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Numerical study on dual-frequency ultrasonic enhancing cavitation effect based on bubble dynamic evolution
Ultrasonics - Sonochemistry 59 (2019) 104744
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Numerical study on dual-frequency ultrasonic enhancing cavitation effect based on bubble dynamic evolution
T
⁎
Linzheng Ye , Xijing Zhu, Yao Liu Shanxi Key Laboratory of Advanced Manufacturing Technology, North University of China, Taiyuan 030051, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Ultrasonic cavitation Dual-frequency Cavitation enhancement effect Bubble dynamic evolution
Ultrasonic cavitation is a physical dynamic phenomenon of bubbles inflation, compression, and collapse in liquid. A dual-frequency ultrasonic cavitation dynamics model is established in this paper to investigate dynamic evolution of bubble under single and dual frequency ultrasonic modes. The variation of bubble radius, pressure, energy, temperature, and number of water vapor molecules inside the bubble in single and dual frequency ultrasonic modes are analyzed, respectively. The results show the oscillation of cavitation bubbles is more unstable and easier to collapse in dual-frequency ultrasound field than those in single-frequency ultrasound field. With the increase of the ultrasonic frequency, cavitation effect is weakened due to the shortage of oscillation period. Under the same ultrasonic power, the maximums of bubble radius, pressure, and water vapor molecules number inside the bubble in the dual-frequency mode are larger than those in the single-frequency mode. Under the ultrasonic excited by 50 kHz + 70 kHz, the maximum bubble radius and pressure can reach 36.061 μm and 2285.9 MPa, respectively, which are much larger than 18.183 μm, 730.61 MPa at 50 kHz and 14.576 μm, 332.25 MPa at 70 kHz. The calculation results of three different frequency combinations (30 kHz + 50 kHz, 40 kHz + 60 kHz and 50 kHz + 70 kHz) indicate dual-frequency ultrasound can significantly enhance the cavitation effect.
1. Introduction Cavitation is a process of phase transition between liquid and vapor or at liquid-solid interface caused by hydrodynamics. According to the mode of formation, cavitation can be divided into two categories: energy deposition induced cavitation and pressure drop induced cavitation. Energy deposition induced cavitation includes particle cavitation, photoinduced cavitation, and laser induced cavitation. Laser induces the change of liquid properties in the focusing region to produce bubbles. Based on laser energy densities, bubbles can be divided into vapor bubbles and plasma bubbles [1]. Pressure drop induced cavitation mainly consists of ultrasonic cavitation and hydraulic cavitation, which produce cavitation through ultrasonic pressure load and the special flow structure, respectively, due to the changing of the liquid ambient pressure. It has been reported that ultrasonic cavitation has higher collapse temperature and pressure than those in hydraulic cavitation [2]. Ultrasonic cavitation is a non-linear and complex acoustic phenomenon in liquid medium activated by ultrasound. When the ultrasonic amplitude is larger than the liquid cavitation threshold, the “weak link” in the connections between liquid molecules will be torn, resulting
⁎
in a large number of gas nuclei. The periodic vibration of ultrasonic pressure due to the ultrasound propagation, cause the micro bubbles or nuclei in liquid medium to grow and break [3–6]. Cavitation can be divided into transient cavitation and steady-state cavitation according to their dynamic characteristics. Transient cavitation usually occurs at relatively high pressure, and cavitation bubbles expand at a high speed in the negative pressure phase of ultrasound and collapse rapidly in the positive pressure phase. Steady-state cavitation refers to the non-linear oscillation of cavitation bubbles for several periods under the action of small alternating sound pressure, accompanied by mass diffusion of gases and radiation force. Ultrasonic cavitation is characterized by formation and collapse of bubbles, accompanied by secondary phenomena such as micro-jet and shock wave [7], resulting in local high temperature and pressure. The oscillating motion of cavitation bubbles, especially the extreme conditions such as high temperature and high pressure caused by collapse, can cause a series of effects: (1) High-pressure effect. Bubble collapse produces micro-jet with velocity up to several hundred meters per second, which cause the high impaction pressure to the material surface. The high-pressure
Corresponding author. E-mail address: [email protected] (L. Ye).
https://doi.org/10.1016/j.ultsonch.2019.104744 Received 12 May 2019; Received in revised form 24 July 2019; Accepted 22 August 2019 Available online 22 August 2019 1350-4177/ © 2019 Elsevier B.V. All rights reserved.
Ultrasonics - Sonochemistry 59 (2019) 104744
L. Ye, et al.
effect may form cavitation pits damage on the surface [8,9], which can also promote material removal [10]. (2) High-temperature effect. Cavitation bubbles in liquids usually contain a certain number of gas molecules. In the collapse process, the volume of bubble decreases sharply, resulting in the instantaneous rise of internal gas temperature, which cannot fully exchange with the outside in a short time. These high-temperature gases will act on material surface and form a potential local hightemperature region. (3) Activation effect. Ultrasonic cavitation will induce high temperature and pressure environment during bubble collapse, which will produce strong oxidizing free radical ions, such as H2 O ↔ OH + H , OH ↔ O + H , O + O ↔ O2 , etc. This can accelerate the chemical reaction rate and oxidize the organic matter, realizing ultrasonic decontamination in the liquid [11]. (4) Sonoluminescence effect [12,13]. Under conditions of cavitation collapse, extremely high temperature in the bubble will radiate light pulses. In different solutions, the intensity of sonoluminescence varies greatly.
