Modelagem Matemática da Oxidação Anódica

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Environmental Chemistry Letters (2024) 22:1521–1561 
https://doi.org/10.1007/s10311-023-01693-0
REVIEW ARTICLE
Mathematical modeling of the anodic oxidation of organic pollutants: 
a review
Ekaterina Skolotneva1  · Andrey Kislyi2 · Anastasiia Klevtsova2 · Davide Clematis1 · Semyon Mareev2 · 
Marco Panizza1
Received: 15 May 2023 / Accepted: 28 December 2023 / Published online: 27 February 2024 
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024
Abstract
Anodic oxidation is a promising method for removing organic pollutants from water due to its high nonselectivity and 
effectiveness. Nevertheless, its widespread application is limited due to its low current efficiency, high energy consumption 
and low treatment rates. These problems may be overcome by the optimization of the process parameters, reactor design and 
electrode geometry, by coupling the experimental investigations with mathematical modeling. Here we review the modeling 
of anodic oxidation with focus on basics of this process, the competition phenomenon in real wastewater, flow cells and batch 
cells, historical aspects, general modeling equations, modeling with plate electrodes, modeling with porous 3-dimension 
electrodes and the density functional theory. Mathematical modeling can provide current, voltage and concentration 
distributions in the system. Mathematical modeling can also determine the effects on the performance of parameters such as 
diffusion layer thickness, flow velocity, applied current density, solution treatment time, initial concentration and diffusion 
coefficients of organic pollutants, electrode surface area, and oxidation reaction rate constant. Mathematical models allow 
to determine whether the limiting factor of the process is kinetics or diffusion, and to study the impact of competition of 
phenomena. The density functional theory provides information on probable reaction pathways and by-products.
Keywords Electrochemical oxidation · Organic pollutant · Mathematical model · Hydroxyl radical · Mass transport · 
Density functional theory
Abbreviations
Blue-TiO2 Blue-titanium dioxide
C6H6 Benzene
CH3COCH3 Acetone
(C2H5)3N Triethylamine
CH3OH Methanol
Cl− Chloride ions
ClCH꞊CCl2 Trichloroethylene
ClO3
− Chlorate
CO2 Carbon dioxide
H2 Hydrogen
H2CO3 Carbonic acid
H2O Water
IrO2-Ta2O5 Iridium dioxide-tantalum pen-
toxide electrode
NaClO4 Sodium perchlorate
NO2 Nitrogen dioxide
O3 Ozone
O3/H2O2 Ozone/hydrogen peroxide
PO4
3− Phosphate
RuO2-TiO2 Ruthenium oxide-titania 
electrode
S2O8
2− Peroxodisulfate
Ti4O7 Sub-stoichiometric titanium 
oxide
Ti/Pt Titanium covered by platinum 
electrode
B Boron
Carbon/Graphite Carbon-coated graphite 
electrode
CH3CH3 Ethane
CH3COOH Acetic acid
C6H5NH2 Aniline
 * Ekaterina Skolotneva 
 ekaterina.skolotneva@edu.unige.it
 * Semyon Mareev 
 mareev-semyon@bk.ru
1 Department of Civil, Chemical and Environmental 
Engineering, University of Genoa, Via All’Opera Pia, 15, 
16145 Genoa, Italy
2 Physical Chemistry Department, Kuban State University, 149 
Stavropolskaya Str., Krasnodar, Russia 350040
http://crossmark.crossref.org/dialog/?doi=10.1007/s10311-023-01693-0&domain=pdf
http://orcid.org/0000-0003-1101-9447
1522 Environmental Chemistry Letters (2024) 22:1521–1561
C6H5OH Phenol
Cl2 Chlorine
ClO− Hypochlorite
ClO4
− Perchlorate
C2O6
2− Peroxodicarbonate
HCO3
− Hydrogen carbonate
HClO Hypochlorous acid
HCOOH Formic acid
NaCl Sodium chloride
Na2SO4 Sodium sulfate
O2 Oxygen
OCH3 Methoxy groups
·OH Hydroxyl radicals ()
P2O8
4− Peroxodiphosphate
SO4
2− Sulfate
TinO2n−1 Magnéli phases of sub-
stoichiometric titanium oxides
Ti/PbO2 Titanium-coated lead dioxide 
electrode
Ti/SnO2 Titanium-coated tin dioxide 
electrode
List of symbols
A Electrode area (m2)
c Concentration (mol m−3)
cs Concentration at the electrode 
surface (mol m−3)
CR Concentration of anodic 
reactants (mol m−3)
ctp Tracer particles concentration 
(mol m−3)
COD Chemical oxygen demand 
(mol O2 m−3)
D Diffusion coefficient of the 
compound (m2 s−1)
dreac Reaction zone thickness (m)
f Body force (N kg−1)
i Current density (A cm−2)
ilim Limiting current density 
(A  m−2)
ilim,ne− Limiting current density of 
DET(A  m−2)
i0 Exchange current density(A 
 cm−2)
j Flux density (mol m−2  s−1)
k·OH ·OH recombination rate 
constant (m3 mol−1  s−1)
P Given loading (mol COD s−1)
Pe Peclet number
R Reactive term (mol m−3  s−1)
Rc Electrolyte ohmic resistance 
(Ω)
ri Oxidation rate of each 
compound in the reaction zone 
(mol m−2  s−1)
Sh Sherwood number
t Time (s)
ts Special time (s)
u Linear fluid velocity (m s−1)
VR Reservoir volume (m3)
ΔVwork Cell potential (V)
X COD conversion (%)
Xcr Critical conversion
Α Electron transfer coefficient
Areq Required electrode area (m2)
cb Bulk concentration (mol m−3)
C0 Concentration of cathodic 
reactants (mol m−3)
C Dimensionless tracer particles 
concentration
ctp
0 Initial tracer particles 
concentration (mol m−3)
COD0 Initial chemical oxygen 
demand (mol O2 m−3)
Esp Specific energy consumption 
(kW h kg COD−1)
F Faraday’s constant(C mol−1)
i Current intensity (A)
iappl Applied current density (A 
 m−2)
ilim
0 Initial limiting current density 
(A m−2)
i·OH Initial limiting current density, 
corresponding to the total 
mineralization of organic 
compounds (A m−2)
J Flux (mol s−1)
km Mass transfer coefficient (m 
 s−1)
Lx Axial length (m)
p Applied pressure (Pa)
Qcr Critical specific charge (Ah 
 m−3)
R Universal gas constant (J 
 mol−1 K −1)
Re Reynolds number
Sc Smidt number
T Temperature (K)
tcr Critical time (s)
uint Interstitial liquid velocity (m 
 s−1)
V Volume of electrolyte (dm3)
ΔVi Oxidation potential of each 
process (V)
1523Environmental Chemistry Letters (2024) 22:1521–1561 
Xl Dimensionless axial length 
(m)
x Axis coordinate along the 
distance (m)
z Charge
Greek letters
δ [delta] Diffusion layer thickness (m)
ε [epsilon] Effectiveness factor
εCl− [epsilon Cl] Faradic yield as a function of 
chloride (Cl−) concentration
ηa [eta a] Overpotential of anodic 
reaction (V)
θ [theta] Dimensionless time
μ [mu] Dynamic viscosity (Pa s)
τ [tau] Electrolysis time (s)
φ [phi, small letter] Electric potential (V)
αi [alpha i] Proportion of electrons 
involved in a particular 
electrochemical process 
corresponds to each process i
α·OH [alpha OH] Term accounts for the fraction 
of current directed toward ·OH 
production
δ(exp) [delta exp] Diffusion layer thickness 
obtained experimentally (m)
εi [epsilon i] Faradaic yield
ηc [eta c] Overpotential of cathodic 
reaction (V)
θi [theta i] Parameter represents the 
oxidation efficiency
ρ[rho] Liquid density (kg  m−3)
ϕ [phi] Dimensionless parameter 
expressing the ratio between 
the chemical reaction rate and 
the mass transfer coefficient
φ [phi, small bold letter] Normalized current efficiency
Introduction
Electrochemical advanced oxidation processes are 
increasingly being used to treat wastewater from 
organic pollutants. Electrochemical advanced oxidation 
processes are defined as purification processes occurring 
at temperatures and pressures close to their values in the 
environment, at which the generation of hydroxyl radicals 
(·OH) occurs with a sufficient rate to promote the oxidation 
of organic pollutants (Chaplin 2014; Sirés et  al. 2014; 
Moreira et al. 2017; Brosler et al. 2023). In recent years, 
electrochemical advanced oxidation processes have attracted 
increasing attention from researchers, as evidenced by the 
growing number of well-cited review articles on the topic 
(Cuerda-Correa et al. 2019; Seibert et al. 2020; da Silva et al. 
2021; Ganiyu et al. 2021; Titchou et al. 2021). One such 
process is anodic oxidation, which allows providing reagent-
free removal of contaminantsby completely oxidizing them 
to inorganic substances (McBeath et al. 2019; Yang 2020; 
Hu et al. 2021; Fu et al. 2023).
Many organic pollutants have been effectively removed 
using anodic oxidation: aromatic compounds (Polcaro et al. 
2003; Borrás et al. 2004; Mascia et al. 2007; Lin et al. 2020), 
dyes (Galus and Adams 1964; Brillas and Martínez-Huitle 
2015; Cruz-Díaz et al. 2018), pharmaceuticals (Lan et al. 
