Modeling of sonochemistry in water in the presence of dissolved carbon dioxide

Ultrasonics - Sonochemistry 45 (2018) 17–28

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Ultrasonics - Sonochemistry journal homepage: www.elsevier.com/locate/ultson

Modeling of sonochemistry in water in the presence of dissolved carbon dioxide

T

⁎

Olivier Authiera, , Hind Ouhabaza, Stefano Bedognia,b a b

EDF R&D Lab Chatou, 6 quai Watier, 78400 Chatou, France Edison, Foro Buonaparte 31, 20121 Milan, Italy

A R T I C LE I N FO

A B S T R A C T

Keywords: Modeling Water Carbon dioxide reduction

CO2 capture and utilization (CCU) is a process that captures CO2 emissions from sources such as fossil fuel power plants and reuses them so that they will not enter the atmosphere. Among the various ways of recycling CO2, reduction reactions are extensively studied at lab-scale. However, CO2 reduction by standard methods is difficult. Sonochemistry may be used in CO2 gas mixtures bubbled through water subjected to ultrasound waves. Indeed, the sonochemical reduction of CO2 in water has been already investigated by some authors, showing that fuel species (CO and H2) are obtained in the final products. The aim of this work is to model, for a single bubble, the close coupling of the mechanisms of bubble dynamics with the kinetics of gas phase reactions in the bubble that can lead to CO2 reduction. An estimation of time-scales is used to define the controlling steps and consequently to solve a reduced model. The calculation of the concentration of free radicals and gases formed in the bubble is undertaken over many cycles to look at the effects of ultrasound frequency, pressure amplitude, initial bubble radius and bubble composition in CO2. The strong effect of bubble composition on the CO2 reduction rate is confirmed in accordance with experimental data from the literature. When the initial fraction of CO2 in the bubble is low, bubble growth and collapse are slightly modified with respect to simulation without CO2, and chemical reactions leading to CO2 reduction are promoted. However, the peak collapse temperature depends on the thermal properties of the CO2 and greatly decreases as the CO2 increases in the bubble. The model shows that initial bubble radius, ultrasound frequency and pressure amplitude play a critical role in CO2 reduction. Hence, in the case of a bubble with an initial radius of around 5 μm, CO2 reduction appears to be more favorable at a frequency around 300 kHz than at a low frequency of around 20 kHz. Finally, the industrial application of ultrasound to CO2 reduction in water would be largely dependent on sonochemical efficiency. Under the conditions tested, this process does not seem to be sufficiently efficient.

1. Introduction Due to increased concern over carbon dioxide (CO2) emissions and their effect on the environment and climate change, many methods have been devised to reduce or control them over recent decades. The most common technique currently undergoing global research and development is CO2 capture and storage (CCS), whereby CO2 is captured and commonly stored in deep geological formations [1]. Another method that has more recently been considered is CO2 capture and utilization (CCU), which involves, by various means, recycling CO2 into fuel gases, e.g. CO or CH4 [2]. However, CO2 reduction by standard methods (hydrogenation, electrochemical reduction, photocatalytic reduction, etc.) is difficult, since CO2 is a very stable molecule. Sonochemistry may be able to contribute to the scientific and technical studies on this issue. The experimental approach may employ a gas

⁎

mixture that contains the CO2 to be bubbled through water and subjected to ultrasound waves. Ultrasound through water involves the use of sound waves of a high frequency ranging from 20 kHz (i.e. with a cycle of 50 μs) to several MHz (i.e. with a cycle lower than 1 μs) that consist of rarefaction and compression cycles. Low-to-medium-frequency ultrasound is typically used for sonochemistry to reach higher localized temperatures and pressures [3]. The chemical effect of ultrasound comes from acoustic cavitation phenomena, i.e. after expansion to many times their initial size, the oscillation and collapse of bubbles filled with dissolved gases and with vapor from the liquid. When ultrasound is applied to liquid water, the liquid directly in contact with the bubble interface may be displaced during stable oscillating cycles (i.e. with bubbles oscillating around a mean radius for many acoustic cycles) or transient collapsing cycles (i.e. with bubbles existing for only a short time before collapsing). Bubbles can collapse after

Corresponding author. E-mail address: [email protected] (O. Authier).

https://doi.org/10.1016/j.ultsonch.2018.02.044 Received 27 October 2017; Received in revised form 11 February 2018; Accepted 26 February 2018 Available online 06 March 2018 1350-4177/ © 2018 Elsevier B.V. All rights reserved.

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Nomenclature

Greek letters

a A c C Cp D e ė E f h I k LHV m N P Q̇ r

β δR ΔC γ η λ μ ρ σ τ υ ω

Ṙ R¨ RG t T V w W Ẇ X Y

NASA coefficient preexponential factor speed of sound in the liquid phase, (m s−1) molar concentration, (mol m−3) thermal capacity, (J kg−1 K−1) mass diffusivity, (m2 s−1) internal energy, (J kg−1) time derivative of internal energy, (W kg−1) energy activation, (J mol−1) frequency, (Hz) enthalpy, (J kg−1) intensity, (W m−2) rate constant low heating value, (J kg−1) mass, (kg) number density, (# m−3) pressure, (Pa) heat loss, (W m−3) overall rate, (mol m−3 s−1) bubble radius, (m) bubble velocity, (m s−1) bubble acceleration, (m s−2) gas constant, (8.314 J K−1 mol−1) time, (s) temperature, (K) volume, (m3) molecular weight, (kg mol−1) energy density, (J m−3) work, (W m−3) conversion per reactant mass fraction

generalized temperature exponent thickness of thermal boundary layer, (m) molar concentration difference, (mol m−3) polytropic exponent chemical efficiency thermal conductivity, (W m−1 K−1) dynamic viscosity, (Pa s) density, (kg m−3) surface tension, (J m−2) characteristic time, (s) stoichiometric coefficient angular frequency, (rad s−1)

Subscripts

A B d di f G H2 O k L n r

0 ∞

acoustic bubble diffusion of gas in liquid phase diffusion of gas within the bubble forward gas phase water specie in gas phase liquid phase natural reverse at the bubble interface Initial static ambient pressure

sonochemistry of dissolved gases and CO2 in water, because direct measurement in a cavitation bubble is extremely difficult using traditional approaches [18]. The hot spot temperature and pressure as well as the composition of radicals and gas species in an oscillating bubble can be estimated more easily by modeling. The aim of this work is to model, for a single bubble, the close coupling of the mechanisms of bubble dynamics with the kinetics of gas phase reactions in the bubble that can lead to CO2 reduction. The calculation of the concentration of free radicals and gases formed in the bubble is undertaken mainly in order to look at the effects of ultrasound frequency, pressure amplitude, initial bubble radius and bubble composition. Finally, a comparison is made between theoretical predictions and literature measurements.

