Sono-electrolysis performance based on indirect continuous sonication and membraneless alkaline electrolysis: Experiment, modelling and analysis

Highlights

• •Hydrogen rate and energy conversion efficiency are studied in sono-electrolysis.
• •Polarization curves are plotted and analysed under silent and ultrasound conditions.
• •The acoustic cavitation bubble is modelled and simulated.
• •Microjet and shockwave powers are quantified at collapse for bubbles composing the population.
• •Effect of bubble coverage of electrodes on the Ohmic resistance is studied.

Abstract

In the present study, experiments of membraneless alkaline sono-electrolysis are combined to a mathematical model describing the performance of a sono-electrolyzer based on the electrochemical resistances and overpotentials (activation, Ohmic and concentration) and the oscillation of the acoustic cavitation bubble, and its related sono-physical and sonochemical effects, as a single unit and within population. The study aims to elucidate the mechanism of action of acoustic cavitation when coupled to alkaline electrolysis, using a membraneless H-cell configuration and indirect continuous sonication (40 kHz, 60 W_e). The calorimetric characterization constituted the bridge between experimental results and the numerical and simulation approach, while the quantification of the rate of produced hydrogen both experimentally and numerically highlighted the absence of the contribution of sonochemistry, and explained the role of ultrasounds by the action of shockwaves and microjets. Finally, the energetic sono-physical approach allowed an estimation of the predominance of the shockwaves and microjets effects according to the bubble size distribution within the population corresponding to the acoustic conditions of the study. The resulting macroscopic effect in sono-electrolysis process has been assessed considering the induced degassing. A reduction in the fraction of electrodes’ coverage by bubbles from 76% to 42% has been recorded, corresponding to a decrease of 7.2% in Ohmic resistance and 62.35% in bubble resistance.

Nomenclature

Symbol

Signification

${\sigma }_{\mathbf{a}}$

Anodic conductivity [S.cm⁻¹]

${\sigma }_{\mathbf{c}}$

Cathodic conductivity [S.cm⁻¹]

${\delta }_{\mathbf{c}}$

Cathode thickness [cm]

${\delta }_{\mathbf{a}}$

Anode thickness [cm]

${\sigma }_{\mathbf{l}}$

Diffusion layer thickness around the electrode [cm]

${\delta }_{\mathbf{e}\mathbf{l}}$

Thickness of the electrolyte layer [cm]

${\sigma }_{\mathbf{e}\mathbf{l}}$

Ionic conductivity of alkaline solution [S.cm⁻¹]

${\mathbf{a}}_{\mathbf{a}}$

Anodic charge transfer coefficient [A.cm⁻²]

${\mathbf{a}}_{\mathbf{c}}$

cathodic charge transfer coefficient [A.cm⁻²]

${\epsilon }_{{\mathbf{H}}_2}$

Energy conversion efficiency [%]

${\eta }_{\mathbf{F}}$

Faradaic efficiency

μ

Dynamic viscosity [Pa s]

${\rho }_{\mathbf{e}\mathbf{l}}$

Electrolyte density [kg/m³]

σ

Surface tension [N/m]

α

Accommodation coefficient

$\xi$

Thickness of the heat diffusion interface [m]

$\lambda$

Diffusion coefficient [W/m².K]

$\mathbf{v}$

Jet velocity [m/s]

${\vartheta }_{\mathbf{k}\mathbf{i}}$

stoichiometric coefficient of the k^th reactant involved in the i^th elementary reaction

ΔH_i

Reaction heat of the ith reaction [J/mol]

A

Area of the working electrode [cm²]

A_f

conductor cross-sectional area [m²]

c

Sound celerity [m/s]

C

Concentration of species in the electrolyte [mol.cm⁻³]

${\mathbf{C}}_{\mathbf{v}}$

Heat capacity [J/mol.K]

${\mathbf{D}}_0$

Diffusion coefficient [cm².s⁻¹.°C]

${\mathbf{d}}_{\mathbf{a}}$

Distance from anode to membrane [m]

${\mathbf{d}}_{\mathbf{c}}$

Distance from cathode to membrane [m]

e

Percentage of the electrodes surface covered by bubbles [J/mol]

${\mathbf{E}}_{\mathbf{i}}$

Activation energy of the i^th reaction

${\mathbf{E}}_{{\mathbf{H}}_2}$

Power stored in hydrogen flow [W]

E_rev

Reversible voltage [V]

f

Frequency [Hz]

F

Faraday’s constant

${\mathbf{F}}_{\mathbf{i}}$

Number fraction

$\mathbf{h}$

Jet nozzle exit diameter [m]

${\mathbf{H}}_{{\mathbf{H}}_2}$

Heat of combustion of hydrogen [MJ.kg^−1]

I

Cell current [A]

${\mathbf{I}}_{\mathbf{a}}$

Anodic current [A]

${\mathbf{I}}_{\mathbf{c}}$

Cathodic current [A]

${\mathbf{I}}_{\mathbf{a}\mathbf{c}}$

Acoustic intensity [W/m²]

${\mathbf{I}}_{\mathbf{l}\mathbf{i}\mathbf{m}}$

Limit current of diffusion [A]

${\mathbf{I}}_{0\mathbf{a}}$

Anodic exchange current [A]

${\mathbf{I}}_{0\mathbf{c}}$

Cathodic exchange current [A]

$\mathbf{l}$

Length of electron path [m]

${\mathbf{L}}_{\mathbf{a}}$

Anode thickness [cm]

${\mathbf{L}}_{\mathbf{c}}$

Cathode thickness [cm]

${{\mathbf{j}}_0}_{\mathbf{a}}$

Anodic exchange current [A.cm⁻²]

${{\mathbf{j}}_0}_{\mathbf{c}}$

Cathodic exchange current [A.cm⁻²]

M

Molar mass [kg/mol]

${\mathbf{m}}_{\mathbf{K}\mathbf{O}\mathbf{H}}$

Electrolyte molality [mol.kg⁻¹ _H2O]

m

Mass flow of liquid due to evaporation–condensation [kg/m²]

${\mathbf{m}}_{{\mathbf{H}}_2}$

Mass of produced hydrogen [g]

$\dot{\mathbf{m}}$

Rate of evaporation–condensation of liquid [kg/m² s]

$\ddot{\mathbf{m}}$

Acceleration of mass flow of liquid due to evaporation–condensation [kg/m² s²]

${\dot{\mathbf{m}}}_{{\mathbf{H}}_2}$

The mass flow of hydrogen [mol.s⁻¹]

${\mathbf{M}}_{{\mathbf{H}}_2}$

Molar mass of hydrogen [g.mol⁻¹]

${\mathbf{M}}_{{\mathbf{H}}_2\mathbf{O}}$

Molar mass of water [g.mol⁻¹]

N

Number density of bubbles [Bubbles/m³]

$\dot{\mathbf{n}}$

Molar rate [mol/s]

${\mathbf{P}}_{\mathit{A}}$

Acoustic amplitude [Pa]

${\mathbf{P}}_{\mathit{a}\mathit{c}}$

Acoustic power [W]

${\mathbf{P}}_{\mathbf{B}}$

Pressure on bubble wall [Pa]

${\mathbf{P}}_{\mathit{c}\mathit{e}\mathit{l}\mathit{l}}$

Cell power [W]

