusion, pressure reduction, or by rise in temperature. However, there are several parameters which are directly involved in transient cavitation. Transient cavitation is nothing but the growth of bubbles extensively over time scales of the order of the acoustic cycle and then undergoes an implosive/energetic collapse resulting in either fragmentation, decaying oscillation, or a repeat performance [19]. Cavitation can also be the result of the enlargement of cavities that are already present in bulk liquid. Sometimes cavitation bubbles are suspended/trapped in tinny cracks in the interface of solid and liquid. These cavitation bubbles grow and collapse in the medium based on the pressure variation inside the bubble. On the basis of pressure variation, cavitation can be categorized into four subsections: hydrodynamic cavitation, acoustic cavitation, optic cavitation, and particle cavitation [20]. Hydrodynamic cavitation is the result of pressure variation in the liquid medium due to the flow variation of liquid through the medium, while acoustic cavitation is due to the passing of acoustic waves in the liquid medium. Generally, acoustic cavitation occurs in the frequency range of 20 kHz to 1 MHz, that is, high power low frequency [20]. In acoustic cavitation, very high temperature and pressure (~ 5000 K and ~ 500 bar) are produced inside the bubbles due to transient collapse of cavitation bubbles. The bubble collapse occurs within as small time as 50 ns, and it is almost adiabatic. During the collapse of the bubble, many species are formed such as H_{2}, O_{2}, H_{2}O_{2}, O_{3}, ^{•}OH, ^{•}H, ^{•}O, and . Among these species ^{•}OH radical is the most predominant radical species [21, 22] having an oxidizing potential of 2.33 V [9, 23]. For greater details in cavitation concepts, we would like to refer the readers to see the state of art reviews by Xu et al. [4], Apfel [24], Gong and Hart [25], Suslick et al. [26], and Gogate and Pandit [27].

#### 2. Cavitation Bubble Dynamics Model

In order to investigate the various effects of cavitation, we have considered a model for a single cavitation bubble to simulate the radial motion of cavitation bubble. Although the single bubble analysis for radial motion of cavitation bubble does not give the entire phenomenon of the sonochemical process, but it can give a qualitative physical insight into the mechanism. The resultant effect of sonochemical reaction is the collective oscillations and collapse of cavitation bubbles, with strong interaction between them. For simulation of radial motion, the single bubble model is the most popular model to present the effects of cavitation, namely, physical and chemical effect. The physical and chemical effects induced by ultrasound and cavitation bubbles in the system can be determined using diffusion limited ordinary differential equation (ODE) model. This model is basically based on the boundary layer approximation proposed by Toegel et al. [28]. This radial motion of cavitation bubble model is derived from the partial differential equation (PDE) model of Storey and Szeri [29], which showed that water vapor entrapment in the cavitation bubble is mainly a diffusion limited process and not the condensation limited process. The main components of the radial motion of cavitation bubble dynamics are: Keller-Miksis equation for the radial motion of the bubble [30], equation for the diffusive flux of water vapor and heat conduction through bubble wall. The overall diffusion coefficient in ternary mixture (e.g. N_{2}–O_{2}–H_{2}O in case of air bubbles) or binary mixture (e.g., N_{2}–H_{2}O or O_{2}–H_{2}O or Ar–H_{2}O) has been determined using Blanc’s law [31], equation for heat conduction through cavitation bubble wall, and overall energy balance treating the cavitation bubble as an open system. The set of 4 ODEs are also given below.(1)Radial motion of the cavitation bubble:
Internal pressure in the bubble: Pressure in bulk liquid medium: . (2)Diffusive flux of water molecules: . Instantaneous diffusive penetration depth: .(3)Heat conduction across bubble wall: . Thermal diffusion length: . (4)Overall energy balance: . Mixture heat capacity: (where, ). Molecular properties of water: Enthalpy: . Internal energy: . Heat capacity of various species ():