having grown to an unstable size during the rarefaction cycle, creating cavitation especially for high acoustic pressure amplitudes of about 1 bar and more [4]. Collapsing bubbles can emit light by sonoluminescence [3]. The collapse of bubbles near an adiabatic regime results in the generation of extreme conditions, e.g. temperatures of the order of several thousand degrees Kelvin and pressures of the order of several hundred atmospheres in localized zones, where reactions can occur by formation and recombination of free radicals (e.g. H, HO and HO2 during the sonolysis of water), while the overall liquid environment remains near ambient conditions. Thus, phenomena caused by ultrasonic techniques have been proven to be useful in many liquids to enhance reactions traditionally implemented with high temperature and pressure processes [5]. Many parameters affect cavitation, e.g. acoustic parameters (ultrasonic frequency, acoustic intensity), solvent properties (surface tension, viscosity, vapor pressure, thermal conductivity, speed of sound) and operating conditions (temperature, pressure, dissolved gases). The physicochemical properties of CO2 can change the pressure and the temperature associated with a collapse. Furthermore, CO2 can change the reaction scheme inside the bubble. Sonochemistry of CO2 dissolved in liquid water has been studied by some authors from 20 to 2400 kHz at lab scale in closed vessels with a volume in the 10–1000 cm3 range [6–10]. An interesting finding for CO2 reduction is that the final products formed in CO2 bubbles are mainly CO, H2, O2 and H2O2. In parallel to experimental studies with sonochemistry, some models based on the hot-spot theory have been developed to study bubble dynamics and the factors that affect sonochemistry [4,11–16]. However, very few models have studied the effect of dissolved CO2 on the aqueous medium in the presence of sonochemical activity [17]. There are still several fundamental issues concerning the

2. Previous works in literature The effects of several parameters on sonochemistry of CO2 in liquid water have been studied by several authors [6–10] in controlled conditions (Table 1), e.g. gas dissolved as nuclei in water (especially N2, Ar and He), temperature of liquid water, CO2 concentration in gas phase and ultrasonic frequency. At lab scale, Harada [7] and Harada and Ono [10] have shown that the decreasing rate of CO2 follows the order Ar > He > N2. Indeed, sonochemistry hardly occurs under polyatomic gases and does not proceed in general in a pure CO2 atmosphere. Generally, the capacity of Ar to support sonochemistry has been explained by both its thermal capacity and its thermal conductivity compared to polyatomic gases. Thus, an insulating gas with low thermal conductivity contributes to the slowing down of thermal dissipation from cavitation zones to liquid water. Furthermore, using a dissolved gas that has a high specific heat capacity ratio is efficient for achieving higher temperatures during 18

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Table 1 Experimental studies at lab scale on sonochemical reduction of CO2 in water. Authors Reference Frequency (kHz) Intensity (W/cm2) Power (W) Sonication time (min) Water temperature (K) CO2 in gas phase (%) Main gaseous products

Henglein [6] 300 3.5 n/a 2–15 293 0–7 CO, H2, O2

Harada [7] 200 n/a 200 60–1140 278–318 0–4 CO, H2, O2

Harada et al. [8] 200 n/a 200 120–300 298 0.4–5 CO, H2, O2

Koblov [9] 20 40–95 1000 60 288 5–75 CO, H2

Harada and Ono [10] 28, 200, 2400 n/a 15–100 10–180 298 0–25 n/a

3.1. Assumptions

bubble collapse. The temperature achieved in pure CO2 cavitation bubbles is lower because of the specific heat capacity ratio of 1.28 for CO2 against 1.66 for Ar at ambient conditions. So, Ar is an efficient atmospheric gas for promoting CO2 reduction under ultrasound. Cavitation also seems to be facilitated by gases with high solubility providing more nucleation sites in water and lowering the threshold for nucleation [3]. With regard to the effect of liquid water temperature, the CO2 reaction rate is lower with liquid temperature increase in the 278–318 K range [7]. Indeed, as the liquid temperature is raised, the vapor pressure of water in bubble is also increased, leading to lower sonochemical effects because of less violent collapses. Another significant finding is that the CO2 reduction rate strongly depends on CO2 concentration. The effect of CO2 concentration in the gas phase on product yields shows optimal peak values with a low percentage concentration of CO2 in the gas phase, typically in a range between 1 and 3 per cent in mole [7,19]. The product yields then decrease with higher CO2 concentrations in the gas phase. So, for large amounts of CO2, the reactions are inhibited, while small amounts below 3 per cent result in an improvement of sonochemical rates. Henglein [6] and Harada et al. [7,8] have shown that the final products in the gas phase are mainly CO, H2 and a small amount of O2. The formation of hydrogen peroxide (H2O2), which is highly soluble in water, is also reported [6–8]. As a first approximation, the CO yield depends on the rate of CO2 reduction, whereas the H2 yield depends on the rate of water sonolysis. Some modeling studies have attempted to understand the coupling between bubble dynamics and the kinetics of the gas phase reactions in the bubble [11–13,15–17,20]. The main phenomena are considered in detailed models, e.g. bubble motion, mass and thermal transport, phase change, chemical reactions, etc. However, the comparison of numerical calculations with experimental data is difficult, partly because the number of bubbles in the liquid evolves during the experiments. The most insightful study on the effect of dissolved CO2 on cavitation in water is the one by Gireesan and Pandit, based on a single bubble cavitation model coupled with 11 chemical reactions involving 8 species [17]. In the reaction scheme, CO2 is involved in 3 chemical reactions, i.e. dissociation and reactions with O and H radical. The simulations for an Ar-CO2 bubble have been conducted at different ultrasound frequencies (from 213 to 1000 kHz) and acoustic pressure amplitudes (from 3.22 to 10 bar). The results have shown that the collapse temperature and the production of HO radicals decrease as the CO2 fraction of the bubble increases. This finding confirms that the use of CO2 is more beneficial at low concentrations. Besides, the production of HO radicals is the highest at the lowest ultrasonic frequency (i.e. 213 kHz) and at the highest acoustic pressure amplitude (i.e. 10 bar) that lead to bubble intense collapse. Hence, this model constitutes the first comprehensive study of the effect of frequency and pressure amplitude during the cavitation of bubbles containing CO2.

