Acoustic frequency and optimum sonochemical production at single and multi-bubble scales: A modeling answer to the scaling dilemma.

The present work consists of an innovative approach aiming to address the scalability dilemma of the sonochemical activity dependency of acoustic frequency. The study originates from the discordance of observations between the theoretical investigations of the sonochemical activity of the single acoustic cavitation bubble in function of the acoustic frequency, in one hand, and the experimental findings regarding the optimal frequency condition, mainly in terms of pollutant degradation, in the other hand. A single bubble and an up-scaled model of the sonochemical activity are suggested and simulations were conducted based on both of them over the frequencies 20, 200, 300, 360, 443, 500, 600 and 800 kHz under an oxygen atmosphere. The results reveal that the sonochemical production at single bubble scale is monotonously decreasing with the increase of frequency, while all the products demonstrate an absolute optimum of sonochemical production at 200 kHz, except HO• that attains its maximum molar yield under 300 kHz. Besides, the production of the predominant species, namely HO2•, HO• and O3, manifests a clear rebound at 500 kHz. All the present results were compared to and confirmed by experimental findings, while the scalability of the concentrations of sonochemically produced species was discussed using a parameter we introduced as "the mass focusing factor".

Accommodation coefficient

Thermal layer width (m)

Thermal conductivity (W/m·K)

Stoichiometric coefficient related to the kth species in the ith reaction in the left side of the chemical reaction

Stoichiometric coefficient related to the kth species in the ith reaction in the right side of the chemical reaction

Density of liquid (kg/m3)

Surface tension (N/m)

Dynamic viscosity (Pa·s)

Area of the basis of the sonochemical reactor (m2)

Pre-exponential factor (m3/mol·s) for two body reaction and (m6/mol2·s) for a three-body reaction

Sound celerity (m/s)

Molar concentration of the kth species within the single acoustic cavitation bubble (mol/m3)

Molar concentration of the kth species within the control volume (mol/m3)

Molar concentration of the kth species within the sonochemical reactor (mol/m3)

Activation energy (J/mol)

Frequency (Hz)

Mass focusing factor

Reaction heat of the ith reaction (J/mol)

Acoustic intensity (W/cm2)

Wave number (m−1)

Yield of evaporation and condensation (kg/ m2)

Rate of evaporation and condensation (kg/m2·s)

Acceleration of evaporation and condensation (kg/m2·s2)

Molar mass of water (kg/mol)

Total molar yield (mol)

Number density of bubbles (m−3)

Time derivate of the number density of bubbles (m−3·s−1)

Second time derivate of the number density of bubbles (m−3·s−2)

Molar yield of the kth species within the single acoustic cavitation bubble (mol)

Molar yield of the kth species within the sonochemical reactor (mol)

Acoustic amplitude (Pa)

Partial pressure (Pa)

Pressure of gas (Pa)

Saturating pressure (Pa)

Ambient pressure (Pa)

Bubble radius (m)

Bubble wall velocity (m/s)

Bubble wall acceleration (m/s2)

Reaction rate of the ith reaction (mol/s m3)

Ideal gas constant (J/mol·K)

Section of the bubble wall (m2)

Temperature within the bubble (K)

Time (s)

Ambient temperature (K)

Volume of the bubble (m3)

Volume of the sonicated liquid in the sonochemical reactor (m3)

The coordinate of the jth position

Refers to the kth species ()

Refers to the ith reaction ()

Refers to the jth iteration in spatial discretization ()

Introduction

Sonoprocessing refers to the use of sonic and ultrasonic waves in chemical processes, with a wide range of possible applications in environmental engineering, green chemical synthesis, and processing. The chemical effect induced by ultrasound, namely “sonochemistry” is actually an indirect consequence of the wave, that harnesses an intermediate purely physical phenomenon called acoustic cavitation [1]. Gases dissolved in the liquid medium form microscopic germs, i.e. microscopic gaseous inclusions within the liquid or simply nuclei, that evolve to create bubbles under the effect of ultrasound. The dissolution of gases is the principal phenomenon behind bubble inception [2]. It is then enough that the liquid medium be irradiated with ultrasound of high intensity, a sound energy field is formed inducing the molecules to oscillate at their mean position, if the intensity is high enough, it breaks the interaction of molecules and the acoustic cavitation is initiated. Since the acoustic pressure oscillates sinusoidally at each position, succession and repetition of compression and rarefaction phases result in the growth of created cavitation by diffusion and coalescence, unless they are dissolved, then occurs a violent collapse starting from a maximum size the bubbles reach [3]. During the time slot surrounding the collapse, extreme conditions of very high temperatures and pressures develop inside cavitation [4]. These conditions are closely dependent of the applied acoustic frequency [5], they activate a panoply of precursor chemical reaction for free radicals formation, such as the pyrolysis of water, dissolved gases and other hydrophobic molecules [6], [7]. At the last compression period, the bubble collapses, releasing into the surroundings highly concentrated energy and extremely reactive radicals such as O, HO2• and •OH. These free radicals are expected to attack and oxidize the organic pollutants [8].

Sonochemistry is nowadays a rapidly growing research field with promising applications in environmental remediation [9]. It offers the opportunity for cheaper reagents, shorter reaction cycles, and less extreme operational conditions, leading to less expensive and probably smaller plants. However, some theoretical and engineering aspects related to environmental sonochemistry remain today not fully understood [10].

During the three last decades, the theoretical comprehension of environmental sonochemistry has been actively supported by numerous studies [11], [12]. To illustrate, Yasui [12] suggested an early model to describe the chemical mechanism evolving within an air bubble and quantified the molar rates of chemical reactions assuming a single acoustic cavitation bubble oscillating in water. Ferkous et al. [11] supported their experimental results in terms of the sonolytic degradation of naphthol blue black in water using a numerical approach describing and simulating the sonochemical production of hydroxyl radicals at single bubble scale. Interestingly, the studies based on theoretical approach and fundamental phenomena agreed on the generation of radicals from the cleavage of H2O and dissolved gases [13], then the reactivity of the formed oxidants with organic contaminants such as phenols and organochlorines [8], however, their parametric investigation of the sonochemical activity was critically limited to the single bubble scale [14], [15], [16]. In the other hand, Dewulf et al. [17] proceeded to the modeling of the ultrasonic degradation of trichloroethylene and chlorobenzene using a pseudo-first order kinetic model, while Chiha et al. [18] described the degradation of non-volatile organic compounds by Langmuir-type equations. The modeling studies which treated the reactor scale kinetics were all of semi-empirical nature and led to apparent kinetic constant, intrinsically related to the experiments’ conditions.