2018; Trellu et al. 2018b; Zhang et al. 2021), pesticides 
(Trellu et  al. 2021), contaminants of emerging concern 
(Shahid et  al. 2021) and microplastics (Kiendrebeogo 
et  al. 2021; Ricardo et  al. 2021). The use of anodic 
oxidation can significantly reduce the damage caused to 
the ecosystem by wastewater from some industries, such as 
the pharmaceutical, textile, petroleum, paper and tannery 
industries (Garcia-Segura et al. 2018).
It is rather difficult to achieve high current efficiencies in 
the anodic oxidation process due to the kinetic and diffusion 
limitations of the process (Panizza and Cerisola 2009). 
The influence of these restrictions is reduced in two main 
directions: optimization of the anode structure, i.e., design 
of porous anodes with specified characteristics (porosity 
and pore size), as well as the use of promising materials for 
its manufacture, e.g., boron-doped diamond and TinO2n−1 
(Magnéli phases of sub-stoichiometric titanium oxides) 
(Ganiyu et al. 2019; He et al. 2019; Hu et al. 2021; Cui 
et al. 2022; Kumar et al. 2022; Ma et al. 2022). Hybrid plants 
are being developed that combine anodic oxidation with 
other organic pollutant removal processes (Hu et al. 2021). 
Energy consumption can be reduced by using renewable 
energy sources, application of microbial fuel cells, and 
photocatalysis (Gude 2016; Ge et al. 2017; Ganiyu et al. 
2020). At the same time, the understanding of this process 
needs to be improved. Thereby, mathematical models play 
an extremely important role.
In the literature, there are various mathematical models 
developed to describe the anodic oxidation process of 
organic compounds (Newman and Tiedemann 1975; 
Comninellis 1994; Simond et al. 1997; Panizza et al. 2001a; 
Kapałka et al. 2008; Rodriguez et al. 2012; Trellu et al. 
2016; Misal et al. 2020; Skolotneva et al. 2021). In these 
models is described the process on a plate (two-dimensional) 
electrodes (Comninellis 1994; Simond et al. 1997; Kapałka 
et  al. 2008; Trellu et  al. 2016). There are also models 
that describe the process on porous (three-dimensional) 
electrodes (Newman and Tiedemann 1975; Misal et  al. 
2020; Skolotneva et al. 2021). Some models make it possible 
to obtain an analytical solution of the problem and can be 
a convenient tool for qualitative analysis of the influence 
of the main process parameters on the system behavior 
1524 Environmental Chemistry Letters (2024) 22:1521–1561
(Comninellis 1994; Simond et al. 1997; Trellu et al. 2016). 
These models rarely accurately account for all factors, but 
they greatly simplify the understanding of the basic patterns 
of the process. Other problems are solved numerically, 
allowing to obtain a more accurate solution taking into 
account hydrodynamic characteristics of experimental 
system that can be applied to specific experimental system 
(Skolotneva et al. 2021; Monteil et al. 2021). Such models 
are more cumbersome and require certain computing 
resources but can give a more detailed idea of the process. 
Finally, there are empirical models, that make it possible to 
study the influence of various factors on the anodic oxidation 
process occurring in a particular experimental system 
(Ghazouani et al. 2016; Kothari and Shah 2020).
As regards the state of the art in the field of modeling the 
electrochemical oxidation process of organic pollutants, a 
relatively small number of publications should be noted here, 
compared, for example, with the number of publications in 
the field of fuel cell or electrode plating modeling. The first 
anodic oxidation models were presented by Comninellis 
(1994). Since then, several other models have been 
developed to improve understanding of the anodic oxidation 
process. It is also noticeable that over the past decade only a 
few essentially new mathematical models in this area have 
been presented (Marshall and Herritsch 2018; Skolotneva 
et al. 2020; Misal et al. 2020).
Review articles that consider anodic oxidation models in 
general or for predicting the behavior of specific experimental 
systems are very useful for the scientific community. These 
articles provide an opportunity for a quick and relatively sim-
ple formation of an idea about the anodic oxidation knowledge 
area. Review articles can also help the reader take a new per-
spective at the description of the study object and can contrib-
ute to the development of new model ideas. At the same time, 
it should be noted that the number of review articles in which 
various aspects of anodic oxidation modeling are considered 
in detail is extremely small (Russo 2021).
This review article is devoted to a detailed description of 
the theoretical aspects of the anodic oxidation process. A 
lot of effort has gone into making this paper a starting point 
in modeling electrochemical oxidation processes for those 
who are interested. In this review, a simple description of the 
most common models is presented, their main advantages 
and disadvantages are indicated, and various approaches for 
the anodic oxidation modeling are considered.
Basics of the anodic oxidation process
The application of the anodic oxidation process to remove 
organic pollutants is possible due to partial degradation or 
complete mineralization using electrochemical oxidation 
reactions. The electrocatalytic properties of anodic materials 
play an undeniable role in the organic removal efficiency of 
the anodic oxidation process.
Anodic oxidation of organic compounds for wastewater 
treatment is implemented in two main ways (Fig. 1):
Direct anodic oxidation—a process that involves direct 
electron transfer reactions between the anode surface 
and organic pollutants, i.e., electron transfer occurs on 
the electrode surface without the participation of other 
substances (Martínez-Huitle and Ferro 2006; Panizza and 
Cerisola 2009). Electrons are capable of oxidizing some 
organic pollutants at lower potentials than the oxygen 
evolution reaction (Panizza and Cerisola 2009; Garcia-
Segura and Brillas 2011). The direct oxidation process 
usually requires the adsorption of pollutants onto the anode 
surface (see scheme in Fig. 1), which limits the process 
rate. It does not lead to the complete combustion of organic 
pollutants, R (Eq. 1), and thus surface deactivation of an 
electrode may occur (Rodgers et al. 1999; Rodrigo et al. 
2001).
Indirect anodic oxidation—a process in which organic 
pollutants are oxidized under the effect of highly oxidizing 
species generated on the anode surface, which act as 
intermediaries for the movement of electrons between 
the electrode and organic compounds (Martínez-Huitle 
and Ferro 2006; Brillas et al. 2009; Panizza and Cerisola 
2009; Sirés et al. 2014; Brillas and Martínez-Huitle 2015; 
Martínez-Huitle et al. 2015). Different kinds of oxidizing 
species can be generated by the anodic oxidation process 
(Fig.  1b,c,d). Some of the most important are reactive 
oxygen species, such as ·OH. The generation of large 
quantities of ·OH from the water dissociation onto the 
surface of the anode material, M, with a high-oxygen 
overpotential proceeds as follows:
With consequent oxidation of organic pollutants:
Degradation products may be carbon dioxide (CO2), 
water (H2O) and other inorganic oxides of heteroatoms 
contained in the initial organic molecule.
Theoretically, anodic oxidation is possible at low 
potentials before oxygen evolution (direct anodic oxidation), 
but under these conditions, the anode surface is rapidly 
deactivated due to the deposition of a polymer layer on it 
(fouling).The fouling depends on the adsorption properties of 
the anode surface, as well as on the concentration and 
nature of organic compounds. This effect can be avoided 
(1)R → (R⋅)+ + e−
(2)M + H2O → M(⋅OH) + H+ + e−
(3)R +M(⋅OH) → degradation byproducts
1525Environmental Chemistry Letters (2024) 22:1521–1561 
by conducting anodic oxidation in the range of the water 
dissociation potentials, due to the intermediate products of 
the oxygen evolution reaction (indirect anodic oxidation, 
Fig.1 b,c,d).
The efficiency of the process depends on the operating 
conditions and primarily on the nature of electrode mate-
rial. In particular, anodes with a low oxygen evolution 
overpotential, such as electroactive ruthenium oxide with 
titanium oxide nanotube array (RuO2–TiO2), oxide mixture 
of iridium dioxide and tantalum pentoxide (IrO2–Ta2O5), 
titanium covered by platinum (Ti/Pt), carbon-coated graphite 
(Carbon/Graphite), are referred to as "active" (Fig. 2), as 
they are involved in “chemical” adsorption of ·OH (Fig. 1b). 
These anodes contribute to the partial and selective oxida-
tion of pollutants, i.e., electrochemical conversion, whereas 
anodes with a high-oxygen evolution overpotential, such as 
titanium-coated lead dioxide (Ti/PbO2), titanium-coated tin 
dioxide (Ti/SnO2), boron-doped diamond or sub-stoichio-
metric titanium oxide (Ti4O7), exhibit "non-active" behavior 
and therefore are ideal electrodes for electrochemical incin-
eration of organic pollutants to CO2 in wastewater treatment.