${\mathbf{P}}_{\mathbf{i}}$

Partial pressure [Pa]

${\mathbf{P}}_{\mathbf{j}}$

Exit jet power [W]

${\mathbf{P}}_{\mathbf{g}}$

Gas pressure [Pa]

P_v

Vapor pressures of electrolyte solution [atm]

P_v*

Vapor pressure of purified water [atm]

${\mathrm{P}}_{\mathrm{s}}$

Power of shockwave [W]

${\mathit{P}}_{\mathit{\infty }}$

Static pressure [Pa]

R

Bubble radius [m]

$\dot{\mathbf{R}}$

Bubble wall velocity [m/s]

$\ddot{\mathbf{R}}$

Bubble wall acceleration [m/s²]

${\mathbf{R}}_{\mathbf{b}\mathbf{f}}$

Bubble-free electrolyte resistance [Ω]

${\mathbf{R}}_{\mathbf{b}}$

Bubble resistance [Ω]

${\mathbf{r}}_{\mathbf{i}}$

Reaction rate of the i^th reaction [mol/s m³]

${\mathbf{R}}_{\mathbf{g}}$

Ideal gas constant [J/mol K]

${\mathbf{R}}_{\mathbf{o}\mathbf{h}\mathbf{m}}$

Ohmic resistance [Ω]

${\mathbf{R}}_{\mathbf{e}\mathbf{l}\mathbf{e}\mathbf{c}\mathbf{t}\mathbf{r}\mathbf{o}\mathbf{l}\mathbf{y}\mathbf{t}\mathbf{e}}$

Electrolyte resistance [Ω]

${\mathbf{R}}_{\mathbf{a}}$

Anodic resistance [Ω]

${\mathbf{R}}_{\mathbf{c}}$

Cathodic resistance [Ω]

$\mathbf{S}$

Section [m²]

${\mathbf{S}}_{\mathbf{a}}$

Anode cross section [m²]

${\mathbf{S}}_{\mathbf{c}}$

cathode cross section [m²]

T_b

Temperature within the bubble [K]

t

Time [S]

U_cell

Cell voltage [V]

U_ohm

Ohmic voltage [V]

U_conc

Concentration voltage [V]

${\mathbf{U}}_{\mathbf{a}\mathbf{c}\mathbf{t}}$

Activation voltage [V]

${\mathbf{U}}_{\mathbf{a}\mathbf{c}\mathbf{t}\mathbf{a}}$

Anodic activation voltage [V]

${\mathbf{U}}_{\mathbf{a}\mathbf{c}\mathbf{t}\mathbf{c}}$

Cathodic activation voltage [V]

$\mathbf{V}$

Volume [m³]

${\mathbf{w}}_{\mathbf{K}\mathbf{O}\mathbf{H}}$

Mass fraction of KOH in the electrolyte

Z

number of moles of electrons transferred in the reaction

1. Introduction

The acoustic cavitation bubble [1], [2], [3] has been widely investigated experimentally and through modelling; as a unique event of microscopic concentration of energy [4] within a liquid flow, induced by the passage of an ultrasonic wave at appropriate acoustic conditions. Most of the scientific works retrieved in the literature deal with sonochemistry and ultrasound assisted processes to harness the chemical [5] and physical [6] consequences of the oscillation and implosion of the acoustic cavitation. As an isolated process, sonication is generally used in liquid phase to induce harsh microscopic chemical conditions, able to reproduce pyrolysis and/or combustion reactions within the bubble volumes, at their interface or even within the liquid effluent. This is known as “sonochemistry” [7] and is intensively used in water remediation [8] as an advanced oxidation process based on the generation of free radicals from water decomposition. Besides, sonication is also combined to other processes, in order to increase the performance of the isolated processes by the addition of their effects, or create a synergy by the interaction of the physical–chemical actions of acoustic cavitation bubble with the action of the original process. Several examples can be cited in this regards [9], in sono-catalysis [10], sono-photolysis [11], sono-crystallisation [12] and sono-electrolysis [13].

The action of sonication has been mathematically modelled in terms of the chemical and physical effects. For instance, the sonochemical process of generation of free radicals [14], [15], degradation of organic contaminants [16], [17], or even production of hydrogen [18], [19], [20] has been elucidated through chemical mechanisms associated to the thermodynamics of the oscillation of the single acoustic cavitation bubble. The population dimension has been also integrated in the modelling of sonochemistry [21], [22], through the number density of bubbles [23], [24] and their size distribution [25]. On the other hand, the physical effects associated with the oscillation of the single acoustic cavitation bubble have been studied via mathematical modelling and simulation in a number of scientific papers dealing with microjets [26], shockwaves [27], acoustic streaming [28] and microstreaming [29]. The physical effects have been advanced as a rational for the observed effects under sonication, however, modelling has been rarely associated to experiments of ultrasound assisted processes to explain the action of sonication.

Sono-electrolysis is one of the hybrid processes attracting an increasing attention of researchers in the recent times [13], [30], [31], [32], [33]. For instance, Islam et al. [34] explained the influence of ultrasounds on the electrolytic production of hydrogen through the enhancement of mass transport and electrode cleaning due to the combined effect of microjets and microstreaming, as well as the drop of the overall potential, caused by the degassing effects and/or the modifications of the surface of the electrode [35]. Budischak [36] and McMurray [37] proposed similar setups without a physical separation between the anodic and cathodic compartments for the production of hydrogen and oxygen mixture known as HHO. They demonstrated through the voltammetry measurements at the cathode and the anode, an improvement in the working current, which was explained by the enhancement of mass transport and the cleaning of electrodes’ surface from bubbles to create more active sites.

On the other hand, Zadeh et al. [38] adopted a membraneless electrolyzer configuration with a physical separation of the anodic and cathodic compartments, in order to improve the purity of the produced hydrogen. The configuration integrated a 20 kHz sonication of alkaline solutions (KOH and NaOH) with different electrode materials (platinum, glassy carbon, industrial carbon, 316 stainless steel, and nickel-based electrodes). The results showed an enhancement in hydrogen production due to the increase in mass transport, electrode cleaning, and degassing. Zadeh [38] observed an improvement in H₂ generation of 14% and 25% with NaOH and KOH, respectively, when coupling ultrasound to electrolysis process, as bubble detachment from the electrode prepares the surface for new electrochemical reactions.

In the present study, experiments are performed at a laboratory scale with a membraneless H-cell configuration of a sono-electrolyzer based on alkaline electrolysis and indirect continuous sonication. The performance of the sono-electrolyzer is monitored in terms of the kinetics of hydrogen production and energy conversion efficiency under silent conditions and when integrating ultrasounds. The configuration is also characterized calorimetrically to assess the energetic acoustic efficiency and hence simulate the behaviour of the single acoustic cavitation bubble and the bubble population under the corresponding conditions of frequency and acoustic amplitude. The combination of the mathematical model of the sono-electrolyzer with the performed experiments aims, on one hand, to elucidate the extent of the chemical and physical effects induced by sonication in the ultrasound assisted electrolysis, and on the other hand, approach the macroscopic effect of ultrasounds on the performance of the alkaline electrolysis through the mathematical assessment of the Ohmic resistance of the cell and explain quantitatively its reduction, if demonstrated, by investigating the percentage of utilization of the electrodes surface.