We consider a single bubble submitted to ultrasound waves in unbounded pure liquid water, i.e. without considering solid particles and the effects of the bubbles on one another. In a simple approach, the dissolved gases present initially in liquid water are Ar and CO2. The main model assumptions and characteristics are discussed below. If the rate-controlling phenomena are not identified, the model must consider sophisticated couplings between conservation of mass, momentum and energy. Thus, an estimation of time-scales is used to define the controlling steps and consequently to solve a reduced model. However, such a preliminary analysis is limited because many parameters vary greatly as a function of temperature. Consequently, this preliminary analysis is sometimes restricted to qualitative effect discussion. – The interactions between the bubble and its surroundings are neglected. Since the work of Bjerknes on the force of interaction between two pulsating bubbles, the bubble–bubble interactions in an acoustic field is well known [21]. The interaction effects may be significant, e.g. when the bubbles’ radii and the spacing between them are small in comparison to the wavelength of the sound wave. It is of smaller importance when the bubbles are at a large distance from each other. – The bubble shape remains spherically symmetric. An isolated bubble is spherical, i.e. with the minimization of the surface area for a given volume, notably when the surface tension dominates over other forces and the pressure is equal all around the bubble interface. When a bubble is moving through a liquid phase and is interacting with nearby bubbles, it can lose its spherical shape because the dominant forces on the bubble change. Other instability mechanisms, e.g. Rayleigh-Taylor instability, may cause deviations from sphericity [22]. However, the impact of asymmetric bubbles on sonochemistry is not clear. – The temperature in the bubble is spatially uniform. The external heat transfer time scale, the internal heat transfer time scale and the representative time scale of the bubble are in competition. Firstly, the ratio of external and internal heat transfer resistances is the Biot number, which determines if the temperature inside the bubble will vary in space. In general, the rate-controlling phenomenon is external heat transfer when the Biot number is lower than 1, i.e. uniformity of temperature fields within the bubble. Secondly, when the external heat transfer time scale is longer than the dynamic time scale, the bubble behaves adiabatically. The representative time scale of the bubble is here close to the time scale of the ultrasound wave. It can be noticed that the temperature gradient between the bubble and the liquid water can simply be modeled by a heat diffusion term loss in the energy equation for the non-adiabatic process. In this case, the temperature of the liquid remains undisturbed. Finally, the thermal

3. Modeling The present model is based on the following equations: bubble dynamic equation, energy and mass conservation in the bubble, and the equation of state for the gases in the bubble. 19

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[11–13,15,16]. Since we wish to study mainly the chemical reactions occurring in a few bubble oscillation cycles, initial bubble vapor composition is assumed and mass transfer due to evaporation and condensation is neglected.

diffusivity in the bubble can be compared to the mass diffusivity produced by the Lewis number, assumed to be close to unity. – The distribution of species in the bubble is spatially uniform.

– Ionization effects, i.e. the formation of ions and electrons during bubble collapse, are not considered.

When the time scale of internal diffusion is lower than the time scale of bubble dynamics, the bubble composition is uniform. The time scale for the diffusion of gas within the bubble is given by:

τdi =

R2 DG

The ionization energy for Ar (15.8 eV), CO2 (13.8 eV) and H2O (12.6 eV) is larger than the typical equivalent temperatures during bubble collapse (104 K–0.9 eV).

(1)

For representative values such as 1 μm for bubble radius and 10−5 m2 s−1 for mass diffusivity in the gas phase at normal conditions, the time scale for gas diffusion is about 0.1 μs, which is lower than time scale of bubble dynamics, so that a uniform distribution of species can be justified.

– The gas in the bubble is ideal. This assumption is very useful and widely used due to its simplicity [4,12,13]. Intermolecular forces in the gas are neglected as a first approximation. However, they are likely to play a role in the conditions of extreme gas temperature and pressure. The compressibility factor is a correction factor that describes the deviation of real gas from ideal gas behavior. Its value (1 for ideal gas) generally increases with pressure and decreases with temperature.

– The pressure in the bubble is spatially uniform. The dynamics of a gas bubble strongly depends on the gas pressure. A simplification consists of the approximation of a spatially uniform pressure in the bubble, especially at low excitation magnitudes, and not very violent collapses. It requires the bubble radius to be smaller than the sound wavelength in the gas and the Mach number of the motion of the bubble interface (representing the ratio of velocity of the bubble interface to the speed of sound in the bubble) to be small:

|R|̇ <1 c

– The chemical reactions that obey the Arrhenius form occur within the bubble and not in the liquid phase. – The energy dissipation by viscous shear stress is neglected. 3.2. Bubble dynamic equation – energy and mass balances

(2) The dynamic behavior of the bubble interface is described by the equation obtained by Keller (and his co-workers) from the continuity and momentum balance equations in compressible liquid [28]:

The assumption of the spatial uniformity of pressure inside a bubble is out of order in the case of a Mach number larger than one.

̇ ̇ ̇ ⎛1− R ⎞ RR¨ + 3 ⎛1− R ⎞ Ṙ2 = 1 ⎛1 + R + R d ⎞ (PR−P∞ + PA) 2 ⎝ 3c ⎠ ρL ⎝ c c dt ⎠ ⎝ c⎠

– The mass transfer across the bubble interface and the mass interaction with the liquid phase (e.g. condensation, evaporation, absorption, etc.) are not considered.

⎜

ρG R2 2DL ΔC

⎜

⎟

⎜

⎟

(4)

where is the bubble radius, Ṙ and R¨ represent respectively the first(velocity) and second-order (acceleration) time derivatives of the bubble radius, c is the speed of sound in the liquid water far from the bubble, ρL is the density of liquid water, PR is the pressure at the bubble interface (i.e. on the liquid side of the interface), PA is the acoustic pressure and P∞ is the static ambient pressure (usually equal to 1 atm). At the initial time,

During the bubble collapse, gas may gradually dissolve into liquid water and water vapor may condense at the bubble interface. The amount of gas and water vapor trapped in the bubble is a consequence of diffusion as well as of water-phase change by condensation and evaporation [23]. When the dynamic time-scale of the bubble is lower than the external mass transfer time-scale or water-phase change (e.g. in the case of a high speed of bubble collapse) then no mass can escape the bubble, and the diffusion of gas species in the liquid can be disregarded. The characteristic time required for complete diffusion of the mass into liquid can be estimated by [24]:

τd =

⎟

R (t = 0) = R 0

(5)

Ṙ (t = 0) = 0

(6)

As mass transfer is neglected at the interface, pressure at the bubble interface is controlled by the surface tension (the Laplace pressure) and the liquid viscosity according to:

(3)

.

Typical values are higher than 0.05 s under the assumptions that the ratio ΔC ρG is about 0.01 coupled with a value of mass diffusivity in the liquid phase of about 10−9 m2 s−1 at normal temperatures and with a bubble radius of 1 μm. Within a few bubble oscillation cycles, these time scales are higher than the time scale of bubble dynamics. Thus, dissolution of the gaseous compounds into the surrounding liquid has a negligible effect. The effect of condensation and evaporation at the bubble interface has been studied in detail e.g. by Yasui et al. [25–27]. Non-equilibrium evaporation and condensation of water at the bubble wall can have a significant effect on the dynamic behavior of a bubble. For accurate study over multiple cycles, the physics of mass transfer has to be coupled to the chemical reactions, especially at the collapse of the bubble. However, the partial pressure of water vapor is almost identical to the saturated vapor pressure (at ambient liquid temperature), except at the strongest collapses [26]. Moreover, the effects of evaporation and condensation are commonly neglected to facilitate the analysis