Table 1 Main parameters. Parameters
Values
Ultrasonic pressure amplitude Pa/Pa Speed of sound in water c/m·s−1 Surface tension of water σ/N·m−1 Viscosity of water μ/Pa·s Universal gas constant Rg/J·mol−1·K−1 Density of water ρ/Kg·m−3 Liquid pressure at infinity P0/Pa
1.3 × 105 1483 7.275 × 10−2 1 × 10−3 8.3145 998.2 1 × 105
sound waves on the secondary Bjerknes force, and found it changed both the sign and the value of the Bjerknes force coefficient for the cases near the boundaries significantly. Moholkar [29] explored the effects of different phase differences on spatially uniform acoustic field, and results showed it had great influence on overall sonochemical effect in the processor. Wrede [31] has achieved the spatial control and induced cavitation of the targeted microbubble through a dual-frequency transducer. Although controlled cavitation under dual-frequency ultrasound has been confirmed in some laboratories, the research is still in infancy and its universality needs to be improved, and its theoretical mechanism needs to be further explored. Cavitation dynamics is the most effective way to understand cavitation effect from a theoretical perspective. Since Rayleigh first established the single spherical cavitation dynamics model in ideal fluid in 1917, the research on the cavitation dynamics has been continuing. The famous cavitation dynamics models include Rayleigh-Plesset model, Keller-Miksis model, and Gilmore model and so on [32]. Kyuichi [33–35] studied ultrasonic cavitation dynamics by building a bubble dynamics model considering the condensation, evaporation of water vapor and thermal effect. Lu [36] carried out thermodynamic analysis and established a heat conduction-radiation model of cavitation. Based on the pressure difference between inside and outside the bubble, Gao [37] obtained a mass transfer model between inside and outside of cavitation bubble and discussed the parameter correlation under the stable equilibrium radius. In 2011, Suslick [38] used the spectral analysis method to quantitatively study the high temperature and high pressure generated during bubble collapse. The temperature in the cavitation bubble was controlled by the vapor pressure and thermal conductivity of the liquid volatiles and gases, respectively. In 2012, Dular [39] experimentally investigated the growth and collapse of a single cavitation bubble using a high-speed thermal imaging camera. The results demonstrated that the temperature of liquid-vapor boundary layer decreases continuously during bubble growth, while the opposite consequence occurs when the bubble collapses, and it is found that evaporation and condensation are the two main factors driving heat conduction. Brujan [40] used a high-speed camera to observe the near-wall cavitation bubbles generated by high-intensity focused ultrasound pulses, and found that the distance between the ultrasonic focusing point and the rigid wall plays a decisive role in the collapse characteristics of cavitation bubbles. The velocity of micro-jet and the pressure of shock wave increase with the increase of the distance in a certain range, and the maximum values are 130 m/s and 0.65 MPa, respectively. In 2016, Cogné [41] established an inertial cavitation model for single cavitation bubble to accurately calculate the temperature and pressure at the instant of bubble collapse near the solid wall. Two heat transfer models between liquid and bubbles were proposed, and the changes of bubble radius, gas temperature, interface temperature and pressure under the two models were compared, and their application ranges were defined, respectively. In this paper, to understand the dynamic evolution and enhancement effect of cavitation in multi-frequency ultrasonic theoretically, a dual-frequency ultrasonic bubble dynamics model is established. This model was built based on Kyuichi's bubble dynamics model and considering heat exchange inside and outside the bubble. Then, the
These characteristics of ultrasonic cavitation make it present good application prospects in the fields of ultrasonic cleaning, surface modification, cancer treatment, petroleum exploitation and so on. Toh [14] and Chen [15] used ultrasonic cavitation for peening and polishing, respectively. In the grinding and modification of semiconductor materials [16], Savkina [17] applied ultrasonic cavitation to surface functionalization of silicon wafer. However, the application of ultrasonic cavitation in industry is not mature, due to the insufficient energy and cavitation shielding [18]. Cavitation shielding is a phenomenon to form cavitation near the end face of the horn and prevents sound energy propagation. At this time, simply increasing the ultrasonic power cannot effectively break the cavitation shielding. To solve this problem, Moholkar [19] adjusted the parameters of two sound sources and found that the bubbles are mainly generated in the liquid far away from the sound source. Therefore, the use of dual-frequency ultrasound can overcome cavitation shielding. In addition, it can avoid the defects of single-frequency ultrasound irradiation, such as inhomogeneous sound field, standing waves, fewer cavitation incidents and so on. Many scholars at domestic and abroad have proved that the dual-frequency ultrasound can improve the sonochemical yield [20,21]. Multi-frequency ultrasound can generate inertial cavitation easier than the single-frequency ultrasound by reducing the threshold. Suo [22] studied the inertial cavitation threshold under multi-frequency ultrasound and built a theoretical model for cavitation threshold under dual-frequency. Guédra [23] used the asymptotic method to study the nonlinear vibration of bubbles under dual-frequency ultrasonic excitation and obtained a larger bubble response amplitude. To quantify the dual-frequency ultrasonic cavitation effect, Ebrahiminia [24] measured the absorbance of potassium iodide dosimeter solution at 350 nm wavelength after ultrasonic treatment, and the results showed that the absorbance of dual-frequency (40 kHz + 1 MHz) mode is about 1.8 times of the algebraic sum of single-frequency phonons. Chen [25] refined the as-cast structure of ZK60 magnesium alloy by single and dual frequency ultrasound respectively, and found that the yield strength, ultimate tensile strength, and elongation of the latter were improved by 20.5%, 20.7%, and 30.0%, respectively, compared with those of the former. This is because dual-frequency ultrasound can produce larger cavitation bubbles and make more bubbles instantaneous, thus improving the refining efficiency. Zhang [26] studied the power spectrum and response curve of oscillating bubbles excited by dual-frequency ultrasound (5.5 MHz and 6.6 MHz), and revealed two unique characteristics of bubble oscillation: combination resonance and simultaneous resonance. In addition, the initial phase and phase difference of dual-frequency are important factors affecting cavitation [19,27–29]. Zhang [30] investigated the influence of phase difference between 2
Ultrasonics - Sonochemistry 59 (2019) 104744
L. Ye, et al.
Fig. 1. Dynamic evolution of bubble under different single-frequency ultrasonic modes. (a) Bubble radius, (b) pressure inside the bubble, (c) energy inside the bubble, (d) temperature inside the bubble, and (e) number of water vapor molecules inside the bubble.
under different ultrasonic modes are studied.
cavitation behaviors under dual-frequency mode (50 kHz + 70 kHz) and single-frequency mode (50 kHz, 70 kHz) are compared under the same ultrasonic power. Finally, three groups of dual-frequency (30 kHz + 50 kHz, 40 kHz + 60 kHz and 50 kHz + 70 kHz) and corresponding single-frequency are selected and their results were contrasted. The dynamic variation rules of bubble radius, pressure, energy, temperature, and number of water vapor molecules inside the bubble
2. Modeling In the actual ultrasonic cavitation, the physical quantities inside and outside the cavitation bubble are always in the dynamic change process. There are evaporation and condensation behaviors of gas-liquid 3
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L. Ye, et al.
where a and b are Van Der Waals constants, which change with the number of water vapor molecules, Rg is universal gas constant, T is temperature inside the bubble, and ν is molar volume. The values of a and b can be obtained by Eqs. (5) and (6).
Table 2 Maxima of bubble physical quantities in different single-frequency ultrasonic fields. Frequency (kHz)
Rmax (μm)
30 40 50 60 70 80
Emax (J)
Pinmax (MPa)
27.488 21.527 18.183 16.105 14.576 13.851
Tmax (K)
nH2Omax
2
−9
1176.1 1037.8 730.61 441.39 332.25 329.42
1.5767 × 10 1.2766 × 10−9 1.0964 × 10−9 9.5501 × 10−10 8.3229 × 10−10 7.6106 × 10−10
1140.2 1401.3 1504.8 1480.5 1407.1 1357.0
⎜
1.0374 × 10 4.6821 × 1010 2.6834 × 1010 1.7902 × 1010 1.2809 × 1010 1.1294 × 1010
(
=
1 ρ
Ṙ c
+
ṁ cρ
) + R ̇ (1 − 3 2
( 1 + ) {P Ṙ c
¨ mR + ρ
out
(1 −
2
Ṙ 3c
+
2ṁ 3cρ
+
ṁ cρ
) + (R ̇ + ṁ ρ
+
̇ ̇ Rm 2ρc
)+
T=
(
(
(
(
)
Pa =
Rg T
R dPout cρ dt
a ν2
⎜
⎟
(6)
(7)
103NA 2 ̇ Δt 4π me MH2O
(8)
103NA 4πR2ṁ Δt MH2O
(9)
2ρcI
For single − frequency ultrasonic: For dual − frequency ultrasonic:
)
−
⎟
(10)
where Pa is ultrasonic pressure amplitude and I is ultrasonic intensity. In order to compare the dynamic evolution of cavitation under single and dual-frequency, the ultrasonic intensity of single-frequency should be the same as that of dual-frequency.
