To describe the dynamics of the cluster and the bubble in cluster theoretically in an acoustic field, the model to be established includes some assumptions as follows. 1) A bubble cluster is considered as a large spherical drop that contains liquid and a large quantity of microbubbles. 2) The bubble cluster and cavitation bubble in cluster oscillate spherically during expansion and compression, in the case with only radial motion but without taking into account the influence of gravity, or buoyancy or other forces on cavitation bubbles. 3) The cluster size is much smaller than the acoustic wavelength. 4) A finite volume of an unbounded slightly compressible viscous liquid and surface tension are taken into account. 5) Radiation damping is considered. 6) The scattering cross sections of the neighboring bubbles do not overlap with each other. 7) The gas pressure inside the bubbles is spatially uniform for each bubble. 8) We shall study in detail the specific case of an oxygen bubble in water at 20 °C and constant density. 9) Heat transfer, mass transfer, diffusion of water vapor, and chemical reaction inside the bubble is considered. 10) Heat transfer, the effect of chemical reaction on internal energy, is not taken into consideration. So, a simple model describing the oscillations of the cluster from Nigmatulin is used:^[44]

The dynamic equation of cavitation bubble oscillation in the cluster is written as follows:^[45–47]

Equations (1) and (2) are complemented with the following equation of conservation of liquid volume in the cluster (in this paper, the initial bubble radii in cluster are assumed to be equal to each other):

In this paper, both oxygen and water vapor diffuse across the bubble wall during radial bubble motion.

Mass transfer of oxygen:^[48]

The mass transfer of water vapor is calculated from the following equation^[39]

The temperature inside the bubble satisfies the following equation:

The substances inside the bubble will react to each other chemically at high temperature, and thus produce a series of strong oxidants such as free radicals. The concentration of radicals and the concentration of other oxidants are calculated by using chemical kinetic model. In order to illustrate the model, the water molecule is split and thus produces ⋅H and ⋅OH at an extremely high temperature and pressure under micro-bubble cavitation conditions. Here is an example:

Reaction (9) in the forward direction (from left to right) produces ⋅OH, and therefore contributes to the first sum in Eq. (10) (production = A_fT^b_f exp (−C_f/T)[H₂O]). On the other hand, the backward process (from right to left) corresponds to a consumption of ⋅OH and thus contributes to the second term in Eq. (10) (destruction = A_bT^b_b exp (−C_b/T)[⋅H][⋅OH]). Because the cavitation bubble is oxygen bubble, the important chemical reactions related to oxygen in the bubble are shown in Table 1. The parameters (e.g., A_f and A_b) in Eq. (10) are described in Refs. [32,50].

Table 1.

Table 1.

Table 1. Important reactions inside bubble. . Chemical reaction Chemical reaction Chemical reaction Chemical reaction H2O + M = OH + H + M O2 + M = O + O + M H2O + O = OH + OH OH + M = O + H + M H + OH = O + H2 H2O + H = OH + H2 HO2 + H = OH + OH OH + OH + M = H2O2 + M O + OH = H + O2 H + H = H2 H2O + OH = H2O2 + H H + O2 + M = HO2 + M H2O2 + H = H2 + HO2 HO2 + OH = H2O2 + O O + O2 + M = O3 + M OH + O2 = O3 + H OH + O2 = O + HO2 H + O3 = O + HO2 H + O3 = OH + O2 HO2 + O = OH + O2 H2O + HO2 = H2O2 + OH O2 + O2 = O3 + O H2 + HO2 = H2O2 + H HO2 + HO2 = H2O2 + O2 HO2 + H = H2 + O2 M denotes any third body molecules regarded as a catalyst in the reaction.

Table 1. Important reactions inside bubble. .

The mathematical model forms a set of coupled, highly nonlinear and stiff differential equations. Therefore, these equations are solved by using the Runge–Kutta method with help of Matlab.