2σ 4μ R PR = PB− L − L R R

(7)

where σL is the surface tension, μL is the dynamic viscosity of liquid water and PB is the pressure within the bubble. The pressure far from the bubble due to the acoustic field is described by the sinusoidal function:

PA = PA0 sin(ωt ) = PA0 sin(2πft )

(8)

where PA0 is the ultrasound pressure amplitude, ω is the ultrasound angular frequency and f is the ultrasound frequency. In addition to frequency, the main characteristics of ultrasound are acoustic intensity (expressed in W/m2) and acoustic pressure amplitude. The relationship between them is the following:

PA0 = (2IA ρL c )0.5

(9) 3

The acoustic energy density (expressed in J/m ) is: 20

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WA =

1 PA0 2 I ⎛ ⎞ = A 2ρL ⎝ c ⎠ c

X = 1− (10)

– Energy conservation

̇ Ẇ ρG e ̇ = Q−

When bubbles oscillate in a sound field, water vapor and CO2 may dissociate and produce radicals during bubble collapse. The main radicals are HO and H during sonolysis in liquid water [14]. CO2 may, on the one hand, react with the radicals from the water vapor sonolysis and, on the other hand, be directly decomposed during bubble collapse. The chemical reduction of CO2 involves H, O and HO radicals which are obtained during the sonolysis of water. Many reaction mechanisms are available for describing high temperature reactions in the gas phase, especially from studies in flames and shock tubes. However, the kinetic scheme in the bubble is extremely difficult to validate during collapse conditions, at very high pressure and temperature. A kinetic mechanism based on 19 reversible elementary chemical reactions is used to describe the reactions of hydrogen and oxygen compounds as the H2/O2 mechanism between stable species H2O, O2, H2, H2O2 and radicals H, O, HO (hydroxyl), HO2 (hydroperoxyl) [29]. A mechanism based on 12 reversible elementary chemical reactions describes the reactions of carbon, hydrogen and oxygen compounds as the CO/H2/O2 mechanism between stable species CO2, CO and radicals H, O, HO, HO2, HCO (formyl) [30]. The H2/O2 mechanism is a sub-mechanism of the CO/H2/O2 mechanism, which is also a sub-mechanism of CH4 oxidation. The kinetic parameters for both mechanisms are given in Table 2. For a reversible reaction, the overall rate is the difference between the forward and reverse reaction rates as follows:

(11)

ė represents the time derivative of internal energy e : P

e = h− ρB = ∑ Yk hk− ρB G

G

k

T

where hk = hk ref + ∫ Cpk dT (12)

Tref

k represent the species in the gas phase (H2O, O2, H2, H2O2, Ar, CO, CO2 and radicals H, HO, O, HO2, HCO), hk is the enthalpy of the gas specie, Yk is the mass fraction of the gaseous product, Cpk is the thermal capacity at constant pressure of the gaseous specie and ρG is the density of gas phase. Q̇ is the heat loss by conduction across the bubble wall and may be written:

Q̇ = 3λ

T∞−T RδR

(13)

where λ is the thermal conductivity, T∞ is the water temperature and δR is the thickness of the bubble-water thermal boundary layer. Ẇ is the work done by the bubble:

P Ẇ = B V̇ V

r = kf

∑ (hk Yk̇ + Yk Cpk Ṫ )−3λ k

E ⎞ k = AT β exp ⎛− ⎝ RG T ⎠

(15)

⎜

At the initial time, a static bubble is considered in the absence of ultrasound wave:

2σ PB (t = 0) = P∞ + L R0

(16)

T (t = 0) = T0

(17)

Yk (t = 0) = Y0

(18)

(22)

j

where C are the molar concentrations of the species involved in the reactions and ν are the net stoichiometric coefficients. The subscripts f and r denote the forward and the reverse reactions respectively. According to the Arrhenius law, the rate constant can be expressed as:

(14)

T∞−T RδR

∏ Cifνi−kr ∏ C jrνj i

By combining Eqs. (11)–(14), the following form is obtained:

PḂ = ρG

(21)

3.3. Chemical mechanism and kinetics

The energy equation in a control volume of bubble is obtained from the first law of thermodynamics, using the following assumptions: spatially uniform pressure and temperature inside the bubble, constant mass of gas inside the bubble, negligible viscous dissipation function and mass transfer, and linearization of the heat loss term. The energy balance can be written as follows:

P

Y Y0

⎟

(23)

where A is the frequency factor, β is the generalized temperature exponent and E is the activation energy. When a third body (M in Table 2) participates in a reaction, the reaction rate is modified by including the third body efficiency [29,30]. 3.4. Physical parameters used in the model – Initial composition of the bubble

The ideal gas law is used as the equation of state:

PB =

ρG RG T 1/ ∑ Yk / wk k

The initial composition of the bubble in water vapor can be estimated by the thermodynamic equilibrium at the initial temperature of the surrounding liquid. The saturated vapor pressure is estimated by the Antoine equation:

(19)

where wk is the molecular weight of species k and RG is the gas constant.

log PH2 O = a− – Mass conservation

wk r ρG

(24)

where P is the vapor pressure in bar, T is the water temperature in K and a (= 6.20963), b (= 2354.731) and c (= 7.559) are water specific constants. Thus, at the initial time, the mass fraction of vapor in the bubble is:

The mass conservation for each specie in the gas phase leads to the following equation:

Yk̇ =

b c+T

(20)

YH2 O (t = 0) =

where r is the overall rate. The conversion per reactant (H2O and CO2) is calculated according to:

PH2 O (T0 ) PB (t = 0)

(25)

It can also be noticed that the amount of dissolved gas in water is proportional to its partial pressure in the gas phase according to the 21

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constants of Henry’s law. Henry solubility, defined as the ratio of the concentration in the aqueous phase to the partial pressure in the gas phase under equilibrium conditions, is 3.4 × 10−4 mol m−3 Pa−1 for CO2 and 1.4 × 10−5 mol m−3 Pa−1 for Ar at 298.15 K. Gas solubility into liquid water also needs to be considered because the dissolved gas bubbles may serve as nucleation sites, especially in saturated liquid. Furthermore, Henry’s law constants for the main gaseous products CO, H2 and H2O2 at 298.15 K are around 9.4 × 10−6, 7.7 × 10−6 and 9.0 × 102 mol m−3 Pa−1 respectively.