P2 =
Pa sin(2πf2 t ) 2
P1 = Pa sin(2πf1 t ) P1 =
(11)
Pa sin(2πf1 t ), 2 (12)
Table 1 shows the values of the parameters in this paper. To obtain the motion of the cavitation bubble, the initial conditions of the above equations are also needed. The specific parameters are as follows: R|t=0 = R0, Ṙ t = 0 = 0 , T|t=0 = T0, Pout|t=0 = P0, dPout/ ¨ t = 0 = 0 . The initial radius of bubble R0 is dt|t=0 = 0, and ṁ t = 0 = m 4.5 μm, the initial temperature of water T0 is 293.15 K, the initial numbers of water vapor molecules and air molecules inside the bubble are 2.20415 × 108 and 1.23822 × 1010, respectively. In addition, according to the frequency bands commonly used in sonochemistry and industry, the ultrasonic frequency f is selected as 30, 40, 50, 60, 70, and 80 kHz.
(3)
where Pin is pressure inside the bubble, μ is viscosity of water, and ρa is average density of gas inside the bubble. Pin can be calculated based on the Van Der Waals equation of state:
(ν − b)
⎟⎜
Eq. (1) can be solved and physical parameters, such as R, Pin, T, E, and n H2O all can be obtained. For convenience of study, the effect of initial phase of ultrasound is not considered, so φ1 = φ2 = 0. The relationship between the ultrasonic pressure amplitude and the ultrasonic intensity is as follows [42]:
The external pressure of bubble, Pout, is related to the pressure inside the bubble and liquid properties [33].
Pin (t ) =
(5)
2
⎜
)
4μ ⎛ ̇ 2σ ṁ ⎞ 2⎛1 − 1 ⎞ − ⎜R − ⎟ − ṁ ⎜ ⎟ R R ⎝ ρ⎠ ρa ⎠ ⎝ρ
⎟
NA2 EV + (n H2O + nair ) a (nH2O CV , H2O + nair CV , air ) NA V
n H2O (t + Δt ) = n H2O (t ) +
(2)
Pout (t ) = Pin (t ) −
⎜
where MH2O is molar quantity of water. The first and second terms on the right side of Eq. (8) represent the variation of energy inside the bubble caused by the change of pressure and the number of water vapor molecules inside the bubble during Δt, respectively. The detail of e and ṁ calculation has been given in the literature [6]. The number of water vapor molecules at t + Δt is the sum of the number of the molecules at t and the number of the change during Δt.
)
)
⎟
ΔE (t ) = −Pin (t )ΔV (t ) +
(1) where Ẋ and X¨ are the first and second derivative of X , R is bubble radius, c is the speed of sound in water, ρ is water density, σ is surface tension of water, Pout is the external pressure of bubble, P0 is liquid ¨ indicate velocity and acceleration of water pressure at infinity, ṁ and m vapor evaporation and condensation, P1, P2, f1, f2, φ1, and φ2 are pressure amplitude, frequency and initial phase of ultrasound with different frequencies, respectively. Eq. (1) is a typical second-order nonlinear differential equation, which is hard to obtain the analytical solution directly. Therefore, the fourth-order Runge-Kutta method is used to obtain the numerical solution. For the convenience of programming, Eq. (1) is transformed into Eq. (2).
(
⎟
where NA is Avogadro constant, E is energy inside the bubble, CV, air and CV ,H2O are molar heat capacities at constant volume of air and water vapor, respectively. The energy change inside the bubble during Δt is as follows:
R dPout cρ dt
Ṙ = R2 ⎧ ⎪ Ṙ = R¨ ⎪ 2 ⎪ 1 Ṙ 1 + c {Pout − [P1 sin(2πf1 t + φ1) ⎪ ρ ⎪ ⎪ + P2 sin(2πf2 t + φ2)] − P0} ̇ ̇ ¨ mR Ṙ ṁ ṁ ṁ Rm ⎨ = + ρ 1 − c + cρ + ρ Ṙ + 2ρ + 2ρc + ⎪ ⎪ 2ṁ 3 Ṙ ⎪ − 2 Ṙ2 1 − 3c + 3cρ ⎪ ⎪ ̇ ⎡R 1 − R + ṁ ⎤ ⎪ c cρ ⎣ ⎦ ⎩
⎟⎜
where aair and bair (aH2O and bH2O) are the values of Van Der Waals of air (water vapor), aair - H2O = aair × aH2O , 3 bair - H2O = ( 3 bair + 3 bH2O ) 2 . nair, n H2O , and nt are the numbers of air, water vapor, and total vapor molecules, respectively. The temperature inside the bubble T can be expressed as:
) ṁ 2ρ
⎜
2
⎜
− [P1 sin(2πf1 t + φ1) + P2 sin(2πf2 t + φ2)] − P0}
Ṙ c
⎟
nH O nH O n n b = bair ⎛ air ⎞ + 2bair − H2O ⎛ air ⎞ ⎛ 2 ⎞ + bH2O ⎛ 2 ⎞ n n n ⎝ t ⎠ ⎝ t ⎠⎝ t ⎠ ⎝ nt ⎠
transition and material exchange at the bubble interface. Based on Kyuichi Yasui's work, this paper makes the following assumptions: (1) the cavitation bubble remains spherical with a fixed center and only vibrates radially; (2) the number of air molecules inside the bubble remains constant; (3) the variation of density and sound velocity in liquid is ignored; (4) the surface tension and viscous resistance of liquid cannot be neglected. A single bubble dynamics model considering heat exchange in a dual-frequency ultrasound field is established:
RR¨ 1 −
2
nH O nH O n n a = aair ⎛ air ⎞ + 2aair − H2O ⎛ air ⎞ ⎛ 2 ⎞ + aH2O ⎛ 2 ⎞ ⎝ nt ⎠ ⎝ nt ⎠ ⎝ nt ⎠ ⎝ nt ⎠
11
(4) 4
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L. Ye, et al.
Fig. 2. Contrast curves of bubble dynamic evolution in single and dual frequency ultrasonic fields. (a) Ultrasonic pressure, (b) bubble radius, (c) pressure inside the bubble, (d) energy inside the bubble, (e) temperature inside the bubble, and (f) number of water vapor molecules inside the bubble.