2. Material and method

2.1. Experimental procedure

2.1.1. Membraneless sono-electrolysis

A volume of 300 mL of aqueous solution of KOH (25% w/w) is electrolyzed within an H-cell, while being sonicated in an ultrasonic bath. A DC power supply is used as a source of direct current feeding the electrolyzer. An ultrasonic bath of 40 kHz frequency ($f=40kHz$) is used as a source of indirect continuous sonication. The input electric power of the sonicator is equal to 60 W ($P_e=60W$). The configuration of the H-cell electrolysis vessel consists of two cylindrical compartments connected through two horizontal tubes attached to each other using a plier and taking an H form, as shown in Fig. 1. Nickel plate electrodes are placed in the anodic and cathodic compartments of the cell at a distance of 4 cm. The same effective contact surface of 13.5 cm² per electrode is used for all the experiments. The cell is equipped with a temperature probe and placed in a water bath in order to maintain a stable temperature. The kinetics of hydrogen production and the energy efficiency of the sono-electrolysis are monitored. The experiments are carried out under silent and ultrasound modes for comparison. The Ohmic resistance is determined experimentally using the decomposition voltage method [39], and the polarization curve (U vs. I) under silent and ultrasound conditions.

Fig. 1.

Experimental setups of sono-electrolysis based on indirect continuous sonication and membraneless alkaline electrolysis (H-cell, on the left), and calorimetric measurement of the acoustic power received by the electrolyte (electrolysis off mode, on the right).

2.1.2. Calorimetric characterization

In order to assess the acoustic power transferred to the bulk volume of the electrolyte, a blank test has been performed using the ultrasonic bath (40 kHz, 60 W). The electrolyzer vessel (H-cell) has been emerged within the ultrasonic bath filled of water, a temperature probe has been introduced within the sealed electrolyzer and immersed into 300 mL of KOH electrolyte (25% w/w) as illustrated in Fig. 1 (electrolysis off mode). Continuous sonication has been turned on while the chronometer has been launched. The increase of the electrolyte temperature has been recorded as a function of time in the absence of electrolysis. This increase is supposed to originate from the sonication, with the assumption that all the energy carried by the ultrasonic wave and implicated in acoustic cavitation phenomenon is recovered as heat [40], [41], [42]. Hence, the ultrasonic power transmitted to the electrolyte is evaluated through the equation [19], [40], [42]:

$$P_{\mathit{ac}}=m_{\mathit{el}}{C}_p\frac{\mathit{dT}}{\mathit{dt}}$$ (1)

where $C_p$ is the heat capacity of the electrolyte at constant pressure and $m_{\mathit{el}}$ is the mass of ultrasonicated electrolyte.

The acoustic power $P_{\mathit{ac}}$ is related to the acoustic amplitude $P_A$ through Eq. (2) [22], [43], [44].

$$P_{\mathit{ac}}=\frac{{P_A}^{2}S}{2{\rho }_{\mathit{el}}c}$$ (2)

where $S$ is the surface of the basis of the H-cell electrolyzer crossed by the ultrasonic wave ($S=25.12{\mathit{cm}}^2$) and $c$ is the sound velocity in m/s.

The acoustic intensity received by the H-cell vessel is then expressed as indicated in Eq. (3).

$$I_{\mathit{ac}}=\frac{P_{\mathit{ac}}}{S}$$ (3)

The acoustic efficiency ${\epsilon }_a$ is evaluated in continuous mode according to Eq. (4).

$${\epsilon }_a=\frac{P_{\mathit{ac}}}{P_e}$$ (4)

2.2. Numerical approach

2.2.1. Membraneless sono-electrolysis

2.2.1.1. Membraneless electrolysis

In aqueous solution, the passage of the direct current between both electrodes (anode and cathode) leads to the dissociation of water molecules into hydrogen and oxygen [45]. In alkaline conditions, the reduction of water takes place at the cathode through the Eq. (5). While hydroxide oxidation occurs at the anode through the Eq. (6). [46]

$$\mathrm{C}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{d}\mathrm{e}:{2H}_{2}O+2e^{-}\to H_2+2{\mathit{OH}}^{-}$$ (5)

$$\mathrm{A}\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}:2{\mathit{OH}}^{-}\to \frac{1}{2}{O}_2+2e^{-}+H_{2}O$$ (6)

The cell voltage presents the sum of different voltages involved in the cell and is expressed according to the Eq. (7) [47].

$$U_{\mathit{cell}}=E_{\mathit{rev}}(T,P)+U_{\mathit{Ohm}}+U_{\mathit{Conc}}+U_{\mathit{act}}$$ (7)

where $E_{\mathit{rev}}(T,P)$ represents the reversible voltage at the operating temperature $T$ and operating static pressure $P$ which is the lowest theoretical voltage required for the electrolysis to occur, it is expressed by Eq. (8) [48].

$$E_{\mathit{rev}}(T,P)=E_{\mathit{rev}}\left(T\right)+\frac{R_{\mathit{g}}T}{\mathit{ZF}}\mathrm{l}\mathrm{n}\left(\frac{{P_v}^{\ast }{\left(P-P_v\right)}^{1.5}}{P_v}\right)$$ (8)

$Z$ is the number of moles of electrons transferred in the reaction and $F$ is the Faraday’s constant.

$P_v$ and ${P_v}^{\ast }$ are the vapor pressure of electrolyte solution and the vapor pressure of purified water, expressed respectively by Eqs. (9), (10) [49].

$$P_v=e^{\left(0.01621-0.1280\times m_{\mathit{KOH}}+0.1933\times {m_{\mathit{KOH}}}^{0.5}+1.024\times \mathrm{l}\mathrm{n}({P_v}^{\ast })\right)}$$ (9)

$${P_v}^{\ast }=e^{\left(37.043-\frac{6275.7}{T}-3.4159\times \mathrm{l}\mathrm{n}(T)\right)}$$ (10)

$E_{\mathit{rev}}(T)$ is the reversible voltage that depends on the temperature and can be written according to Eq. (11) [50].

$$E_{\mathit{rev}}\left(T\right)=1.5184-1.5423\times {10}^{-3}T-9.524\times {10}^{-5}TlnT+9.84\times {10}^{-8}{T}^2$$ (11)

The Ohmic voltage $U_{\mathit{Ohm}}$ is due to the Ohmic losses during water electrolysis, caused by the formation of the gas bubble, the wastage of electrical energy and the passage of ions in the electrolyte. The Ohmic voltage is governed by the Ohmic resistance, which is the sum of intrinsic resistances of the electrolyte (bubble free), the bubbles formed on the electrodes’ surfaces, the electrodes (anode and cathode), and the membrane (if present), as indicated in Eq. (12).

$$R_{\mathit{ohm}}=R_{\mathit{electrolyte}}+R_{\mathit{bubble}}+R_a+R_c$$ (12)

The ionic resistance of the bubble free electrolyte can be obtained using Eq. (13) [51], [52].

$$R_{\mathit{electrolyte}}=\frac{{\delta }_{\mathit{el}}}{{\sigma }_{\mathit{el}}}$$ (13)

where δ_el and σ_el are the thickness of the electrolyte layer and the ionic conductivity of the alkaline solution, respectively, given by Eqs. (14), (15) (S.m⁻¹) [51], [52], [53].