Furthermore, the boron-doped diamond electrodes are the 
most suitable non-active anodes due to good chemical and 
electrochemical stability, long lifetime, and a wide range 
of water dissociation potentials. Thereby, boron-doped 
diamond electrodes are promising anodes for industrial-
scale wastewater treatment. It is known that when using 
boron-doped diamond electrodes, many water contaminants 
are completely mineralized, and in some cases (namely, at 
kinetic limitations) the current efficiency of the process can 
R (pollutant)
Rох(product) 
di
re
ct
 e
le
ct
ro
n 
tra
ns
fe
r
adsorption
desorption
a
Oxidant 
precursor 
(H2O)
½ O2
H+
H+
Rox(product) 
R(pollutant) 
*chems-chemisorption
b
Oxidant 
precursor 
(H2O)
½ O2+ H2
H+
Rox(product) 
R(pollutant) 
c
Oxidant 
precursor 
Stable 
oxidants
Rox(product) 
R(pollutant) 
2 3 2
4 4 3SO Cl PO CO− − − −( , , , )
Rox(product) 
R(pollutant) 
Rox(product) 
R(pollutant) 
Activated
oxidants 
Activation
d
Fig. 1 Processes involved in the anodic oxidation: a Direct oxidation: 
the molecule of organic pollutant (R (pollutant)) is first adsorbed on 
the electrode surface (Rads), and then oxidized (Roxads) by direct elec-
tron transfer (e−); Indirect oxidation: b Generation of reactive oxy-
gen (O2) on active anode: hydroxyl radicals (·OH) formed from the 
discharge of water (H2O) is adsorbed on the active site and interacts 
with the material of electrode (·OHads), which leads to the formation 
of higher oxide. The reactive oxygen in this case is chemisorbed (–
Ochems); c Generation of O2 on non-active anode:·OH formed from the 
discharge of H2O is adsorbed on the active site (·OHads), but it cannot 
interact with the material of electrode, thus, the O2 in this case is phy-
sisorbed; d Generation of other reactive species: from the oxidation 
of common electrolytes such as sulfate (SO4
2−), chloride (Cl−), phos-
phate (PO4
3−) and carbonate (CO3
2−) many stable oxidant agents can 
be formed. Rox(product)—organic product of oxidation, H+—hydro-
gen ion, Stable oxidants: peroxodisulfate (S2O8
2−), active chlorine 
 (Cl2), peroxodiphosphate (P2O8
4−), peroxodicarbonate (C2O6
2−)
1526 Environmental Chemistry Letters (2024) 22:1521–1561
reach nearly 100% (Martínez-Huitle and Ferro 2006; Panizza 
and Cerisola 2009; Sirés et al. 2014; Brillas and Martínez-
Huitle 2015; Martínez-Huitle et al. 2015, 2018; Ganiyu et al. 
2018).
Recently, TinO2n−1 has been proposed as a new economic 
anode material for the electrocatalytic oxidation of organic 
pollutants (Ganiyu et  al. 2019). Nevertheless, plate 
 TinO2n−1 has been achieved slightly less efficiency in the 
electrochemical oxidation of organics than in the boron-
doped diamond anode (Ma et al. 2023b). But it is possible 
to prepare 3D porous electrodes made of TinO2n−1, and in 
this case the efficiency increases significantly (Ma et al. 
2023a). Another promising material based on titanium 
oxides, namely, blue-titanium dioxide (blue-TiO2) nanotube 
arrays, has been proposed for use as an anode material in 
the anodic oxidation process (Kim et al. 2014). According 
to Cai et al. (2019), blue-TiO2 nanotube anode compared 
to the boron-doped diamond anode had a comparable and 
even better characteristics, such as ·OH production activity 
and total organic carbon (TOC), chemical oxygen demand 
(COD) removal, with a lower energy consumption. Reactive 
electrochemical membranes based on blue-TiO2 nanotube 
arrays are also known, which make it possible to achieve 
complete removal of organics in a single-pass flow-through 
mode (Wang et al. 2022).
Indirect oxidation is used to prevent fouling of the 
electrode by eliminating the direct electron transfer between 
the organic compounds and the anode surface. Therefore, 
the oxidizing species generated electrochemically at the 
anode oxidize the contaminants in the bulk solution. 
Among the oxidizing species generated at the anode, 
active chlorine (Cl2) is the most common and widely used 
for wastewater treatment (Garcia-Segura et al. 2018). The 
probable mechanism for the electrogeneration of active Cl2 
species mediated by reactive oxygen species is proposed by 
Bonfatti et al. (2000), Neodo et al. (2012) and Rosestolato 
et  al. (2014). The oxygen transfer reactions are carried 
out by adsorbed oxychlorinated species formed according 
to reaction (Eq. 4), as an intermediate of the Cl2 release 
(Eq. 5) as in Fig. 1d. Formed hypochlorous acid (HClO) is a 
weak acid (pKa 7.5), that is in equilibrium with hypochlorite 
 (ClO−) (Eq. 6). Therefore, pH solution value significantly 
affects the concentration of Cl2 compounds and thus the 
efficiency of the oxidation process (Scialdone et al. 2021; 
Hao et  al. 2022). Indeed, Cl2 prevails at very low pH, 
HClO—in moderate acidic conditions, and ClO−—in basic 
conditions. However, the formation of other intermediate 
oxidants, such as chlorate (ClO3
−) and perchlorate (ClO4
−), 
is possible (Eqs. 7–12), which are less active compared to 
 ClO− (Titchou et al. 2021). Therefore, their formation is 
an undesirable process. The generation rate of ClO3
− and 
 ClO4
− can be reduced by the irradiation of the solution 
(Kiwi et al. 2000). It should be noted that Cl2 can lead to the 
formation of chlorinated by-products which could be toxic 
(de Moura et al. 2014; Mostafa et al. 2018).
(4)M(⋅OH) + Cl− → M(HOCl)
(5)M(HOCl) → M + 1∕2Cl2 + OH−
(6)HClO ⇄ H+ + ClO−
(7)Cl− + ⋅OH → ClOH−
⋅
Oxidation power
Oxygen evolution overpotential (V)
Chemisorbed ·OH Physisorbed ·OH 
1.4 – 1.7
RuO2-TiO2
1.5 – 1.8
IrO2-Ta2O5
1.7 – 1.9
Ti/Pt
1.7
Carbon/
Graphite
1.8 – 2.0
Ti/PbO2
1.9 – 2.2
Ti/SnO2
BDD,
2.2 – 2.6
Ti4O7
Active Non-active
Fig. 2 Classification of the electrode materials used in anodic oxi-
dation. The value of oxygen evolution overpotential determines the 
mechanism of O-transfer reaction for each electrode. Electrodes with 
the value of oxygen evolution overpotential (V) > 1.8 V can be clas-
sified as active anodes: ruthenium dioxide (RuO2)-titanium dioxide 
 (TiO2), iridium dioxide (IrO2)-tantalum pentoxide (Ta2O5), titanium 
(Ti)-platinum (Pt), carbon/graphite; and the ones with the value of 
oxygen evolution overpotential < 1.8 V can be classified as non-active 
anodes: titanium (Ti)/lead dioxide (PbO2), titanium (Ti)/tin dioxide 
 (SnO2), boron-doped diamond (BDD) or sub-stoichiometric titanium 
oxide (Ti4O7); hydroxyl radicals(·OH). Adapted with the permission 
of Taylor & Francis from Garcia-Rodriguez et al. (2022)
1527Environmental Chemistry Letters (2024) 22:1521–1561 
Most often, the electrodes used to produce active Cl2 
consist of Pt or a mixture of metal oxides, for example RuO2, 
 TiO2 and IrO2. These electrodes have good electrocatalytic 
properties, long-term stability, low price and may be applied 
to a wide range of pollutants, such as olive oil, textile and 
tannery wastewaters (Martínez-Huitle and Ferro 2006; 
Brillas et al. 2009; Panizza and Cerisola 2009; Sirés et al. 
2014; Brillas and Martínez-Huitle 2015; Martínez-Huitle 
et al. 2015, 2018; Chung et al. 2018; Ganiyu et al. 2018).
Other oxidizing species are electrogenerated during the 
oxidation of common electrolytes such as sulfate (SO4
2−), 
phosphate (PO4
3−) and hydrogen carbonate (HCO3
−) 
yielding peroxodisulfate (S2O8
2−), peroxodiphosphate 
 (P2O8
4−) and peroxodicarbonate (C2O6
2−) according 
to reactions (13–22) (Serrano et  al. 2002; Velazquez-
Peña et  al. 2013; de Paiva Barreto et  al. 2015; Ganiyu 
and Gamal El-Din 2020). In comparison, these species 
are weaker oxidants than ·OH and active Cl2 and are not 
capable of completely mineralizing the organic pollutants. 
Nevertheless, they could facilitate the oxidation of some 
organic molecules, i.e., S2O8
2− accelerates the degradation 
rate of polystyrene microplastics (Kiendrebeogo et al. 2022).
(8)HOCl + ⋅OH → ClO ⋅ +H2O
(9)ClO− + ⋅OH → ClO ⋅ +HO−
(10)ClO−
2
+ ⋅OH → ClO ⋅2 +HO
−
(11)ClO ⋅2 + ⋅ OH → ClO−
3
+ H+
(12)ClO−
3
+ ⋅OH → ClO−
4
+ H+ + e−
(13)2HSO−
4
→ S2O
2−
8
+ 2H+ + 2e−
(14)HSO−
4
→ SO−
4
⋅ +H+ + e−
(15)2PO3−
4
→ P2O
4−
8
+ 2e−
(16)HPO2−
4
→ PO2−
4
⋅ +H+ + e−
(17)SO2−
4
+ ⋅OH → SO−
4
⋅ +HO−
(18)HSO−
4
+ ⋅OH → SO−
4
⋅ +H2O
(19)HPO2−
4
+ ⋅OH → PO2−
4
⋅ +H2O
(20)PO3−
4
+ ⋅OH → PO2−
4
⋅ +HO−
Thus, in the indirect oxidation, the supporting electrolyte 
has a significant effect on the oxidation kinetics. In this 
regard, for accurate mathematical description of the anodic 
oxidation process, it is necessary to take into account 
reactions involving the inorganic matrix. In the anodic 
oxidation process, in addition to active species and oxidation 
products of organic pollutants, gaseous products such as 
oxygen (O2) and hydrogen (H2) are also formed according 
to Eqs. 23–24.