Ultrasonic degradation of organic pollutants requires chemical reactions, but they must occur on the basis of acoustic cavitation background. During the collapse of acoustic cavitation bubbles, the strong acoustic effects such as transient high temperature and pressure in the bubbles, acoustic microjets and shock waves outside the bubbles are formed. These strong acoustic effects occur mainly inside the cavitation bubble and near the bubble wall.^[51] These strong sound effects are related to the dynamic behavior of the cavitation bubble, so the understanding of the dynamic behavior of acoustic cavitation bubbles is helpful in exploring the ways to improve the degradation of organic pollutants by acoustic cavitation. Figure 1 shows the behavior of a bubble cluster and cavitation bubbles in a cluster at a frequency of 50 kHz and an acoustic pressure amplitude of 1.5 atm (The unit 1 atm = 1.01325 × 10⁵ Pa). The initial cluster radius (R₀) is 1 mm, the number of the cavitation bubbles is N = 300, and the initial cavitation bubble radius (r_0a) in the cluster is 4 μm. In this paper, the driving waveform is the square wave signal (P_us sign(sin (2πft))). In Fig. 1(a), bubble cluster radius (R) is shown as a function of time. Under the action of driving acoustic pressure (P_t), the cluster radius changes very little. But, at collapse of a bubble cluster, the liquid pressure (P_c) in cluster increases suddenly up to 14.9 times the static ambient pressure, followed by small oscillations due to the very small bounces of cluster radius (see Fig. 1(b)). The oscillation of liquid pressure in cluster also causes the cavitation bubbles in cluster to oscillate (see Fig. 1(c)). The dash-dotted line (left scale, in Fig. 1(c)) represents the relationship between the radius of the cavitation bubble and time. It shows that the radius of cavitation bubble is almost unchanged in the positive half (0–10 μs) of the first acoustic cycle (20 μs). In the negative half cycle (10–20 μs), cavitation bubble begins to expand till its maximum value. Then, in the positive half of the second acoustic cycle, the bubble is compressed and collapsed rapidly, and then expanded and collapsed repeatedly. In this period, it produces high temperatures in the previous collapse stages, and soon afterwards the temperature is negligible because the collapse process is relatively weak as shown with the solid line (right scale). In Fig. 1(d), the number of molecules inside a bubble is shown as a function of time. The solid line represents the total number of molecules inside a bubble, the dash-dotted line denotes the number of water vapor molecules (n_H2O) inside a bubble, and the dashed line refers to the number of oxygen molecules (n_O2) inside a bubble. On the one hand it is seen from Fig. 1(d) that the n_H2O changes drastically with time due to the evaporation and condensation. At the slow expansion phase in bubble oscillation, n_H2O increases drastically due to the evaporation because the partial pressure of water vapor (p_v) in the bubble decreases considerably (see Fig. 1(e)). On the other hand, at the collapse of a bubble, n_H2O decreases drastically due to the condensation because p_v increases significantly.^[31] The number of oxygen molecules does not change. Because the time scale for diffusion of oxygen is where r_0a is the initial bubble radius (4 μm) and D_O2 is the diffusion coefficient of oxygen in water (10⁻⁹ m²/s). The time scale for oxygen diffusion is 0.01 s, which is far higher than the time scale of bubble dynamics (1/50 kHz = 20 μs), therefore the transport of oxygen across the bubble can be ignored. In Fig. 1(e), the partial pressure of water vapor (p_v) in the bubble is shown as a function of time with logarithmic vertical axis. At the slow expansion phase in a bubble oscillation, the pressure of water vapor inside the bubble is 3.35 × 10³ Pa, it is the saturated vapor pressure at 20 °C. On the other hand, at the collapse of cavitation bubble, the p_v increases suddenly up to 8.62 × 10⁶ Pa, which is 3 orders of magnitude larger than the saturated vapor pressure at 20 °C, followed by small oscillation due to the small bounces of bubble radius. Its change reflects the variation of the quantity of water vapor inside a bubble (see Fig. 1(d)). Figure 1(f) shows the liquid pressure at the bubble interface as a function of time. At the slow expansion phase in a bubble oscillation, the pressure (P_wall) at bubble wall is almost identical to the liquid pressure (P_c) (see the inset in Fig. 1(f)) in the cluster. On the other hand, at collapse of a bubble, the pressure (P_wall) increases dramatically up to 1.6 × 10⁹ Pa, followed by small oscillations due to the small bounces of bubble radius (see Fig. 1(c)).