Acoustic frequency is one of those most influencing operational parameters posing a flagrant discordance between theoretical research and experimental findings. Indeed, while single bubble studies revealed a regular evolution of sonochemical yields in function of frequency [11], [19], [20], most of experimental works pointed out optimums as reported in Table 1. The “scaling dilemma” is about the selection of the acoustic frequency condition according to both research paths that deliver different results. Indeed, the numerical studies suggest opting for the lowest frequency in order to enhance the sonochemical performance, while the experimental works prove the existence of an optimum operating frequency. The divaricated results regarding to the same aspect of the sonochemical reactions, namely the maximization of the sonochemical yield, constitute a scientific “dilemma” that creates a confusion when setting the operating conditions, in particular the acoustic frequency. This dilemma is due to the different scales from which both suggested results emanate. Thus, in order to explain the discordance point and lead to a consensus of experimental and modeling results, it is essential to proceed to the scaling-up of the model in order to describe the sonochemical activity at the reactor scale. This is exactly the objective of the present study.

Table 1

Bibliographic review of experimental studies that examined the frequency dependency of sonochemical production.

Author	Study	Range of frequencies (kHz)	Observed optimum
Chiha et al. [63]	Degradation of 4-cumylphenol	300, 600	300 kHz
Pétrier et al. [49]	Degradation of phenol	20, 200, 500, 800	200 kHz
Kanthale et al. [48]	Production of hydrogen peroxide	213, 355, 647, 1056	213 and 355 kHz
Pétrier and Francony [49]	Production of hydrogen peroxide	20, 200, 500, 800	200 kHz
Jiang et al. [50]	Production of hydrogen peroxide	20, 200, 500, 800	200 kHz
Jiang et al. [50]	Degradation of 4-chlorophenol	20, 200, 500, 800	200 kHz
Suzuki et al. [61]	I3− formation	24.1, 200, 500, 1170	200 kHz
Pétrier et al. [52]	I3− formation	20, 514	514 kHz
Son et al. [64]	Degradation of bisphenol A	36, 262	262 kHz
Lim et al. [51]	Degradation of chlorobenzene	35, 74, 170, 300, 1000	300 kHz
Lim et al. [51]	Degradation of chloroform	35, 74, 170, 300, 1000	300 kHz
Lim et al. [51]	Degradation of carbon tetrahloride	35, 74, 170, 300, 1000	300 kHz
Lim et al. [51]	Formation of hydrogen peroxide	35, 74, 170, 300, 1000	300 kHz
Wayment et al. [53]	Degradation of alachlor	20, 300	300 kHz
Wayment et al. [53]	Formation of hydrogen peroxide	20, 300, 446	300 kHz
Koda et al. [43]	I3− formation	19.5, 25, 40, 45, 96, 130, 200, 300, 400, 500, 1200	300 kHz
Mark et al. [44]	Formation of hydrogen peroxide	80, 200, 300, 400, 500, 600, 700, 900, 1000, 1100	300 kHz

The scalability of the models of sonochemical production, which consists in elaborating a numerical model that describes the sonochemical activity at the “reactor scale”, starting from a predefined model describing it at the single acoustic cavitation scale, remains an unsolved problem of sonochemistry [21]. Actually, the rarity of attempts to up-scale the developed models [22] prevents a concise parametric theoretical study of sonochemistry, including the effect of acoustic frequency on the yields of the main produced oxidants.

In the present numerical modeling work, an attempt is made to approach the problem of acoustic frequency dependency of sonochemical activity at both single bubble and reactor scales. The objective is to bring a comprehensive answer to the scaling dilemma of sonochemical production and figure out the reported optimums revealed by experiments, versus no optimum recorded in previous theoretical works.

Numerical modeling

The present numerical model is based on combining both sonochemical activity of single acoustic cavitation bubble and kinetic evolution of number density of bubbles within a control volume of a sonochemical reactor filed of water saturated with oxygen. The single bubble dynamics is governed by the modified Keller-Miksis equation accounting for the non-equilibrium of evaporation and condensation of water molecules at the gas–liquid interface [23], [24], as expressed in Eq. (1).

The active bubble population is supposed to be composed of bubbles having similar equilibrium radius . The value of is frequency dependent, ambient radii were selected in respect to the theoretical intervals defined by Yasui [25] and reported by Brotchie et al. [26]. These values are limited by Blake [27] and Minnaert [28], [29] thresholds under the different acoustic frequencies i.e. 20, 200, 300, 360, 443, 500, 600 and 800 kHz. The reactor scale is defined as the macroscopic scale delimited by a water volume of 1 dm3 contained within a vessel of cylindrical design of 50 cm2 of base area A, i.e., 4 cm radius, and 20 cm of height, sonicated in the longitudinal direction. The acoustic amplitude is supposed to be constant at 1.5 atm within the distance travelled by the wave, i.e. the water height as shown in Fig. 1(b). The assumed acoustic amplitude is equivalent to an acoustic intensity 0.77 W/cm2 according to Eq. (2) [15], and a power density of 38.5 W/dm3 according to Eq. (3).

Fig. 1

Schema of heat and mass transfer phenomena related to the single acoustic cavitation bubble acting as a micro-reactor (a) and spatial discretization of the sonochemical reactor (b).

Each single bubble constitutes a microreactor of spherical form and varying volume, the volume variation is then expressed by Eq. (4).

While the volume of each microreactor varies in function of time, temperature and pressure within it evolves according to Eqs. (5), (6), respectively.