The initial conditions of the ordinary differential equations are taken as: at , , , , , and . Thermodynamic data for various species and physicochemical properties of various solvents, that is, cavitation medium, are given in Tables 1 and 2. The thermal conductivity and diffusion coefficient (transport parameters for the heat and mass transfer) are estimated using Chapman-Enskog theory using Lennard-Jones 12-6 potential at the bulk temperature of the liquid medium [1–3, 31]. This model ignores the diffusion of gas across bubble interface, because the time scale for the diffusion of gases is much higher than the time scale of cavitation bubble model. The ODEs in the bubble dynamics model have been solved simultaneously using Runge-Kutta 4th- and 5th-order adaptive step size method [32]. Various parameters used in the numerical solution of radial motion of cavitation bubble dynamics and their numerical values are given in Tables 1 and 2. The other parameters are as follows: ultrasound frequency kHz, ultrasound pressure amplitude kPa, initial bubble radius *μ*m, sound velocity in water m/s, sound velocity in toluene m/s, and sound velocity in n-hexane m/s. All these parameters remain unchanged unless otherwise stated.

*Physical Effect of Ultrasound.* Ultrasound propagates through the medium in the form of longitudinal wave with series of compression and rarefaction and causes rapid oscillatory motion of fluid elements called microstreaming. This motion gives rise to intense micromixing in the medium. The magnitude of the microstreaming velocity is given by . Substituting values of as 1.5 × 10^{5} Pa (or 1.5 bar), kg/m^{3}, and m/s gives m/s.

*Sonochemical Effect of Cavitation Bubbles.* The numerical solution of bubble dynamics model predicts the temperature, pressure, and the number of gas and solvent molecules in the bubble at transient collapse. At transient collapse, temperature and pressure reach extreme conditions, which result in dissociation of molecules to form various chemical species. To calculate the composition of the bubble at the time of collapse, we assume that thermodynamic equilibrium is attained [9]. The equilibrium mole fraction of different chemical species in the bubble at the peak conditions reached at transient collapse is estimated using Gibbs free-energy minimization technique [33]. The radial motion of cavitation bubbles generates intense convection in the medium through two phenomena. (i)Microconvection [20]: .(ii)Shock (or acoustic) waves [34, 35]: ,where is the volume of the bubble. A representative value of is taken as 1 mm. In bubble dynamic model, direct estimation of initial bubble radius is very difficult. Several phenomena such as rectified diffusion and fragmentation of the bubble cause continuous change in this parameter. In a multibubble system, a large variation in this parameter may be expected. In order to investigate the influence of this parameter, two numerical values for this parameter, namely, 5 and 10 *μ*m, are selected for this present study. The equilibrium composition of the bubble contents was determined using the FactSage online software [36], which is based on the free-energy minimization algorithm same as proposed by Eriksson [33].

#### 3. Numerical Solution of Mathematical Model

The simulation results of radial motion of cavitation bubble dynamics are depicted in Figures 1, 2, 3, 4, and 5 and Tables 3, 4, 5, 6, and 7. Tables 3–7 present the temperature () and pressure () peaks reached at the moment of transient collapse of cavitation bubble, the number of water molecules in the bubble at collapse, and the equilibrium composition of the various species resulting from dissociation of entrapped molecules in the bubble when the temperature and pressure reached to its extreme conditions during transient collapse of cavitation bubbles.

##### 3.1. Initial Bubble Size in the Liquid Medium

Bubbles with different initial radius present in the liquid medium play an important role in cavitation. Kumar and Moholkar [37] have reported that the smaller bubble has higher Laplace pressure, which results in higher expansion and more intense collapse during the transient collapse. In this present study, a single bubble model with different initial radius has been chosen to present the effects of bubble radius in sonochemical process. From Table 3, it could be seen that the ^{•}OH radical formation during transient collapse of cavitation bubble is higher with the initial bubble radius of 5 *μ*m followed by 25 *μ*m. This is due to the bubble with small bubble radius can undergo many acoustic cycles before reaching its critical radius when it collapses, and as a result the highest temperature peak is reached at the moment of transient collapse of the cavitation bubble with 5 *μ*m radius bubble as shown in Figure 1 and Table 3.