5
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temperature, and number of water vapor molecules inside the bubble under different ultrasonic frequencies. All parameters of cavitation bubble show periodic oscillation. The cavitation bubble is relatively stable in the initial ultrasonic positive pressure phase. When the negative pressure phase arrives, the bubble expands slowly due to the pressure difference between inside and outside the bubble, which induces the pressure, energy, temperature, and water vapor molecules inside the bubble decrease. The increase of the absolute value of ultrasonic pressure accelerates the expansion of the bubble. Correspondingly, the pressure, energy, temperature, and number of water vapor molecules inside the bubble also change faster. The bubble radius reaches the maximum when negative pressure phase transfers into the positive pressure phase, and the pressure, energy, and temperature inside the bubble reach the minimum while the number of water vapor molecules reaches the maximum at the same time. Due to the pressure inside lower than that outside, the bubble shrinks sharply. After the ultrasonic positive pressure phase coming, the bubble radius oscillates under the influence of pressure discrepancy. When the bubble is compressed to the minimum, the pressure, energy, and temperature inside the bubble reach the maximum, while the number of water vapor molecules decreases to the minimum. Similarly, these physical parameters continue to oscillate periodically under unstable conditions. If the bubble collapses at this time, it will release huge energy, produce sonoluminescence, and even may destroy the near wall. The reasons for the dynamic evolution of bubble state parameters are complex. First, water vapor molecules can leave or enter the bubble through the cavity wall by condensation and evaporation. When the water vapor molecule condenses, it changes from gaseous to liquid and enters the liquid environment through the cavity wall, during which heat is released. So the energy inside the bubble increases and the number of water vapor molecules decreases. On the contrary, when the water vapor molecule evaporates, it changes from liquid to gaseous and enters in the bubble, which is an endothermic process and will reduce the energy inside the bubble and increase the number of water vapor molecules. Therefore, the change of number of water vapor molecules inside the bubble represents the gas-liquid phase change of water vapor during the dynamic evolution of bubbles, and also reflects the oscillation behavior of bubbles laterally. Furthermore, it can be seen that the physical quantities of the bubble are coupled and correlated from the above mathematical model. Although the physical quantities present reasonable periodic oscillation law, the time to reach the extreme point is slightly different. For example, energy inside the bubble, E, as shown in Eq. (8), is not only related to the pressure inside the bubble, but also to the number of water vapor molecules. Therefore, the evolution of E is a comprehensive result of pressure and number of water vapor molecules changing inside the bubble. As shown in Fig. 1(a) and (b), with the increase of ultrasonic frequency, the oscillation period of cavitation decreases and the maximum of bubble radius decreases, and the maximum pressure inside the bubble reduces. Bubble radius and pressure inside the bubble can reflect the strength of cavitation effect to a certain extent. The larger the radius and the higher the pressure, the stronger the cavitation effect. Therefore, when the ultrasonic frequency increases, the cavitation effect recedes instead. This is because the time of negative pressure phase is shortened at high frequencies, which makes the cavitation bubble unable to fully undergo the growth and expansion process, resulting in smaller radius, shorter collapse process, and lower cavitation intensity. The results also validate the phenomenon that the cavitation effect is weakened when the ultrasonic frequency exceeds a certain threshold. It can be seen from Fig. 1(b)–(d) that the higher the ultrasonic frequency, the smaller the maximum pressure and energy inside the bubble, which also provides evidence for the decrease of cavitation effect when the ultrasonic frequency increases. With the increase of ultrasonic frequency, the minimum values of pressure and temperature inside the bubble increase. Similarly, due to multiple factors, there is no
Fig. 3. Bubble radius in different ultrasonic modes.
Fig. 4. Pressure inside the bubble under different ultrasonic modes.
Fig. 5. Number of water vapor molecules inside the bubble under different ultrasonic modes.
3. Results and discussions 3.1. Cavitation dynamic evolution for single-frequency Fig. 1 shows the variation of bubble radius, pressure, energy,
6
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4. Conclusions
obvious variation rule in the maximum temperature and the minimum energy inside the bubble. Fig. 1(e) shows the variation of the number of water vapor molecules inside the bubble with time, which indicates that the higher the ultrasonic frequency, the less the number of water vapor molecules. This is because the higher the ultrasonic frequency, the shorter the time of cavitation bubble under negative pressure, the shorter the duration of low pressure inside the bubble, and the shorter the time of high evaporation rate of water vapor molecules, resulting in less water vapor molecules accumulated in the bubble. In the mathematical model presented in this paper, the maximum pressure and temperature inside the bubble can reach up to thousands of MPa and thousands of K in the ultrasonic field as seen from Fig. 1(b) and (d). The maximum values of the bubble physical quantities under the ultrasonic field are extracted and filled in Table 2.
In this paper, based on Kyuichi Yasui's equation, a dual-frequency ultrasonic bubble dynamics model considering thermal effect is established. The numerical solution is obtained by the fourth-order RungeKutta method to clarify the dynamic evolution of bubble radius, pressure, energy, temperature, and number of water vapor molecules inside the bubble. The results of dual and single frequency modes are compared under the same ultrasonic power. The conclusions are as follows: (1) The state parameters of cavitation bubbles in single-frequency ultrasound field exhibit periodic oscillation. With the increase of ultrasonic frequency, the oscillation period of cavitation bubbles is shortened and the cavitation effect is weakened, which is represented by the decrease of bubble radius, pressure, energy, and number of water vapor molecules inside the bubble. In this paper, the maximum pressure and temperature in single-frequency ultrasound field can reach 1176.1 MPa and 1504.8 K, which illustrates the high energy of a cavitation bubble. (2) Dual-frequency ultrasound can destroy the stable oscillation and accelerate the collapse of the bubble. Under the same ultrasonic power, the bubble radius and pressure inside the bubble can reach 36.061 μm and 2285.9 MPa, respectively, under the excitation of dual-frequency ultrasound (50 kHz + 70 kHz), which are much larger than 18.183 μm, 730.61 MPa at 50 kHz and 14.576 μm, 332.25 MPa at 70 kHz. The results of three groups of different frequency combination (30 kHz + 50 kHz, 40 kHz + 60 kHz, and 50 kHz + 70 kHz) also indicate that dual-frequency ultrasound can significantly enhance the cavitation effect. Therefore, dual-frequency ultrasound is an effective means in the field of ultrasonic cavitation industry applications.
3.2. Cavitation effect enhanced by dual-frequency The frequency combination of 50 kHz + 70 kHz is selected and compared with single frequency 50 kHz and 70 kHz, respectively. Comparisons of the bubble characteristics are shown in Fig. 2. Fig. 2(a) is the dynamic variation curve of ultrasonic pressure in different frequency modes. It can be seen that the period of ultrasonic pressure in dual-frequency mode of 50 kHz + 70 kHz becomes 100 μs, which is much larger than that in corresponding single-frequency mode. The change of ultrasonic pressure during one cycle in dual-frequency mode is more complex. The bubble collapses in less than one cycle as seen from Fig. 2(b)–(f). The periodic dynamic evolution of cavitation bubbles in single-frequency ultrasound field is relatively stable and has the characteristics of steady-state cavitation. Under the irradiation of dual-frequency ultrasound, the dynamic evolution of cavitation bubbles has changed significantly, and the physical quantities exhibit special changes that a larger oscillation occurs and then bubble collapses after two relatively small oscillations. Cavitation bubbles are more unstable and easy to collapse under the disturbance of dual-frequency ultrasound, so dual-frequency ultrasound is conducive to destroying the stable oscillation of cavitation bubbles and accelerating their collapse. As shown in Fig. 2(b), under the same ultrasonic power, the maximum bubble radius can reach 36.061 μm under dual-frequency ultrasound excitation, which is much larger than 18.183 μm at 50 kHz and 14.576 μm at 70 kHz. In dual-frequency mode, the pressure inside the bubble reaches a maximum of 2285.9 MPa immediately before bubble collapse, which is much higher than 730.61 MPa at 50 kHz and 332.25 MPa at 70 kHz from Fig. 2(c). These results illustrate that the dual-frequency mode intensifies the cavitation effect. After bubble collapse, the pressure and energy inside the bubble decrease sharply. The minimum values of pressure and temperature inside the bubble in dual-frequency mode are lower than those in single-frequency mode, and the maximum number of water vapor molecules inside the bubble is much larger. Then, different frequency combinations are selected under the same ultrasonic power: 30 kHz + 50 kHz, 40 kHz + 60 kHz, and 50 kHz + 70 kHz. The maxima of bubble radius, pressure and number of water vapor molecules inside the bubble in dual-frequency mode are obtained. The results are compared with those of single-frequency mode, and the histograms are drawn, as shown in Figs. 3–5, respectively. It can be found that the maximum values of bubble radius, pressure and number of water vapor molecules in dual-frequency mode are all larger than those in the corresponding single-frequency mode, which confirms the phenomenon of dual-frequency ultrasound enhancing cavitation. Therefore, dual-frequency ultrasound is a feasible and effective means in the field of ultrasonic cavitation industry applications.