Eq. (5) represents the heat balance applied to the single bubble volume. This heat balance considers the single acoustic cavitation bubble as opened to heat exchange through thermal diffusion across the thin heat transfer layer at the bubble interface, and evaporation and condensation processes carrying water molecules enthalpy toward and outward the bubble volume as illustrated in Fig. 1(a). Each microreactor is also considered open to mass transfer through physical phenomenon of simultaneous evaporation and condensation of water molecules at bubble interface as shown in Fig. 1(a), whose kinetics is governed by Hertz-Knudsen equation shown in Eq. (7) [30].

The oscillation of active acoustic cavitation bubbles is particularly characterized by spectacular elevations of pressures and temperatures attaining 1000 bar and 6000 K, respectively, under 1.5 atm of acoustic amplitude [20], [31] and occurring when the bubble contracts to its minimal size and collapses. The harsh conditions then attained activate a considerable number of elementary reactions initiated by the thermal decomposition of H2O into hydroxyl and hydrogen radicals, and the reaction of H2O and O2 as shown below.

H2O → H● + ●OH

O2 + H2O → ●OH + HO2●

In the present model, 45 elementary reactions are expected to occur according to the schema reported in Table 2 and initially suggested by Yasui [23] and inspired from [32], [33], [34], [35]. The molar rate related to ith reaction is expressed in Eq. (8). Hence, the molar rate due to chemical reactions and related to the kth species among the nine involved in the model is represented by Eq. (9).

Table 2

Adopted scheme of the possible reactions occurring inside an O2/H2O collapsing bubble [23]. M is the third body. A is expressed in (m3/mol.s) for two body reaction (m6/mol2.s) for a three-body reaction.

i	Reaction i	Ai	bi	Ei/Rg (K)	ΔHi (kJ/mol)
1	H + O2 ⇒ O + ●OH	1.92 × 108	0	8270	69,17
2	O + H2 ⇒ H● + ●OH	5.08 × 10−2	2.67	3166	8,23
3	●OH + H2 ⇒ H● + H2O	2.18 × 102	1.51	1726	−64,35
4	●OH + ●OH ⇒ H2O + O	2.1 × 102	1.4	200	−72,59
5	H2 + M ⇒ H● + H● + M; Coef. H2: 2.5, H2O: 16.0	4.58 × 1013	−1.4	52,500	444,47
6	O + O + M ⇒ O2 + M; Coef. H2: 2.5, H2O: 16.0	6.17 × 103	−0.5	0	−505,4
7	O + H● + M ⇒ ●OH + M; Coef·H2O: 5.0	4.72 × 105	−1.0	0	−436,23
8	H● + ●OH + M ⇒ H2O + M; Coef. H2: 2.5, H2O: 16.0	2.25 × 1010	−2.0	0	−508,82
9	H● + O2 + M ⇒ HO2● + M; Coef. H2: 2.5, H2O: 16.0	2.00 × 103	0	−500	−204,8
10	H● + HO2● ⇒ O2 + H2	6.63 × 107	0	1070	−239,67
11	H● + HO2● ⇒ ●OH + ●OH	1.69 × 108	0	440	−162,26
12	O + HO2● ⇒ O2 + ●OH	1.81 × 107	0	−200	−231,85
13	●OH + HO2● ⇒ O2 + H2O	1.45 × 1010	−1.0	0	−304,44
14	HO2● + HO2● ⇒ O2 + H2O2	3.0 × 106	0	700	−175,35
15	H2O2 + M ⇒ ●OH + ●OH + M; Coef. H2: 2.5, H2O: 16.0	1.2 × 1011	0	22,900	217,89
16	H2O2 + H● ⇒ H2O + ●OH	3.2 × 108	0	4510	−290,93
17	H2O2 + H● ⇒ H2 + HO2●	4.82 × 107	0	4000	−64,32
18	H2O2 + O ⇒ ●OH + HO2●	9.55	2	2000	−56,08
19	H2O2 + ●OH ⇒ H2O + HO2●	1.00 × 107	0	900	−128,67
20	O3 + M ⇒ O2 + O + M; Coef. O2: 1.64; Coef. O2: 1.63, H2O: 15	2.48 × 108	0	11,430	109,27
21	O3 + O ⇒ O2 + O2	5.2 × 106	0	2090	−396,14
22	O3 + ●OH ⇒ O2 + HO2●	7.8 × 105	0	960	−164,92
23	O3 + HO2● ⇒ O2 + O2 + ●OH	1 × 105	0	1410	−121,92
24	O3 + H● ⇒ HO2● + O	9 × 106	0.5	2010	−135,65
25	O3 + H● ⇒ O2 + ●OH	1.6 × 107	0	0	−96,2
26	O + ●OH ⇒ H + O2	7.18 × 105	0.36	−342	−69,17
27	H● + ●OH ⇒ O + H2	2.64 × 10−2	2.65	2245	−8,23
28	H● + H2O ⇒ ●OH + H2	1.02 × 103	1.51	9370	64,35
29	H2O + O ⇒●OH + ●OH	2.21 × 103	1.4	8368	72,59
30	H● + H● + M ⇒ H2 + M; Coef. H2: 2.5, H2O: 16.0	2.45 × 108	−1.78	480	−444,47
31	O2 + M ⇒ O + O + M; Coef. H2: 2.5, H2O: 16.0	1.58 × 1011	−0.5	59,472	505,4
32	●OH + M ⇒ O + H● + M; Coef·H2O: 5.0	4.66 × 1011	−0.65	51,200	436,23
33	H2O + M ⇒ H● + ●OH + M; Coef. H2: 2.5, H2O: 16.0	1.96 × 1016	−1.62	59,700	508,82
34	HO2● + M ⇒ H● + O2 + M; Coef. H2: 2.5, H2O: 16.0	2.46 × 109	0	24,300	204,8
35	O2 + H2 ⇒ H● + HO2●	2.19 × 107	0.28	28,390	239,67
36	●OH + ●OH ⇒ H● + HO2●	1.08 × 105	0.61	18,230	162,26
37	O2 + ●OH ⇒ O + HO2●	3.1 × 106	0.26	26,083	231,85
38	O2 + H2O ⇒ ●OH + HO2●	2.18 × 1010	−0.72	34,813	304,44
39	O2 + H2O2 ⇒ HO2● + HO2●	4.53 × 108	−0.39	19,700	175,35
40	●OH + ●OH + M ⇒ H2O2 + M; Coef. H2: 2.5, H2O: 16.0	9.0 × 10−1	0.90	−3050	−217,89
41	H2O + ●OH ⇒ H2O2 + H●	1.14 × 103	1.36	38,180	290,93
42	H2 + HO2● ⇒ H2O2 + H●	1.41 × 105	0.66	12,320	64,32
43	●OH + HO2● ⇒ H2O2 + O	4.62 × 10−3	2.75	9277	56,08
44	H2O + HO2● ⇒ H2O2 + ●OH	2.8 × 107	0	16,500	128,67
45	O2 + O + M ⇒ O3 + M; Coef. O2: 1.64; Coef. O2: 1.63, H2O: 15	4.1	0	−1057	−109,27