Table 2 Reaction mechanism used in the simulations for CO2/Ar/H2O system and adapted from [29] and [30] (Units: cm3, mol, s, kcal, K and efficiency factors aH2 = 2.5 and H2O = 12, b H2 = 2.5, H2O = 12 and Ar = 0.83, cH2 = 2.5, H2O = 12 and Ar = 0.75, dH2 = 0.73, H2O = 12 and Ar = 0.38, eH2 = 1.3, H2O = 14 and Ar = 0.67, fH2 = 2.5, H2O = 12 and Ar = 0.64, gH2 = 2, H2O = 6, Ar = 0.5, O2 = 6, CO = 1.5 and CO2 = 3.5, hH2 = 2, H2O = 12, Ar = 1.5 and CO2 = 2). Reaction H2/O2 chain reactions 1 H + O2 ↔ O + HO 2

O + H2 ↔ H + HO

3

HO + H2 ↔ H + H2O

4

O + H2O ↔ HO + HO

H2/O2 dissociation/recombination reactions 5a H2 + M ↔ H + H + M b

6

O2 + M ↔ O + O + M

7c

HO + M ↔ O + H + M

8d

H2O + M ↔ H + HO + M

HO2 reactions 9e H + O2 + M ↔ HO2 + M 10

HO2 + H ↔ H2 + O2

11

HO2 + H ↔ HO + HO

12

HO2 + O ↔ HO + O2

13

HO2 + HO ↔ H2O + O2

H2O2 reactions 14 H2O2 + O2 ↔ HO2 + HO2 f

15

H2O2 + M ↔ HO + HO + M

16

H2O2 + H ↔ H2O + HO

17

H2O2 + H ↔ H2 + HO2

18

H2O2 + O ↔ HO + HO2

19

H2O2 + HO ↔ H2O + HO2

CO2 and CO reactions 20g CO2 + M ↔ CO + O + M 21

CO2 + O ↔ CO + O2

22

CO2 + H ↔ CO + HO

23

CO2 + HO ↔ CO + HO2

HCO reactions 24h HCO + M ↔ H + CO + M 25

HCO + O ↔ CO2 + H

26

HCO + O ↔ CO + OH

27

HCO + OH ↔ CO + H2 O

28

HCO + HO2 ↔ CO2 + OH + H

29

HCO + H ↔ CO + H2

30

HCO + O2 ↔ CO + HO2

31

HCO + HCO ↔ H2 + CO + CO

A

β

E

1.915 × 1014 5.481 × 1011 5.080 × 104 2.667 × 104 2.160 × 108 2.298 × 109 2.970 × 106 1.465 × 105

0 0.39 2.67 2.65 1.51 1.40 2.02 2.11

16.44 −0.293 6.292 4.880 3.430 18.32 13.40 −2.904

4.577 × 1019 1.146 × 1020 4.515 × 1017 6.165 × 1015 9.880 × 1017 4.714 × 1018 1.912 × 1023 4.500 × 1022

−1.40 −1.68 −0.64 −0.50 −0.74 −1.00 −1.83 −2.00

104.40 0.820 118.9 0 102.10 0 118.5 0

1.475 × 1012 3.090 × 1012 1.660 × 1013 3.164 × 1012 7.079 × 1013 2.027 × 1010 3.250 × 1013 3.252 × 1012 2.890 × 1013 5.861 × 1013

0.60 0.53 0 0.35 0 0.72 0 0.33 0 0.24

0 48.87 0.823 55.51 0.295 36.84 0 53.28 −0.497 69.08

1.434 × 10 1.300 × 1011 2.951 × 1014 3.656 × 108 2.410 × 1013 1.269 × 108 6.025 × 1013 1.041 × 1011 9.550 × 106 8.660 × 103 1.000 × 1012 1.838 × 1010

−0.35 0 0 1.14 0 1.31 0 0.70 2.00 2.68 0 0.59

37.06 −1.629 48.43 −2.584 3.97 71.41 7.95 23.95 3.97 18.56 0 30.89

9.874 × 1015 1.800 × 1010 8.035 × 1015 1.050 × 1012 5.896 × 1011 2.230 × 105 2.280 × 1016 3.010 × 1013

−0.934 0 −0.800 0 0.699 1.890 −0.47 0

130.0 2.384 51.23 42.54 24.26 −1.158 84.97 23.00

11

0.660 1.041 0 −0.553 0 0.638 0 0.551 0 0 0 0.656 0 0.309 0 0

14.87 −0.457 0 112.2 0 86.82 0 103.1 0 0 0 88.23 0.410 33.95 0 0

13

4.750 × 10 3.582 × 1010 3.000 × 1013 1.241 × 1018 3.020 × 1013 4.725 × 1011 1.020 × 1014 3.259× 1013 3.000 × 1013 0 7.340 × 1013 2.212 × 1012 7.580 × 1012 1.198 × 1012 3.000 × 1012 0

– Initial radius of the bubble The choice of the initial radius of the bubble is somewhat arbitrary, because a multi-bubble field is generated during experiments. Furthermore, bubble size distribution has not been widely studied under all operational conditions. However, the initial bubble radius is crucial in predicting the pressure and temperature of the collapse. Different techniques have been used to estimate the size and the distribution of bubbles experimentally, e.g. laser light diffraction, active cavitation detection and phase-Doppler methods. Representative experimental values from 0.9 to 5.0 μm are reported from 20 to 1100 kHz [31,32]. Other values from the literature in the case of Ar dissolved in water (roughly 20–30 μm at 20 kHz) are also used in the simulations [11,13,20]. An initial radius of up to 100 μm at around 20 kHz has been used in modeling studies [12]. It can be noticed that the initial bubble radius is also related to the natural bubble frequency that can be estimated from [24]:

fn =

1 ⎛ 3γP∞ 2σL ⎞ − ⎜ ⎟ 2π ⎝ ρL R 0 2 ρL R 03 ⎠

0.5

(26)

where fn is the natural frequency and γ is the polytropic exponent. – Thermal capacity, enthalpy and other parameters The NASA coefficients available in a temperature range from 200 K to 6000 K are used to calculate the following parameters [33]:

Cpk =

RG wk

5

∑ aik T i −1

(27)

i=1

5

hk =

RG T ⎛ a a ∑ ik T i −1 + T6 ⎞⎟ wk ⎜ i = 1 i ⎝ ⎠

(28)

where aik are numerical coefficients supplied in NASA thermochemical data for a low-temperature range (below 1000 K) and a high-temperature range (above 1000 K). The values of other parameters used in the model are given in Table 3. 3.5. Numerical model A Matlab-based program is used to solve the system of first-order differential equations (Matlab solver ode15s adapted to stiff problem). Table 3 Physical parameters used in the simulations. aThe thermal conductivity is assumed to be the one of Ar that is the predominant gas within the bubble. Parameter

22

Value

Thermal conductivity of gasa

W/m K

Speed of sound in liquid water Density of liquid water Surface tension of liquid water Viscosity of liquid water Thickness of bubble-water thermal boundary layer

m/s kg/m3 J/m2 Pa s m

λ = 9 × 10−3 + 3.2 × 10−5T c = 1500 ρL = 1000 σL = 0.073 μL = 0.001 δR = 0.2R

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The bubble dynamic equation is turned into two first-order differential equations. The relative error tolerance, which measures the error relative to the magnitude of each solution component, is reduced to 10−8 (default value of 10−3). The numerical model has been validated by comparison with previous simulation results obtained for Ar-H2O and H2-O2 mixture without CO2 [13,20]. Furthermore, the mass balance is checked at each time step by validating that:

∑ Yk = 1 k

capacity (the ratio of CO2 to Ar thermal capacities is about 2.7 at 3200 K). Fig. 2(C) shows the production of main radicals during the same cycle. The orders of magnitude follow the values in Fig. 1(C). Fig. 2(D) shows the production of gaseous species (H2, CO and O2) and the consumption of reactants (H2O and CO2). The total number of product molecules increases slightly during the first collapse. The results also demonstrate that the decrease of reactants is very small under these operating conditions.