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Numerical study on dual-frequency ultrasonic enhancing cavitation effect based on bubble dynamic evolution
T
⁎
Linzheng Ye , Xijing Zhu, Yao Liu Shanxi Key Laboratory of Advanced Manufacturing Technology, North University of China, Taiyuan 030051, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Ultrasonic cavitation Dual-frequency Cavitation enhancement effect Bubble dynamic evolution
Ultrasonic cavitation is a physical dynamic phenomenon of bubbles inflation, compression, and collapse in liquid. A dual-frequency ultrasonic cavitation dynamics model is established in this paper to investigate dynamic evolution of bubble under single and dual frequency ultrasonic modes. The variation of bubble radius, pressure, energy, temperature, and number of water vapor molecules inside the bubble in single and dual frequency ultrasonic modes are analyzed, respectively. The results show the oscillation of cavitation bubbles is more unstable and easier to collapse in dual-frequency ultrasound field than those in single-frequency ultrasound field. With the increase of the ultrasonic frequency, cavitation effect is weakened due to the shortage of oscillation period. Under the same ultrasonic power, the maximums of bubble radius, pressure, and water vapor molecules number inside the bubble in the dual-frequency mode are larger than those in the single-frequency mode. Under the ultrasonic excited by 50 kHz + 70 kHz, the maximum bubble radius and pressure can reach 36.061 μm and 2285.9 MPa, respectively, which are much larger than 18.183 μm, 730.61 MPa at 50 kHz and 14.576 μm, 332.25 MPa at 70 kHz. The calculation results of three different frequency combinations (30 kHz + 50 kHz, 40 kHz + 60 kHz and 50 kHz + 70 kHz) indicate dual-frequency ultrasound can significantly enhance the cavitation effect.
1. Introduction Cavitation is a process of phase transition between liquid and vapor or at liquid-solid interface caused by hydrodynamics. According to the mode of formation, cavitation can be divided into two categories: energy deposition induced cavitation and pressure drop induced cavitation. Energy deposition induced cavitation includes particle cavitation, photoinduced cavitation, and laser induced cavitation. Laser induces the change of liquid properties in the focusing region to produce bubbles. Based on laser energy densities, bubbles can be divided into vapor bubbles and plasma bubbles [1]. Pressure drop induced cavitation mainly consists of ultrasonic cavitation and hydraulic cavitation, which produce cavitation through ultrasonic pressure load and the special flow structure, respectively, due to the changing of the liquid ambient pressure. It has been reported that ultrasonic cavitation has higher collapse temperature and pressure than those in hydraulic cavitation [2]. Ultrasonic cavitation is a non-linear and complex acoustic phenomenon in liquid medium activated by ultrasound. When the ultrasonic amplitude is larger than the liquid cavitation threshold, the “weak link” in the connections between liquid molecules will be torn, resulting
⁎
in a large number of gas nuclei. The periodic vibration of ultrasonic pressure due to the ultrasound propagation, cause the micro bubbles or nuclei in liquid medium to grow and break [3–6]. Cavitation can be divided into transient cavitation and steady-state cavitation according to their dynamic characteristics. Transient cavitation usually occurs at relatively high pressure, and cavitation bubbles expand at a high speed in the negative pressure phase of ultrasound and collapse rapidly in the positive pressure phase. Steady-state cavitation refers to the non-linear oscillation of cavitation bubbles for several periods under the action of small alternating sound pressure, accompanied by mass diffusion of gases and radiation force. Ultrasonic cavitation is characterized by formation and collapse of bubbles, accompanied by secondary phenomena such as micro-jet and shock wave [7], resulting in local high temperature and pressure. The oscillating motion of cavitation bubbles, especially the extreme conditions such as high temperature and high pressure caused by collapse, can cause a series of effects: (1) High-pressure effect. Bubble collapse produces micro-jet with velocity up to several hundred meters per second, which cause the high impaction pressure to the material surface. The high-pressure
Corresponding author. E-mail address: [email protected] (L. Ye).
https://doi.org/10.1016/j.ultsonch.2019.104744 Received 12 May 2019; Received in revised form 24 July 2019; Accepted 22 August 2019 Available online 22 August 2019 1350-4177/ © 2019 Elsevier B.V. All rights reserved.
Ultrasonics - Sonochemistry 59 (2019) 104744
L. Ye, et al.
effect may form cavitation pits damage on the surface [8,9], which can also promote material removal [10]. (2) High-temperature effect. Cavitation bubbles in liquids usually contain a certain number of gas molecules. In the collapse process, the volume of bubble decreases sharply, resulting in the instantaneous rise of internal gas temperature, which cannot fully exchange with the outside in a short time. These high-temperature gases will act on material surface and form a potential local hightemperature region. (3) Activation effect. Ultrasonic cavitation will induce high temperature and pressure environment during bubble collapse, which will produce strong oxidizing free radical ions, such as H2 O ↔ OH + H , OH ↔ O + H , O + O ↔ O2 , etc. This can accelerate the chemical reaction rate and oxidize the organic matter, realizing ultrasonic decontamination in the liquid [11]. (4) Sonoluminescence effect [12,13]. Under conditions of cavitation collapse, extremely high temperature in the bubble will radiate light pulses. In different solutions, the intensity of sonoluminescence varies greatly.