According to Eqs. (7), (8), (9), the molar yields of the nine species involved in the chemical schema can be expressed in terms of molar concentration as shown in Eqs. (10), (11), (12), (13), (14), (15), (16), (17), (18), (19). These concentrations represent the molar yields expected to emerge within the active volume.

At the macroscopic scale, the sonochemical activity of bubble population is observed within a cylindrical control volume at the coordinate . The axis is oriented in the direction of propagation of acoustic wave along the water height. The water height is discretized into small distances equal to as reported in Table 3. As indicated in this table, the number of iterations depends on the frequency and the value of the step. The acoustic pressure is supposed to be stable around the coordinate along this distance, or in other words between the coordinates and . At the coordinate , we assume that at , the acoustic wave starts a rarefaction phase, this supposes that at other coordinates , the liquid medium undergoes the acoustic perturbation induced by previous acoustic cycles. The energy balance applied to the control volume leads to Eq. (19), already demonstrated in a previous work of our research group [24].

Table 3

Parameters related to the spatial discretization and volume integration in function of frequency.

Frequency (kHz)	20	200	300	360	443	500	600	800
Wavelength (cm)	7.50	0.75	0.50	0.42	0.34	0.30	0.25	0.19
Step (mm)	0.750	0.075	0.050	0.042	0.034	0.030	0.025	0.019
Number of iterations: n	267	2667	4000	4800	5907	6667	8000	10,667
Control volume (cm3)	3.750	0.375	0.250	0.208	0.169	0.150	0.125	0.094

In this equation, represents the number density of bubbles at instant t and coordinate . Since all the active bubbles at xaround xj are supposed to undergo the same acoustic perturbation , the total sonochemical production of bubbles falling within the control volume around the coordinate xj is assumed to be the sum of molar yields emerging from identical acoustic cavitation bubbles. Hence, the molar rates related to the sonochemical products (, , , , , ), emerging from the reactions reported in Table 2, are expressed at reactor scale according to Eq. (20).

represents the molar concentration of the kth species (among , , , , , ) within the control volume around the coordinate , while and are the parameters relative to the chemical kinetics, explained and explicited in Table 2.

The concentration within the water volume is then calculated at each instant t by Eq. (21).

The molar yield of the kth species within the water volume is then given at each instant by Eq. (22).

In the present model, the molar concentrations of nine species involved in the chemical schema within the water volume around a coordinate are governed by the differential equation ranging from Eqs. (23), (24), (25), (26), (27), (28), (29), (30), (31).

In the present paper, concentrations of chemical species within active and reacting volumes are compared using a factor that we called mass focusing factor and defined as the ratio of concentration within the active volume to the concentration in reacting volume , as represented in Eq. (32).

Results and discussion

From single bubble to reactor scale

Molar yields of sonochemical emerging species are first estimated within a control volume around the coordinate x1 under the different acoustic frequencies for the same irradiation duration of 50 µs. As illustrative case, Fig. 2 presents the results obtained under 20 kHz frequency; we highlight however that similar trends of species predominance are observed under all the treated frequencies. At first step, we focus on the molar yields of the seven produced chemical species within a single acoustic cavitation bubble, as shown in Fig. 2(a), hence, the volume of the bubble is neither discussed nor involved at this stage. In other words, the considered space unit is the single bubble, while at the reactor scale, control volume around the coordinate x1 is considered. The results retrieved under 20 kHz frequency are reported in Fig. 2(b). This figure demonstrates that the predominant sonochemical products are HO2•, HO•, O and O3, these species are typically observed when sonicating aqueous medium saturated with oxygen as denoted experimentally by Kohno et al. [36] and demonstrated numerically by Merouani et al. [19] and Kerboua and Hamdaoui [31]. Considerable emergence of O and O3 is explained by the saturation with oxygen. However, the most important products remain free radicals. At single bubble scale, HO2• and HO• attain by the end of 50 µs 9.2210−16 and 3.5810−17 mol, respectively. It is noticeable that the spectacular increase in these molar yields occurs at 39.7 µs, they remain stable until 50 µs, i.e. the end of the acoustic cycle under 20 kHz.

Fig. 2

Molar yields of emerging species within single acoustic cavitation bubble (a) and control volume around x1 of within sonochemical reactor (b) under 20 kHz frequency and 1.5 atm amplitude by the end of 50 µs.

At reactor scale, the curves are quietly different since no pallier is observed. The molar yields of products start increasing at 39.7 µs, however, the time slot comprised between 40 and 46 µs is characterized by a gradual increase of the molar yields of the emerging products until reaching a maximum at 46 µs. At this instant, the control volume of 3.75 cm3 around the coordinate x1 = 0.375 mm contains 4.7010−14 mol of HO2• and 1.83 10−14 mol of HO•. The apparition of a maximum at reactor scale is explained by the simultaneous evolution of the sonochemical production at single bubble scale and the gradual decrease of the number density of bubbles by the end of the acoustic cycle after 46 µs within the control volume around x1, as demonstrated in a previous study of our research group [24].