(29) 4.2. Results at high frequency

4. Results and discussion

The second simulations (cases 2a and 2b) are conducted for comparison in a sound field of 3.20 atm and 300 kHz, using the same ultrasonic parameters as in the study of Henglein [6]. As before, case 2b is simulated with a low percentage of CO2 (2% in volume) in the bubble, while case 2a is undertaken without CO2. Furthermore, the natural frequency of a 5 μm Ar bubble is around 1.3 MHz, which is four times the driving frequency (300 kHz). Fig. 3(A) and (B) show the normalized radius, the normalized internal pressure and the bubble temperature as a function of time during three cycles of the acoustic field. For each cycle, the bubble undergoes a 2 μs expansion followed by a 1 μs rapid collapse. The maximum expansion ratio is 5.6 times the initial radius and the peak collapse temperature reaches 13,700 K during the most violent collapse. Fig. 3(C) and (D) show the production of main radicals (HO, O and H), the production of gaseous species (H2 and O2) and the consumption of reactant (H2O) under the same conditions. A significant amount of H2O is consumed during the collapses, while a corresponding production of O2, H2 and radicals is observed. The concentrations remain almost unchanged after the collapses, since the temperatures are kept lower to maintain the reactions. In the presence of CO2, Fig. 4(A) and (B) show that the maximum expansion ratio is unchanged while the peak collapse temperature during the second collapse decreases about 500 K in comparison to case 2a. The production of main radicals (HO, O and H), the production of gaseous species (H2, CO and O2) and the consumption of reactants (H2O and CO2) are shown in Fig. 4(C) and (D). The decrease of reactants is significant under these operating conditions, especially during the first collapse. This is due to the effects of both temperature and species concentration. The CO2 reduction is clearly accompanied by CO, H2, O2 and the formation of radicals. The results also confirm that the non-linear response and maximum bubble expansion are stronger when the ultrasonic frequency is lower than the natural bubble frequency.

The simulations are carried out using the parameters presented in Table 4 to determine the time profiles for the normalized radius, normalized internal pressure, bubble temperature and production of radicals and gases. The focus is placed on very slightly soluble fuel products, i.e. CO and H2, rather than on products that are highly soluble in water, e.g. H2O2. 4.1. Results at low frequency The first simulations (cases 1a and 1b) are undertaken in a sound field of 1.03 atm and 20 kHz, using the same ultrasonic parameters as in the study of Gong [20]. Case 1b is simulated with a low percentage of CO2 (2% in volume) in the bubble, while case 1a is carried out without CO2. Indeed, the use of CO2 seems to be more beneficial at low concentrations. Furthermore, the natural frequency of a 29.7 μm Ar bubble is around 122 kHz, which is six times the driving frequency (20 kHz), so a bubble in free oscillation would oscillate six times in one driving cycle. Fig. 1(A) shows the normalized radius and the normalized internal pressure as a function of time during one cycle of the acoustic field. Fig. 1(B) shows the bubble temperature during the same cycle. The bubble expands slowly before reaching a maximum radius of 2.70 times the initial radius. Then the bubble collapses rapidly and the pressure peak reaches 185 times the initial pressure, while the bubble radius reaches a minimum value of 0.39 times the initial radius. At this point, the predicted maximum temperature is close to 3300 K. Thereafter, the bubble temperature drops very rapidly. After the first half of the cycle, the pressure peaks and the collapse largely decrease during the next three rebounds. Simultaneously, the temperature peaks decrease until the bubble returns to its equilibrium radius. Fig. 1(C) shows the production of main radicals (HO, O and H) during the same cycle. The production of HO radicals from H2O dissociation mainly occurs during the first violent collapse, in line with the maximum temperature peak. During the second half of the cycle at lower temperatures, the recombination of HO is promoted and the quantity of HO decreases, apart from a small peak during the second collapse. The concentrations of O and H radicals significantly decrease during the first bubble collapse. However, they exhibit fewer variations between the first and second collapse, and remain relatively high. Fig. 1(D) shows the production of gaseous species (H2 and O2) and the limited consumption of reactant (H2O) under the same conditions. In a similar way, the production of H2 is controlled by the first collapse at maximum peak temperature, while the production of O2 that occurs during the second half of the cycle is less significant. These results globally agree with the previous theoretical study of Gong under the same conditions [13,20]. The effect of CO2 on bubble dynamic, pressure, temperature and species production is presented in Fig. 2. The general trends in Fig. 2(A) are similar for the normalized radius and internal pressure as shown in Fig. 1(A). The pressure peak reaches 189 times the initial pressure, while the bubble radius reaches a minimum value of 0.39 times the initial radius. Fig. 2(B) shows a small decrease (about 100 K) of the predicted maximum temperature close to 3200 K. So, a bubble with CO2 reaches a lower collapse temperature as a result of a higher thermal

4.3. Effect of bubble initial radius Fig. 5 shows the normalized maximum radius attained by a bubble and the conversion per reactant (H2O and CO2) after 350 μs as a function of the initial bubble radius and in the same conditions as case 1. At the 20 kHz frequency used in the simulations, the natural resonant radius is around 180 μm. The two main peaks in conversion per reactant correspond to the resonances occurring at around 1/7 and 1/6 of the time of the resonant radius. It is clear that H2O conversion is higher than CO2 conversion under these conditions. Furthermore, no chemical Table 4 Parameters used in the simulations at low frequency (case 1) and high frequency (case 2).

23

Case

PA0 atm

f kHz

R0 μm

yAr

y H2 O

yCO2

1a 1b 2a 2b

1.030 1.030 3.198 3.198

20 20 300 300

29.7 29.7 5.0 5.0

0.978 0.958 0.982 0.982

0.022 0.022 0.018 0.018

0 0.020 0 0.020

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Fig. 2. Simulation results of case 1b (low frequency, with CO2) as a function of time. (A) Normalized radius (solid line, left scale) and normalized internal pressure (dotted line, right scale); (B) Normalized radius (solid line, left scale) and bubble temperature (dotted line, right scale); (C) Production of radicals (HO, O and H); (D) Production of gaseous species (H2, CO and O2) and consumption of reactants (H2O and CO2).

Fig. 1. Simulation results of case 1a (low frequency, without CO2) as a function of time. (A) Normalized radius (solid line, left scale) and normalized internal pressure (dotted line, right scale); (B) Normalized radius (solid line, left scale) and bubble temperature (dotted line, right scale); (C) Production of radicals (HO, O and H); (D) Production of gaseous species (H2 and O2) and consumption of reactant (H2O).