Fig. 1.

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	Fig. 1. Dynamic behavior of acoustic cavitation bubble versus time over one cycle of acoustic, when acoustic frequency is 50 kHz, acoustic pressure amplitude is 1.5 atm, number of the cavitation bubbles in cluster is 300, driving waveforms is square wave, duty cycle is 50%, bubble cluster radius is 1 mm, and initial cavitation bubble radius is 4 μm.

In order to explore the effect of acoustic frequency on the behavior of cavitation bubbles and the quantity of strong oxidants produced such as free radicals inside bubbles, in this subsection, the temperature characteristics inside the bubble and variation characteristics of the quantity of strong oxidizing substances produced by the strong acoustic effect are considered, when acoustic pressure amplitude is 1.5 atm, the number of the cavitation bubbles in the cluster is 300, and the acoustic frequencies are 25 kHz, 50 kHz, and 100 kHz, respectively. Equation (10) is a modified Arrhenius formula, which shows that the higher the temperature, the higher the reaction rate is. Acoustic cavitation accelerates chemical reactions due to the high temperature produced by the collapse of cavitation bubbles. So, it is necessary to compare the temperature characteristics of cavitation bubble under the action of different acoustic frequencies.

Figure 2(a) shows the time-dependent center temperatures at 25 kHz (dotted line), 50 kHz (solid line), and 100 kHz (dash line). It is clear that the maximum center temperatures are 7537.1 K (25 kHz), 3913.5 K (50 kHz), and 2025.6 K (100 kHz), respectively. These results show that the higher the acoustic frequency, the lower the temperature generated by the collapse of cavitation bubbles is. The variations of the number of molecules inside bubble with time are shown for ⋅O, ⋅H, ⋅OH, ⋅HO₂, H₂O₂, and O₃ during one acoustic cycle at 25 kHz in Fig. 2(b), 50 kHz in Fig. 2(c), and 100 kHz in Fig. 2(d). These oxidants play a major role in sonochemical reaction. The change of the quantity of oxidants inside a bubble reflects the change of the quantity of oxidants released into surrounding solution from the interior of bubble. It can be seen from the three figures that the higher the frequency, the less the quantity of strong oxidants produced inside the bubble is. It is also shown from the above figures that the number of molecules of each oxidant inside the bubble is nearly constant before the end of the first half of the second acoustic cycle. In other words, the quantity of strong oxidants inside the bubble does not change any more after ending all the collapse phases during over one acoustic cycle. Another common feature is that the strong oxidants produced at the collapse are not released into the bulk solution substantially, which results in the degradation of organic pollutants in the bulk solution, which is dependent on the nature of the organic matter. When the pollutants are volatilized, they will turn into the bubbles during the expansion of the cavitation bubbles. On the one hand, they will be burned off due to high temperature and pressure inside the bubbles. On the other hand, they will be reacted with strong oxidizing substances generated in the bubble, and also, a small part will be reacted with supercritical water generated by the gas–liquid interface. For non-volatile organic compounds (non-vocs), it mainly depends on the quantity of the oxidizing substances released into the bulk solution. So, compared with volatile organic compounds (vocs), the degradation effect of non-vocs is very poor, and the experiment also confirms this theoretical evidence.^[9,18,52] By contrast, at 25 kHz, the number of ⋅H, ⋅O, ⋅OH, and ⋅HO₂ radicals inside bubble decrease, and the quantity and species of oxides dissolving into the surrounding liquid from the interior of the bubble are more than those at 50 kHz and 100 kHz, In conclusion, non-vocs in the bulk liquid are easily degraded at 25 kHz, and vocs are easily degraded at 50 kHz and 100 kHz. Table 2 shows the quantity of oxidants inside the bubble at different frequencies.

Fig. 2.