The gas formation on the electrodes can significantly 
reduce the process efficiency for the following reasons:
• Gas bubbles released on the electrode surface can lead 
to undesired blockage of the electroactive electrode 
surface, resulting in energy losses and redistribution 
of current density in the system (Angulo et al. 2020). 
Energy losses and redistribution of current density are 
due to the fact that gas bubbles have an extremely low 
electrical conductivity, which leads to an increase in the 
ohmic resistance of the solution.
• The bubbles are a steric obstacle to the delivery of the 
contaminant to the electrode surface. It can also lead 
to blocking of reaction centers and a decrease in anode 
reactivity (Liu et al. 2013).
• The oxygen evolution reaction consumes part of the 
electric current in the system and therefore reduces the 
current efficiency.
• In systems with porous electrodes, gas bubbles can block 
the pores, which leads to a decrease in the hydrodynamic 
permeability of the system (Sun et al. 2013; Geng and 
Chen 2017).
At the same time, the gas bubbles formation can have a 
positive effect. According to Wu et al. (2008) and Ahmed 
et  al. (2016), it was reported that gas bubbles can be 
used to prevent fouling and can increase the efficiency of 
electrochemical backwashing by physically removing the 
contaminant layer on the electrode surface.
O2 and H2 are not involved in the oxidation process, so 
the rate of their generation is reduced as much as possible. 
The O2 evolution rate can be reduced by selecting the 
anode material and optimizing the current regimes. The H2 
evolution rate directly depends on the current density. H2 
(21)HCO−
3
+ ⋅OH → CO−
3
⋅ +H2O
(22)CO2−
3
+ ⋅OH → CO−
3
⋅ +HO−
(23)M(⋅OH) → M + 1∕2O2 + H+ + e−
(24)H2O → HO− + 1∕2H2
1528 Environmental Chemistry Letters (2024) 22:1521–1561
current efficiency in most cases is about 90% (Roy Ghatak 
2020). This means that most of the current consumption of 
the cathode is flowed on H2 evolution rate. In addition, the 
released H2 can be recuperated to part of the spent energy 
using fuel cells or gas turbines. At the same time, the process 
could potentially recover 70% of the energy (total in the form 
of heat and in the form of electricity) (Roy Ghatak 2020). 
It should be noted that the implementation of such energy 
recovery is easier in the case of separate collection of gaseous 
products. This means the use of cells with separation of the 
cathode and anode chambers using porous partitions or 
membranes.
At present, mathematical modeling of the bubble formation 
on the surface of plate electrodes is quite well developed. 
The work of Taqieddin et  al. (2018) provides a detailed 
discussion of model approaches to describing the processes 
of nucleation, growth of bubbles and their detachment from 
the surface. In review on modeling of bubble formation, the 
interfacial supersaturation and surface coverage, models for 
calculating the ohmic resistance of gas dispersions in aqueous 
solutions and the influence of the gas evolution rate on the 
mass transfer coefficient, km, are discussed (Zhao et al. 2019).
There are only few models describing gas formation 
inside the pores of porous electrodes. Ateya and El-Anadouli 
(1991) considered the electrode kinetics using the Butler-
Volmer equation and the change in the resistivity of the gas-
electrolyte dispersion that fills the pore space, as a result of 
a change in the ratio of gas and electrolyte volume fractions, 
hydrodynamic characteristics, porosity, thickness and 
specific surface area of the electrode. Several dimensionless 
groups of parameters have been proposed that describe the 
behavior of the system. Saleh et al. (2006) and Saleh (2007, 
2009) improved the Ateya’s group model. In their work, the 
electrical conductivity of the electrode matrix was taken into 
account.
It should be noted that the modeling approaches described 
in these articles are common to all electrolysis systems 
and do not take into account the specific features of the 
anodic oxidation process. For the best of our knowledge, 
currently, there is only one mathematical model simulating 
the gas bubbles formation during anodic oxidation process. 
In a study by Mareev et  al. (2021), a one-dimensional 
nonstationary model was proposed to describe the anodic 
oxidation process in a system with reactive electrochemical 
membranes. The authors introduced the function of the 
dependence of the gas volume fraction on the concentration 
of O2. This allowed the authors to take into account the 
influence of the gas fraction on the electrical conductivity 
of the solution and the hydrodynamic permeability of the 
porous anode. The effects of undesired blockage of the 
surface and pore blocking were also investigated.
Competition phenomena in real wastewater 
treatment
Nonselectivity is considered the main advantage of anodic 
oxidation, as it allows the treatment of raw wastewaters 
which are usually a mixture of different organic compounds 
without online composition control and choice of specific 
purification technology for each compound. However, most 
laboratory studies are performed with single-contaminant 
solutions, and consequently, their results may not be relevant 
for mixtures. This section aims to clarify which competitive 
phenomena between mixture components require attention 
when treating real wastewaters.
First of all, there is competition between the oxidation 
of two or more organic compounds: An organic compound 
with a higher degradation rate constant is oxidized first, and 
the difference inremoval rates depends on the difference 
in the values of the degradation rate constants (Groenen-
Serrano et al. 2013). It should also be born in mind that 
relative reaction rates measured in a single-component 
solution cannot be used to predict the oxidation process in a 
mixture, since the presence of additional organic compounds 
may interfere with each other’s degradation rates (Chaplin 
2014). In general, interfering compounds have little effect 
on the degradation rate of strongly adsorbed contaminants, 
while their effect on the degradation rate of weakly adsorbed 
contaminants can be significant. It should be noted that 
by-products formed in the process of mineralization of 
organic pollutants are also involved in the competition 
for oxidizing agents and, therefore, interfere with the 
degradation rates of initial contaminants.
Another important point is the fact that real wastewaters 
may contain low concentrations of bio-refractory or toxic 
compounds together with high concentrations of non-toxic 
or biodegradable compounds that can be removed by other 
more conventional and cheaper methods. Since the initial 
concentration has an effect on the oxidation efficiency, 
this results in a lower removal efficiency of target toxic 
compound as the non-toxic compounds with higher initial 
concentration are preferentially degraded (Moreira et al. 
2017; Najafinejad et al. 2023). Moreover, it is pointed out 
that in laboratory studies of anodic oxidation target pollutant 
has a concentration an order of magnitude higher than in 
the environment, which also leads to the overestimation of 
removal efficiency in laboratory conditions (Garcia-Segura 
et al. 2020).
Since high energy consumption is the main disadvantage 
of electrochemical oxidation, it is always necessary to pay 
attention to the value of electrical conductivity of treated 
solutions. The higher electrical conductivity of the solu-
tion leads to a lower ohmic voltage drop and, consequently, 
to a lower the required cell voltage. However, many real 
1529Environmental Chemistry Letters (2024) 22:1521–1561 
wastewaters have low electrical conductivity, e.g., pharma-
ceutical industries, food industries, hospital wastewaters, 
resulting in the need to add supporting electrolyte, i.e., usu-
ally sodium sulfate (Na2SO4), sodium chloride (NaCl) and 
sodium perchlorate (NaClO4) (Clematis and Panizza 2021). 
Nevertheless, this poses a range of related issues such as cost 
and transportation of reagents, the need for the authorization 
procedure. On the other hand, there is wastewater contain-
ing more than one electrolyte, e.g., textile dyeing, tannery 
petroleum effluents (Garcia-Segura et al. 2018). In this case, 
the interaction of these electrolytes with each other can have 
both synergetic and inhibition effects on the efficiency of 
degradation. Inorganic salts in the anodic oxidation process 
can act as precursors of various radicals and other oxidiz-
ing species, which can lead to both an increase in oxidation 
efficiency and the formation of toxic by-products (see the 
section above). At the same time, some types of electrolytes, 
e.g., nitrates, do not form oxidizing agents, but can act as 
scavengers of ·OH, thus reducing the oxidation efficiency of 
target organic compounds.
In addition to the competitive effects described above, 
which are mainly inherent in multi-component systems and 
real wastewater, there are competitive phenomena, which are 
observed even in single-component systems, for example, 
the well-known competition between target reaction and 
the parasitic reaction of oxygen evolution. If the system 
parameters are not properly selected, the second reaction 
will preferentially occur, thereby reducing the current 
efficiency. In this review, much attention is paid to the 
competition between the reaction rate and the mass transfer 
rate, which plays a key role in determining the efficiency of 
the process. Anodic oxidation occurs most effectively when 
the mass transfer rate of the pollutant to the reaction zone is 
equal to the rate of its removal.
As it is seen from above, to understand which param-
eters need to be improved to achieve greater efficiency of 
the anodic oxidation process of organic pollutants and to 
enable its optimization, it is necessary to distinguish the 
contribution of different competitive phenomena. It is often 
difficult to perform such investigations by experimental 
methods; therefore, mathematical modeling is required. 
It allows to determine the limiting stage of the process, to 
obtain a detailed description of the degradation mecha-
nism, to analyze the influence of various parameters on the 
system behavior and to predict the most optimal operating 
conditions. Application of mathematical modeling is highly 
advised at development of systems for treatment of real 
wastewater by anodic oxidation.