Table 1 Main parameters. Parameters
Values
Ultrasonic pressure amplitude Pa/Pa Speed of sound in water c/m·s−1 Surface tension of water σ/N·m−1 Viscosity of water μ/Pa·s Universal gas constant Rg/J·mol−1·K−1 Density of water ρ/Kg·m−3 Liquid pressure at infinity P0/Pa
1.3 × 105 1483 7.275 × 10−2 1 × 10−3 8.3145 998.2 1 × 105
sound waves on the secondary Bjerknes force, and found it changed both the sign and the value of the Bjerknes force coefficient for the cases near the boundaries significantly. Moholkar [29] explored the effects of different phase differences on spatially uniform acoustic field, and results showed it had great influence on overall sonochemical effect in the processor. Wrede [31] has achieved the spatial control and induced cavitation of the targeted microbubble through a dual-frequency transducer. Although controlled cavitation under dual-frequency ultrasound has been confirmed in some laboratories, the research is still in infancy and its universality needs to be improved, and its theoretical mechanism needs to be further explored. Cavitation dynamics is the most effective way to understand cavitation effect from a theoretical perspective. Since Rayleigh first established the single spherical cavitation dynamics model in ideal fluid in 1917, the research on the cavitation dynamics has been continuing. The famous cavitation dynamics models include Rayleigh-Plesset model, Keller-Miksis model, and Gilmore model and so on [32]. Kyuichi [33–35] studied ultrasonic cavitation dynamics by building a bubble dynamics model considering the condensation, evaporation of water vapor and thermal effect. Lu [36] carried out thermodynamic analysis and established a heat conduction-radiation model of cavitation. Based on the pressure difference between inside and outside the bubble, Gao [37] obtained a mass transfer model between inside and outside of cavitation bubble and discussed the parameter correlation under the stable equilibrium radius. In 2011, Suslick [38] used the spectral analysis method to quantitatively study the high temperature and high pressure generated during bubble collapse. The temperature in the cavitation bubble was controlled by the vapor pressure and thermal conductivity of the liquid volatiles and gases, respectively. In 2012, Dular [39] experimentally investigated the growth and collapse of a single cavitation bubble using a high-speed thermal imaging camera. The results demonstrated that the temperature of liquid-vapor boundary layer decreases continuously during bubble growth, while the opposite consequence occurs when the bubble collapses, and it is found that evaporation and condensation are the two main factors driving heat conduction. Brujan [40] used a high-speed camera to observe the near-wall cavitation bubbles generated by high-intensity focused ultrasound pulses, and found that the distance between the ultrasonic focusing point and the rigid wall plays a decisive role in the collapse characteristics of cavitation bubbles. The velocity of micro-jet and the pressure of shock wave increase with the increase of the distance in a certain range, and the maximum values are 130 m/s and 0.65 MPa, respectively. In 2016, Cogné [41] established an inertial cavitation model for single cavitation bubble to accurately calculate the temperature and pressure at the instant of bubble collapse near the solid wall. Two heat transfer models between liquid and bubbles were proposed, and the changes of bubble radius, gas temperature, interface temperature and pressure under the two models were compared, and their application ranges were defined, respectively. In this paper, to understand the dynamic evolution and enhancement effect of cavitation in multi-frequency ultrasonic theoretically, a dual-frequency ultrasonic bubble dynamics model is established. This model was built based on Kyuichi's bubble dynamics model and considering heat exchange inside and outside the bubble. Then, the
These characteristics of ultrasonic cavitation make it present good application prospects in the fields of ultrasonic cleaning, surface modification, cancer treatment, petroleum exploitation and so on. Toh [14] and Chen [15] used ultrasonic cavitation for peening and polishing, respectively. In the grinding and modification of semiconductor materials [16], Savkina [17] applied ultrasonic cavitation to surface functionalization of silicon wafer. However, the application of ultrasonic cavitation in industry is not mature, due to the insufficient energy and cavitation shielding [18]. Cavitation shielding is a phenomenon to form cavitation near the end face of the horn and prevents sound energy propagation. At this time, simply increasing the ultrasonic power cannot effectively break the cavitation shielding. To solve this problem, Moholkar [19] adjusted the parameters of two sound sources and found that the bubbles are mainly generated in the liquid far away from the sound source. Therefore, the use of dual-frequency ultrasound can overcome cavitation shielding. In addition, it can avoid the defects of single-frequency ultrasound irradiation, such as inhomogeneous sound field, standing waves, fewer cavitation incidents and so on. Many scholars at domestic and abroad have proved that the dual-frequency ultrasound can improve the sonochemical yield [20,21]. Multi-frequency ultrasound can generate inertial cavitation easier than the single-frequency ultrasound by reducing the threshold. Suo [22] studied the inertial cavitation threshold under multi-frequency ultrasound and built a theoretical model for cavitation threshold under dual-frequency. Guédra [23] used the asymptotic method to study the nonlinear vibration of bubbles under dual-frequency ultrasonic excitation and obtained a larger bubble response amplitude. To quantify the dual-frequency ultrasonic cavitation effect, Ebrahiminia [24] measured the absorbance of potassium iodide dosimeter solution at 350 nm wavelength after ultrasonic treatment, and the results showed that the absorbance of dual-frequency (40 kHz + 1 MHz) mode is about 1.8 times of the algebraic sum of single-frequency phonons. Chen [25] refined the as-cast structure of ZK60 magnesium alloy by single and dual frequency ultrasound respectively, and found that the yield strength, ultimate tensile strength, and elongation of the latter were improved by 20.5%, 20.7%, and 30.0%, respectively, compared with those of the former. This is because dual-frequency ultrasound can produce larger cavitation bubbles and make more bubbles instantaneous, thus improving the refining efficiency. Zhang [26] studied the power spectrum and response curve of oscillating bubbles excited by dual-frequency ultrasound (5.5 MHz and 6.6 MHz), and revealed two unique characteristics of bubble oscillation: combination resonance and simultaneous resonance. In addition, the initial phase and phase difference of dual-frequency are important factors affecting cavitation [19,27–29]. Zhang [30] investigated the influence of phase difference between 2
Ultrasonics - Sonochemistry 59 (2019) 104744
L. Ye, et al.
Fig. 1. Dynamic evolution of bubble under different single-frequency ultrasonic modes. (a) Bubble radius, (b) pressure inside the bubble, (c) energy inside the bubble, (d) temperature inside the bubble, and (e) number of water vapor molecules inside the bubble.
under different ultrasonic modes are studied.
cavitation behaviors under dual-frequency mode (50 kHz + 70 kHz) and single-frequency mode (50 kHz, 70 kHz) are compared under the same ultrasonic power. Finally, three groups of dual-frequency (30 kHz + 50 kHz, 40 kHz + 60 kHz and 50 kHz + 70 kHz) and corresponding single-frequency are selected and their results were contrasted. The dynamic variation rules of bubble radius, pressure, energy, temperature, and number of water vapor molecules inside the bubble
2. Modeling In the actual ultrasonic cavitation, the physical quantities inside and outside the cavitation bubble are always in the dynamic change process. There are evaporation and condensation behaviors of gas-liquid 3
Ultrasonics - Sonochemistry 59 (2019) 104744
L. Ye, et al.
where a and b are Van Der Waals constants, which change with the number of water vapor molecules, Rg is universal gas constant, T is temperature inside the bubble, and ν is molar volume. The values of a and b can be obtained by Eqs. (5) and (6).
Table 2 Maxima of bubble physical quantities in different single-frequency ultrasonic fields. Frequency (kHz)
Rmax (μm)
30 40 50 60 70 80
Emax (J)
Pinmax (MPa)
27.488 21.527 18.183 16.105 14.576 13.851
Tmax (K)
nH2Omax
2
−9
1176.1 1037.8 730.61 441.39 332.25 329.42
1.5767 × 10 1.2766 × 10−9 1.0964 × 10−9 9.5501 × 10−10 8.3229 × 10−10 7.6106 × 10−10
1140.2 1401.3 1504.8 1480.5 1407.1 1357.0
⎜
1.0374 × 10 4.6821 × 1010 2.6834 × 1010 1.7902 × 1010 1.2809 × 1010 1.1294 × 1010
(
=
1 ρ
Ṙ c
+
ṁ cρ
) + R ̇ (1 − 3 2
( 1 + ) {P Ṙ c
¨ mR + ρ
out
(1 −
2
Ṙ 3c
+
2ṁ 3cρ
+
ṁ cρ
) + (R ̇ + ṁ ρ
+
̇ ̇ Rm 2ρc
)+
T=
(
(
(
(
)
Pa =
Rg T
R dPout cρ dt
a ν2
⎜
⎟
(6)
(7)
103NA 2 ̇ Δt 4π me MH2O
(8)
103NA 4πR2ṁ Δt MH2O
(9)
2ρcI
For single − frequency ultrasonic: For dual − frequency ultrasonic:
)
−
⎟
(10)
where Pa is ultrasonic pressure amplitude and I is ultrasonic intensity. In order to compare the dynamic evolution of cavitation under single and dual-frequency, the ultrasonic intensity of single-frequency should be the same as that of dual-frequency.
