An alternative technique for determining the number density of acoustic cavitation bubbles in sonochemical reactors

Graphical abstract

Highlights

• •A novel technique is developed for determining the active bubbles’ number in sonochemical reactor.
• •The technique is based on determining CCl₄ conversions per bubble and in the whole solution.
• •An advanced cavitation model was applied for predicting the single bubble sono-conversion.
• •The number density of bubbles increased monotonically with ultrasound frequency increase.
• •High level of consistency between our findings and those found in the literature was obtained.

Abstract

The present paper introduces a novel semi-empirical technique for the determination of active bubbles’ number in sonicated solutions. This method links the chemistry of a single bubble to that taking place over the whole sonochemical reactor (solution). The probe compound is CCl₄, where its eliminated amount within a single bubble (though pyrolysis) is determined via a cavitation model which takes into account the non-equilibrium condensation/evaporation of water vapor and heat exchange across the bubble wall, reactions heats and liquid compressibility and viscosity, all along the bubble oscillation under the temporal perturbation of the ultrasonic wave. The CCl₄ degradation data in aqueous solution (available in literature) are used to determine the number density through dividing the degradation yield of CCl₄ to that predicted by a single bubble model (at the same experimental condition of the aqueous data). The impact of ultrasonic frequency on the number density of bubbles is shown and compared with data from the literature, where a high level of consistency is found.

Nomenclature

A_f (A_r)

Pre-exponential factor of the forward (reverse) reaction, [(cm³ mol⁻¹ s⁻¹) for two body reaction and (cm⁶ mol⁻² s⁻¹) for three body reaction].

b_f (b_r)

Temperature exponent of the forward (reverse) reaction.

c

Speed of sound in the liquid medium, (m s⁻¹).

E_af (E_ar)

Activation energy of the forward (reverse) reaction, (cal mol⁻¹).

f

Frequency of ultrasonic wave, (Hz).

I_a

Acoustic intensity of ultrasonic irradiation, (W m⁻²).

k_f (k_r)

Forward (reverse) reaction constant, [(cm³ mol⁻¹ s⁻¹) for two body reaction and (cm⁶ mol⁻² s⁻¹) for three body reaction].

p

Pressure inside a bubble, (Pa).

p_max

Maximum pressure inside a bubble (Pa).

p_∞

Ambient static pressure, (Pa).

P_A

Amplitude of the acoustic pressure, (Pa).

P_v

Vapor pressure of water, (Pa).

P_g0

Initial gas pressure, (Pa).

R

Radius of the bubble, (m).

R_max

Maximum radius of the bubble, (m).

R₀

Ambient bubble radius, (m).

t

Time, (s).

T

Temperature inside a bubble, (K).

T_max

Maximum temperature inside a bubble, (K).

T_∞

Bulk liquid temperature, (K).

x_i

Solubility (in mole fraction) of the gas i in water.

y_H2O

Mole fraction of water vapor trapped at the collapse.

Greek letters

γ

Specific heat ratio (c_p/c_v) of the gas mixture.

σ

Surface tension of liquid water, (N m⁻¹).

$\rho$

Density of liquid water, (kg m⁻³).

$\lambda$

Gas thermal conductivity (W m⁻² K).

1. Introduction

The irradiation of a solution with ultrasonic waves causes acoustic cavitation, a transitory phenomenon that increases chemical activity [1]. Through the growth of previously formed nuclei during the alternating expansion and compression cycles of ultrasonic waves, acoustic cavities are obtained [2]. As a result, at collapse, peculiar conditions are created, where a temperature as high as 5000 K and pressure exceeding 1000 atm are believed to exist within the acoustic bubble (hot spot) [3], [4], [5], [6], [7]. These tremendous conditions lead to the thermolysis of water vapor and non-condensable gas present inside the hot spots, therefore, a variety of radicals (e.g. ^●OH, HO₂^● and H^●) and reactive species (e.g·H₂O₂) are created [8]. These species are able to initiate other secondary chemical reactions and emit light [9], [10], [11]. The sonochemical process has found a variety of applications in different fields such as the synthesis of nanomaterials, polymers, degradation of organic pollutants, biomedical and food science applications, etc. [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. On the other hand, several works have been oriented towards the enhancement of our understanding regarding the effect of the different operational conditions (i.e. frequency, acoustic intensity, liquid temperature and viscosity, irradiation mode…etc.) on the acoustic bubble activity as well as the efficacy of sonochemical reactors [23], [24], [25], [26], [27], [28], [29], [30].

In the last two decades, a number of theoretical and experimental (and semi-empirical) works [26], [31], [32], [33], [34] have been focused on the different characteristics of bubbles population (i.e. distribution of acoustic bubbles and number density). To illustrate, Lee et al. [35] have used a pulsed sonoluminescence technique in order to determine the size distribution of SL bubbles (at 515 kHz) by using the total dissolution time of bubbles and Epstein-Plesset equation. The same method has been adopted by Brotchie et al. [36] to correlate the ultrasound pulse separation with the ambient bubble radius in absence of ultrasound in a dual-frequency system (355 and 20 kHz). It has been demonstrated that the pulsed mechanism increases the bubble density, improving coalescence rates and increasing the bubble size, contrary to the continuous operation, which has the opposite effect. Furthermore, Brotchie et al. [36] have indicated the strong correlation between the relative coalescence extent and the enhancement in sonochemical reaction rates in a dual-frequency system. On the other hand, Burdin et al. [37] have employed two laser techniques (Laser diffraction (LD) and phase Doppler (PD)) in order to evaluate the effect of acoustic power on the mean radii and the void fraction of bubbles population at 20 kHz. The determination of the number of active bubbles (number density) according to the experimental works of Labouret and Frohly [38], [39] and Iida et al. [40] is based on the evaluation of void (total volume of bubbles) fraction. In the experimental study of Iida et al. [40] the void rate is determined via a capillary system, whereas, the size distribution of active bubbles is evaluated by a pulsed laser diffraction method. Labouret and Frohly [38], [39] determined the void fraction through electromagnetic resonance; therefore, the size distribution of bubbles is obtained from the analysis of its void rate dissipation evolution. Sivakumar and Gedanken [41] have linked the sonochemical decomposition of volatile Fe(CO)₅ within a single bubble to the overall conversion yield obtained in decalin solvent at 20 kHz sonication. The single bubble yield was calculated assuming 100% conversion of all Fe(CO)₅ molecules present within the bubble at it maximum size (R_max, taken as 150 µm), where the ideal gas-law was used in these calculation. Following this approach, the obtained number of collapsing bubbles was 7.75 × 10⁵ s⁻¹ [41]. However, the Sivakumar and Gedanken method enclosed limitations concerning the assumption of 100% conversion of Fe(CO)₅ in the single bubble, the non-considering of the impact of mass and heat transfer as well as the reaction heat on the single bubble conversion, in addition to the fact that their results are for decalin solvent (at 40 kHz only). On the other hand, through the semi-empirical work of Merouani et al. [42], the production rate of active bubbles’ number is determined based on the determination of the molar yield of H₂O₂, ^●OH and HO₂^● of a single bubble in addition to the production rate of H₂O₂ measured experimentally (in solution). Recently, Kerboua et al. [43], [44] have developed a theoretical method for the evaluation of number density (and void fraction), where their model is built on an energetic analysis of the macroscopic (control volume) and microscopic (acoustic bubble) systems under the action of ultrasonic perturbation.