Implementation of anodic oxidation devices
The cell design has an extremely large impact on the 
efficiency of the anodic oxidation process (Sandoval et al. 
2022). In mathematical models, the cell design determines 
the fluid dynamics parameters used in the calculations: 
the distribution of fluid flow rates and the diffusion layer 
thickness. The more precisely these parameters are defined, 
the more accurately it is possible to describe mass transfer 
and, consequently, the efficiency of the system.
There are several main principles by which electrochemi-
cal reactors can be classified (Fig. 3). The primary method of 
classification may be the operation mode. In batch cells, the 
portion of solution is placed into the reactor before the reac-
tion starts and there is no addition or withdrawal of material 
during the operation process. In continuous flow cells, the 
solution is pumped through the cell (Foutch and Johannes 
2003). It should be noted that some researchers use the term 
“batch mode” to describe the recirculation regime of liq-
uid flow. This can create some confusion, because then the 
flow cell can be operated in a batch mode (Martínez-Huitle 
et al. 2015). This approach emphasizes the characteristics of 
Fig. 3 Classification of reactors 
used in anodic oxidation. There 
are four main principles of clas-
sification: operation mode, flow 
mode, reactor architecture and 
electrode geometry. Adapted 
with the permission of MDPI 
from Liu et al. (2022)
Operation mode
• Batch mode
• Continuous mode
Reactor architecture
• Mixed-tank reactor
• Plate frame/Filter press 
reactor
• Tubular reactor
Flow mode
• Flow-by
• Flow-through
Electrode geometry
• Plate electrode
• Mesh electrode
• Porous electrode
• Particle electrode
2D
3D
Anodic oxidation 
reactor design
1530 Environmental Chemistry Letters (2024) 22:1521–1561
the process: The feed solution is passed through the reactor 
more than once, therefore, the same portion of the solution 
is treated. However, since the primary interest in this section 
is the essential differences in the hydrodynamics of flow and 
non-flow cells, the term "batch cell" is used only for cells 
through which no solution is pumped.
Here the short review of reactors design used in anodic 
oxidation is presented. The more comprehensive and 
detailed reviews for in-depth reading are aimed at those 
who are interested: (Martínez-Huitle et al. 2015; Cornejo 
et al. 2020; Rivera et al. 2021; Sandoval et al. 2022; Liu 
et al. 2022).
Batch cells
Mixed-tank cells are one of the most used batch cells due to 
the simplicity of its design and application (Martínez-Huitle 
et al. 2015). A scheme of a typical one is shown in Fig. 4a. 
The main advantage of such cell is their extreme flexibility 
and simplicity compared to other reactors design. The disad-
vantages of this design are its markedly lower efficiency due 
to the big volume of dead zones and poor mass transfer com-
pared to flow cells and the lack of scalability. It is believed 
that this cell type is applicable only for the preliminary labo-
ratory studies and forthe organic contaminants oxidation in 
solutions with very high concentrations, where cell design 
limitations are not significant to the anodic oxidation process 
(Martínez-Huitle et al. 2015).
Mixed-tank cell is mostly used with plate electrodes in 
parallel configuration (Magro et al. 2020; Salvestrini et al. 
2020; Periyasamy et al. 2022; Ma et al. 2023b). However, in 
the literature there are also examples of implementation of 
this cell with mesh, foam and porous electrodes (Hao et al. 
2022; Ma et al. 2022, 2023a).
Flow cells
The main advantage of flow cells is enhanced mass 
transport properties, which makes it possible to work with 
solutions with rather low concentrations of pollutants, 
as well as the possibility to scale-up the plant for the 
industrial applications. Three main architectures of flow 
electrochemical reactors are discussed below.
Mixed-tank cells can be also used in a continuous flow 
mode (Fig. 4b). As all advantages of mixed-tank reactors 
described above (flexibility and simplicity of application) 
are remained in flow mode and as this reactor design is 
suitable for the treatment of large volumes requiring high 
contact time, they are the most often used cells for anodic 
oxidation processes (Martínez-Huitle et al. 2015). The main 
drawback of this type of cells is the poor mass transport 
characteristics (big dead zones volume); therefore, it is hard 
to scale-up this type of reactors and stirring conditions in 
such cells are one of the most important parameter for the 
anodic oxidation process efficiency (de Oliveira et al. 2011).
Plate frame (Fig. 5a) and filter press (Fig. 5b) reactors 
consist of electrodes fitted in a parallel plate assembly held 
by a frame, and they are commonly used configuration in 
anodic oxidation processes. Reactors can be accompanied 
by electrodes of different geometry, i.e., plate, mesh, porous; 
also, additional elements can be added: membrane to sepa-
rate electrode chambers or turbulence promoters to enhance 
mass transfer. The main advantage of this cell type is rela-
tively uniform current and potential distribution and well-
defined fluid flow in a rectangular channel which are good 
for the scale-up (Frías-Ferrer et al. 2011). However, dead 
zones are the major problem of these reactors and the ideal 
mixing conditions cannot be achieved, which reduces the 
mass transfer.
In tubular reactor, the solution is continuously input-
ted and outputted through a tube. The configuration with 
one tubular electrode and one rod electrode (Fig. 5c) is the 
Fig. 4 Mixed-tank reactor (cell) 
in: a conventional batch mode, 
when withdraw or addition of 
material are not stipulated by 
the reactor design and in: b 
continuous flow mode when 
there are inlet and outlet in the 
walls of reactor. Redrawn with 
the permission of Elsevier from 
Santos et al. (2020)
Inlet
Outlet
Cathode Anode
Magnetic stirring
Cathode Anode
Magnetic stirring
a b
1531Environmental Chemistry Letters (2024) 22:1521–1561 
Rod 
cathode
Porous 
anode
Feed water
Permeate
c
a
outletinlet
anode cathode
e−e−
Anode Outlet
Gasket
b
Conductive tape
Cathode
BezelElectrolytic 
cell
Bezel
Inlet
Inlet OutletAnode Cathode
d
R + H2O
RO + O2 H2
Anode
Cathode
Nafion
e
Fig. 5 Types of main flow cells: a traditional plate frame reactor 
(adapted with the permission of MDPI from Liu et al. (2022)), e−—
electron, b filter press reactor (reprinted with the permission of Else-
vier from Zhang (2022)), c tubular reactor with tubular anode and 
rod cathode (reprinted with the permission of MDPI from Skolot-
neva (2020)), d tubular reactor with electrodes placed perpendicu-
larly to the flux (reprinted with the permission of Elsevier from Wang 
(2015)), e reactor with ion-exchange membrane, oxygen (O2), hydro-
gen (H2), water (H2O), organic pollutant (R), oxidized organic pollut-
ant (RO), Nafion—Nafion™ ion exchange membrane
Fig. 6 Types of flow modes: a 
flow-through mode—current is 
parallel to liquid flow, b flow-by 
mode—current is perpendicular 
to liquid flow, e − – electron. 
Reprinted with the permission 
of MDPI from Liu et al. (2022)
1532 Environmental Chemistry Letters (2024) 22:1521–1561
most common, but the placement of assembly of mesh or 
porous electrodes perpendicular to the liquid flux (Fig. 5d) 
is also possible. This type of reactor has fewer dead zones 
and achieves the same output as a filter press reactor at a 
smaller reactor volume. The drawback of tubular reactor is 
its complexity regarding operating conditions compared to 
ones described above.
Flow cells can be realized in two different configurations: 
(i) flow-through, i.e., current is parallel to the liquid flow 
(Fig. 6a) and (ii) flow-by, i.e., current is perpendicular to the 
liquid flow (Fig. 6b). The use of flow-through cells is pre-
ferred because in this mode the mass transfer coefficient, km, 
is 2–6 times higher than in flow-by mode (10−6–10−5 m  s−1 
in flow-by mode and 10−5–10−4 m  s−1 in flow-through) 
(Chaplin 2014). In addition, flowing the solution toward the 
electrode can significantly reduce the thickness of the diffu-
sion layer, which decreases the diffusion length of organic 
molecules and is therefore favorable for overcoming diffu-
sion limitations.
Flow cells with plate electrodes
Plate electrodes in a parallel configuration are the most often 
used type of electrodes applied in anodic oxidation processes 
(Cornejo et al. 2020). This is due to the simplicity of their 
manufacturing. These electrodes can be fitted in all main 
types of electrochemical cells. As it has been said above, 
mixed-tank cell is the most used in anodic oxidation, and 
many studies have been implemented with this cell in flow 
mode (Pillai and Gupta 2015; Rivera et al. 2015a; Magro 
et al. 2020; Monteil et al. 2021).
The plate electrodes are also used in plate frame and filter 
press reactors. The commercial and well-studied reactor 
FM01-LC (ICI Chemicals & Polymers Ltd, Electrochemical 
Technology, Cheshire, UK) is broadly used in laboratory 
investigations and as a pre-pilot plant of anodic oxidation 
process (Butrón et al. 2007; Nava et al. 2007, 2014). Another 
type of commercial cell applied for the anodic oxidation 
process is DiaCell®, which is the cell with disk electrodes 
(area 70 cm2) operated in flow-by mode (Chatzisymeon 
et  al. 2009; Cano et  al. 2016; Gomez-Ruiz et  al. 2017; 
Armijos-Alcocer et al. 2017). There are also many reports 
of implementation of home-made plate frame and filter 
press reactors (Costa et al. 2009; García et al. 2013; Degaki 
et al. 2014; Farinos and Ruotolo 2017; Barbosa et al. 2018; 
Ghazouani et  al. 2019). To improve mass transfer, the 
turbulence promoters can be installed (Mascia et al. 2013).