$${\delta }_{\mathit{el}}=\left(\left(\frac{d_a}{S_a}\right)+\left(\frac{d_c}{S_c}\right)\right)$$ (14)

$${\sigma }_{\mathit{el}}=K_1\left(100\times w_{\mathit{KOH}}\right)+K_2\times \mathrm{T}+K_3\times T^2+K_4\times T\times \left(100\times w_{\mathit{KOH}}\right)+K_5\times T^2\times {\left(100\times w_{\mathit{KOH}}\right)}^{K_6}+K_7\frac{T}{\left(100\times w_{\mathit{KOH}}\right)}+K_8\frac{\left(100\times w_{\mathit{KOH}}\right)}{T}$$ (15)

$S_a$ and $S_c$ are the anodic and cathodic crosse sections, respectively, and $d_a$ and $d_c$ are the distances from to anode (cathode, respectively) to the separator zone. $w_{\mathit{KOH}}$ represents the mass fraction of $\mathit{KOH}$ in the electrolyte, while $K_1$ to $K_8$ are constants given in Table 1.

Table 1.

Values of parameters and constants.

Parameter	Symbol	Value	Unit	Reference
Anode cross section	$S_a$	13.5	cm2	Experiment
Cathode cross section	$S_c$	13.5	cm2	Experiment
Distance from anode to membrane	$d_a$	2	cm	Experiment
Distance from cathode to membrane	$d_c$	2	cm	Experiment
Anode thickness	$L_a$	0.035	cm
Cathode thickness	$L_c$	0.035	cm
Anodic exchange current	${j_0}_a$	2.82 × 10−6	A.cm−2	[58]
Cathodic exchange current	${j_0}_c$	3.55 × 10−5	A.cm−2	[59]
Constants for the calculation of the conductivity of the electrolyte	$K_1$	27.9844803	S.m−1	[53]
	$K_2$	−0.924129482	S.m−1.K−1
	$K_3$	−0.0149660371	S.m−1.K−2
	$K_4$	−0.0905209551	–
	$K_5$	0.0114933252	S.m−1.K−1
	$K_6$	0.1765	S.m−1.K−2
	$K_7$	6.96648518	S.m−1.K−1
	$K_8$	−2898.15658	S.m−1.K
Conductivity of Nickel	${\sigma }_{\mathit{Ni}}$	1.43 × 107	S.m−1	[60]
Molality of the electrolyte solution	$m_{\mathit{KOH}}$	5.94	mol.kg−1H2O	Experiment
Mass fraction of KOH in the electrolyte	$w_{\mathit{KOH}}$	0.25	–	Experiment

The bubble resistance $R_{\mathit{bubble}}$ is calculated based on the bubble free electrolyte resistance $R_{\mathit{electrolyte}}$ according to Eq. (16) [54].

$$R_{\mathit{bubble}}=R_{\mathit{electrolyte}}\left(\left(\frac{1}{{\left(1-\frac{2}{3}e\right)}^{1.5}}\right)-1\right)$$ (16)

$e$ is the fraction of the electrodes’ surface covered by bubbles.

The resistances of the anode $R_a$ and cathode $R_c$ are related to the used material. When considering Nickel, both resistances are expressed respectively by Eqs. (17), (18) [51].

$$R_a=\frac{1}{{\sigma }_{\mathit{Ni}}}\left(\frac{L_a}{S_a}\right)$$ (17)

$$R_c=\frac{1}{{\sigma }_{\mathit{Ni}}}\left(\frac{L_c}{S_c}\right)$$ (18)

$L_a$ and $L_c$ are the anode and cathode thicknesses, respectively, while ${\sigma }_{\mathit{Ni}}$ is the conductivity of Nickel material given in Table 1.

The concentration voltage $U_{\mathit{con}}$ is due to the variation in the concentration of reactant on the electrode surface that is caused by the mass transport. The concentration voltage is expressed according to Eq. (19) [47].

$$U_{\mathit{conc}}=\frac{R_{\mathit{g}}T}{\mathit{ZF}}\left(\ln \left(1-\left(\frac{I}{I_{\mathrm{l}\mathrm{i}\mathrm{m}}}\right)\right)\right)$$ (19)

where $I_{\mathit{lim}}$ is the limit current of diffusion given as indicated in Eq. (20) [39].

$$I_{\mathit{lim}}=\frac{Z\times F\times A\times D_0\times C^{\ast }}{{\sigma }_l}$$ (20)

$D_0$ is the diffusion coefficient cm²/s, $C^{\ast }$ is the is the concentration of the species in the bulk solution in mol/cm³, ${\sigma }_l$ is the diffusion layer thickness and A is the area of the working electrode (cm²), which is equal to the cross section.

The last term of the cell voltage in Eq. (1) is the activation voltage $U_{\mathit{act}}$ that is caused by the kinetics at the electrodes, it is the sum of the anodic and cathodic activation overpotential. The anodic and cathodic activation voltages can be obtained using the Tafel equation as shown in Eqs. (21), (22).

$${U_{\mathit{act}}}_a=\frac{2.3026\times R_g\times T}{Z\times F\times a_a}\mathrm{l}\mathrm{o}\mathrm{g}\left(\frac{\left(I/S_a\right)}{{j_0}_a}\right)$$ (21)

$${U_{\mathit{act}}}_c=\frac{2.3026\times R_g\times T}{Z\times F\times a_c}\mathrm{l}\mathrm{o}\mathrm{g}\left(\frac{\left(I/S_c\right)}{{j_0}_c}\right)$$ (22)

${j_0}_a$ and ${j_0}_c$ refer to the anodic and cathodic exchange current densities, respectively, given in Table 1. The anode and cathode are activated only when both current densities at the anode and the cathode exceed ${j_0}_a$ and ${j_0}_c$ [51]. $a_a$ and $a_c$ are the charge transfer coefficients given in Eqs. (23), (24), respectively [55].

$$a_a=0.0675+0.00095\times T$$ (23)

$$a_c=0.1175+0.00095\times T$$ (24)

The rate of hydrogen gas produced by water electrolysis is expressed by Faraday’s law as shown in Eq. (25) [56].

$${\dot{m}}_{H_2}=\frac{I}{Z{M}_{H_2}F}{\eta }_F$$ (25)

where ${\dot{m}}_{H_2}$ is the hydrogen mass flow of produced hydrogen (g/s), ${\eta }_F$ is the Faraday efficiency.

The measurement of the effectiveness of the real cell is performed by comparing the charges fed to the system, and the amount of energy stored in produced hydrogen, indicated in Eq. (26) [57].

$$E_{H_2}={\dot{m}}_{H_2}{H}_{H_2}$$ (26)

where $H_{H_2}$ is the upper energy density of hydrogen, estimated at $142MJ/Kg$.