Carbon tetrachloride is a highly volatile and hazardous chemical that has been efficiently degraded by sonication in aqueous solution over a wide range of operating frequency [45], [46], [47], [48], [49], acoustic intensity [50], [51], [52], [53], bulk solution temperature [51], [52], [53], [54], solution pH [53], saturating gases [52], static pressure [55], initial CCl₄ concentration [53], [56], [57] and presence of other contaminants [57], [58]. Degradation experiments confirm that pyrolysis/combustion of CCl₄ within the acoustic bubbles is the main degradation pathway of CCl₄; conclusion obtained via the analysis of the degradation by-products (e.g.: :CCl₂, C₂Cl₄, C₂Cl₆, HOCl, HCl, etc.[56], [48]) in addition to the fact that the addition of an excess of alcohol (efficient ^●OH-scavenges in the gas phase of the bubble) did not affect the degradation rate of CCl₄, over a wide range of operating ultrasound frequency [45], [46], [47], [48], [49]. In fact, the reactivity of CCl₄ toward hydroxyl radical is negligible [49], [59]. Recently, our research group has published two interesting computation works on CCl₄ degradation by ultrasound [60], [61], where a new reaction scheme for CCl₄ has been used to identify the main reactive chlorine species (RCS: ^●CCl₃, :CCl₂ and ^●Cl and HOCl) resulted from the pyrolysis of CCl₄ within a single acoustic bubble. According to the results of these works [60], [61], CCl₄ degradation within the bubble not only produces RCS but also accelerates the generation of hydroxyl radical through its strong scavenging effect on H^● radical (CCl₄+H^● → HCl+^●CCl₃).

In the present work, a new technique is proposed for evaluating the active bubble number generated within a sonochemical reactor per unit volume and unit time (i.e. number density). This method is based on the linkage between the single bubble chemistry (resulting from CCl₄ pyrolysis) to that developed inside the sonoreactor (in solution). The single bubble sonochemistry model developed early for CCl₄ has been used for the prediction of the single bubble degradation of CCl₄, under some experimental data revealing the degradation of CCl₄ in aqueous solutions (principally the works of Pétrier and Francony [62] and of Hung and Hoffmann [48] which provided the effect of frequency on the sonochemical degradation of carbon tetrachloride). A qualitative and quantitative comparison has been made between our results and those found in the literature.

2. Theoretical package

2.1. The single bubble system

For the simulation of the bubble dynamics and the internal bubble chemistry, the mathematical model developed early by our research group [60], [61] for studying the CCl₄ sonochemisty is used. A detailed discussion about these reaction schemes is available in [60], [61]. The model is based on a set of ordinary differential equations that account for non-equilibrium evaporation and condensation of water vapor at the bubble wall, heat transfer between the bubble interior and the surrounding liquid, chemical reactions heat, and CCl₄ pyrolytic reactions. Table 1 summarizes the equations that govern the model. All numerical simulations were carried out for a single bubble oscillating in argon or oxygen-saturated water containing varying concentrations of carbon tetrachloride. The bubble–bubble interactions are ignored. This assumption has been made due to the complicated nature of multibubble systems (clusters) which makes the process modeling more and more complex. The bubbles interaction in sonicating medium may increase the coalescence of bubbles or reducing their intensity of implosion, which make the sonochemical process less efficient. The involvement of this phenomenon in computation will be studied in future project. However, many leading research groups in sonochemistry have adopted the single bubble approach (without bubbles interaction) for understanding the overall observed sonochemical effects (sonoluminescence and sonochemistry) in aqueous solutions, with relation to influencing factors [42], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79]. Table 2, Table 3 show the reaction mechanism used to study the internal bubble chemistry for an Ar-CCl₄-bubble (Table 2, 31 reversible reactions) and an O₂-CCl₄-bubble (Table 3, 29 reversible reactions). A detailed discussion about these reaction schemes is available in [60], [61]. Table 1 contains the following main equations:

• 1.Eq. (1) (the modified Keller-Miksis equation [80]) describes the radial dynamics, R(t), of the bubble during its oscillation in compressible medium (water) saturated with Ar or O₂ in presence of a determined concentration of CCl₄.
• 2.Eqs. (3) and (4) provide the internal bubble pressure and temperature during oscillation.
• 3.Eq. (5) (the Hertz-Knudsen formula [81]) describes the mass flux, dm/dt, of water evaporation and condensation at the bubble-solution interface.
• 4.Eqs. ((6)–(8)) (heat dissipation by conduction [75]) describe the heat exchange dQ/dt inside and outside the bubble during oscillation. According to Eq. 6, the heat exchange is given as a function of the interior bubble temperature and the bubble surface temperature, which is assumed constant (T_S = T_liq). This approach has been adopted from the work of Toegel et al. [75] (i.e. which is one of the pioneers in sonoluminescence and sonochemistry fields) and it is largely used in several theoretical studies giving interesting results in sonochemistry [60], [65], [67], [75], [79], [82], [83]. This assumption is supported by the short lifetime of the bubble and the very short collapsing time especially at higher ultrasound frequencies, which is the case of our study (200–1078 kHz). For example at 205 kHz (I_n = 1.48 W/cm²), the collapse time is 1.24 µs, representing 24.8% of the whole bubble lifetime (5 µs). At 1078 kHz (I_n = 1.48 W/cm²), the collapse time is 0.24 µs, giving 26.08% of the bubble lifetime (0.92 µs), which is already very short. Therefore, the use of a constant temperature on the liquid shell of the bubble will not largely affect the results of our numerical simulations.
• 5.Eq. (9) describes the internal bubble energy change with time.
• 6.Eqs. ((10)–(15)) describe the change, with time, in quantities of H₂O and all other species ‘k’ within the bubble during oscillation.