P2 =
Pa sin(2πf2 t ) 2
P1 = Pa sin(2πf1 t ) P1 =
(11)
Pa sin(2πf1 t ), 2 (12)
Table 1 shows the values of the parameters in this paper. To obtain the motion of the cavitation bubble, the initial conditions of the above equations are also needed. The specific parameters are as follows: R|t=0 = R0, Ṙ t = 0 = 0 , T|t=0 = T0, Pout|t=0 = P0, dPout/ ¨ t = 0 = 0 . The initial radius of bubble R0 is dt|t=0 = 0, and ṁ t = 0 = m 4.5 μm, the initial temperature of water T0 is 293.15 K, the initial numbers of water vapor molecules and air molecules inside the bubble are 2.20415 × 108 and 1.23822 × 1010, respectively. In addition, according to the frequency bands commonly used in sonochemistry and industry, the ultrasonic frequency f is selected as 30, 40, 50, 60, 70, and 80 kHz.
(3)
where Pin is pressure inside the bubble, μ is viscosity of water, and ρa is average density of gas inside the bubble. Pin can be calculated based on the Van Der Waals equation of state:
(ν − b)
⎟⎜
Eq. (1) can be solved and physical parameters, such as R, Pin, T, E, and n H2O all can be obtained. For convenience of study, the effect of initial phase of ultrasound is not considered, so φ1 = φ2 = 0. The relationship between the ultrasonic pressure amplitude and the ultrasonic intensity is as follows [42]:
The external pressure of bubble, Pout, is related to the pressure inside the bubble and liquid properties [33].
Pin (t ) =
(5)
2
⎜
)
4μ ⎛ ̇ 2σ ṁ ⎞ 2⎛1 − 1 ⎞ − ⎜R − ⎟ − ṁ ⎜ ⎟ R R ⎝ ρ⎠ ρa ⎠ ⎝ρ
⎟
NA2 EV + (n H2O + nair ) a (nH2O CV , H2O + nair CV , air ) NA V
n H2O (t + Δt ) = n H2O (t ) +
(2)
Pout (t ) = Pin (t ) −
⎜
where MH2O is molar quantity of water. The first and second terms on the right side of Eq. (8) represent the variation of energy inside the bubble caused by the change of pressure and the number of water vapor molecules inside the bubble during Δt, respectively. The detail of e and ṁ calculation has been given in the literature [6]. The number of water vapor molecules at t + Δt is the sum of the number of the molecules at t and the number of the change during Δt.
)
)
⎟
ΔE (t ) = −Pin (t )ΔV (t ) +
(1) where Ẋ and X¨ are the first and second derivative of X , R is bubble radius, c is the speed of sound in water, ρ is water density, σ is surface tension of water, Pout is the external pressure of bubble, P0 is liquid ¨ indicate velocity and acceleration of water pressure at infinity, ṁ and m vapor evaporation and condensation, P1, P2, f1, f2, φ1, and φ2 are pressure amplitude, frequency and initial phase of ultrasound with different frequencies, respectively. Eq. (1) is a typical second-order nonlinear differential equation, which is hard to obtain the analytical solution directly. Therefore, the fourth-order Runge-Kutta method is used to obtain the numerical solution. For the convenience of programming, Eq. (1) is transformed into Eq. (2).
(
⎟
where NA is Avogadro constant, E is energy inside the bubble, CV, air and CV ,H2O are molar heat capacities at constant volume of air and water vapor, respectively. The energy change inside the bubble during Δt is as follows:
R dPout cρ dt
Ṙ = R2 ⎧ ⎪ Ṙ = R¨ ⎪ 2 ⎪ 1 Ṙ 1 + c {Pout − [P1 sin(2πf1 t + φ1) ⎪ ρ ⎪ ⎪ + P2 sin(2πf2 t + φ2)] − P0} ̇ ̇ ¨ mR Ṙ ṁ ṁ ṁ Rm ⎨ = + ρ 1 − c + cρ + ρ Ṙ + 2ρ + 2ρc + ⎪ ⎪ 2ṁ 3 Ṙ ⎪ − 2 Ṙ2 1 − 3c + 3cρ ⎪ ⎪ ̇ ⎡R 1 − R + ṁ ⎤ ⎪ c cρ ⎣ ⎦ ⎩
⎟⎜
where aair and bair (aH2O and bH2O) are the values of Van Der Waals of air (water vapor), aair - H2O = aair × aH2O , 3 bair - H2O = ( 3 bair + 3 bH2O ) 2 . nair, n H2O , and nt are the numbers of air, water vapor, and total vapor molecules, respectively. The temperature inside the bubble T can be expressed as:
) ṁ 2ρ
⎜
2
⎜
− [P1 sin(2πf1 t + φ1) + P2 sin(2πf2 t + φ2)] − P0}
Ṙ c
⎟
nH O nH O n n b = bair ⎛ air ⎞ + 2bair − H2O ⎛ air ⎞ ⎛ 2 ⎞ + bH2O ⎛ 2 ⎞ n n n ⎝ t ⎠ ⎝ t ⎠⎝ t ⎠ ⎝ nt ⎠
transition and material exchange at the bubble interface. Based on Kyuichi Yasui's work, this paper makes the following assumptions: (1) the cavitation bubble remains spherical with a fixed center and only vibrates radially; (2) the number of air molecules inside the bubble remains constant; (3) the variation of density and sound velocity in liquid is ignored; (4) the surface tension and viscous resistance of liquid cannot be neglected. A single bubble dynamics model considering heat exchange in a dual-frequency ultrasound field is established:
RR¨ 1 −
2
nH O nH O n n a = aair ⎛ air ⎞ + 2aair − H2O ⎛ air ⎞ ⎛ 2 ⎞ + aH2O ⎛ 2 ⎞ ⎝ nt ⎠ ⎝ nt ⎠ ⎝ nt ⎠ ⎝ nt ⎠
11
(4) 4
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Fig. 2. Contrast curves of bubble dynamic evolution in single and dual frequency ultrasonic fields. (a) Ultrasonic pressure, (b) bubble radius, (c) pressure inside the bubble, (d) energy inside the bubble, (e) temperature inside the bubble, and (f) number of water vapor molecules inside the bubble.