24

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Fig. 3. Simulation results of case 2a (high frequency, without CO2) as a function of time. (A) Normalized radius (solid line, left scale) and normalized bubble pressure (dotted line, right scale); (B) Normalized radius (solid line, left scale) and bubble temperature (dotted line, right scale); (C) Production of radicals (HO, O and H); (D) Production of gaseous species (H2 and O2) and consumption of reactant (H2O).

Fig. 4. Simulation results of case 2b (high frequency, with CO2) as a function of time. (A) Normalized radius (solid line, left scale) and normalized bubble pressure (dotted line, right scale); (B) Normalized radius (solid line, left scale) and bubble temperature (dotted line, right scale); (C) Production of radicals (HO, O and H); (D) Production of gaseous species (H2 and O2) and consumption of reactant (H2O).

25

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Fig. 5. Normalized maximum radius (filled circle: case 1a and empty circle: case 1b, left scale) and conversion per reactant (filled diamond: H2O – case 1a, empty diamond: H2O – case 1b and cross: CO2 – case 1b, right scale) as a function of initial radius. The simulation time is 350 μs.

Fig. 8. Normalized maximum radius (filled circle: 3.20 atm and empty circle: 9.60 atm, left scale) and conversion per reactant (filled diamond: H2O – 3.20 atm, empty diamond: H2O – 9.60 atm, large cross: CO2 – 3.20 atm and small cross: CO2 – 9.60 atm, right scale) as a function of frequency. The simulation time is 20 μs.

bubble initial radius higher than 10 μm that corresponds to around 1/2 the resonant radius. Contrary to the case 1, CO2 conversion is higher than H2O conversion under these conditions. 4.4. Effect of bubble composition in CO2, ultrasound frequency and pressure amplitude Fig. 7 shows the predicted conversion per reactant (H2O and CO2) as a function of the CO2 mole fraction in the same conditions as case 2. The experimental yield of iodine in a sonicated iodine solution (0.1 M KI) at various CO2-Ar with composition from the Henglein study [6] is also plotted in Fig. 7. The experimental method is based on iodine ion (I−) oxidation in aqueous solution to form iodine (I2). The comparison between experimental and simulated data cannot be direct, because experimental data have been obtained in a multi-bubble system, while the simulation is carried out only on one bubble. However, the general trends on bubble composition in CO2 are compared. The addition of CO2 during experiments clearly inhibits iodine formation. Similarly, the predicted conversions of H2O and CO2 also decrease as the CO2 fraction increases. The conversion decrease is more pronounced for H2O than CO2 under these conditions. Similar modeling results have been obtained by Gireesan and Pandit [17], which have shown that a reduction of the collapse temperature reduces the production of HO radicals and iodine liberation. Fig. 8 shows the maximum normalized radius attained by a bubble, and the conversion per reactant (H2O and CO2) as a function of ultrasound frequency (from 100 kHz to 2 MHz) and for two pressure amplitudes (3.20 atm and 9.60 atm). The maximum normalized radius globally decreases with the increase in frequency and with the decrease in pressure amplitude. The conversion per reactant follows the same trend and declines dramatically beyond 900 kHz. Thus, low frequency ultrasound may provide adequate conditions to facilitate CO2 reduction, while the creation of cavitation is more difficult, leading to an increase in ultrasound frequency. Indeed, low frequencies have a long rarefactions phase, which results in a high temperature inside the bubble. At high frequencies, the bubble does not have time to expand and collapse.

Fig. 6. Normalized maximum radius (filled circle: case 2a and empty circle: case 2b, left scale) and conversion per reactant (filled diamond: H2O – case 2a, empty diamond: H2O – case 2b and cross: CO2 – case 2b, right scale) as a function of initial radius. The simulation time is 10 μs.

Fig. 7. Conversion per reactant (solid line: CO2 and dotted line: H2O, left scale) and iodine yield (empty circle, right scale, experimental data from [6]) as a function of CO2 mole fraction. The simulation time is 10 μs.

4.5. Sonochemical efficiency reaction occurs with an initial bubble radius of less than 15 μm. In addition, Fig. 6 shows the normalized maximum radius and the conversion per reactant after 10 μs as in the case 2. Under this scenario, the natural resonant radius is around 20 μm at the 300 kHz frequency used in the simulations. H2O and CO2 conversion greatly decreases for

The efficiency of the ultrasonic system is crucial in power applications. During the sonochemical process, energy is converted into various forms (mainly mechanical, acoustic, thermal and chemical energies). Energy losses at the bubble scale occur mainly through thermal losses, viscous dissipation, diffusion of soluble products in water (e.g. 26

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Table 5 Chemical efficiency. aThe bubble number density used in the calculation (cases 1b and 2b) is 3 × 107 bubbles per cubic meter at first approximation. Case or reference Frequency (kHz) Intensity (W/cm2) Power Sonication time Water temperature (K) CO2 in gas phase (%) CO in gaseous products H2 in gaseous products Chemical efficiencya

1b 20 0.36 n/a 350 μs 293 2 2.6 × 10−16 mol 2.3 × 10−15 mol 7.9 × 10−3

2b 300 3.50 n/a 20 μs 293 2 5.2 × 10−16 mol 8.8 × 10−17 mol 2.3 × 10−4

η=

∑

References

Yk LHVk

k = CO, H2

WA NB

[9] 20 60 410 W/L 60 min 288 5 5.54 μmol/L 1.07 μmol/L 1.3 × 10−6

reactions in the bubble are advantaged when the ultrasonic frequency is lower than the natural bubble frequency. Hence, in the case of a bubble with an initial radius of around 5 μm, CO2 reduction appears to be more favorable at a frequency of around 300 kHz than at a low frequency of around 20 kHz. The industrial application of ultrasound to CO2 reduction in water would be largely dependent on sonochemical efficiency. This process does not seem to be sufficiently efficient under the conditions tested.