Table 1.

Principal equations of the model (see detail in Refs. [60], [61])*

Bubble dynamics:
$\left(1-\frac{\dot{R}}{C}+\frac{\dot{m}}{C{\rho }_L}\right)R\ddot{R}+\frac{3}{2}{\dot{R}}^2\left(1-\frac{\dot{R}}{3C}+\frac{2\dot{m}}{3C{\rho }_L}\right)=\frac{1}{{\rho }_L}\left(1+\frac{\dot{R}}{C}\right)\left[P_B\left(t\right)-P_{A}sin\left(2\pi f\left(t+\frac{R}{C}\right)\right)-P_{\infty }\right]+\frac{\ddot{m}R}{{\rho }_L}\left(1-\frac{\dot{R}}{C}+\frac{\dot{m}}{C{\rho }_L}\right)+\frac{\dot{m}}{{\rho }_L}\left(\dot{R}+\frac{\dot{m}}{2{\rho }_L}+\frac{\dot{R}\dot{m}}{2C{\rho }_L}\right)+\frac{R}{C{\rho }_L}\frac{d{P}_B}{\mathit{dt}}$ (Eq.1)
- Pressure at the external bubble wall:
$P_B\left(t\right)=P\left(t\right)-\frac{2\sigma }{R}-\frac{4\mu \dot{R}}{R}\left(\text{Eq}.2\right)$
- Bubble pressure and Temperature:
$P\left(t\right)=\frac{n{R}_{g}T}{V-nb}+\frac{a{n}^2}{V^2}\left(\mathit{Eq}.3\right)$
$T=\frac{\left(E+\frac{a{n}^2}{V}\right)}{C_v{n}_t}\left(\mathit{Eq}.4\right)$
Mass transfer (water vapor condensation and evaporation):
$\dot{m}=\alpha \frac{\left\{P_{\mathit{sat}}[R-P_v]\right\}}{\sqrt{\frac{2\pi T_{[R]}{R}_g}{M_{H_{2}O}}}}\left(\mathit{Eq}.5\right)$
Heat transfer (thermal conduction):
$\dot{Q}=4\pi R^2{\lambda }_{\mathit{mix}}\frac{\left(T_{\mathit{liq}}-T\right)}{L_{\mathit{th}}}\left(\mathit{Eq}.6\right)$
$L_{\mathit{th}}=min\left\{\frac{R}{\pi },\sqrt{\frac{R\chi }{\dot{R}}}\right\}$ (Eq.7)
${\lambda }_{\mathit{mix}}={\lambda }_{H_{2}O}\left(T\right)\left(\frac{n_{H_{2}O}}{n_t}\right)+{\lambda }_i\left(T\right)\left(\frac{n_i}{n_t}\right)+{\lambda }_{\mathit{CC}{l}_4}\left(T\right)\left(\frac{n_{\mathit{CC}{l}_4}}{n_t}\right)i=ArorO2\left(\mathit{Eq}.8\right)$
Internal bubble energy:
$\mathrm{\Delta }E=-P\left(t\right)\mathrm{\Delta }V\left(t\right)+4\pi R^2\mathrm{\Delta }t\frac{\dot{m}}{M_{H_{2}O}}{e}_{H_{2}O}+4\pi R^2\mathrm{\Delta }t\lambda \frac{\left(T_{\mathit{liq}}-T\right)}{L_{\mathit{th}}}-\frac{4}{3}\pi R^3\mathrm{\Delta }t\sum _{i=1}^n\mathrm{\Delta }{H}_i{r}_i$ (Eq.9)
Change in species quantities (mol)
- For H2O:
$n_{H_{2}O}\left(T+\mathrm{\Delta }t\right)=n_{H_{2}O}\left(T\right)+4\pi R^2\mathrm{\Delta }t\frac{\dot{m}}{M_{H_{2}O}}+V\mathrm{\Delta }t{\dot{U}}_{H_{2}O}$ (Eq.10)
- For other species k (except Ar):
$n_k\left(T+\mathrm{\Delta }t\right)=n_k\left(T\right)+V\mathrm{\Delta }t{\dot{U}}_k$ (Eq.11)
Where:
${\dot{U}}_k=\frac{1}{V}\frac{d{n}_k}{\mathit{dt}}\sum _{i=1}^I\left({\upsilon }^{″}-{\upsilon }^{\prime }\right)r_i\left(k=1,\dots ,K\right)$ (Eq.12)
$r_i=k_{\mathit{fi}}\prod _{k=1}^K{\left[X_k\right]}^{{\upsilon }_{\mathit{ki}}^{\prime }}-k_{\mathit{ri}}\prod _{k=1}^K{\left[X_k\right]}^{{\upsilon }_{\mathit{ki}}^{″}}$ (Eq.13)
$k_{\mathit{fi}}=A_{\mathit{fi}}{T}^{b_{\mathit{fi}}}exp\left(-\frac{E_{a_{\mathit{fi}}}}{R_{g}T}\right)\left(\mathit{Eq}.14\right)$
$k_{\mathit{ri}}=A_{\mathit{ri}}{T}^{b_{\mathit{ri}}}exp\left(-\frac{E_{a_{\mathit{ri}}}}{R_{g}T}\right)\left(\mathit{Eq}.15\right)$