##### 3.2. Static Pressure of the System

As stated earlier in the previous section, the cavitation is nothing but the nucleation, growth, and implosive collapse of bubbles, but the growth/expansion of the bubble depends on the system’s static pressure. So, when the bubble contains vapor, cavitation may occur due to reduced static pressure sufficiently at constant temperature. To investigate the effects of pressure, five different static pressures ranging from 100–300 kPa have been chosen. The results showed that as the static pressure increases the radical formation decreases drastically; even it becomes zero at higher ambient static pressure. This is due to at elevated static pressure, the sonochemical effects get eliminated which directly affects the generation of radical formation, provided the applied pressure must be greater than or equal to the acoustic pressure amplitude () [9, 38]. The highest ^{•}OH radical production was seen at nearly ambient static pressure (i.e., 100 kPa), and the production of ^{•}OH radical decreases as the static pressure approaches to the magnitude of acoustic pressure, that is, 150 kPa. The production of ^{•}OH radical at 100 kPa is typically 1.5 fold higher than the ^{•}OH radical production at 120 kPa, while at 150 kPa or higher static pressure no ^{•}OH radical formation was seen. This result can be attributed to the temperature peak reached at the moment of collapse of the bubble. At the time of transient collapse, the highest temperature peak (3999 K) was seen when the pressure was 100 kPa, while the lowest temperature peak (~313 K) was observed with elevated static pressure of 300 kPa.

##### 3.3. System Operating Temperature

Another important factor that affects the transient cavitation is system temperature. To study the temperature effect on sonochemical process, the numerical simulations were run in the temperature range from 283 to 373 K (viz. 283, 303, 323, 343, and 373 K). The simulation results are depicted in Figure 3. As the temperature in the system increases the formation of oxidizing species due to transient cavitation of bubbles reduces drastically. This could be attributed to the effects of bubble radius as discussed in the earlier section. In this case, vapor bubbles grow with increasing the temperature through the phenomenon of boiling [20]. So, the lower the temperature means the higher the cavitation effects in the system. Table 5 shows that the production of ^{•}OH radical is highest with the lowest temperature, that is, at 283 K, while at the boiling point of water no ^{•}OH radical formation has been observed. The overall trend of the ^{•}OH radical production under various operating temperature is 283 K > 303 K > 323 K > 343 K.

##### 3.4. Type of Dissolved Gas Content

The type of dissolved gas has an impact on bubble nucleation rate as described in the literatures [15, 37]. The simulation results for four gases are shown in Table 6 and Figure 4. The production of ^{•}OH radical as well as H_{2}O_{2} is highest for O_{2} bubble followed by air, Ar, and N_{2} bubble. Also the generation of O_{3} per bubble for oxygen bubble is one order of magnitude higher than the O_{3} generation by air bubble. The radicals by air bubble are formed through the reactions (3)–(22), while (24) and (25) represent the radical formation by O_{2} bubble [39]. The production of radicals is least with N_{2} bubble as compared to other gas bubbles. This is due to the scavenging of radicals by N_{2} molecules through the reactions (3)–(16). Moreover, there is no regeneration of ^{•}OH and O^{•} radicals by O_{2} when the bubble contents are N_{2} gas. So the total number of radicals per bubble in N_{2} bubbling is always lower than the other gas bubbles. The trend of production of ^{•}OH radical per bubble with various gases bubbling is as follows O_{2} > Air > Ar > N_{2}.

##### 3.5. Type of Solvent or Cavitation Medium

The solvent or cavitation medium is another important factor that affects directly on the sonochemical process. In order to assess the suitability of solvent for cavitation, we have chosen three solvents with different physicochemical properties, namely, water, toluene, and n-hexane, as shown in Table 2. To address this issue, we have presented the numerical solution with various solvent as cavitation medium in Figure 5 and Table 7, which lists the temperature and pressure peak reached at the moment of collapse with various equilibrium compositions of species. The number of ^{•}OH radical production is highest when water is used as cavitation medium. This can be attributed to the lower surface tension and higher vapor pressure of the organic solvent. The production of ^{•}OH radical in water is typically two orders of magnitude higher than that in toluene. So, due to the low intensity of bubble collapse, organic solvents are not able to generate sufficient numbers of radical *in-situ* for sonochemical reactions.