H2O2) and sonoluminescence. Further losses appear at a mesoscale and macroscale evaluation, e.g. by conversion of electrical energy to mechanical energy in the ultrasonic system, the effect of reactor walls, attenuation, reflection and thermal losses, etc. The sonochemical efficiency (SE-value) is often assessed by the concentration of the species (expressed in mol/m3) divided by the acoustic energy density (expressed in J/m3) [34]. In this study, chemical efficiency is defined as the ratio of chemical energy in fuel gases (i.e. CO and H2) to acoustic energy supplied by ultrasound in water:

mB

[8] 200 n/a 200 W 120 min 298 5 114 μmol 11 μmol 2.3 × 10−5

[1] M. Kanniche, Y. Le Moullec, O. Authier, H. Hagi, D. Bontemps, T. Neveux, M. LouisLouisy, Up-to-date CO2 capture in thermal power plants, Energy Procedia 114 (2017) 95–103, http://dx.doi.org/10.1016/j.egypro.2017.03.1152. [2] T.P. Senftle, E.A. Carter, The holy grail: chemistry enabling an economically viable CO2 capture, utilization, and storage strategy, Acc. Chem. Res. 50 (2017) 472–475, http://dx.doi.org/10.1021/acs.accounts.6b00479. [3] L.H. Thompson, L.K. Doraiswamy, Sonochemistry: science and engineering, Ind. Eng. Chem. Res. 38 (1999) 1215–1249, http://dx.doi.org/10.1021/ie9804172. [4] S. Sochard, A.-M. Wilhelm, H. Delmas, Gas-vapour bubble dynamics and homogeneous sonochemistry, Chem. Eng. Sci. 53 (1998) 239–254, http://dx.doi.org/10. 1016/S0009-2509(97)85744-2. [5] K.S. Suslick, Y. Didenko, M.M. Fang, T. Hyeon, K.J. Kolbeck, W.B. McNamara, M.M. Mdleleni, M. Wong, Acoustic cavitation and its chemical consequences, Philos. Trans. R. Soc. London Math. Phys. Eng. Sci. 357 (1999) 335–353, http://dx. doi.org/10.1098/rsta.1999.0330. [6] A. Henglein, Sonolysis of carbon dioxide, nitrous oxide and methane in aqueous solution, Z. Für Naturforschung B 40 (1985) 100–107, http://dx.doi.org/10.1515/ znb-1985-0119. [7] H. Harada, Sonochemical reduction of carbon dioxide, Ultrason. Sonochem. 5 (1998) 73–77, http://dx.doi.org/10.1016/S1350-4177(98)00015-7. [8] H. Harada, C. Hosoki, M. Ishikane, Sonophotocatalysis of water in a CO2–Ar atmosphere, J. Photochem. Photobiol. Chem. 160 (2003) 11–17, http://dx.doi.org/ 10.1016/S1010-6030(03)00214-4. [9] A. Koblov, Sonochemical reduction of carbon dioxide. < http://espace.library. curtin.edu.au/R?func=dbin-jump-full&local_base=gen01-era02&object_id= 183473 > , 2011 (accessed March 2, 2016). [10] H. Harada, Y. Ono, Improvement of the rate of sono-oxidation in the presence of CO2, Jpn. J. Appl. Phys. 54 (2015) 07HE10, http://dx.doi.org/10.7567/JJAP.54. 07HE10. [11] V. Kamath, A. Prosperetti, F.N. Egolfopoulos, A theoretical study of sonoluminescence, J. Acoust. Soc. Am. 94 (1993) 248–260, http://dx.doi.org/10.1121/1. 407083. [12] S. Sochard, A.M. Wilhelm, H. Delmas, Modelling of free radicals production in a collapsing gas-vapour bubble, Ultrason. Sonochem. 4 (1997) 77–84, http://dx.doi. org/10.1016/S1350-4177(97)00021-7. [13] C. Gong, D.P. Hart, Ultrasound induced cavitation and sonochemical yields, J. Acoust. Soc. Am. 104 (1998) 2675–2682, http://dx.doi.org/10.1121/1.423851. [14] K. Yasui, T. Tuziuti, M. Sivakumar, Y. Iida, Theoretical study of single-bubble sonochemistry, J. Chem. Phys. 122 (2005) 224706, http://dx.doi.org/10.1063/1. 1925607. [15] S. Merouani, H. Ferkous, O. Hamdaoui, Y. Rezgui, M. Guemini, New interpretation of the effects of argon-saturating gas toward sonochemical reactions, Ultrason. Sonochem. 23 (2015) 37–45, http://dx.doi.org/10.1016/j.ultsonch.2014.09.009. [16] S. Merouani, H. Ferkous, O. Hamdaoui, Y. Rezgui, M. Guemini, A method for predicting the number of active bubbles in sonochemical reactors, Ultrason. Sonochem. 22 (2015) 51–58, http://dx.doi.org/10.1016/j.ultsonch.2014.07.015. [17] S. Gireesan, A.B. Pandit, Modeling the effect of carbon-dioxide gas on cavitation, Ultrason. Sonochem. 34 (2017) 721–728, http://dx.doi.org/10.1016/j.ultsonch. 2016.07.005. [18] Y.T. Didenko, K.S. Suslick, The energy efficiency of formation of photons, radicals and ions during single-bubble cavitation, Nature 418 (2002) 394–397, http://dx. doi.org/10.1038/nature00895. [19] A. Henglein, Sonolysis of carbon dioxide, nitrous oxide and methane in aqueous solution, Z. Für Naturforschung B 40 (2014) 100–107, http://dx.doi.org/10.1515/ znb-1985-0119.

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where mB is the bubble mass, LHVk is the lower heating value of gas product (10.9 MJ kg−1 and 120 MJ kg−1 for CO and H2 respectively) and NB is the bubble number density. The bubble number density is a critical parameter to assess chemical efficiency. However, the bubble population in a cavitating medium is very complex (nucleation, coalescence, fragmentation, etc.). The bubble number density was about 1.6–3.9 × 107 bubbles per cubic meter at a size of D50 (i.e. the median diameter of bubble size distribution) in the 20–60 μm range when a Fraunhofer laser diffraction technique was used to characterize the bubble population in the size from 0.5 to 355 μm in water at 443 kHz [35]. Thus, a rough value of 3 × 107 active bubbles per cubic meter that react to the application of ultrasound is used as a first approximation. The chemical efficiency is given in Table 5 for cases 1b and 2b. Values obtained from the studies of Harada et al. [8] and Koblov [9] in multi-bubble systems are also presented in Table 5. Experimental values show that CO2 reduction to produce fuel gases is quite ineffective from an energetic point of view at both low and high frequencies. Theoretical calculations at the bubble scale confirm that the process is energy inefficient under the conditions tested. Consequently, the efficient conversion of CO2 to fuel gas (syngas) by sonochemical reduction in water seems too limited for practical application. 5. Conclusion The main contribution of this study is to couple bubble dynamics in a sound field with the chemical kinetics of CO2 dissolved with Ar in water. The production of free radicals and light fuel gases formed during the collapse phases is obtained as a function of time over many cycles. The strong effect of bubble composition on the CO2 reduction rate is confirmed, in accordance with the experimental data from literature. When the initial fraction of CO2 in a bubble is low, bubble growth and collapse are slightly modified with respect to simulation without CO2, and chemical reactions leading to CO2 reduction are thus promoted. Among the different products, fuel species like CO and H2 are obtained in the gas phase. However, the peak collapse temperature depends on CO2 thermal properties and greatly decreases with the increased presence of CO2 in the bubble, leading to lower CO2 reduction. The model shows that initial bubble radius, ultrasound pressure amplitude and frequency play a critical role in CO2 reduction. Chemical 27

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