*Variables description: dots denoted here time derivative (d/dt), R is the bubble radius, C is the sound speed in the medium (water), ρ_L is the liquid density, $\dot{\text{m}}$ is the net rate of evaporation per unit area and unit time and P_∞ is the ambient static pressure. P_A is the acoustic amplitude (linked to the acoustic intensity I_a by: P_A=(2I_a${\rho }_L$C)^1/2). P_B(t) is the liquid pressure at the liquid side of the bubble, P(t) is the pressure inside the bubble, σ is the surface tension, μ is the liquid viscosity. f is the sound frequency, P_v is the vapor pressure within the bubble, a and b (in Eqs. (3) and (4)) are the Van de Waals constants (given in [95]). R_g is the universal gas constant, V is the volume of the bubble [V = 4/3(πR³)], T is the temperature inside the bubble. E is the internal bubble energy, P_sat[R] is the saturated vapor pressure (calculated by using Antoine’s equation) at the interface temperature T_[R] = T_liq. M_H2O is the molecular weight of water vapor and ‘α’ is the accommodation coefficient (given in [69]). λ_mix, $\chi$ and ${\mathrm{L}}_{\mathrm{t}\mathrm{h}}$ are the heat conductivity, thermal diffusivity of the gas mixture and the thickness of the thermal boundary layer, respectively [Individual λ_i of gases[95], [96]: λ_H2O(T) = 9.967213 × 10⁻⁵T − 1.1705 × 10⁻², λ_Ar(T) = 3.5887 × 10⁻⁵T + 6.81277 × 10⁻³, λ_CCl4(T) = 3.0 × 10⁻⁵T-2.2 × 10⁻³ and λ_O2(T) = 6.4478 × 10⁻⁵T + 7.2211 × 10⁻³, χ = λ_mix/Cp]. C_p is the heat capacity concentration (J m⁻³ K⁻¹) for H₂O, CCl₄, Ar, or O₂ mixture (given in [75]). C_v is the molar heat of gases and vapor in the bubble [C_v = (3/2)R_g for monoatomic gases (Ar, H, …), (5/2)R_g for diatomic gases (O₂, Cl₂, …) and (6/2)R_g for triatomic gases]. ΔH_i and r_i are the enthalpy change and the rate of the i^th reaction, respectively, and e_H2O is the energy transported by 1 mol of an evaporating or condensing water vapor [e_H2O = C_v,H2OT]. ${\dot{\text{U}}}_{\text{k}}\text{and}{\dot{\text{U}}}_{{\text{H}}_{\text{2}}\text{O}}$ are the production rate of H₂O and k^th species within the bubble.

Table 2.

Scheme of the possible chemical reactions inside a collapsing Ar-bubble in the presence of CCl₄[76], [97], [98], [99]. M is the third Body. Subscript “f” denotes the forward reaction and “r” denotes the reverse reaction. A is in (m³ mol⁻¹ s⁻¹) for two body reaction [(m⁶ mol⁻² s⁻¹) for a three body reaction], and E_a is in (kJ mol⁻¹) and ΔH in (kJ mol⁻¹).

n°	Reaction	Af	nf	Eaf	Ar	nr	Ear	ΔH
1	H2O + M ⇌ H●+●OH + M	1.912 × 107	−1.83	28.35	2.2 × 1010	−2.0	0.0	508.82
2	●OH + M ⇌ O + H●+M	9.88 × 1011	−0.74	24.43	4.714 × 106	−1.0	0.0	436.23
3	O + O + M ⇌ O2 + M	4.515 × 1011	−0.64	28.44	6.165 × 103	−0.5	0.0	505.4
4	H●+O2 ⇌ O+●OH	1.915 × 108	0.0	3.93	5.481 × 105	0.39	−7.01 × 10−2	69.17
5	H•+O2 + M ⇌ HO2● +M	1.475	0.6	0.0	3.09 × 106	0.53	11.7	−204.80
6	O + H2O ⇌ ●OH+●OH	2.97	2.02	3.21	1.465 × 10−1	2.11	−6.94 × 10−1	72.59
7	HO2●+H● ⇌ H2 + O2	1.66 × 107	0.0	1.97 × 10−1	3.164 × 106	0.35	13.3	−239.67
8	HO2●+H● ⇌ ●OH+●OH	7.079 × 107	0.0	7.06 × 10−2	2.027 × 104	0.72	8.8	−162.26
9	HO2●+O ⇌ •OH + O2	3.25 × 107	0.0	0.0	3.252 × 106	0.33	12.75	−231.85
10	HO2●+●OH ⇌ H2O + O2	2.89 × 107	0.0	− 1.19 × 10−1	5.861 × 107	0.24	16.53	−304.44
11	H2 + M ⇌ H●+H●+M	4.577 × 1013	−1.4	24.98	1.146 × 108	−1.68	1.96 × 10−1	444.47
12	O + H2 ⇌ H●+●OH	3.82 × 106	0.0	1.9	2.667 × 10−2	2.65	1.17	8.23
13	●OH + H2 ⇌ H●+H2O	2.16 × 102	1.52	8.25 × 10−1	2.298 × 103	1.40	4.38	−64.35
14	H2O2 + O2 ⇌ HO2●+HO2●	4.634 × 1010	−0.35	12.12	4.2 × 108	0.0	2.87	175.35
15	H2O2 + M ⇌ ●OH+●OH + M	2.951 × 108	0.0	11.59	1.0 × 102	−0.37	0.0	217.89
16	H2O2 + H● ⇌ H2O+●OH	2.410 × 107	0.0	9.5 × 10−1	1.269 × 102	1.31	17.08	−290.93
17	H2O2 + H● ⇌ H2 + HO2●	6.025 × 107	0.0	1.9	1.041 × 105	0.70	5.74	−64.32
18	H2O2 + O ⇌ ●OH + HO2●	9.550	2.0	9.5 × 10−1	8.66 × 10−3	2.68	4.45	−56.08
19	H2O2+●OH ⇌ H2O + HO2●	1.0 × 106	0.0	0.0	1.838 × 104	0.59	7.4	−128.67
20	CCl4 ⇌ ●CCl3 + ●Cl	3.236 × 1022	−2.29	288.9	3.388 × 1010	−1.29	−5.9	288.245
21	●CCl3 ⇌ :CCl2+●Cl	1.09 × 109	0.0	228.1	1.0 × 104	0.0	0.0	281.874
22	2●CCl3 ⇌ C2Cl6	4.55 × 106	−1.6	1.5	3.83 × 1026	− 4.5	288	−304.4
23	2●Cl + M ⇌ Cl2 + M	1.0	0.23	− 45.9	3.981 × 107	0.0	198.5	−242.604
24	C2Cl6+●CCl3 ⇌ CCl4 + C2Cl5	7.94 × 105	0	59.9	2.95 × 102	0.8	39.5	34.167
25	C2Cl5 + Cl2 ⇌ C2Cl6+●Cl	2.04 × 105	0.0	9.9	3.51 × 1011	−1.1	64.7	−79.81
26	C2Cl5+●Cl ⇌ C2Cl4 + Cl2	2.45 × 107	0.0	0.0	6.37 × 109	−0.4	195	−184.502
27	●Cl + H2 ⇌ HCl + H●	94.56	1.72	12.837	55.99	1.72	9.09	4.4
28	CCl4 + H● ⇌ HCl+●CCl3	172	1.8	17.14	1.67 × 10−7	3.72	155.36	− 143.4
29	●Cl + H2O ⇌ ●OH + HCl	1.17 × 102	1.67	63.84	4.12 × 10−1	2.12	−5.37	65.514
30	2C2Cl5 ⇌ C2Cl6 + C2Cl4	8.13 × 105	0.0	59.03	3.023 × 102	0.76	38.63	− 264.31
31	Cl2 + H2O ⇌ HOCl + HCl	64.56	0.72	10.8	43.1	1.03	8.7	73.775

Table 3.