The ion-exchange membrane can be implemented in plate 
frame reactors to separate anode and cathode chambers and 
to greatly increase the conductivity of the system. Home-
made reactors and commercial CabECO® cell were realized 
in this configuration (Vasconcelos et al. 2016; Isidro et al. 
2018, 2019; Mora-Gómez et al. 2020; Carrillo-Abad et al. 
2020). The use of such cells seems promising in poorly con-
ducting solutions (Clematis and Panizza 2021).
Implementation of plate electrodes in tubular reactors 
are rare due to the poor hydrodynamic properties of this 
configuration. Nevertheless, there a few studies in which 
plate electrodes are placed in tubular reactor perpendicularly 
to the liquid flux (Brito et al. 2018; Ghazouani et al. 2020). 
Regarding mass transfer characteristics and flow regime, 
these are essentially round-shaped filter press reactors.
The most significant disadvantage of all cells with plate 
electrodes is the mass transport limitations through the 
diffusion layer thickness (δ). The thickness of the stationary 
diffusion layer in electrochemical flow cells can reach 
100 μm, depending on the velocity of the forced flow of the 
solution, the length of the channel and the distance betweenthe electrodes. That is, the efficiency of the oxidation process 
is highly dependent on the rate of diffusion of organic 
pollutants through diffusion layer. The use of electrodes 
with a large surface area only slightly increases the values 
of the mass transfer coefficient, since the characteristics of 
the surface roughness of the electrodes are smaller than the 
diffusion layer thickness. The use of porous electrodes in a 
flow-through configuration makes it possible to overcome 
these diffusion limitations.
Flow cells with mesh electrodes
Mesh electrodes have an extended electroactive area than 
plate electrodes and are suitable for use in a flow-through 
configuration. They require less pressure drop for solution 
pumping than porous electrodes, which corresponds to 
lower energy consumption. These features make it possible 
to produce mesh electrodes from cheaper materials with high 
performance. Thus, there are several studies proved that the 
performance of mesh electrodes made of cheaper material 
under some conditions is compared with the one of plate 
boron-doped diamond as they promote mass transfer (Degaki 
et al. 2014; Nava et al. 2014; Farinos and Ruotolo 2017). 
Mesh electrodes are mostly used in filter press and tubular 
reactors (Nava et al. 2008, 2014; Skban Ibrahim et al. 2014; 
Degaki et al. 2014; Wang et al. 2015; Vijayakumar et al. 
2016; Xu et al. 2016; Farinos and Ruotolo 2017). Although 
there are examples of mesh electrodes implementation in 
mixed-tank cells operated in both bath and flow modes 
(Santos et al. 2020; Hao et al. 2022), in tubular reactors 
they usually form a tube and work together with the rode 
cathode (Skban Ibrahim et al. 2014; Vijayakumar et al. 
2016; Xu et al. 2016). However, there are configurations 
in which mesh electrodes are placed perpendicularly to the 
flux (Wang et al. 2015). It should be noted that boron-doped 
diamond can be synthesized as a mesh, and this shape is 
advised (Nava et al. 2008; Mascia et al. 2016).
1533Environmental Chemistry Letters (2024) 22:1521–1561 
Flow cells with porous electrodes
Porous electrodes have a several of advantages over mesh 
electrodes. They allow to combine separation and removing 
of organic pollutants. And they have even a more developed 
electroactive surface area, several times greater than the one 
of mesh electrodes. The main drawback of porous electrodes 
is fouling.
Porous electrodes can be fabricated in two main 
shapes for implementation in different types of reactors. 
They can be flat for installation in plate frame and filter 
press reactors or tubular to form a tube of tubular reactor 
(Vecitis et al. 2011; Gao et al. 2014; Li et al. 2016; Zhang 
et al. 2016, 2022; Duan et al. 2016; Trellu et al. 2018b).
As the mass transport limitation through the diffusion 
layer is the main problem that porous electrodes aim 
to solve, pore size is the key parameter that determines 
the efficiency of such electrodes. To overcome diffusion 
limitations average pore size should be comparable or 
less than the diffusion layer thickness (which is around 
100 μm). However, it should bear in mind that the lower 
the pore radius the lower permeability and the higher 
pressure drop across the porous electrode is required 
to pump the solution. Thus, the trade-off between short 
diffusion distance and low permeability should be well 
optimized.
Flow cells with particle electrodes
Particle electrodes are made up of many granules of 
conductive material (carbon, metal, metal oxide) filling the 
space between plate electrodes in traditional plate frame 
reactor. Under applied electrical bias, these particles are 
polarized and form a large number of microelectrodes. At 
the same time, electrochemical reaction can occur at the 
surface of each particle. Thus, particle electrodes have an 
enhanced electroactive surface area and a short diffusion 
length to the electrode surface, comparable to porous 
electrodes. Moreover, as particle electrodes usually fill 
the whole reactor volume the conductivity of the system 
is increased which reduces the ohmic losses. Additional 
advantage of using such electrodes is adsorption which 
can increase the degradation efficiency due to the increase 
in the concentration of pollutants on the electrode surface 
(Ma et al. 2021).
Many materials were used as particle electrodes 
for water treatment in recent years. The most common 
ones are carbon-based materials (Sowmiya et al. 2016; 
Alighardashi et  al. 2018; Mengelizadeh et  al. 2019). 
Catalyst-loaded particles are also used (Yan et al. 2011; 
Wang et al. 2019; Zhang et al. 2019). Recently, Ti4O7 
particle electrode was first implemented (Kislyi et  al. 
2023). They showed excellent removing efficiency closed 
to 100%.
Particle electrodes can be used in two main cell 
types: fixed bed and fluidized bed. In fixed bed reactor, 
the particles do not move as the solution passes through 
the cell, whereas in fluidized bed reactor the solution 
flows upward and, therefore, the particles are constantly 
moving and mixed. The hydrodynamic conditions in fixed 
bed reactors are closed to the ones of porous electrodes, 
while the mathematical description of flow pattern in 
fluidized bed is much more complicated and requires more 
computational resources.
Historical aspects
In this subsection, we briefly presented the main works in 
the field of anodic oxidation, concerning the development of 
ideas about this process, and the emergence of new materials 
and models. Date-linked historical references provide a 
sense of the path taken to the existing understanding of 
this process, and a schematic representation (Fig. 7) helps 
readers to consolidate the information.
• 1820s: Reinhold and Erman were among the first to use 
electricity as an oxidizing or reducing agent (Piersma and 
Gileadi 1966).
• 1830s: Ludersdorff investigated products obtained using 
various electrodes for the oxidation of alcohol (Piersma 
and Gileadi 1966).
• 1840s: Kolbe was the first to obtain ethane (CH3CH3) 
by electrolysis of alkali acetates, which led to extensive 
research on the electrolysis of aromatic hydrocarbons and 
their derivatives (Piersma and Gileadi 1966).
• 1850s: Friedel, during the electrolytic oxidation of 
acetone (CH3COCH3), found a mixture of formic, acetic 
and carbonic acids (HCOOH, CH3COOH, H2CO3) with 
the release of O2 and CO2 at the anode (Piersma and 
Gileadi 1966).
• 1880s: The first anodic oxidation of benzene (C6H6) was 
performed (Piersma and Gileadi 1966).
• 1900s: Many works concerning the electrolytic oxidation 
of organic substances existed in early 1900. However, 
most of the works did not fully cover the topic and were 
chaotic (Law and Perkin 1905).
• 1900–1950s: The anodic oxidation of a species such 
as CH3CH3, or many other organic species, has been 
extensively studied (Bockris 1972).
• 1960s: There have been attempts to use electrogenerated 
ozone (O3) for the treatment of municipal and industrial 
wastewater and experiments have been carried out on the 
anodic oxidation of various organic compounds: 1963—
anodic oxidation of triethylamine ((C2H5)3N) (used in the 
production of mineral fertilizers, herbicides, medicines, 
1534 Environmental Chemistry Letters (2024) 22:1521–1561
1535Environmental Chemistry Letters (2024) 22:1521–1561 
paints) (Dapo and Mann 1963), 1964—anodic oxidation 
of methanol (CH3OH) (Oxley et al. 1964), 1964—anodic 
oxidation of CH3OH of triphenylmethane dyes (used 
chiefly in copying papers, in hectograph and printing 
inks, and in textile applications) (Galus and Adams 1964)
• 1970s: The improved methods of ozonation began to be 
investigated, this made it possible to completely oxidize 
refractory organic matter. Extensive investigation of 
this technology commenced in the 70 s, when Nilsson 
et  al. (1973) investigated the anodic oxidation of 
phenolic compounds, Kuhn (1971)—anodic oxidation 
of cyanide, Papouchado et  al. (1975)—anodic 
oxidationpathways of phenolic compounds, Mieluch 
et al. (1975)—electrochemical oxidation of phenolic 
compounds in aqueous solutions.