#### 4. Conclusions

The lower is the system temperature the higher is the cavitation effects, that is, when the system temperature is moderately low the effect of cavitation is more. In this present study, the radical formation due to cavitation is highest when the system temperature is 283 K. So, the system with low operating temperature and cavitation medium contents with small initial bubble radius is always preferable for sonochemical reactions since this type of sonochemical reaction systems are able to generate large amount of highly oxidizing species. Due to the high vapor pressure and low surface tension, organic solvents cannot produce large amount of radicals in the reaction medium; thus, they are not suitable for sonochemical reactions. However, they could be good solvents for extraction or other processes using ultrasound. Moreover, depending on the process requirement one can choose the system with different gas bubbling and aqueous or organic solvents for conducting experiments.

#### Nomenclature

: | Heat capacity at constant volume, J kg^{−1} K^{−1} |

: | Velocity of sound in liquid medium, m s^{−1} |

: | Velocity of sound in water, m s^{−1} |

: | Velocity of sound in toluene, m s^{−1} |

: | Velocity of sound in n-hexane, m s^{−1} |

: | Diffusion coefficient of solvent vapor, m^{2} s^{−1} |

: | Frequency of ultrasound wave, Hz |

: | Translational and rotational degrees of freedom |

: | Van der Waal’s hard core radius, m |

: | Instantaneous diffusive penetration depth for water molecules |

: | Boltzmann constant, J K^{−1} |

: | Number of water molecules trapped in the bubble |

: | Number of N_{2} molecules in the bubble |

: | Number of oxygen molecules in the bubble |

: | Number of toluene molecules in the bubble |

: | Number of n-hexane molecules in the bubble |

: | Number of Ar molecules in the bubble |

: | Total number of molecules (gas + vapor) in the bubble |

: | Static pressure in the liquid medium, Pa |

: | Pressure peak reached in the bubble at the time of first collapse, Pa |

: | Pressure amplitude of the acoustic wave generated by the cavitation bubble, Pa |

: | Heat conducted across bubble wall, J s^{−1} |

: | Radius of the bubble, m |

: | Initial radius of the cavitation bubble, m |

Bubble wall velocity, m s^{−1} | |

: | Time, s |

: | Temperature of the bubble contents, K |

: | Ambient (or bulk liquid medium) temperature, K |

: | Temperature peak reached in the bubble at the time of first collapse, K |

: | Average velocity of the microturbulence in the medium generated by ultrasound and cavitation in the medium (estimated at 1 mm distance from bubble center), m s^{−1} |

: | Density of the liquid, kg m^{−3} |

: | Kinematic viscosity of liquid, m^{2} s^{−1} |

: | Surface tension of liquid, N m^{−1} |

: | Thermal conductivity of bubble contents, W m^{−1} K^{−1} |

: | Thermal diffusivity of bubble contents, m^{2} s^{−1} |

: | Characteristic vibrational temperature(s) of the species, K. |

#### Conflict of Interests

The authors of the paper do not have any direct financial relation that might lead to a conflict of interest for any of the authors.