Reaction scheme inside a collapsing CCl₄-O₂ bubble [60], [61], [97], [99]. M is the third Body. Subscript “f” denotes the forward reaction and “r” denotes the reverse reaction. A is in (m³ mol⁻¹ s⁻¹) for two body reaction [(m⁶ mol⁻² s⁻¹) for a three body reaction], and E_a is in (KJ mol⁻¹) and ΔH in (kJ mol⁻¹).

	Reaction	Af	nf	Eaf	Ar	nr	Ear	ΔH
1	H2O + M ⇌ H●+●OH + M	1.912 × 107	−1.83	28.35	2.2 × 1010	−2.0	0.0	508.82
2	O2 + M ⇌ O + O + M	4.515 × 1011	−0.64	28.44	6.165 × 103	−0.5	0.0	505.4
3	●OH + M ⇌ O + H●+M	9.88 × 1011	−0.74	24.43	4.714 × 106	−1.0	0.0	436.23
4	H●+O2 ⇌ O+●OH	1.915 × 108	0.0	3.93	5.481 × 105	0.39	−7.01 × 10−2	69.17
5	H●+O2 + M ⇌ HO2● +M	1.475	0.6	0.0	3.09 × 106	0.53	11.7	−204.80
6	O + H2O ⇌ ●OH+●OH	2.97	2.02	3.21	1.465 × 10−1	2.11	−6.94 × 10−1	72.59
7	HO2●+H●⇌H2 + O2	1.66 × 107	0.0	1.97 × 10−1	3.164 × 106	0.35	13.3	−239.67
8	HO2●+H●⇌●OH+●OH	7.079 × 107	0.0	7.06 × 10−2	2.027 × 104	0.72	8.8	−162.26
9	H2 + M ⇌ H●+H●+M	4.577 × 1013	−1.4	24.98	1.146 × 108	−1.68	1.96 × 10−1	444.47
10	O + H2 ⇌ H●+●OH	3.82 × 106	0.0	1.9	2.667 × 10−2	2.65	1.17	8.23
11	●OH + H2 ⇌ H●+H2O	2.16 × 102	1.52	8.25 × 10−1	2.298 × 103	1.40	4.38	−64.35
12	H2O2 + M⇌●OH+●OH + M	2.951 × 108	0.0	11.59	1.0 × 102	−0.37	0.0	217.89
13	H2O2 + O ⇌ ●OH + HO2●	9.550	2.0	9.5 × 10−1	8.66 × 10−3	2.68	4.45	−56.08
14	O3 + M ⇌ O2 + O + M	1.6 × 107	0	0	–	–	–	−96.20
15	CCl4 ⇌ ●CCl3+●Cl	3.236 × 1022	−2.29	288.9	3.388 × 1010	−1.29	−5.9	288.245
16	●CCl3⇌ :CCl2+●Cl	1.09 × 109	0.0	228.1	1.0 × 104	0.0	0.0	281.874
17	2●CCl3 ⇌ C2Cl6	4.55 × 106	−1.6	1.5	3.83 × 1026	− 4.5	288	−304.4
18	2●Cl + M ⇌ Cl2 + M	1.0	0.23	− 45.9	3.981 × 107	0.0	198.5	−242.604
19	C2Cl6+●CCl3 ⇌ CCl4 + C2Cl5	7.94 × 105	0	59.9	2.95 × 102	0.8	39.5	34.167
20	C2Cl5 + Cl2 ⇌ C2Cl6+●Cl	2.04 × 105	0.0	9.9	3.51 × 1011	−1.1	64.7	−79.81
21	C2Cl5+●Cl ⇌ C2Cl4 + Cl2	2.45 × 107	0.0	0.0	6.37 × 109	−0.4	195	−184.502
22	●Cl + H2 ⇌ HCl + H●	94.56	1.72	12.837	55.99	1.72	9.09	4.4
23	CCl4 + H●⇌HCl+●CCl3	172	1.8	17.14	1.67 × 10−7	3.72	155.36	− 143.4
24	●Cl + H2O⇌ ●OH + HCl	1.17 × 102	1.67	63.84	4.12 × 10−1	2.12	−5.37	65.514
25	2C2Cl5 ⇌ C2Cl6 + C2Cl4	8.13 × 105	0.0	59.03	3.023 × 102	0.76	38.63	− 264.31
26	Cl2 + H2O ⇌ HOCl + HCl	64.56	0.72	10.8	43.1	1.03	8.7	73.775
27	COCl2 ⇌ COCl + Cl●	5.71 × 1015	0.0	36.44	1.006 × 102	1.425	−2.196	277.646
28	CCl3● + O ⇌ COCl2 + Cl●	1.4 × 107	0.0	0.198	1.02 × 108	−0.019	51.15	−418.15
29	COCl + Cl●⇌ CO + Cl2	1.3 × 109	0.0	1.67	3.36 × 1010	−0.026	27.57	−169.08

The resolution approach of the set of differential equations shown in Table 1 is described in detail in our earlier work [60], [61]. In order to simulate the bubble chemistry as well as the CCl₄ degradation inside a single bubble under the experimental conditions of Pétrier and Francony [62] and Hung and Hoffmann [48], a collection of ambient bubble radii (R₀) is used. These radii were selected as a function of frequency according to the experimental determinations. The used ambient radii are: 8 µm for 200 and 205 kHz [84], 3.2 µm for 358 kHz [85], 3 µm for 500 kHz [35], 2.9 µm for 618 kHz [85], 2.7 µm for 800 kHz [85] and 2 µm for 1078 kHz [85]. In reality, bubbles population in the acoustical field is an interval rather than a single value; which is well proven experimentally through several techniques [37], [38], [84], [85], [86], [87] and theoretically through different models [24], [37], [38], [63], [77], [84], [85], [86], [87]. This means that bubbles of different ambient radii (not a single radius) are present in the cavitating medium. However, these intervals that have Gaussian-curves form are rather narrow and are enclosed around a single bubble radius, which is known as the mean ambient bubble size of active bubbles. The ambient bubble radii provided above (used through our simulation) represents the mean ambient bubble radii observed experimentally.