• 1980s: The ozone/hydrogen peroxide (O3/H2O2) 
system was investigated by Nakayama et al. (1979) 
for wastewater treatment, and more recently by Brunet 
and Dore (1984) and Duguet et  al. (1985). Duguet 
and coauthors showed that the addition of peroxide 
enhanced the efficiency of oxidation of several organic 
substances, trihalomethane precursors, and also 
increased the rate of O3 transfer. Kirk et al. (1985)—
anodic oxidation of aniline (C6H5NH2) for waste water 
treatment. Sharifian and Kirk (1986)—electrochemical 
oxidation of phenol (C6H5OH). Chettiar and Watkinson 
(1983) studied the anodic oxidation of phenolics 
found in coal conversion effluents. Glaze et al. (1987) 
defined advanced oxidation processes as water 
treatment processes. These processes are based on 
the in situ generation of a powerful oxidizing agent, 
such as ·OH, at a concentration sufficient to effectively 
decontaminate waters. In the above studies, the 
influence of the nature of the electrode material during 
anodic mineralization of organics was studied in detail; 
it was found that the optimal process conditions are 
achieved at high-oxygen overpotential anodes.
• 1990s: The potential of electrochemical conversion or 
destruction of organic substrates in wastewater remains 
relevant in the 1990s (Kötz et al. 1991; Comninellis and 
Pulgarin 1993; Comninellis 1994). 1991—Comninel-
lis studied the electrochemical oxidation of C6H5OH 
for waste water treatment using a Pt anode (Comninel-
lis and Pulgarin 1991), and in 1994, he was the first to 
propose an “active” electrode mechanism for organic 
oxidation (Comninellis 1994). First mathematical mod-
els of anodic oxidation processes were proposed in the 
following works (Simond and Comninellis 1997; Simond 
et al. 1997; Cañizares et al. 1999; Chen et al. 1999). Beck 
et al. (1998) and Fisher et al. (1998) investigated a new 
electrode material with very promising characteristics: It 
consists of a silicon support coated by a layer of synthetic 
diamond, heavily doped with boron (B) to obtain accept-
able electrical conductivity. Chen et al. (1999) found 
that Ebonex® porous ceramics (Ti4O7) is applicable for 
anodic oxidation of trichloroethylene (ClCH꞊CCl2). Fur-
ther, this material was very popular in the field of anodic 
oxidation.
• 2000s–2010s: The boron-doped diamond and Ti4O7 
were recognized as the most promising materials for the 
anodic oxidation process. The decade was plenty by the 
different models of anodic oxidation (Rodrigo et al. 2001; 
Panizza et al. 2001a; Cañizares et al. 2002, 2003; Xu 
2016) with numerical (Mascia et al. 2007, 2012; Panizza 
et al. 2008; Kapałka et al. 2009; Polcaro et al. 2009; 
Donaghue and Chaplin 2013) and analytical solutions 
(Panizza et al. 2001a; Kapałka et al. 2008). Most of them 
are considered in the following sections.
• 2020s: Two interesting mathematical models were 
presented by Misal et al. (2020) for anodic oxidation 
system with porous electrode and by Monteil et  al. 
(2021) for flow cells with plate electrodes in serial mode. 
Ma et al. (2023a, b) developed a 3D-printed electrode 
made of TinO2n−1. This is the starting point for the new 
development of anodic oxidation.
General equations used for anodic oxidation 
modeling
Material balance law
The fundamental equation used in almost all models 
described below is the material balance law (Eq. 25). This 
equation allows to relate the change in the concentration of 
a chemical compound over time to its causes: emergence 
or escape of a substance from a volume as a result of the 
incoming fluxes of this substance (the first term) and the 
formation or decomposition of this substance in a reaction 
(the second term).
here c is the concentration, t is the time, j is the flux density, 
and R is the reactive term.
Equation (25) can be applied to describe the change in 
concentrations of all substances present in the solution: 
target component, by-products, and reactive oxygen species. 
If one writes down this equation for each considering 
compound, one obtains a system of equations related only 
by reaction terms. The more precisely the reactions are 
(25)
�c
�t
= −∇j + R
Fig. 7 Development of anodic oxidation of organic pollutants, 
 TinO2n−1 (Magnéli phases of sub-stoichiometric titanium oxides). The 
progression in mathematical modeling in this area began in the late 
1990s
◂
1536 Environmental Chemistry Letters (2024) 22:1521–1561
described, the more accurately the relationship between 
the concentrations of all components of the system can be 
investigated. However, increasing the number of reactions 
and considering more components significantly complicates 
the mathematical problem. For simplification, usually only 
the most important components of the system are considered: 
the target organic compound and reactive species, while 
by-products are excluded from consideration. The influence 
of by-products mineralization on the performance can be 
taken into account applying lamped constant (Kapałka et al. 
2009; Trellu et al. 2016; Ma et al. 2023a).
Flux density equations
To calculate the first summand of Eq. (25), it is necessary 
to know the equation for the flux density. For this case, 
there are several options, the choice of which depends on 
the physical properties of simulated system and the aim of 
the model: Fick's law, the Nernst-Planck equation and the 
use of the mass transfer coefficient.
Most often, the first Fick's law is used to describe the flux 
density (Eq. 26). This equation gives an expression for the 
diffusion flux of matter and does not consider the migration 
and convection components of mass transfer. Indeed, in 
most cases for the simulation of anodic oxidation process 
the consideration of migration is redundant. A background 
electrolyte is added to reduce the resistance of the solution, 
and the transport number of the target organic compound 
(as well as by-products, and many uncharged radicals) is 
often negligible compared to the transport number of the 
background electrolyte (Bard and Faulkner 2001).
here j is the flux density, D is the diffusion coefficient of the 
compound, c is the concentration.
Some researchers attempt to consider the convection 
using Fick's equation with a convective term (Eq. 27) 
(Rivero et al. 2018; Skolotneva et al. 2020). However, 
this greatly complicates the mathematical problem as 
it becomes necessary to determine the velocity field, 
which is often difficult as hydrodynamics calculations are 
required.
here j is the flux density, D is the diffusion coefficient of the 
compound, c is the concentration, and u is the linear fluid 
velocity.
In cases where the considered compound is charged 
organics or radical is charged, and no background 
electrolyte is used, migration cannot be neglected and the 
Nernst-Planck equation should be used to describe the flux 
density (Eq. 28) (Geng and Chen 2016):
(26)j = −D∇c
(27)j = −D∇c + cu
here j is the flux density, D is the diffusion coefficient of the 
compound, c is the concentration, z is the charge, F is the 
Faraday’s constant, R is the universal gas constant, T is the 
temperature, φ is the electric potential.
Most researchers simplify the problem of convection 
accounting by using an equation containing the mass 
transfer coefficient to describe the flux density (Eq. 29) 
(Gherardini et al. 2001; Cañizares et al. 2003; Lan et al. 
2018; Monteil et al. 2021). The mass transfer coefficient 
provides of proportionality between the flux density 
and the difference in the concentration of substance 
in the zones between which transfer occurs, and thus, 
it reflects the co-transport of matter by diffusion and 
convection (in contrast to Eq. (27)). This constant can 
be measured in an independent experiment using a 
standardized ferrocyanide-ferricyanide limiting current 
method (Cañizareset al. 2006). There disadvantages of 
this approach are obvious: (i) The mass transfer coefficient 
depends on each experimental setup; (ii) it is impossible 
to distinguish the influence of diffusion and convection. 
Nevertheless, this approach can be justified especially 
in cases when the process under kinetic limitations is 
modeled.
here j is the flux density, km is the mass transfer coefficient, 
cb is the bulk concentration, and cs is the concentration on 
the electrode surface.
Electrochemical and chemical reactions
There are two types of reactions in anodic oxidation 
processes: chemical and electrochemical reactions. 
Electrochemical reactions are those occurring directly 
on the electrode surface: formation of reactive oxygen 
species, oxidation of organics by direct electron transfer 
and formation of gases. Chemical reactions are oxidation 
reactions of organic molecules by radicals in solution within 
the reaction zone.
To model electrochemical reactions, Faraday's 
fundamental law is applied. It relates the current density 
and the flux of reactants or products of the reaction that 
allow this current to flow (Eq. 30).
here j is the flux density, i is the current density, z is the 
charge, and F is the Faraday’s constant.
(28)j = −D(∇c + zc
F
RT
∇�)
(29)j = −km(cb − cs)
(30)j = −
i
zF
1537Environmental Chemistry Letters (2024) 22:1521–1561 
When describing the anodic oxidation process, it is often 
sufficient to apply only Faraday's law, since the reactions in 
many cases occur under mass transfer limitation, and thus 
the reaction rate can be considered as infinite. Nevertheless, 
with this approach one has to make the assumption that only 
one electrochemical reaction takes place, or the reactions 
occur sequentially, otherwise, the current density distribu-
tion between different reactions has to be calculated which 
requires the application of additional equations. The advan-
tage of this approach is the simplicity of the mathematical 
model; the disadvantage is the inability to take into account 
the properties of the electrode material.
To model several reactions occurring in parallel or to 
describe the kinetics in the process performed under current 
control, the Butler-Volmer equation is used (Eq. 31) (Bard 
and Faulkner 2001). This equation relates the rate of a 
chemical reaction to the electrode potential. It reflects the 
properties of electrode material by kinetic parameters: an 
exchange current density and electron transfer coefficient. 
Nevertheless, the application of this equation complicates 
the mathematical problem, and the kinetic parameters are 
often difficult to determine experimentally, therefore, they 
become fitting parameters of the model.
here i is the current intensity, i0 is the exchange current 
density, C0 and CR are the concentrations of cathodic and 
anodic reactants, respectively, Α is the electron transfer 
coefficient, ηc and ηa are overpotentials of cathodic and 
anodic reactions, respectively, F is the Faraday’s constant, 
R is the universal gas constant, and T is the temperature.