Another noteworthy observation is that at reactor scale, all the chemical species, even those emerging in low molar yields, seem to attain their maximum amount at the same instant, i.e. 46 µs, for an irradiation time of 50 µs under 20 kHz. For instance, H2O2, H• and H2 attain at this instant within the control volume around x1 respective molar yields of 1.7310−16, 1.3410−18 and 3.1110−20 mol.

Effect of the acoustic frequency on the sonochemical molar yields

In order to inspect the effect of the acoustic frequency on the quantitative evolution of the sonochemical species, the molar yield of each emerging species is simulated at the single bubble scale, according to Eq. (9), and within the reactor volume, according to Eq. (22). The values depicted at the end of the 50 µs of irradiation time are reported versus the frequency in Fig. 3 for (a), (b), (c), (d), (e), (f) and (g).

Fig. 3

Variation of produced molar yields of emerging species in function of frequency within single acoustic cavitation bubble (orange line and circular markers) and sonochemical reactor volume (black line and square markers) under 1.5 atm amplitude, during 50 µs: (a), (b), (c), H• (d), HO2• (e), HO• (f) and (g). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The instantaneous evolution of the molar yields of the seven sonochemical products under oxygen atmosphere is the result of simultaneous evolution of the sonochemical activity within single bubbles trapped in each control volume and the variation of the number density of bubbles within the same elementary volumes in function of time. This latter point has been previously discussed in a numerical work of our research group [24], some insights into the variation of the average number density of bubbles in function of the acoustic frequency are given in Fig. 4.

Fig. 4

Evolution of the average number density of bubbles within the sonochemical reactor volume in function of the acoustic frequency during one acoustic cycle.

The first direct observation concerns the trend of variation according to scale. The variation of the molar yields of each of the seven emerging sonochemical products in function of increasing frequencies presents a clear monotonous decreasing trend at single bubble scale. Contrariwise, the variation of the same sonochemical species at reactor scale in function of acoustic frequency demonstrates optimums that manifest at 200 kHz for six of the seven products, and at 300 kHz for HO•. We suggest to closely examine the molar yields orders and the variation ratios in function of frequency at both scales for H2, H2O2, O3, H•, HO2•, HO• and O, the discussion will particularly focus on hydrogen, hydrogen peroxide and hydroxyl radical, owing to their relatively higher presence in the literature.

At single bubble scale, hydrogen molar yield, presented in Fig. 3(a), decreases from a value of 6.13 × 10−23 mol under an acoustic frequency of 20 kHz to 7.94 × 10−33 mol under 800 kHz, which reveals a 1010 order of decrease due to the increase of acoustic frequency in the range 20–800 kHz. In fact, the previous trend has been already reported by Merouani et al. [37] and Kerboua and Hamdaoui [38]. However, usual single bubble studies consider as simulation timeframe the acoustic period, i.e. non timely comparable results. Since the present study accounts for 50 µs as common irradiation duration, the variation presented in Fig. 3(a) suggests that the monotonous decreasing production in function of increasing frequency is not directly explainable by the extension in irradiation time under lower frequencies, but rather by the more violent collapse conditions, i.e. harsher core temperature and pressure, attained inside the bubble during collapse as demonstrated by Yasui et al. [39], [40] within an oxygen bubble. At reactor scale, however, an optimum hydrogen production is observed under 200 kHz with a molar yield of 8.58 × 10−19 mol. This value is 7 and 5 times higher than those retrieved under 20 and 300 kHz, respectively, the decrease ratio exceeds 105 when passing from 200 to 800 kHz, at reactor scale. To the best of our knowledge, the experimental results that dealt with the sonochemical production of hydrogen [41], [42] did not analyze its frequency dependency under comparable operational conditions.

Hydrogen peroxide, whose molar yield’s variation is represented in Fig. 3(b), was considerably dosed in sonochemistry experiments using Fricke mechanism [43], [44], [45], terephthalate dosimetry [44] and KI oxidation [43], [46], [47]. At single bubble scale, Fig. 3(b) demonstrates a monotonous decreasing trend from a value of 4.08 × 10−19 to 7.95 × 10−32 mol in the range 20–800 kHz, however, it is noticeable that the highest decrease occurs first when rising the frequency from 20 to 200 kHz where it attains 2.08 × 10−22 mol. The molar yield of H2O2 continues decreasing when the frequency increases from 200 to 500 kHz but in slower rate. An important decrease manifests again between 500 and 800 kHz, the hydrogen peroxide amounts passes from 3.52 × 10−24 to 7.95 × 10−32 mol. Unfortunately, most of simulation studies that treated the sonochemical production of hydrogen peroxide did not particularly examined the variation in function of frequency. The numerical study of Merouani et al. [19] demonstrated that at single bubble scale, the molar yield of hydrogen peroxide decreases gradually as the frequency increases. Kanthale et al. [48] proposed a numerical model of cavitation dynamics to explain sonochemical dosimetry of H2O2 in function of frequency, since their model did not involve the chemical mechanism of H2O2 formation, the analysis was performed based on dynamics and temperature results of simulation. The authors clearly stated that the dependence of H2O2 yield on frequency could only be speculated upon due to complexity of implicated phenomenon, according to them, the dependency relies on the average bubble temperature, heat and mass transfer effects and the number of cavitation bubbles. Actually, Kanthale et al. [48] raised in the previous work the scaling dilemma of sonochemical production of hydrogen peroxide in function of frequency.

Returning back to the results of the present work at reactor scale, Fig. 3(b) demonstrates an optimum production of hydrogen peroxide at 200 kHz, with a molar yield that equals 1.59 × 10−15 mol. This value is 2.3 times higher than that retrieved under 20 kHz, but only 1.51 times higher than that under 300 kHz. The highest decrease is observed when rising the acoustic frequency from 500 to 800 kHz, bringing the hydrogen peroxide molar yield to 7.95 × 10−32 mol.