2.2. The number density of active bubbles

Our technique is simply based on the combination of a single bubble system (microscopic system) to the whole sonochemical reactor (macroscopic system). This linkage is ensured via carbon tetrachloride sono-degradation. Due to its high volatility, the decomposition of carbon tetrachloride is expected to take place only within the bubble [46], [56]. Based on this important property, we build our method. If the initial rate of CCl₄ breakdown (in mol L⁻¹ s⁻¹) inside the sonoreactor (solution/macroscopic system) is known, therefore, the determination of the amount (in mol) of CCl₄ decomposed inside a single bubble (determined via the model) enables us to determine the production rate of active bubbles “N (in L⁻¹ s⁻¹)” (i.e. the number density) in the whole reactor as follow:

$$N\left(L^{-1}{s}^{-1}\right)=\frac{\mathit{Initial}rateofCC{l}_{4}decomposition(mol{L}^{-1}{s}^{-1})[inthesonoreactor]}{\mathit{Conversion}ofCC{l}_4\left(\mathit{mol}\right)[insidea\text{single}bubble]}$$ (16)

The application of this relation is conditioned by some assumptions, given below:

• (i)It is supposed that acoustic bubbles are fragmented during one acoustic cycle (at the first collapse when the R_min is attained).
• (ii)Because of CCl₄′s higher volatility (91 mmHg at 20 °C) and poorer solubility in water (5.2 mM at 20 °C), most of CCl₄ molecules are located within the bubble at t = 0. Therefore, the condensation and evaporation of CCl₄ at the bubble wall is neglected.
• (iii)The presence of CCl₄ within the bulk liquid has no effect on the range of active bubbles, which can be ensured by the use of low concentrations of CCl₄ in the irradiated solution [61]. It should be noted that according to our previous paper [61], the effect of CCl₄ on the range of active bubbles is remarkably reduced (or eliminated) with the decrease of its concentration in the solution or with the increase of ultrasound frequency.

It should be noted that the obtained results (number density) found in the present study are fundamentally based on the experimental works of Hung and Hoffmann [48] and Pétrier and Francony [62], who investigated the degradation of CCl₄ over a wide range of frequency of ultrasound. The initial rates of CCl₄ degradation (mol L⁻¹ s⁻¹) inferred from these studies are regrouped in Table 4, Table 5(a) and (b). The experimental work of Pétrier and Francony [62] is conducted under an atmosphere of oxygen, whereas, the type of saturating gas was not specified in the work of Hung and Hoffmann [48]. Nevertheless, the predominance of C₂Cl₆ and C₂Cl₄ upon the sonolysis of carbon tetrachloride indicates the possibility of using argon (during CCl₄ sonolysis) as a dissolved gas in our study in the purpose to simulate the experimental findings obtained by Hung and Hoffmann [48]. This choice is supported by the experimental results obtained by Hua and Hoffmann [56] for the decomposition of CCl₄ in presence of argon, where the main species resulting from CCl₄ decomposition are chloride ion, Hypochlorous acid and the low concentrations of hexachloroethane and tetrachloroethylene. In contrast, for example, the sonolysis of CCl₄ under an atmosphere of oxygen generates preferably CO₂, CO, Cl⁻ [46], [56]. Consequently, in order to simulate the decomposition of CCl₄ in presence of O₂ under the experimental conditions of Pétrier and Francony [62], the chemical reactions occurring within a bubble initially composed of oxygen and CCl₄ is followed according to the scheme given in Table 3. On the other hand, the decomposition of CCl₄ under an Ar atmosphere under the experimental conditions of Hung and Hoffmann [48], is simulated according to the chemical kinetics shown in Table 2.

Table 4.

[a] Number of active bubbles under the experimental conditions of Hung and Hoffmann[48] (conditions: [CCl₄]_0,liq = 0.2 mM, P_a = 2.05 atm for the range of ultrasound frequency from 205 to 1078 kHz, with the exception of 500 kHz where P_a = 2.31 atm, T_liq = 13 °C). [b] Maximal bubble radius (R_max, µm) achieved during oscillation, bubble lifetime and the acoustic period (µs), under the same conditions of [a].

[a]
f (kHz)	CCl4 removal by a single bubble (mol)	Initial rate of CCl4 removal in solution (mol L−1 s−1)	Production rate of bubbles (L−1 s−1)
205	3,12 × 10−16	1,31654 × 10−07	4,22 × 10+08
358	2,00 × 10−17	1,44864 × 10−07	7,25 × 10+09
500	1,65 × 10−17	1,26277 × 10−07	7,67 × 10+09
618	1,46 × 10−17	1,60285 × 10−07	1,10 × 10+10
1078	5,19 × 10−20	1,1811 × 10−07	2,28 × 10+12

[b]
f (kHz)	Maximum bubble radius, Rmax (µm)	Bubble lifetime (µs)	Acoustic cycle(µs)
205	22.81	4.25	4.87
358	11.01	2.25	2.79
500	9.39	1.71	2.0
618	7.37	1.37	1.61
1078	4.38	0.79	0.92

Table 5.

[a] Predicted number of active bubbles under the experimental conditions of Pétrier and Francony[62] (conditions: [CCl₄]_0,liq 0.43 mM, P_a = 2.62 atm for the range of ultrasound frequency from 200 to 800 kHz, T_liq = 20 °C). [b] Maximal bubble radius (R_max, µm) achieved during oscillation, bubble lifetime and the acoustic period (µs), under the same conditions of [a].

[a]
frequency (kHz)	CCl4 removal by a single bubble (mol)	Initial rate of CCl4 removal in solution(mol L−1 s−1)	Production rate of bubbles (L−1 s−1)
200	7.70 × 10−16	5.50 × 10−07	7.14 × 10+08
500	1.93 × 10−18	6.17 × 10−07	3.20 × 10+11
800	1.03 × 10−21	8.33 × 10−07	8.10 × 10+14

[b]
frequency (kHz)	Maximum bubble radius, Rmax (µm)	Bubble lifetime (µs)	Acoustic cycle(µs)
200	21.91	4.28	5
500	8.04	1.63	2
800	5.83	1.08	1.25

3. Results and discussions

The obtained results of the present developed technique (i.e. based on CCl₄ pyrolysis) are compared to those obtained semi-empirically by Merouani et al. [42] for air and oxygen bubbles and theoretically by Kerboua et al. [44] for oxygen bubbles. The obtained findings of Merouani et al. [42] are based on the determination of the molar yield of H₂O₂, ^●OH and HO₂^● of a single bubble in addition to the production rate of H₂O₂ measured experimentally. On the other hand, Kerboua and Hamdaoui’s [43] work is built on an energetic analysis of the macroscopic (control volume) and microscopic (acoustic bubble) systems under the action of ultrasonic perturbation. It should be noted that the discussed results in this paper are confronted on a range of ultrasound frequencies from ∼20 to 1000 kHz, Table 4, Table 5, Table 6, Table 7.