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temperature, and number of water vapor molecules inside the bubble under different ultrasonic frequencies. All parameters of cavitation bubble show periodic oscillation. The cavitation bubble is relatively stable in the initial ultrasonic positive pressure phase. When the negative pressure phase arrives, the bubble expands slowly due to the pressure difference between inside and outside the bubble, which induces the pressure, energy, temperature, and water vapor molecules inside the bubble decrease. The increase of the absolute value of ultrasonic pressure accelerates the expansion of the bubble. Correspondingly, the pressure, energy, temperature, and number of water vapor molecules inside the bubble also change faster. The bubble radius reaches the maximum when negative pressure phase transfers into the positive pressure phase, and the pressure, energy, and temperature inside the bubble reach the minimum while the number of water vapor molecules reaches the maximum at the same time. Due to the pressure inside lower than that outside, the bubble shrinks sharply. After the ultrasonic positive pressure phase coming, the bubble radius oscillates under the influence of pressure discrepancy. When the bubble is compressed to the minimum, the pressure, energy, and temperature inside the bubble reach the maximum, while the number of water vapor molecules decreases to the minimum. Similarly, these physical parameters continue to oscillate periodically under unstable conditions. If the bubble collapses at this time, it will release huge energy, produce sonoluminescence, and even may destroy the near wall. The reasons for the dynamic evolution of bubble state parameters are complex. First, water vapor molecules can leave or enter the bubble through the cavity wall by condensation and evaporation. When the water vapor molecule condenses, it changes from gaseous to liquid and enters the liquid environment through the cavity wall, during which heat is released. So the energy inside the bubble increases and the number of water vapor molecules decreases. On the contrary, when the water vapor molecule evaporates, it changes from liquid to gaseous and enters in the bubble, which is an endothermic process and will reduce the energy inside the bubble and increase the number of water vapor molecules. Therefore, the change of number of water vapor molecules inside the bubble represents the gas-liquid phase change of water vapor during the dynamic evolution of bubbles, and also reflects the oscillation behavior of bubbles laterally. Furthermore, it can be seen that the physical quantities of the bubble are coupled and correlated from the above mathematical model. Although the physical quantities present reasonable periodic oscillation law, the time to reach the extreme point is slightly different. For example, energy inside the bubble, E, as shown in Eq. (8), is not only related to the pressure inside the bubble, but also to the number of water vapor molecules. Therefore, the evolution of E is a comprehensive result of pressure and number of water vapor molecules changing inside the bubble. As shown in Fig. 1(a) and (b), with the increase of ultrasonic frequency, the oscillation period of cavitation decreases and the maximum of bubble radius decreases, and the maximum pressure inside the bubble reduces. Bubble radius and pressure inside the bubble can reflect the strength of cavitation effect to a certain extent. The larger the radius and the higher the pressure, the stronger the cavitation effect. Therefore, when the ultrasonic frequency increases, the cavitation effect recedes instead. This is because the time of negative pressure phase is shortened at high frequencies, which makes the cavitation bubble unable to fully undergo the growth and expansion process, resulting in smaller radius, shorter collapse process, and lower cavitation intensity. The results also validate the phenomenon that the cavitation effect is weakened when the ultrasonic frequency exceeds a certain threshold. It can be seen from Fig. 1(b)–(d) that the higher the ultrasonic frequency, the smaller the maximum pressure and energy inside the bubble, which also provides evidence for the decrease of cavitation effect when the ultrasonic frequency increases. With the increase of ultrasonic frequency, the minimum values of pressure and temperature inside the bubble increase. Similarly, due to multiple factors, there is no
Fig. 3. Bubble radius in different ultrasonic modes.
Fig. 4. Pressure inside the bubble under different ultrasonic modes.
Fig. 5. Number of water vapor molecules inside the bubble under different ultrasonic modes.
3. Results and discussions 3.1. Cavitation dynamic evolution for single-frequency Fig. 1 shows the variation of bubble radius, pressure, energy,
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4. Conclusions
obvious variation rule in the maximum temperature and the minimum energy inside the bubble. Fig. 1(e) shows the variation of the number of water vapor molecules inside the bubble with time, which indicates that the higher the ultrasonic frequency, the less the number of water vapor molecules. This is because the higher the ultrasonic frequency, the shorter the time of cavitation bubble under negative pressure, the shorter the duration of low pressure inside the bubble, and the shorter the time of high evaporation rate of water vapor molecules, resulting in less water vapor molecules accumulated in the bubble. In the mathematical model presented in this paper, the maximum pressure and temperature inside the bubble can reach up to thousands of MPa and thousands of K in the ultrasonic field as seen from Fig. 1(b) and (d). The maximum values of the bubble physical quantities under the ultrasonic field are extracted and filled in Table 2.
In this paper, based on Kyuichi Yasui's equation, a dual-frequency ultrasonic bubble dynamics model considering thermal effect is established. The numerical solution is obtained by the fourth-order RungeKutta method to clarify the dynamic evolution of bubble radius, pressure, energy, temperature, and number of water vapor molecules inside the bubble. The results of dual and single frequency modes are compared under the same ultrasonic power. The conclusions are as follows: (1) The state parameters of cavitation bubbles in single-frequency ultrasound field exhibit periodic oscillation. With the increase of ultrasonic frequency, the oscillation period of cavitation bubbles is shortened and the cavitation effect is weakened, which is represented by the decrease of bubble radius, pressure, energy, and number of water vapor molecules inside the bubble. In this paper, the maximum pressure and temperature in single-frequency ultrasound field can reach 1176.1 MPa and 1504.8 K, which illustrates the high energy of a cavitation bubble. (2) Dual-frequency ultrasound can destroy the stable oscillation and accelerate the collapse of the bubble. Under the same ultrasonic power, the bubble radius and pressure inside the bubble can reach 36.061 μm and 2285.9 MPa, respectively, under the excitation of dual-frequency ultrasound (50 kHz + 70 kHz), which are much larger than 18.183 μm, 730.61 MPa at 50 kHz and 14.576 μm, 332.25 MPa at 70 kHz. The results of three groups of different frequency combination (30 kHz + 50 kHz, 40 kHz + 60 kHz, and 50 kHz + 70 kHz) also indicate that dual-frequency ultrasound can significantly enhance the cavitation effect. Therefore, dual-frequency ultrasound is an effective means in the field of ultrasonic cavitation industry applications.
3.2. Cavitation effect enhanced by dual-frequency The frequency combination of 50 kHz + 70 kHz is selected and compared with single frequency 50 kHz and 70 kHz, respectively. Comparisons of the bubble characteristics are shown in Fig. 2. Fig. 2(a) is the dynamic variation curve of ultrasonic pressure in different frequency modes. It can be seen that the period of ultrasonic pressure in dual-frequency mode of 50 kHz + 70 kHz becomes 100 μs, which is much larger than that in corresponding single-frequency mode. The change of ultrasonic pressure during one cycle in dual-frequency mode is more complex. The bubble collapses in less than one cycle as seen from Fig. 2(b)–(f). The periodic dynamic evolution of cavitation bubbles in single-frequency ultrasound field is relatively stable and has the characteristics of steady-state cavitation. Under the irradiation of dual-frequency ultrasound, the dynamic evolution of cavitation bubbles has changed significantly, and the physical quantities exhibit special changes that a larger oscillation occurs and then bubble collapses after two relatively small oscillations. Cavitation bubbles are more unstable and easy to collapse under the disturbance of dual-frequency ultrasound, so dual-frequency ultrasound is conducive to destroying the stable oscillation of cavitation bubbles and accelerating their collapse. As shown in Fig. 2(b), under the same ultrasonic power, the maximum bubble radius can reach 36.061 μm under dual-frequency ultrasound excitation, which is much larger than 18.183 μm at 50 kHz and 14.576 μm at 70 kHz. In dual-frequency mode, the pressure inside the bubble reaches a maximum of 2285.9 MPa immediately before bubble collapse, which is much higher than 730.61 MPa at 50 kHz and 332.25 MPa at 70 kHz from Fig. 2(c). These results illustrate that the dual-frequency mode intensifies the cavitation effect. After bubble collapse, the pressure and energy inside the bubble decrease sharply. The minimum values of pressure and temperature inside the bubble in dual-frequency mode are lower than those in single-frequency mode, and the maximum number of water vapor molecules inside the bubble is much larger. Then, different frequency combinations are selected under the same ultrasonic power: 30 kHz + 50 kHz, 40 kHz + 60 kHz, and 50 kHz + 70 kHz. The maxima of bubble radius, pressure and number of water vapor molecules inside the bubble in dual-frequency mode are obtained. The results are compared with those of single-frequency mode, and the histograms are drawn, as shown in Figs. 3–5, respectively. It can be found that the maximum values of bubble radius, pressure and number of water vapor molecules in dual-frequency mode are all larger than those in the corresponding single-frequency mode, which confirms the phenomenon of dual-frequency ultrasound enhancing cavitation. Therefore, dual-frequency ultrasound is a feasible and effective means in the field of ultrasonic cavitation industry applications.
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