The comparison of the simulated variation with experimental findings exhibits a perfect agreement with several works reported in Table 1. For instance, Pétrier and Francony [49] who scanned a large frequency interval ranging from 20 to 800 kHz, retrieved an optimum H2O2 production under 200 kHz. Jiang et al. [50] conducted an experimental study under the same acoustic frequencies as the previous authors, they also demonstrated that optimum H2O2 yield is reached under 200 kHz. Some other researchers who covered as well a large frequency range reported an optimum production of H2O2 under 300 kHz, such as Mark et al. [44]. Some others scanned more restricted values of acoustic frequency as compared to the latter, such as Kanthale et al. [48], Lim et al. [51], Pétrier et al. [52] and Wayment et al. [53]. Kanthale et al. [48] reported an optimum sonochemical production of hydrogen peroxide under 213 and 355 kHz, for a frequency interval comprised between 213 and 1056 kHz. Wayment et al. [53] and Lim et al. [51] retrieved an optimum under 300 kHz, though they scanned the ranges 20–446 kHz and 35–1000 kHz, respectively, the acoustic frequency of 200 kHz was not included in both studies. The results of the present study are then in perfect agreement with the aforementioned experimental results.

Fig. 3(f) reports the simulated evolution of HO• molar yield within single acoustic cavitation bubble and at reactor scale. This figure shows that at single bubble level, the highest production is expected under 20 kHz excitation with a molar yield of 3.58 × 10−17 mol, it then decreases gradually as the frequency increases to attain 4.24 × 10−24 mol under 800 kHz, i.e. a 107 times lower value. The previous trend was pointed out in several numerical works of our research group, indeed, Kerboua and Hamdaoui [20] as well as Ferkous et al. [54] and Merouani et al. [19] demonstrated decreasing variation of hydroxyl radical production at single bubble scale in function of increasing acoustic frequency. Though the previous authors conducted simulations on acoustic period durations corresponding to each frequency, their results are comparable to the present study since they expressed hydroxyl radical molar yields in terms of rates, i.e. as average production reported to the unit of time [19], [54]. In the other hand, Fig. 3(f) reveals a particular variation trend at reactor scale, apparently, HO• radicals attain a maximum sonochemical production under 300 kHz, with a value of 5.12 × 10−13 mol, produced within 50 µs irradiation inside the considered reactor. This value is 1.35 times higher than under 200 kHz and 6.74 times higher than under 20 kHz. HO• molar yield starts decreasing in function of the increasing acoustic frequencies from 300 kHz, however and interestingly, a rebound is observed at 500 kHz, HO• production reaches 3.35 × 10−13 mol, i.e. 1.53 times lower than the absolute maximum observed under 300 kHz. The hydroxyl radical molar yield decreases rapidly then after to attain 3.55 × 10−15 mol under 800 kHz. These results suggest that, quantitatively, HO• is the less sensitive species to frequency modification among the seven emerging sonochemical products. Indeed, the ratio of molar yields reported to the highest yield recorded in function of the acoustic frequency, demonstrates the highest values in the case of HO•, as compared to the six other sonochemical products.

Before comparing the present results to experimental hydroxyl radical production at reactor scale, we deem important to review the principle of sonochemical dosimetry of HO•.

In fact, hydroxyl radical quantification relies on three techniques, KI oxidation [55], [56] based on Bray-Liebhafsky mechanism [57], [58], Fricke reaction [59] and Terephthalate dosimetry [60].

As reported earlier in Table 1, Suzuki et al. [61], Pétrier et al. [52] and Koda et al. [43] assessed the sonochemical performance of their reactors using KI oxidation, the technique is based on the reactivity of iodine ions (supplementary material).

The amount of formed by the reaction of excess of and iodine ion refers to the amount of formed HO• since the KI solution is directly irradiated with ultrasound. Suzuki et al. [61] observed a maximum I3− generation under 200 kHz when they experimented the sonication using 24.1, 200, 500 and 1170 kHz. The present study reveals a similar trend, indeed, 200 kHz leads to 3.78 × 10−13 mol of HO•, while 500 kHz results in 3.34 × 10−13 mol.

The simulation results concord as well with the early study of Pétrier et al. [52] where the authors demonstrated that the irradiation with 514 kHz conducts to higher concentration of than under 20 kHz. The authors qualified that result as an unexpected frequency effect, it appears through the present simulations that 500 kHz excitation generates 4.39 times more HO• as compared to 20 kHz as shown in Fig. 3(f).

The results of the present simulations are also compared to those of the study of Koda et al. [43]. These authors quantified the production of HO• under 19.5, 25, 40, 45, 96, 130, 200, 300, 400, 500 and 1200 kHz through KI oxidation and confirmed that the optimum is attained at 300 kHz, which concords with the absolute optimum observed in Fig. 3(f). Interestingly, they also revealed a rebound of formation under 500 kHz, which is equivalent to another increase of the molar yield of HO• at 500 kHz, this observation confirms the trend observed in Fig. 3(f).

It is worthy to note that when samples are withdrawn from the irradiated solution and then dosed using KI under acidic conditions, ions undergo oxidation by H2O2 molecules according to the reaction:the expected optimum according to the present model can then balance between 200 and 300 kHz, since the technique cannot rout the origin of dosed H2O2, that may come from the reactivity of H•, HO2•, HO•, as shown in reactions 40 to 44 reported in Table 2 and highlighted by Anbar and Pecht [62].

The absolute optimum of the production of HO• at 300 kHz is in good agreement with the finding of Mark et al. [44] who adopted the Fricke dosimetry [59] to assess the variation of the sonochemical production of HO• over the frequencies 80, 200, 300, 400, 500, 600, 700, 900, 1000 and 1100 kHz. The experimental study of Mark et al. [44] demonstrated that the highest amount of Fe3+ is attained under 300 kHz, which is directly proportional to the molar yield of HO• owing to the reaction schema of Fricke dosimetry (supplementary material).

Mark et al. [44] confirmed the aforementioned result using terephthalate dosimetry [44], [60], which is based on terephthalate ions hydroxylation by HO• radicals to give rise to 2-hydroxyterephthalate, dosed through fluorescence. The authors demonstrated that optimum manifests at 300 kHz.

Finally, we suggest comparing the present results to those related to the sonolytic degradation of organic pollutants (RH) [49], [50], [51], [53], [63], [64], which is an indirect quantification of HO• according to the chemical schema of radical attack of HO• on organic molecules RH (supplementary material).