Table 6.

Production rate of active bubbles “N (L⁻¹ s⁻¹)” according to the semi-empirical model of Merouani et al.[42]. [a] under the experimental conditions of Merouani et al. themselves (P_a = 2,41 atm, T_liq = 25 °C and air-saturation gas). [b] Under the operational conditions of Pétrier and Francony[62] (P_a = 2.62 atm for 200, 500 and 800 kHz, T_liq = 20 °C and oxygen-saturation gas) and those of Jiang et al.[93] (same conditions as those of Pétrier and Francony’s work [62]).

[a]
frequency (kHz)	N (L−1 s−1) (Merouani et al. conditions)
300	2,84 × 10+07
585	3,94 × 10+08
860	3,04 × 10+09
1140	3,08 × 10+09

[b]
frequency (kHz)	N ( L−1 s−1)[Pétrier and Francony conditions]	N ( L−1 s−1)(Jiang et al. conditions)
200	5,2249×10+06	5,4339×10+06
500	3,7309×10+07	5,3299×10+07
800	6,3908×10+07	9,1297×10+07

Table 7.

Production rate of active bubbles “N (L⁻¹ s⁻¹)” according to the theoretical work of Kerboua et al.[44] (P_a = 1.5 atm, T_liq = 20 °C and oxygen-saturation gas).

Frequency (kHz)	N (L−1 s−1)
200	1,60 × 10+12
300	1,20 × 10+13
360	1,80 × 10+13
443	2,58 × 10+13
500	5,00 × 10+13
600	2,40 × 10+14
800	4,00 × 10+15

3.1. The single bubble degradation of CCl₄

It is initially critical to demonstrate how to determine the single bubble conversion of CCl₄ using the model described in Section 2. Even though we have done this in detail in our earlier works [61], [82] for many simulation circumstances (frequency, acoustic intensity, and liquid temperature), it is best to describe briefly the topic here to provide the reader a quick grasp of the established approach. The evolution of the bubble dynamics, the bubble temperature and pressure, and the interior bubble chemistry are illustrated in Fig. 1 (for one acoustic cycle in Fig. 1(a) and (b) and around the end of the bubble collapse in Fig. 1(c)). The numerical simulations of Fig. 1 are for a bubble of R₀ = 3 µm oscillating in argon-saturated water containing 0.2 mM of CCl₄, under the following sonication conditions: frequency: 500 kHz, acoustic amplitude P_a = 2.31 atm, liquid temperature 13 °C, and ambient static pressure P_∞= 1 atm. These acoustical conditions are a part of those applied by Hung and Hoffmann [48] for degrading CCl₄ in argon-saturated solution. The Henry law for CCl₄ is given as P_CCl4,0 = K_H,CCl4C_CCl4,0 , where K_H,CCl4 = 1733,06 Pa m³ mol⁻¹ [88], 13 °C) and C_CCl4,0 is the CCl₄ concentration in the aqueous phase, 0.2 mM. By applying this rule, the initial pressure of CCl₄ within the bubble (at t = 0 µs/or R = R₀) could be 0.346 kPa. Given that the initial bubble pressure is P₀ = P_∞ +2σ/R₀ = 150.515 kPa, the initial fraction of CCl₄ with the bubble is P_CCl4,0/P₀ = 2.3028 × 10⁻³ (so 0.23%). Similarly, the initial bubble content on water vapor is 9.79 × 10⁻³. This is calculated as P_v,0/P₀, where P_v,0 is the water vapor (saturated) calculated through Antoine’s equation. The remnant of the initial bubble content [1- P_v,0/ P₀- P_CCl4,0/P₀] is occupied by the saturating gas (i.e. argon in this case).

Fig. 1.

Evolution of the bubble dynamics (a), the bubble temperature and pressure (b) and the interior bubble chemistry (c) during the oscillation of a single bubble of R₀ = 3 µm in an argon-saturated water containing 0.2 mM of CCl₄ (conditions: frequency: 500 kHz, acoustic amplitude P_a = 2.31 atm, liquid temperature 13 °C, and ambient static pressure P_∞ = 1 atm). The y-axis in of Fig. 1(b) “right side” and (c) “left side” is in logarithmic scale.

The bubble oscillation in Fig. 1(a) represents the temporal behavior of the bubble radius, which has been extensively discovered experimentally [89], [90] and numerically by a number of sonochemists, including Yasui [91], Merouani et al. [68], Kerboua et al. [69], and others [79], [92]. During the negative rarefaction cycle of the sound wave, the bubble grows from R₀ to R_max in 1.219 µs and then abruptly collapses in 0.49 µs, producing the occurrence of a peak temperature around the end of the bubble collapse (around t = 1.71 µs), as shown in Fig. 1(b). A maximum temperature of 5155.53 K is attained within the bubble at R_min. This intense temperature enables reaction chemistry to occur within the bubble; this latter is considered as a micro-reactor within which high-energy chemical reactions occur during the final instant of collapse (about R_min), as seen in Fig. 1(c). According to this figure, a part of CCl₄ and H₂O molecules trapped at the collapse are pyrolyzed, yielding a variety of chemicals (argon is unreactive species; its amount could be then unchangeable). Refs. [60], [61] provide a thorough description of the reaction findings that led to Fig. 1(c). This could not be supplied here since the study is aimed primarily to determine the bubbles’ number, which needs knowledge of the quantity of CCl₄ removed (degraded) by one bubble in one acoustic cycle (i.e. from one implosion). This latter is obtained by subtracting the amount of CCl₄ at the end of the bubble collapse (at R = R_min, 9.46 × 10⁻²³ mol) from the initial quantity of CCl₄ within the bubble (at t = 0 µs, 1.647 × 10⁻¹⁷ mol), which yields 1.6446 × 10⁻¹⁷ mol (99.85% of CCl₄ is eliminated).