Finally, the energy efficiency of water sonolysis system is evaluated as given in Eq. (27).

$${\epsilon }_{H_2}=\frac{E_{H_2}}{U\times I}$$ (27)

2.2.1.2. Sonication

The integration of sonication in water electrolysis process contributes to the efficient detachment of bubbles from the electrodes surface, and enhances the mass transfer allowing the reduction of bubble and ion resistances. In order to figure out the mechanism of acoustic cavitation bubbles in the sono-electrolysis process, the dynamics of single acoustic bubble and its associated physical effects are modeled and simulated under the acoustic conditions of the experiments described in Section 2.1 ($f,P_A)$. The motion of the acoustic bubble wall is described by the modified Keller Miksis equation given in Eq. (28) [61], considering the flow of evaporated and condensed water molecules at the bubble interface as illustrated in Fig. 2.

$$\ddot{R}-\frac{\left(c+\dot{R}\right)}{{\rho }_{el}\left(cR-\dot{R}R+\frac{\dot{m}R}{{\rho }_{\mathit{el}}}\right)+4\mu }\left(P_g-\frac{2\sigma }{R}-4\mu \frac{\dot{R}}{R}-P_{\infty }+P_{A}sin\left(2\pi f\left(t+\frac{R}{c}\right)\right)\right)=\frac{R}{{\rho }_{el}\left(cR-\dot{R}R+\frac{\dot{m}R}{{\rho }_{\mathit{el}}}\right)+4\mu }\left(\frac{d{P}_g}{\mathit{dt}}+\frac{2\sigma \dot{R}}{R^2}+4\mu \frac{{\dot{R}}^2}{R^2}\right)-\frac{{\rho }_{el}\left(\left(3c-\dot{R}\right)+\frac{2\dot{m}}{{\rho }_{\mathit{el}}}\right){\dot{R}}^2}{2\left({\rho }_{el}\left(cR-R\dot{R}+\frac{\dot{m}R}{{\rho }_{\mathit{el}}}\right)+4\mu \right)}+\frac{\ddot{m}\left(cR-R\dot{R}+\frac{\dot{m}R}{{\rho }_{\mathit{el}}}\right)}{{\rho }_{el}\left(cR-R\dot{R}+\frac{\dot{m}R}{{\rho }_{\mathit{el}}}\right)+4\mu }+\frac{\dot{m}\left(c\dot{R}+\frac{\dot{m}}{2{\rho }_{\mathit{el}}}\left(c+\dot{R}\right)\right)}{{\rho }_{\mathit{el}}\left(cR-R\dot{R}+\frac{\dot{m}R}{{\rho }_{\mathit{el}}}\right)+4\mu }$$ (28)

Fig. 2.

Key phenomena intervening in the acoustic cavitation bubble during its oscillation.

The physical kinetics of evaporation and condensation of water molecules at the bubble interface evolves in function of time during both expansion and collapsing phases according to the Hertz Knudsen equation as shown in Eq. (29).

$$\dot{m}=\frac{\sqrt{M_{H_{2}O}}}{\sqrt{2\pi R_g}}\alpha \frac{1}{\sqrt{T_b}}\left(P_v-P_{H_{2}O}\right)$$ (29)

where $P_v$ is the saturating pressure of water in the electrolyte and $P_{H_{2}O}$ is the partial pressure of water vapor within the bubble volume (both in Pa), $\alpha$ is the accommodation coefficient [62], presented by Yasui [63] and shown in Eq. (30) in function of ${\mathrm{T}}_{\mathrm{b}}$(K).

$$\mathrm{\alpha }=0.35\mathrm{for}{\mathrm{T}}_{\mathrm{b}}\le 350\mathrm{K}$$

$$\mathrm{\alpha }=0.35-0.05{\left(\frac{{\mathrm{T}}_{\mathrm{b}}}{50}-7\right)}^2+0.025\left(\frac{{\mathrm{T}}_{\mathrm{b}}}{50}-7\right)\left(\frac{{\mathrm{T}}_{\mathrm{b}}}{50}-8\right)\left(\frac{{\mathrm{T}}_{\mathrm{b}}}{50}-9\right)\mathrm{f}\mathrm{o}\mathrm{r}350\mathrm{K}\le {\mathrm{T}}_{\mathrm{b}}\le 500\mathrm{K}$$ (30)

$$\mathrm{\alpha }=0.05\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{T}\ge 500\mathrm{K}$$

The energy balance applied to the acoustic cavitation bubble accounts for the heat exchange due to the flow of evaporated and condensed water molecules at the bubble interface, the heat diffusion across the bubble wall and the endothermic and exothermic nature of the chemical reactions evolving inside the bubble volume at collapse. The temperature and the pressure will be governed by Eqs. (31), (32), respectively.

$$-P_{g}4\pi R^2\dot{R}-\frac{1}{3}\sum _{j=1}^N\mathrm{\Delta }{H}_j{r}_{j}4\pi R^3+4\pi R^2\frac{\dot{m}}{M_{H_{2}O}}{C}_{V_{H_{2}O}}{T}_b=\frac{\lambda }{\xi }4\pi R^2\left(T_b-T_{\infty }\right)+\sum _{i=1}^K{n}_i{C}_V{\dot{T}}_b$$ (31)

$$P_g=\frac{{\sum }_{i=1}^K{n}_i{R}_g{T}_b}{\frac{4}{3}\pi R^3}$$ (32)

Eq. (31) considers the work of pressure forces, the energy intake due to the chemical reactions, the heat exchange related to the non-equilibrium of evaporation and condensation of water molecules (three respective terms of the left side of Eq. (31)), the heat diffusion across the bubble–liquid interface of width $\xi$, according to the diffusion coefficient $\lambda$, and the impact on the variation of the bubble’s internal energy (both terms of the right side of Eq. (31)). The main species $i$ participating in the sums appearing in Eqs. (31), (32) are the water vapor and the saturating gas.

When the bubble collapses, a spherical shockwave is emitted from the bubble in the liquid, and can be responsible of dispersion of molecules in the medium, mass transfer and modification of the transport properties. The power emitted by the single acoustic cavitation bubble in the form of a shockwave $P_s$ is expressed by Eq. (33).

$$P_s=\frac{4\pi {\rho }_{\mathit{el}}}{c}{\left(R^2\ddot{R}+2R\dot{R^2}\right)}^2$$ (33)

Besides, when the bubble collapses near to a solid surface (the electrodes in sono-electrolysis), it loses its sphericity. A change of the shape of the bubble occurs, from the original approximate sphere to a slender shape and then flattened. Finally, a microjet breaks the bubble wall near the solid and impacts towards it. According to the ultrasonic cavitation phenomenon, micro-jet velocity is given by Eq. (34) [64].

$$v=8.97{\left(\frac{R}{R_0}\right)}^2\sqrt{\frac{P_{\infty }+P_A\mathrm{s}\mathrm{i}\mathrm{n}(2\pi ft)-P_v}{{\rho }_{\mathit{el}}}}$$ (34)

The exit jet power $P_j$ is then given in Eq. (35) as a function of the jet velocity $v$ and the jet nozzle exit diameter $h$ [65], [66].

$$P_j=\frac{1}{8}\pi {\rho }_{\mathit{el}}{h}^2{v}^3$$ (35)

The measurement of the effective acoustic power transmitted to the electrolyte due to sonication, performed using calorimetry as explained earlier (Eqs. (1), (2)), allows the assessment of the acoustic amplitude and hence the energetic power released in the form of shockwave and microjet by the collapse of the acoustic cavitation bubble (as indicated in Eqs. (33), (35)).