Pétrier et al. [49] examined the degradation of phenol under 20, 200, 500 and 800 kHz and demonstrated that the highest degradation rate is attained under 200 kHz, while Son et al. [64], who studied the degradation of bisphenol A under both 36 and 262 kHz frequencies reported an optimum degradation under 262 kHz. The results of both studies are in perfect agreement with the trend shown in Fig. 3(f) and the molar yields of HO• mentioned in the previous paragraph. Besides, Chiha et al. [63], Lim et al. [51] and Wayment et al. [53], who accounted for 300 kHz among the examined frequencies for the degradation of 4-cumylphenol, chlorobenzene, chloroform, carbon tetrachloride and alachlor, agreed on the optimal degradation of the treated organic pollutant under 300 kHz, which confirms the absolute optimum of HO• production retrieved in the present study.

In the other hand, Fig. 3(d) and (e) represent the effect of frequency on the molar yield of both H• and HO2•, respectively. H• sonochemical production at reactor scale attains a maximum of 2.9 × 10−16 mol under 200 kHz, a value that is 45.7 times higher than under 20 kHz, and 106 times higher than under 800 kHz. The decrease of the molar yields of both H• in function of acoustic frequency starting from 200 kHz is clearly monotonous and does not show any rebound. Contrariwise, the molar yield variation of HO2• in function of frequency demonstrates an optimum at 200 kHz with 1.51 × 10−12 mol, a value that is 7.8 times higher than under 20 kHz irradiation. However, starting from 200 kHz, the molar yield of HO2• decreases until rebound appears at 500 kHz, it then attains 3.88 × 10−13 mol, which is almost 3 times lower than the absolute optimum. HO2• and HO• sonochemical productions demonstrate similar variation trends in function of acoustic frequency, while the optimums attained under 200 and 300 kHz, respectively, are clearly the results of a compromise between the evolution of sonochemical production at single bubble level and bubble number density. The explanation of the rebound at 500 kHz requires deeper investigation of the intervening factors. In fact, and according to Eq. (22) and Table 3, the variation of the frequency decides as well of the spatial distribution of the molar yield within the reactor, though a unique and same volume is considered in the present study whatever the excitation frequency, the wavelength decreases as the frequency increases. Then, the control volume becomes smaller and the spatial variation of the molar yields of species becomes less pronounced. When integrating over the whole reactor volume, the spatial distribution effect, added to the production at single bubble scale and the number density of bubble within each control volume, induces the apparition of a rebound in the sonochemical production under 500 kHz at reactor scale only for the species which are predominant at single bubble level, namely HO2•, HO•, O3 and O, as depicted previously from Fig. 2.

To confirm the preceding explanation, we suggest to observe the molar yields variation of O3 and O presented in Fig. 3(c) and (g), respectively. The molar yield of O3 reaches an absolute maximum under 200 kHz with a value of 1.51 × 10−12 mol, a rebound appears under 500 kHz with a value of 5.48 × 10−14 mol. The sonochemical production of O exhibits an optimum under 200 kHz, where its molar production attains 3.07 × 10−12 mol, Fig. 3(g) reveals a recovery in the downward curve at 500 kHz but without resulting in another maximum. The rebound at 500 kHz is clearly more pronounced for HO2• (Fig. 3(e)) then HO• (Fig. 3(f)) followed by O3 (Fig. 3(b)), this order is exactly that of their emergence within the single cavitation bubble as demonstrated in Fig. 2(a), which supports the explanation suggested previously.

The results revealed that the present section have two major practical outcomes. Firstly, they point-out the limitations of the single bubble model in forecasting the sonochemical production, in particular the optimal operating frequency under oxygen atmosphere, and present a novel conceptualization of the numerical prediction of the sonochemical activity. Secondly, they give guidelines for conducting a sonochemical reaction under optimal frequency conditions according to the intended application, for instance, they demonstrate that 200 kHz is an optimal acoustic frequency for hydrogen production, while 300 kHz maximizes the generation of hydroxyl radicals and hence is optimal for degradation of pollutants in water.

Effect of the acoustic frequency on the concentrations within active and reacting volumes

At this stage of the present work, we suggest to report the production at single bubble scale to the same space reference by expressing the sonochemical production in terms of concentration. Indeed, the acoustic cavitation bubble volume closely depends of the frequency, in order to analyze and compare the molar yields within the “active volume”, i.e. the internal volume of oscillating bubbles, it is necessary to evaluate their concentrations, reported to the unit of active volume. Fig. 5 presents the temporal evolution of the molar concentrations of H2, H2O2, O3, H•, HO2•, HO• and O during the 50 µs of irradiation time within the active volume and the control volume around x1 in the sonochemical reactor, under a 20 kHz excitation. Further finding regarding frequency dependency are presented later in Fig. 6.

Fig. 5

Evolution of molar concentration of emerging species in function of time within active volume (orange line) and control volume around x1 within sonochemical reactor (black line) under 20 kHz frequency and 1.5 atm amplitude, during 50 µs: (a) , (b), (c), H• (d), HO2• (e), HO• (f) and (g). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6

Maximum molar concentrations in active volume (orange line and circular markers) and reactor volume (black line and square markers) of (a) , (b), (c), H• (d), HO2• (e), HO• (f) and (g). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4 reveals that, under 20 kHz, maximum concentration within the active volume is attained around the instant of the harsh collapse, i.e. at 39.7 µs, as predictable. This is due to the augmentation of species molar yields inside the single bubble volume, in one hand, and the strong contraction of the bubble volume, in the other hand. The maximum concentration in the control volume around x1 occurs as well by the end of the 50 µs, exactly at 45.6 µs. The observation established under 20 kHz is applicable to all the treated frequencies, however, several peaks of concentrations appear according to the acoustic period. The concentrations at both scales are related by the void rate, indeed, while the chemistry due to ultrasound evolves within the bubble volumes qualified of active, we usually refer to the sonochemical reactor by its macroscopic volume, i.e. the irradiated volume. The active volume constitutes a ratio defined as the void fraction, as compared to the reactor volume, this parameter has been deeply studied in our previous paper [24].