#### References

- J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird,
*Molecular Theory of Gases and Liquids*, John Wiley & Sons, New York, NY, USA, 1954. - E. U. Condon and H. Odishaw,
*Handbook of Physics*, McGraw-Hill, New York, NY, USA, 1958. - R. C. Reid, J. M. Prausnitz, and B. E. Poling,
*Properties of Gases and Liquids*, McGraw-Hill, New York, NY, USA, 1987. - H. Xu, B. W. Zeiger, and K. S. Suslick, “Sonochemical synthesis of nanomaterials,”
*Chemical Society Reviews*, vol. 42, pp. 2555–2567, 2013. View at Google Scholar - J. H. Bang and K. S. Suslick, “Applications of ultrasound to the synthesis of nanostructured materials,”
*Advanced Materials*, vol. 22, no. 10, pp. 1039–1059, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Kanmuri and V. S. Moholkar, “Mechanistic aspects of sonochemical copolymerization of butyl acrylate and methyl methacrylate,”
*Polymer*, vol. 51, no. 14, pp. 3249–3261, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Sivakumar, A. Gedanken, W. Zhong et al., “Nanophase formation of strontium hexaferrite fine powder by the sonochemical method using Fe(CO)
_{5},”*Journal of Magnetism and Magnetic Materials*, vol. 268, no. 1-2, pp. 95–104, 2004. View at Publisher · View at Google Scholar · View at Scopus - P. R. Gogate and G. S. Bhosale, “Comparison of effectiveness of acoustic and hydrodynamic cavitation in combined treatment schemes for degradation of dye wastewaters,”
*Chemical Engineering and Processing: Process Intensificationdoi*, 2013. View at Publisher · View at Google Scholar - S. Chakma and V. S. Moholkar, “Physical mechanism of sono-fenton process,”
*AIChE Journal*, 2013. View at Publisher · View at Google Scholar - J. B. Bhasarkar, S. Chakma, and V. S. Moholkar, “Mechanistic features of oxidative desulfurization using sono-fenton-peracetic acid (Ultrasound/Fe
^{2+}-CH_{3}COOH-H_{2}O_{2}) system,”*Industrial & Engineering Chemistry Research*, vol. 52, no. 26, pp. 9038–9047, 2013. View at Publisher · View at Google Scholar - P. Braeutigam, M. Franke, R. J. Schneider, A. Lehmann, A. Stolle, and B. Ondruschka, “Degradation of carbamazepine in environmentally relevant concentrations in water by Hydrodynamic-Acoustic-Cavitation (HAC),”
*Water Research*, vol. 46, no. 7, pp. 2469–2477, 2012. View at Publisher · View at Google Scholar · View at Scopus - J. Madhavan, P. S. S. Kumar, S. Anandan, M. Zhou, F. Grieser, and M. Ashokkumar, “Ultrasound assisted photocatalytic degradation of diclofenac in an aqueous environment,”
*Chemosphere*, vol. 80, no. 7, pp. 747–752, 2010. View at Publisher · View at Google Scholar · View at Scopus - P. A. Parkar, H. A. Choudhary, and V. S. Moholkar, “Mechanistic and kinetic investigations in ultrasound assisted acid catalyzed biodiesel synthesis,”
*Chemical Engineering Journal*, vol. 187, pp. 248–260, 2012. View at Publisher · View at Google Scholar · View at Scopus - H. A. Choudhury, S. Chakma, and V. S. Moholkar, “Mechanistic insight into sonochemical biodiesel synthesis using heterogeneous base catalyst,”
*Ultrasonics Sonochemistry*, 2013. View at Publisher · View at Google Scholar - J. Rooze, E. V. Rebrov, J. C. Schouten, and J. T. F. Keurentjes, “Dissolved gas and ultrasonic cavitation—a review,”
*Ultrasonics Sonochemistry*, vol. 20, pp. 1–11, 2013. View at Publisher · View at Google Scholar - K. D. Bhatte, S.-I. Fujita, M. Arai, A. B. Pandit, and B. M. Bhanage, “Ultrasound assisted additive free synthesis of nanocrystalline zinc oxide,”
*Ultrasonics Sonochemistry*, vol. 18, no. 1, pp. 54–58, 2011. View at Publisher · View at Google Scholar · View at Scopus - K. S. Suslick, M. M. Mdleleni, and J. T. Ries, “Chemistry induced by hydrodynamic cavitation,”
*Journal of the American Chemical Society*, vol. 119, no. 39, pp. 9303–9304, 1997. View at Publisher · View at Google Scholar · View at Scopus - J. S. Krishnan, P. Dwivedi, and V. S. Moholkar, “Numerical investigation into the chemistry induced by hydrodynamic cavitation,”
*Industrial and Engineering Chemistry Research*, vol. 45, no. 4, pp. 1493–1504, 2006. View at Publisher · View at Google Scholar · View at Scopus - T. G. Leight,