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Fig. 2. Time-dependent temperature inside bubble (a) at three different acoustic frequencies (25 kHz, 50 kHz, and 100 kHz) and (b) for different oxidants inside bubble at acoustic frequency 25 kHz, and (c) and (d) time-dependent quantity of advanced oxides in different time ranges at acoustic frequencies of (c) 50 kHz and (d) 100 kHz, when acoustic pressure amplitude is 1.5 atm, number of cavitation bubbles in cluster is 300, and duty cycle is 50%.

Table 2.

Table 2.

Table 2. Quantities of oxidants inside bubble at different frequencies. . Species f = 25 kHz f = 50 kHz f = 100 kHz Initial value Stable value Initial value Stable value Initial value Stable value ⋅OH 1.31 × 1015 2.66 × 1012 1.82 × 1012 3.7 × 1011 8.95 × 108 8.94 × 108 ⋅O 3.6 × 1015 4.35 × 1013 2.87 × 1012 9.5 × 1011 8.3 × 106 1.3 × 106 ⋅H 3.5 × 1014 1.17 × 1010 3.64 × 1011 3.80 × 107 2.9 × 106 0.0 HO2 1.9 × 1014 2.2 × 1012 7.2 × 1012 8.1 × 1011 8.9 × 108 8.92 × 108 H2O2 1.14 × 109 5.34 × 1014 1.2 × 107 8.3 × 1011 8.0 × 103 3.9 × 105 O3 2.83 × 109 7.48 × 1014 1.35 × 107 2.45 × 1012 2.0 × 102 7.0 × 106

Table 2. Quantities of oxidants inside bubble at different frequencies. .

In Fig. 3(a), we present the changes of temperature inside the bubble for oxygen bubble in water at a frequency of 25 kHz and acoustic pressure amplitudes of 1.2 atm (dash line), 1.5 atm (solid line), 1.7 atm (dotted line). The maximum central temperatures reach 4405.9 K (1.2 atm), 7537.1 K (1.5 atm), and 9557.8 K (1.7 atm), respectively, and the first collapse times are 46.89 μs, 52.03 μs, and 54.66 μs, respectively. Clearly, with the increase of acoustic pressure amplitude, the collapse time delays even at the same frequency. Figures 3(b)–3(d) show the curves for the quantity of strong oxidants inside the bubbles as function of time for pressure amplitudes of 1.2 atm, 1.5 atm, and 1.7 atm, respectively. In Fig. 3(b), due to the small amplitude of acoustic pressure and low temperature inside bubble, the quantity of strong oxidants inside bubble is also small. The number of ⋅HO₂ radicals does not change, the numbers of H₂O₂ and O₃ inside bubble increase, whereas the number of ⋅O and ⋅OH radicals slightly decrease, and only ⋅H radicals are released in large quantity into the bulk solution. All these indicate that it is easy to degrade vocs.

Fig. 3.

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	Fig. 3. (a) Variations of temperature with time at different acoustic frequencies (1.2 atm, 1.5 atm, and 1.7 atm), and variations of quantity of advanced oxidants inside bubble with time at (b) 1.2 atm, (c) 1.5 atm, and (d) 1.7 atm quantity for different oxidants, when acoustic frequency is 25 kHz, number of cavitation bubbles in cluster is 300, and duty cycle is 50%.

Comparing with Fig. 3(b), the quantities of oxidants in Figs. 3(c) and 3(d) are clearly increased, because the temperature inside bubble is increased several times. The figures show that the numbers of H₂O₂ and O₃ molecules also increase inside bubbles, and the numbers of ⋅H, ⋅O, ⋅OH, and ⋅HO₂ radicals decrease significantly. This trend shows that plenty of oxidants are released into the ambient liquid from the interior of the bubble, which is helpful in degrading the non-vocs. Especially the hydroxyl radical has high activity and has no selectivity to decompose organic pollutants.^[3]

The initial and stable quantities of advanced oxidants can be compared with each other at different acoustic pressures as shown in Table 3. The theory shows that the higher the acoustic pressure amplitude, the better the degradation effect is, but the experimental result is not like that.^[53] Perhaps, as the acoustic power increases, a large number of extra bubbles will be created in bulk liquid, and when the bubble number increases to a certain extent the bubbles will cause an acoustic screen effect and affect the degradation effect.^[53] Therefore, when the acoustic frequency is fixed and the physic–chemical properties of pollutants are also considered, it is necessary to choose an optimal acoustic pressure amplitude to avoid too low or too high an acoustic pressure, thereby affecting the treatment effect.