The sonochemical pathway of hydrogen production [67] under the conditions of sonication is also investigated in order to assess its contribution in the production of hydrogen. The following chemical mechanism, reported in Table 2, is considered to occur within the single acoustic cavitation bubble, behaving as a microscopic reactor with variable volume, temperature and pressure, described respectively in Eqs. (28), (31) and (32).

Table 2.

Chemical scheme containing the possible elementary reactions resulting of the cleavage of water under air atmosphere inside a single acoustic cavitation bubble [18].

i	Reaction i	Third body coefficient	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	H2: 2.5, H2O: 16.0	4.58 × 1013	−1.4	52,500	444,47
6	O + O + M → O2 + M	H2: 2.5, H2O: 16.0	6.17 × 103	−0.5	0	−505,4
7	O + H• + M → •OH + M	H2O: 5.0	4.72 × 105	−1.0	0	−436,23
8	H• + •OH + M → H2O + M	H2: 2.5, H2O: 16.0	2.25 × 1010	−2.0	0	−508,82
9	H• + O2 + M → HO2• + M	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	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 + M1 → O2 + O + M2	M1 O2: 1.64; M2 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	H2: 2.5, H2O: 16.0	2.45 × 108	−1.78	480	−444,47
31	O2 + M → O + O + M	H2: 2.5, H2O: 16.0	1.58 × 1011	−0.5	59,472	505,4
32	•OH + M → O + H• + M	H2O: 5.0	4.66 × 1011	−0.65	51,200	436,23
33	H2O + M → H• + •OH + M	H2: 2.5, H2O: 16.0	1.96 × 1016	−1.62	59,700	508,82
34	HO2• + M → H• + O2 + M	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	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 + M1 → O3 + M2	M1 O2: 1.64; M2 O2: 1.63, H2O: 15	4.1	0	−1057	−109,27
46	N2 + O2 → N2O + O	–	6.3 × 107	0	55,200	336.05
47	O2 + N → O + NO	–	6.4 × 103	1	3150	133.37
48	O + NO + M1 → O + NO2 + M2	M1 NO: 20; M2 O: 20, N2: 20	1.1 × 103	0	−940	−311.14
49	O + N2 → N + NO	–	7.6 × 1010	0	38,000	314.24
50	HO•+ N → H•+ NO	–	4.5 × 107	0	0	202.53
51	HO• + NO + M → HNO2 + M	H2: 2.3	8 × 103	0	−1000	−208.02
52	HO• + NO2 + M → HNO3 + M	N2: 2	5 × 105	0	0	−206.25
53	HO• + N2 → N2O + H•	–	2.5 × 106	0	39,000	267.32
54	N2O + O → N2 + O2	–	1 × 108	0	14,100	−336.05
55	O + NO → O2 + N	–	1.5 × 103	1	19,500	−133.37
56	O + NO2 + M1 → O + NO + M2	M1 NO: 20; M2 O: 20, N2: 20	1.1 × 1010	0	33,000	311.14
57	N + NO → O + N2	–	1.6 × 107	0	0	−314.24
58	H•+ NO → HO• + N	–	1.7 × 108	0	24,500	−202.53
59	HNO2 + M → HO• + NO + M	H2: 2.3	5.1 × 1011	−1	25,000	208.02
60	HNO3 + M → HO• + NO2 + M	N2: 2	1.6 × 109	0	15,400	206.25
61	N2O + H• → HO• + N2	–	7.6 × 107	0	7600	−267.32

The molar yield of hydrogen due to the sonochemical activity of a single acoustic cavitation bubble is given in Eq. (36) [19].

$$\frac{d{n}_{H_2}}{\mathit{dt}}=\frac{4}{3}\pi R^3\sum _{i=1}^N(\vartheta {\text{'}}_{H_{2}i}-{\vartheta }_{H_{2}i})A_i{e}^{\left(\frac{-E_i}{R_g{T}_b}\right)}\prod _{k=1}^K{\left(c_{\mathit{ki}}\right)}^{{\vartheta }_{\mathit{ki}}}$$ (36)

where ${\vartheta \prime }_{H_{2}i}$ and ${\vartheta }_{H_{2}i}$ represent the stoichiometric coefficients (if existent) of H₂ appearing in the right and left sides, respectively, of the i^th elementary reaction mentioned in Table.2. $c_{\mathit{ki}}$ is the molar concentration of the k^th reactant involved in the i^th elementary reaction and ${\vartheta }_{\mathit{ki}}$ is its corresponding stoichiometric coefficient.

The bubble population is characterized through the number density of bubbles, described in our previous works [21], [24]. The yield of hydrogen produced sonochemically within the electrolyte is then expressed as indicated in Eq. (37) [22], considering the bubble size distribution as reported in Table. 3.

$$n_{H_2}=V\sum _{i=1}^4{F}_i\times N_i\times n_{i_{H_2}}$$ (37)

Table 3.

Size distribution if bubble population based on the ambient radius [68].

Ambient radius $R_0$ (µm)	20	40	60	80
Number fraction $F_i$	18%	42%	22%	18%

3. Results and discussion

3.1. Calorimetric characterization

In order to link the modelling and simulation to the experiments, the actual acoustic power received by the electrolyte is determined using calorimetric measurements, the acoustic pressure is then calculated and used as a simulation condition. Fig. 3 presents the variation of the temperature of the electrolyte due to sonication, as well as the corresponding evolution of the acoustic power in function of time over 10 min of continuous sonication. The evolution of the temperature is quasi-linear, resulting in an average acoustic power of 8.13 W received by the 300 mL of electrolyte solution. Considering the receiving area of the electrolyzer, corresponding to the bases of both anodic and cathodic compartments, perpendicular to the travelling wave, the calculated acoustic power corresponds to an acoustic intensity of 3263.6 W/m², and an acoustic amplitude of 98.95 kPa.

Fig. 3.

Evolution of the electrolyte temperature and acoustic power due to indirect sonication as a function of time (40 kHz, 60 W_e, 300 mL).

Considering the electrical power feeding the sonicator, i.e., 60 W, the effective energy conversion efficiency is equivalent to 13.7%. As the studied setup of sono-electrolysis relies on indirect sonication, the 86.3% of the electrical power supplying the sonicator are dissipated in the form of heat at the transducer itself, the bath vessel, and the sonicated water in which the electrolyzer is immersed.

The acoustic amplitude of 99.95 kPa is adopted in the simulation of the acoustic cavitation and its associated effects, presented in Section 2.3.

3.2. Sono-electrolysis performance

3.2.1. Kinetics of hydrogen production and energy conversion efficiency

The volumes of produced hydrogen and oxygen have been monitored over time during the electrolysis operation under a feeding voltage of 4 V. The mass flow rates have been then assessed, we focus here on the mass flow rate of hydrogen, under silent and ultrasound conditions, Fig. 4 (a) reports the obtained values. The figure shows an average improvement in H₂ rate of 3.93%. On the other hand, Fig. 4 (b), which reports the energy conversion efficiency in both cases, demonstrate a relative gain of 2.76%.

Fig. 4.

Mass flow rate of hydrogen (a) and energy conversion efficiency (b) under silent and ultrasound conditions.