3.2. Production rate of active bubbles “the number density”

In Table 4, Table 5(a), the predicted removal of CCl₄ for a single bubble, the initial rate of CCl₄ removal in the sonoreactors (solution) and the calculated production rates of actives bubbles (i.e. number density) are given under the experimental conditions of Hung and Hoffmann [48] and Pétrier and Francony [62], respectively. The obtained results in Table 4(a) and Table 5(a) are compared to those obtained semi-empirically by Merouani et al. [42] under: (i) their experimental conditions [air-bubble, Table 6(a)] and (ii) under the Pétrier and Francony’s [62] and Jiang et al.'s [93] conditions [O₂-bubble, Table 6(b)]. Additionally, the obtained results in Table 4(a) and Table 5(a) are confronted with those obtained theoretically by Kerboua et al. [44] for an oxygen bubble (Table 7). The obtained results in Table 4, Table 5, Table 6, Table 7 are depicted in Fig. 2.

Fig. 2.

Production rate of active bubbles (L⁻¹ s⁻¹) according to: [A1]Kerboua et al.[44] (Table 7), our model under the experimental circumstances of Pétrier and Francony[62][B1] and Hung and Hoffmann[48][B2] (Table 4(a) and 5(a)), Merouani et al. study [42][C1] (Table 6(a)) and Merouani’s model under the experimental conditions of Pétrier and Francony[62][C2] and Jiang et al.[93][C3], (Table 6(b)).

In general, it is observed that the number of active bubbles is substantially increased with the rise of ultrasound frequency either according to our study (Table 4(a) and Table 5(a)) or as it is obtained by Merouani et al. [42] and Kerboua et al. [44] (Fig. 2). For example, under the experimental conditions of Hung and Hoffmann [48] (Table 4(a), Fig. 2), the production rate of active bubbles goes up from 4.22 × 10⁸ to 2.28 × 10¹² L⁻¹ s⁻¹, when ultrasound frequency increased from 205 to 1078 kHz. This trend (i.e. increase of number density with wave frequency) is in good agreement with those reported qualitatively by Kanthale et al. [94] and Brotchie et al. [36]. The increase of the production rate of active bubbles proportionally with the rise of ultrasound frequency can be explained according to the effect of incident ultrasonic waves on the dynamics of bubbles. The increase of ultrasound frequency causes the lifetime of bubbles to be reduced as well as the maximum radii of oscillating cavities, where this is ascribed to the decrease of acoustic period monotonically with the rise of frequency, as seen in Table 4(b) and Table 5(b). Consequently, the number of active bubbles created per unit time is remarkably increased at higher acoustic frequencies compared to the lower ones. The confrontation of our findings (Table 4(a) and Table 5(a)) to those obtained by Merouani et al. (Table 6(a) and (b)) and Kerboua et al. (Table 7) indicates the good concordance between these results concerning the improvement of the production rate of bubbles as a result to the rise of frequency. However, it is clearly observed that our results (production rate of bubbles) are quantitatively situated between those of Merouani et al. (lower values of N) and Kerboua et al. (higher values of N). This is probably ascribed to the adopted techniques for determining the number density of bubbles [42], [44] as it was indicated previously. It should be noted that the improvement of the number density of bubbles is also obtained by the increase of acoustic intensity, use of the pulsed mode of sono-irradiation, increase of the irradiation time and the application of dual-frequency field [26], [36], [38]. Indeed, as it can be observed in Fig. 2, the number density obtained in our study under the experimental conditions of Pétrier and Francony [62] is greater than that retrieved under the experimental conditions of Hung and Hoffmann [48] for the range of acoustic frequency in common. This behavior is observed especially for wave frequencies greater than ∼ 358 kHz. According to the operational conditions of Pétrier and Francony, the number of active bubbles increases from 8.399 × 10⁵ at 20 kHz to 8.1 × 10¹⁴ L⁻¹ s⁻¹ at 800 kHz [Table 5(a)]. Whereas under the experimental conditions of Hung and Hoffmann, the number of active bubbles goes up from 4.22 × 10⁸ at 205 kHz to 2.28 × 10¹² at 1078 kHz [Table 4(a)]. This trend is owing to the relatively higher acoustic amplitude used by Pétrier and Francony (P_a = 2.62 atm for 200–800 kHz) [62] compared to that of Hung and Hoffmann (P_a = 2.05 atm for 205–1078 kHz, and 2.31 atm for 500 kHz) [48]. Furthermore, the higher number density obtained in our study under the experimental conditions of Pétrier and Francony compared to that of Hung and Hoffmann is probably promoted by the relatively higher CCl₄ concentration and liquid temperature used by Pétrier and Francony [62] (0.43 mM and 20 °C) compared to those of Hung and Hoffmann [48] (0.2 mM and 13 °C). In fact, the increase of CCl₄ concentration and the liquid temperature is expected to induce the activation of more bubbles (increases the number density) even when the maximal bubble temperature is reduced [82].

4. Conclusion

In the present paper, a simple technique is proposed for the determination of the number density of acoustic bubbles in sonicated aqueous solution. This method relies on the combination of single bubble chemistry to that of the whole volume of the solution in the reactor. The obtained results (under the experimental conditions of Pétrier and Francony and those of Hung and Hoffmann) have demonstrated the good concordance between our findings and those found in literature especially that obtained semi-empirically under the operational conditions of Merouani et al (Table 6(a)). It is found that the number of active bubbles is proportionally increased by the rise of ultrasound frequency for all adopted experimental conditions. Additionally, the positive effect of acoustic intensity on the number of active bubbles has been revealed, where the number density is improved monotonically with the increase of acoustic amplitude. This positive impact is clearly observed for higher frequencies (>358 kHz). On the other hand, this technique shows the probable dependence of number density to the CCl₄ concentration and liquid temperature, which means that more investigations are needed (in future works) to determine the impact of these crucial parameters (i.e. CCl₄ dosage and liquid temperature) in addition to the acoustic intensity on the number of active bubbles. Moreover, the present work indicates the possibility of using other volatile compounds in order to determine the number of active bubbles and investigate the efficacy of these species compared to that of carbon tetrachloride.

CRediT authorship contribution statement

Aissa Dehane: Conceptualization, Methodology, Software, Formal analysis, Writing – original draft, Writing – review & editing. Slimane Merouani: Project administration, Conceptualization, Supervision, Visualization, Writing – review & editing, Methodology, Formal analysis, Writing – review & editing. Oualid Hamdaoui: Visualization, Validation, Writing – review & editing. Muthupandian Ashokkumar: Visualization, Formal analysis, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.