Table 3.

Table 3.

Table 3. Quantities of oxidants inside bubble at different acoustic pressure amplitudes. . Species f = 25 kHz f = 50 kHz f = 100 kHz Initial value Stable value Initial value Stable value Initial value Stable value ⋅OH 6.65 × 1012 1.24 × 1012 1.31 × 1015 2.65 × 1012 4.16 × 1015 3.2 × 1012 ⋅O 1.35 × 1013 7.44 × 1011 3.6 × 1015 5.0 × 1013 3.7 × 1016 5.15 × 1014 ⋅H 1.4 × 1012 1.03 × 108 3.5 × 1014 1.15 × 1010 2.64 × 1015 1.20 × 1011 HO2 1.6 × 1012 1.67 × 1012 1.9 × 1014 2.2 × 1012 1.6 × 1014 1.3 × 1012 H2O2 5.0 × 107 5.05 × 1012 8.0 × 1011 5.35 × 1014 5.0 × 1011 6.93 × 1014 O3 5.1 × 109 1.2 × 1013 9.1 × 1011 7.5 × 1014 4.0 × 1012 4.53 × 1015

Table 3. Quantities of oxidants inside bubble at different acoustic pressure amplitudes. .

In Fig. 4(a), the calculated temperature inside bubble is shown as a function of time during the collapse phase. Included in this figure are the plots of bubble temperature versus time for the number of the cavitation bubbles in cluster N = 500 (dotted line), N = 1000 (solid line), and N = 3000 (dash line), respectively. Their maximum central temperatures reach 5320 K (N = 500), 3547.4 K (N = 1000), and 1706.4 K (N = 3000), respectively, and their first collapses occur at 52.26 μs, 52.63 μs, and 53.58 μs, respectively. Clearly, with the increase of the number of cavitation bubbles in cluster, the occurrence of collapse delays even at the same frequency and acoustic pressure amplitude. The possible reason is that the larger the number of cavitation bubbles with the fixed cluster radius, the greater the limitation of expansion and compression of the bubbles will be, which results in the delay of the collapse time of the bubbles and the temperature dropping inside each bubble.^[54]

Fig. 4.

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	Fig. 4. Variations of temperature inside bubble with time for three different numbers of cavitation bubbles in cluster (500, 1000, and 3000), and variations of quantity of advanced oxidants with time for (b) 500, (c) 1000, and (d) 3000 cavitation-bubbles in cluster, when acoustic frequency is 25 kHz, acoustic pressure amplitude is 1.5 atm, and duty cycle is 50%.

Figures 4(b)–4(d) show the quantities of strong oxidants inside bubble for the three different numbers of cavitation bubbles (N = 500, N = 1000, and N = 3000), respectively. It can be seen from the three curves that the higher the number of the cavitation bubbles in cluster, the less the quantity of oxidants (such as free radicals) produced inside bubble is. A large quantity of the oxidants are created at the collapse because the bubble temperature increases up to 5320 K (for the case N = 500), while relatively a small quantity of oxidants are created because the bubble temperature increases only up to 1706 K (N = 3000). It is seen that the quantity of the oxidants created inside bubble is correlated with the bubble temperature at collapse. The three figures show that the numbers of H₂O₂ and O₃ molecules also increase inside bubble, while the number of ⋅HO₂ radicals becomes nearly constant, only ⋅H radical is released in large quantity into the bulk solution. In Fig. 4(b), the number of ⋅O and ⋅OH radicals inside bubble decrease slightly, while the number of ⋅O and ⋅OH radicals in bubble are nearly unchanged as indicated in Figs. 4(c) and 4(d). By compare with Fig. 3(c), it can be seen that the smaller the number of cavitation bubbles in cluster, the more the quantity and species of the oxidants released into the bulk liquid is and the better the degradation of non-vocs.