The improvement of hydrogen flow rate is attributed to the increase in the cell current, due to the drop in the Ohmic resistance. The decrease of the Ohmic resistance can be explained by the surface cleaning of the electrodes in the presence of ultrasounds, accelerating the removal of bubbles and freeing the electrochemical reactional sites on the electrodes’ surfaces. This increases the electron transport and thus improves the cell current and the kinetics of hydrogen production. The energy conversion efficiency results of the competitive phenomena accompanying the increase of hydrogen flow, namely the increase of the consumed power (due to the increase of the cell current for a constant potential), and the augmentation of the potential chemical energy stored in the flow of produced hydrogen. As the effect of sonication is positive, the observed increase is attributed to the predominance of the increase of the flow of recovered hydrogen, as compared to consumed power.

3.2.2. Cell resistance

Polarization curves are plotted under silent and ultrasound conditions by varying the feeding voltage from 0 to 12 V, and recording the corresponding current density. The results are reported in Fig. 5 (a), to highlight the noticed variation in the absence and presence of indirect continuous sonication.

Fig. 5.

Polarization curves under silent and ultrasound conditions (a), and variation of the experimental Ohmic resistance vs. the cell potential (b).

The decomposition voltage method has been adopted to estimate the cell resistance in both cases. The decomposition voltage is defined as the minimum potential difference that must be applied between a pair of electrodes before decomposition occurs and a current flows [39]. The extrapolation of the second portion of the curve back to zero current reveals a decomposition voltage value of 1.6 V under silent and ultrasound conditions. Considering a reversible potential of 1.23 V, according to Eq. (8), the activation overpotential is then experimentally estimated at 0.37 V. Sonication has then no effect on the activation overpotential.

Besides, the obtained curves demonstrate quasi-linear evolutions, and no third portion is distinguishable within the covered range. Hence, the second portion concerns the Ohmic overpotential, the Ohmic resistance is then deduced from the slop of the curves. The average Ohmic resistance is estimated at 9.03 Ω under silent conditions, and 8.38 Ω under ultrasounds, which is equivalent to a drop of 7.2%.

Fig. 5 (b) reports the variation of the resistance of the cell as a function of the cell potential. It is noticed that at low cell potential within the range from 2 to 2.6 V, ultrasounds have no effect and similar values are recorded under silent and ultrasounds conditions. However, a difference is observed starting from 2.8 V, with lower resistance under continuous sonication. The highest gap is attained at 4.8 V, with a relative decrease of 17.11% under ultrasounds. Overall, the induced difference in terms of the cell resistance in the presence of ultrasounds is observable over the interval 3 to 12 V, which corresponds to the portion of the Ohmic overpotential as shown in Fig. 5 (a).

3.3. Acoustic cavitation in sono-electrolysis

The macroscopic effect of ultrasounds in the sono-electrolysis process being observed and measured in the previous sections, we suggest here to simulate the acoustic cavitation bubble evolving in the electrolyte, in order to assess the chemical and physical effects associated to it and elucidate the mechanism of action of ultrasounds at the microscopic scale. The calorimetric characterization presented in Section 3.1 allowed the calculation of the effective acoustic amplitude of the wave travelling across the electrolyte, estimated at 98.95 kPa. The simulation of the oscillation of the single acoustic cavitation bubble is performed through the resolution of the system of non-linear differential equations presented earlier in Section 2.2.1, considering the corresponding acoustic conditions (40 kHz, 98.95 kPa), the initial molar yields of oxygen, nitrogen, and water vapor, and the initial conditions:

$$\frac{\mathit{dR}}{\mathit{dt}}\left(t=0\right)=0$$

$${R\left(t=0\right)=R}_0$$

Oxygen and nitrogen (forming air) are 27 and 12 folds more soluble in water than hydrogen, thus, the considered acoustic cavitation bubbles are deemed to form from gaseous nuclei composed of water vapor and air.

The numerical simulations over one acoustic cycle, i.e., 25 µs, consider the various ambient radii $R_0$ of the bubbles appearing in the population under 40 kHz, fairly represented by 20, 40, 60 and 80 µm, as shown in Table 3. Fig. 6 presents the variation of the bubble radius as a function of time for the representative ambient radii.

Fig. 6.

Variation of the bubble radius vs. time over one acoustic cycle under 40 kHz and 98.95 kPa, for ambient radii of 20, 40, 60 and 80 µm.

It is observed that the small bubbles having an ambient radius of 20 µm show harsher oscillation with several rebounds. The oscillation of the bubble becomes softer as the ambient radius increases from 20 to 80 µm. For instance, the expansion ratio decreases from 2.25 to 1.46 when the ambient radius increases from 20 to 80 µm. similarly, the compression ratio is reduced from 7.57 to 2.89. The maximum and minimum radii achieved during one acoustic cycle, and the related expansion and compression ratios are reported in Table 4 for the treated ambient radii.

Table 4.

Maximum and minimum radii and expansion and compression ratios in function of the ambient radii.

$R_0$(µm)	20	40	60	80
$R_{\mathit{max}}$(µm)	44.97	73.13	95.79	116.56
$R_{\mathit{min}}$(µm)	5.94	13.90	25.20	40.27
Expansion ratio	2.25	1.83	1.60	1.46
Compression ratio	7.57	5.26	3.80	2.89

The compression ratio is an indicator of the severity of the thermodynamic conditions attained inside the bubble volume, which allows an acceptable prediction of the extent of the chemical activity within the bubble as a microscopic reactor. This aspect will be investigated latter in order to estimate the yield of hydrogen produced sonochemically at single bubble level and within the cathodic compartment. However, the physical effects, supposed to be the main responsible of the observed reduction in the Ohmic resistance under ultrasounds, require specific simulations, based on Eqs. (33), (34), (35). Fig. 7 reports the maximum powers associated to the radiated shockwave and the emitted microjet at the bubble collapse. The simulations are performed for each ambient radius in order to elucidate the overall behaviour of the bubble population. Fig. 7 reveals that the exit microjet power increases with the increase of the ambient radius. To illustrate, it equals 0.56 W for an 80 µm bubble, while it is limited to 0.01 W at 20 µm. In contrast, the power associated to the radiated shockwave exhibits a maximum value of 0.18 W attained with bubbles of 40 µm of ambient radius. The comparison of the orders of magnitude of the powers of microjets and shockwaves demonstrate that relatively small bubbles are mainly concerned by the shockwave process and its contribution to the creation of turbulence within the electrolyte, which facilitates the hydrogen bubble detachment from the electrode surface, promotes the degassing process and liberates the reactional sites. Relatively large bubbles, i.e., having ambient radii of 60 and 80 µm, are more concerned by the microjetting process. The exit power of microjet is 1.76 times higher than the power of the radiated shockwave for bubbles of 60 µm of ambient radius, and 11.26 times higher when the ambient radius increases to 80 µm. considering the distribution of the number density of bubbles shown in Table 3, the bubbles population is mainly composed of 40 µm bubbles, the shockwave process is then predominant.

Fig. 7.

Exit power of microjet and power of radiated shockwave at the bubble collapse according to the ambient radius.