Scalability of sonochemical concentration and mass focusing factor

In order to compare the maximum concentrations attained within the active and the reacting volumes submitted to the same pressure field, the values of maximum concentrations of H2, H2O2, O3, H•, HO2•, HO• and O within the single acoustic cavitation bubble, and within the control volume around the coordinate x1 in function of the acoustic frequencies were reported in Fig. 6. The Fig. 6(a)–(g) denote a quasi-parallelism of the curves within the active and reacting volume submitted to the same acoustic field starting from 400 kHz. Since the concentrations at both scales are related by the void fraction, and considering that both values occur almost during the same time slot (but not at the same instant), the previous observation would be explained by the stability of this parameter in function of frequency starting from 400 kHz. Indeed, in a recent numerical investigation of the void fraction due to a population of acoustic cavitation bubbles, Kerboua and Hamdaoui [24] reported that ultrasonic irradiations of 443, 500, 600, 800 and 1000 kHz result in 0.009%, 0.010%, 0.011%, 0.010%, and 0.012% of void. The void fraction is more or less stable around 10−4 within the previous frequency range, which confirms the above explanation. Nevertheless, the void fraction expresses the relationship between active and reacting volumes, instantaneously, and since the maximum concentrations at both scales may occur with a certain time delay, we suggest to concisely assess the ratio of concentrations using the mass focusing factor that we defined in Eq. (32), applied to the maximum concentrations reported in Fig. 6. The obtained results are presented in Table 4. The objective of such procedure is to examine the possibility to predict the upscaled sonochemical production starting from the study of the sonochemical activity of the single acoustic cavitation bubble. As an overall observation, the maximum concentration within the control volume around the coordinate x1, supposed submitted to the same acoustic field as the single acoustic cavitation bubble whose activity is examined in the present paper, is 103 to 106 lower than within the active volume, i.e. the single bubble volume. However, the ratio shows not only a frequency dependency, but it varies also in function of the sonochemical species. The highest ratios are all recorded under the lowest frequency, namely 20 kHz, while the lowest ratios are all recorded under 800 kHz. The chemistry occurring inside the sonochemical reactor is perceived according to this definition as a dilution of mass focused in a small portion of the reactor volume. HO2• and HO•, which are the predominant sonochemically emerging species, present exactly the same orders of mass focusing ratios in function of the acoustic frequency. Besides, the variation in function of frequency reveals that the mass focusing ratio is comprised between the orders of magnitude of 104 to 105 for frequencies ranging from 200 to 600 kHz, however, the variation is neither monotonous in function of frequency, nor similar from one species to another.

Table 4

Variation of the mass focusing factor in function of the acoustic frequency for the seven sonochemical products.

Frequency (kHz)	H2	H2O2	H•	HO2•	HO•	O	O3
20	1.57 × 105	8.35 × 104	1.71 × 105	6.85 × 105	6.74 × 105	1.68 × 106	3.50 × 104
200	2.02 × 105	1.29 × 105	3.21 × 104	3.26 × 104	4.46 × 104	2.84 × 104	5.22 × 104
300	4.69 × 104	4.77 × 104	3.46 × 104	2.90 × 104	1.92 × 104	3.59 × 104	2.21 × 104
360	3.46 × 104	1.10 × 104	9.16 × 104	2.74 × 104	2.53 × 104	4.95 × 104	1.86 × 104
443	4.22 × 104	6.62 × 104	3.15 × 104	4.12 × 104	4.10 × 104	3.22 × 104	4.76 × 104
500	1.06 × 104	7.88 × 103	4.22 × 104	3.01 × 103	2.36 × 103	1.03 × 104	5.02 × 103
600	1.97 × 104	4.15 × 104	1.45 × 104	1.01 × 104	9.91 × 103	1.40 × 104	9.94 × 103
800	5.92 × 103	9.40 × 103	5.74 × 103	1.02 × 103	1.02 × 103	4.86 × 103	2.04 × 103

Conclusion

The present numerical study is an attempt to resolve the scalability dilemma related to the frequency dependency of sonochemical production, based on the mathematical modeling and simulation of the sonochemical activity within a single acoustic cavitation bubble, referred to as active volume, as well as within a sonochemical reactor of 1 dm3 discretized into elementary control volumes.

The qualitative analysis of the sonochemical activity under an oxygen atmosphere revealed that the predominant species are HO2• then HO• followed by O3 and O. The quantitative study in function of frequency revealed that at single bubble scale, the molar yield decreases monotonously when the frequency increases. At reactor scale, the integration of the sonochemical production over the reactor volume demonstrated the apparition of an optimum at 200 kHz for all species, except HO•. This latter attains a maximum production under 300 kHz. The observed trends showed a good agreement with the experimental results based on KI oxidation, Fricke reaction, terephthalate dosimetry and organic pollutants degradation rates.

Interestingly, the molar yields of the predominant species manifested a pronounced rebound at 500 kHz at the reactor scale, the phenomenon was explained by the spatial distribution inside the reactor and confirmed by several experimental studies.

Based on the present results, it appears that the selection of operational acoustic frequency should be in accordance with the objective of sonication, for instance, 300 kHz would lead to the optimal degradation rates of pollutants under O2 atmosphere.

Finally, the variation of the concentration within the active and reacting volumes submitted to the same acoustic field was investigated. Maximum concentrations were depicted at both scales and their ratio, defined according to the present study as “mass focusing factor”, was reported for the seven sonochemical products versus the acoustic frequency. The mass focusing factor varies between 105 and 106 under 20 kHz, 104 and 105 between 200 and 600 kHz, and decreases to 103 under 800 kHz. Interestingly, HO2• and HO• present exactly the same orders of mass focusing ratios in function of the frequency.

CRediT authorship contribution statement

Kaouther Kerboua: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft, Writing - review & editing. Oualid Hamdaoui: Project administration, Visualization, Supervision, Writing - review & editing. Abdulaziz Alghyamah: Visualization, 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.