*The Acoustic Bubble*, Academic Press, London, UK, 1995. - Y. T. Shah, A. B. Pandit, and V. S. Moholkar,
*Cavitation Reaction Engineering*, Kluwer Academic/Plenum, New York, NY, USA, 1999. - K. S. Suslick, “Sonochemistry,”
*Science*, vol. 247, no. 4949, pp. 1439–1445, 1990. View at Google Scholar · View at Scopus - T. Sivasankar and V. S. Moholkar, “Mechanistic approach to intensification of sonochemical degradation of phenol,”
*Chemical Engineering Journal*, vol. 149, no. 1–3, pp. 57–69, 2009. View at Publisher · View at Google Scholar · View at Scopus - R. Andreozzi, V. Caprio, A. Insola, and R. Marotta, “Advanced oxidation processes (AOP) for water purification and recovery,”
*Catalysis Today*, vol. 53, no. 1, pp. 51–59, 1999. View at Google Scholar · View at Scopus - R. E. Apfel, “Acoustic cavitation inception,”
*Ultrasonics*, vol. 22, no. 4, pp. 167–173, 1984. View at Google Scholar · View at Scopus - C. Gong and D. P. Hart, “Ultrasound induced cavitation and sonochemical yields,”
*Journal of the Acoustical Society of America*, vol. 104, no. 5, pp. 2675–2682, 1998. View at Publisher · View at Google Scholar · View at Scopus - K. S. Suslick, Y. Didenko, M. M. Fang et al., “Acoustic cavitation and its chemical consequences,”
*Philosophical Transactions of the Royal Society A*, vol. 357, no. 1751, pp. 335–353, 1999. View at Google Scholar · View at Scopus - P. R. Gogate and A. B. Pandit, “A review of imperative technologies for wastewater treatment II: hybrid methods,”
*Advances in Environmental Research*, vol. 8, no. 3-4, pp. 553–597, 2004. View at Publisher · View at Google Scholar · View at Scopus - R. Toegel, B. Gompf, R. Pecha, and D. Lohse, “Does water vapor prevent upscaling sonoluminescence?”
*Physical Review Letters*, vol. 85, no. 15, pp. 3165–3168, 2000. View at Publisher · View at Google Scholar · View at Scopus - B. D. Storey and A. J. Szeri, “Water vapour, sonoluminescence and sono chemistry,”
*Proceedings of the Royal Society A*, vol. 456, no. 1999, pp. 1685–1709, 2000. View at Google Scholar · View at Scopus - J. B. Keller and M. Miksis, “Bubble oscillations of large amplitude,”
*Journal of the Acoustical Society of America*, vol. 68, no. 2, pp. 628–633, 1980. View at Google Scholar · View at Scopus - R. B. Bird, W. E. Stewart, and E. N. Lightfoot,
*Lightfoot, Transport Phenomena*, John Wiley & Sons, New York, NY, USA, 2nd edition, 2001. - W. H. Press, S. A. Teukolsky, B. P. Flannery, and W. T. Vetterling,
*Numerical Recipes*, Cambridge University Press, New York, NY, USA, 2nd edition, 1992. - G. Eriksson, “Thermodynamic studies of high temperature equilibria—XII: SOLGAMIX, a computer program for calculation of equilibrium composition in multiphase systems,”
*Chemica scripta*, vol. 8, pp. 100–103, 1975. View at Google Scholar - S. Grossmann, S. Hilgenfeldt, M. Zomack, and D. Lohse, “Sound radiation of 3-MHz driven gas bubbles,”
*Journal of the Acoustical Society of America*, vol. 102, no. 2, pp. 1223–1230, 1997. View at Publisher · View at Google Scholar · View at Scopus - V. S. Moholkar and M. M. C. G. Warmoeskerken, “Integrated approach to optimization of an ultrasonic processor,”
*AIChE Journal*, vol. 49, no. 11, pp. 2918–2932, 2003. View at Publisher · View at Google Scholar · View at Scopus - FACTSAGE, http://www.factsage.com/.
- P. Kumar and V. S. Moholkar, “Numerical assessment of hydrodynamic cavitation reactors using organic solvents,”
*Industrial and Engineering Chemistry Research*, vol. 50, no. 8, pp. 4769–4775, 2011. View at Publisher · View at Google Scholar · View at Scopus - S. Chakma and V. S. Moholkar, “Mechanistic features of ultrasonic desorption of aromatic pollutants,”
*Chemical Engineering Journal*, vol. 175, no. 1, pp. 356–367, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. Prosperetti and A. Lezzi, “Bubble Dynamics in a compressible liquid. part 1. first order theory,”
*Journal of Fluid Mechanics*, vol. 168, pp. 457–478, 1986. View at Google Scholar · View at Scopus