The contribution of the sonochemical pathway in the production of hydrogen is investigated through the simulation of the sonochemical production at the single bubble scale considering the covered ambient radii. Fig. 8 (a) presents the evolution of the yield of hydrogen produced sonochemically over one acoustic cycle within a single acoustic cavitation of 20, 40, 60 and 80 µm. Although the small bubbles know the highest production, the yields remain negligible and do not exceed the sonochemical activity threshold fixed conventionally at 10⁸ molecules/s [69], [70]. Considering the number density of bubbles corresponding to each ambient radius, and the size distribution reported in Table 3, Fig. 8(b) shows the yield of hydrogen produced by the end of the acoustic cycle for each ambient radius, it is clearly observed that the macroscopic sonochemical activity is negligible [18], and thus, the mass flow rate of hydrogen retrieved experimentally is only due to the electrochemical reaction presented in Eq. (5).

Fig. 8.

Sonochemical production of hydrogen at the single bubble scale (a) and within the cathodic compartment (b) over one acoustic cycle according to the ambient radius.

3.4. Effect of sonication in sono-electrolysis process

The microscopic analysis of the activity of the single acoustic cavitation bubbles composing the population demonstrated that the physical effects contribute exclusively to the reduction of the Ohmic resistance in the sono-electrolysis process, while the sonochemical activity remains negligible. We suggest in the present section the examination of the impact of the reduction of electrodes coverage by bubbles on the value of the Ohmic resistance, in order to make the link between the microscopic events associated with the acoustic cavitation bubble and its macroscopic role in electrodes’ cleaning. Fig. 5(a) allowed, through the voltage decomposition method, to observe that the Ohmic resistance is predominant within the interval ranging from 3 to 12 V. Lower operational voltages lead to a cell operation governed by the activation overpotential, while higher voltages conduct to the predominance of concentration overpotential (which manifests with high current densities). On the other hand, sonication, through the physical effects induced by the acoustic cavitation bubble (mainly microjet and shockwaves), induces the desorption of bubbles from the electrodes surfaces and hence facilitates the gas recovery and liberates active sites for the electrochemical reactions. The action of ultrasounds concerns then the bubble resistance given in Eq. (16), which is part of the Ohmic resistance as seen in Eq. (12). This explains the observable action of ultrasounds in the potential range from 3 to 12 V in Fig. 5(b), corresponding to the Ohmic resistance.

Fig. 9 reports the results of the numerical simulation of the Ohmic and bubble resistance as functions of the fraction of electrode surface coverage. The simulation demonstrates that when the electrodes’ surfaces are totally accessible (in the absence of bubble coverage), the average Ohmic resistance of the cell equals 8.08 Ω. It increases gradually to 10 Ω when the total surface is supposed to be covered by the bubbles, the bubble resistance neighbours in this case a value of 2 Ω. The highest increase in the bubble resistance, and consequently the Ohmic resistance, is observed starting from a coverage of 60%.

Fig. 9.

Simulated evolutions of the Ohmic and bubble resistances in function of the fraction of electrodes’ coverage.

On the other hand, the experiments returned the values of the Ohmic resistances under silent conditions (9.03 Ω), and under ultrasounds (8.38 Ω), as mentioned in Section 3.2.2. These values have been projected on the curve of Ohmic resistance in Fig. 9 to retrieve the corresponding fractions of electrodes’ coverage. This projection (black triangles) returns 76% and 42%, respectively. It is then revealed that the integration of sonication reduces the electrodes’ coverage by the bubbles from 76% to 42%, which corresponds to a decrease of 7.2% in Ohmic resistance and 62.35% in bubble resistance.

The simulation returns as well the values of the overpotentials and their variations with the electrodes’ coverage by the bubbles as shown in Table 5. The results are in good agreement with the values retrieved experimentally, the activation voltage remains constant around 0.36 to 0.37 V, which is the value reported earlier in Section 3.2.2. The power consumed by the cell decreases with the increase of the bubble coverage, due to the drop of the cell current at a fixed feeding potential. According to the previous analysis of Fig. 9, the current passes from 0.268 to 0.286 when integrating sonication, which is equivalent to an increase of 6.72%.

Table 5.

Simulation results (potential in V, power in W and current in A).

	$E_{\mathit{rev}}(T,P)$	$U_{\mathit{Ohm}}$	$U_{\mathit{Conc}}$	$U_{\mathit{act}}$	$P_{\mathit{cell}}$	$I$
0%	1.23	2.395	0.00033	0.37010	1.186	0.297
10%	1.23	2.396	0.00032	0.36979	1.179	0.295
20%	1.23	2.396	0.00032	0.36941	1.170	0.293
30%	1.23	2.397	0.00032	0.36896	1.160	0.290
40%	1.23	2.397	0.00031	0.36841	1.147	0.287
50%	1.23	2.398	0.00031	0.36772	1.132	0.283
60%	1.23	2.399	0.00030	0.36686	1.112	0.278
70%	1.23	2.400	0.00030	0.36574	1.088	0.272
80%	1.23	2.401	0.00029	0.36424	1.056	0.264
90%	1.23	2.403	0.00028	0.36218	1.014	0.253
100%	1.23	2.406	0.00026	0.35919	0.955	0.239

4. Conclusion

The present study combined an experimental investigation of a sono-electrolytic process based on membraneless alkaline electrolysis using an H-cell configuration, and indirect continuous sonication, with numerical modeling and simulation of physical and chemical microscopic acoustic cavitation events and their macroscopic consequences on electrolysis in terms of electrodes’ coverage by bubbles. The experiments demonstrated an average improvement in H₂ rate of 3.93% and a relative gain of 2.76% in energy conversion efficiency when integrating sonication. The polarization curves obtained experimentally under silent and ultrasounds conditions presented quasi-linear evolutions, related to the Ohmic overpotential. The retrieved average Ohmic resistance was estimated at 9.03 Ω under silent conditions, and 8.38 Ω under ultrasounds, which is equivalent to a drop of 7.2%.

Besides, the calorimetric characterization led to an acoustic intensity of 3263.6 W/m², and an acoustic amplitude of 98.95 kPa. This value was used in the numerical simulation of the acoustic cavitation bubble.

The simulations revealed that the exit microjet power increases with the increase of the ambient radius, while the power associated to the radiated shockwave exhibits a maximum at 40 µm. Moreover, the relatively small bubbles are mainly concerned by the shockwave process and its contribution to the creation of turbulence within the electrolyte, which facilitates the hydrogen bubble detachment from the electrode surface, promotes the degassing process and liberates the reactional sites. In contrast, relatively large bubbles are more concerned by the microjetting process.

The simulation of the sonochemical activity of the single bubble and bubble population proved that the yields of the sonochemically produced hydrogen remains negligible and do not exceed the sonochemical activity threshold fixed conventionally at 10⁸ molecules/s.bubble.

The microscopic analysis of the activity of the single acoustic cavitation bubbles composing the population demonstrated that the physical effects contribute exclusively to the reduction of the Ohmic resistance in the sono-electrolysis process.

Finally, the macroscopic simulation revealed through the projection of the experimental Ohmic resistances retrieved under silent and ultrasound conditions on the simulated curve of the Ohmic resistance as a function of the fraction of electrodes’ coverage, that the integration of sonication reduces the electrodes’ coverage by the bubbles from 76% to 42%, which corresponds to a decrease of 7.2% in Ohmic resistance and 62.35% in bubble resistance.

CRediT authorship contribution statement

Kaouther Kerboua: Project administration, Conceptualization, Methodology, Software, Formal analysis, Writing – original draft, Writing – review & editing. Nour Hane Merabet: Investigation, 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.

Data availability

No data was used for the research described in the article.