Chemical reactions in the anodic oxidation process are 
most often modeled as pseudo-first-order reactions (Pol-
caro et al. 1999; Ghazouani et al. 2016, 2020). It is assumed 
that the concentration of oxidizing species is so large that 
it does not significantly change during the reaction and can 
be included in the reaction rate constant. The study of the 
concentration distribution of the oxidizing species near the 
electrode surface or the modeling of competitive phenom-
ena occurring during the oxidation of several components 
requires the application of a second-order reaction model 
(Kapałka et al. 2009; Donaghue and Chaplin 2013; Groenen-
Serrano et al. 2013).
Simulation of the flow pattern
The hydrodynamic regime strongly has a huge impact on 
the efficiency of the electrochemical system as it determines 
the mass transfer coefficient, which in turn significantly 
affects the performance of the anodic oxidation system. 
For example, velocity field obtained from the fluid 
(31)i = i0
[
CO exp
(
−AF�c
RT
)
− CR exp
(
(1 − A)F�a
RT
)]
dynamics modeling can be inserted in a convection term of 
Nernst-Plank equation (Eq. 32). Here a brief overview of 
approaches to modeling hydrodynamics in anodic oxidation 
systems will be presented; for a more complete and detailed 
understanding, the reader is referred to the following papers 
(Frías-Ferrer et al. 2011; Rivera et al. 2015b, 2021; Zhou 
et al. 2018; Catañeda et al. 2019).
It should be noted here that in most models of anodic 
oxidation the fluid dynamics are not simulated. To 
describe related mass transport, researchers use empirical 
characterization technics such as mass transport coefficient 
(Eq.  33) or classical models of ideal reactors such as 
continuous stirred tank reactor or plug flow reactor model 
(Cañizares et al. 2002, 2004; Polcaro et al. 2009). In latter 
case, to characterize the deviation of flow from ideal plug 
flow behavior the residence time distribution curves are 
usually obtained from experiment and dispersed plug flow 
model is applied (Eq. 31) (Bengoa et al. 2000; Mascia et al. 
2012, 2016). Sometimes the dependence of mass transport 
on hydrodynamics is described using the well-known 
dimensionless group correlation (Reynolds (Re), Sherwood 
(Sh) and Smidt (Sc) numbers) (Eq. 33) (Nava et al. 2007; 
Cruz-Díaz et al. 2018).
here C = ctp/ctp
0 is the dimensionless tracer particles 
concentration, ctp is the tracer particles concentration, ctp
0 
is the initial tracer particles concentration, Pe is the Peclet 
number which describes flow dispersion, θ = tsuint/Lx is the 
dimensionless time, uint is the interstitial liquid velocity, 
ts is the special time, Lx is the axial length, Xl = x/Lx is 
dimensionless axial length, x is the axis coordinate along 
the reactor length, and a, b and c are constants found from 
experimental data (Rivera et al. 2010).
Computational fluid dynamics is a powerful technic to 
obtain precise fluid flow distribution and velocity field in 
a reactor volume. It applies different numerical methods 
(mostly volume element and finite element methods) to 
solve fundamental transport equations within the simulated 
domain. The most complete description of the velocity field 
is given by the fundamental governing law of fluid motion—
Navier–Stokes equations (Eqs. 34–35) (Łukaszewicz and 
Kalita 2016). However, other equations could be applied, 
for example, the Darcy’s law, describing the liquid flow into 
the porous matter (Mareev et al. 2021).
(32)
�C
��
=
1
Pe
�
2C
�X2
l
−
�C
�Xl
(33)Sh = aRebScc
(34)
�u
�t
+ (u ⋅ ∇)u = −
1
�
∇p +
�
�
Δu + f
1538 Environmental Chemistry Letters (2024) 22:1521–1561
here u is the linear fluid velocity, t is the time, ρ is the liquid 
density, p is the applied pressure, μ is the dynamic viscosity 
and f is the volume force.
Modeling of anodic oxidation with plate 
electrodes
Plate electrodes are the most common in the anodic oxidation 
due to their simple implementation. It is convenient to use 
them in batch mode of oxidation processes. Nowadays, 
one of the best “non-active” electrodes is the boron-doped 
diamond, which is usually plate. Thus, most of the models 
refer to anodic oxidation systems with plate electrodes. 
Table 1 presents the classification of models of anodic 
oxidation on plate electrodes proposed by authors and key 
parameters of each group of models. These models and their 
example are described in detail in following sections.
Kinetic models
First group of models that we propose to classify as “kinetic 
models.” They are based on the material balance equations 
that describe the kinetics of several chemical reactions, but 
do not consider in any way the mass transfer mechanisms; 
such models allow obtaining the reaction rate constants from 
the time dependence of the concentration of the components 
in the system. In addition, they make it possibleto take into 
account the appearance of by-products during the conversion 
of organics into CO2, H2O and other inorganic compounds.
(35)div u = 0
Probably, the first kinetic model was proposed by 
Comninellis (1994). This straightforward model utilizes only 
kinetic relations and allows calculating of the instantaneous 
current efficiency of electrochemical oxidation taking into 
account the oxygen evolution reaction. The equation for the 
calculation of instantaneous current efficiency is presented 
as the ratio of the target organic oxidation reaction rate to the 
sum of the rates of this reaction and oxygen evolution reac-
tion. Obtained dependencies for the instantaneous current 
efficiency show that in the case of active anodes, it is inde-
pendent of the anode potential, and in the case of non-active 
anodes, the potential affects the instantaneous current effi-
ciency. Also, for all anode types, the instantaneous current 
efficiency depends on the nature of the organic compound, 
its concentration and the anode material.
Popović and Johnson (1998) developed a simple 
mathematical model that can describe the total current 
resulting from competitive reactions of the anodic 
O-transfer and oxygen evolution. At the stage of the 
problem formulation, oxygen adsorption on the electrode 
surface was taken into account. A simple equation for the 
half-wave potential is also derived. This model allows to 
build the current–voltage characteristics of the system. A 
good comparison between the experimental and theoretical 
data confirms the assumptions made in the problem 
formulation (anodic discharge of H2O is the prerequisite 
for oxidation of the studied organic compound by the 
O-transfer mechanism). The next work of these authors 
improved this model tacking into account the reactant 
adsorption (Popović et al. 1998).
The most used model belonging to this group is the 
pseudo-first-order kinetic model (Fig. 8) (Cañizares et al. 
1999; Polcaro et al. 1999; Ghazouani et al. 2016, 2020; 
Trellu et al. 2016). This model assumes that the concen-
tration of radicals is high enough to make them unrestrict-
edly available for the reaction with molecules of organic 
compounds and to assume that their concentration does not 
change during the oxidation process, so the rate of chemical 
reaction does not depend on their concentration. It should be 
noted that Fig. 8 represents only the most general case of the 
pseudo-first-order model. For example, each compound, Ri, 
can be formed and/or removed in several parallel chemical 
reactions and then the number of terms in the right-hand 
side of the material balance equation for this component 
will obviously equal the number of corresponding reac-
tions. With the pseudo-first-order model, it is also possible 
to describe the adsorption process.
Cañizares et al. (1999) proposed a simple nonstationary 
mathematical model of electrooxidation of C6H5OH 
considering the three reaction pathways at the active sites 
of the anodes: direct degradation or electrochemical cold 
combustion, chemical oxidation, and polymerization. 
This model allows evaluation of the influence of current 
Table 1 Key parameters of models of anodic oxidation on plate elec-
trodes
Model type Key parameters
Kinetic models Chemical reactions rate constants
Two-mode models Initial concentration of organic compound
Mass transfer coefficient
Electrode surface area
Applied current density
Multy-zone models Diffusion layer thickness (as reaction zone 
thickness is assumed to be equal to it)
Applied current density
Chemical reactions rate constants
Mass transfer coefficient
Diffusion-kinetic models Diffusion layer thickness
Diffusion coefficients
Applied current density
Initial concentration of organic compound
Chemical reactions rate constants
1539Environmental Chemistry Letters (2024) 22:1521–1561 
intensity on the process and predicts the time dependencies 
of concentration, Faradic efficiency and electrochemical 
oxidation index. This study shows that kinetic constants 
increase with the current intensity, the fraction of C6H5OH 
processed by the direct oxidation pathway is approximately 
constant and independent of the current intensity, and the 
electrochemical oxidation index decreases with the current 
intensity increase.
Polcaro et al. (1999) investigated the electrochemical 
oxidation of chlorophenol on a plate anode and used a 
kinetic time-dependent model similar to that presented 
in the previous paragraph, which also takes into account 
the degradation of intermediates (Cañizares et al. 1999). 
The model is based on a system of three linear differential 
equations of the material balance, which allows to obtain 
a simple analytical solution. An analysis of the reaction 
constants determined using the model by fitting the 
theoretical and experimental data makes it possible to reveal 
limiting chemical reactions at different anodes: The rate of 
a ring-opening reaction to form aliphatic acids is an order 
of magnitude higher in the case of Ti/SnO2 compared to Ti/
PbO2.
Similar models are also used in another papers by Ghaz-
ouani et al. (2016, 2020) and Trellu et al. (2016) to describe 
the reduction of nitrates and the oxidation or reduction of 
their by-products in the presence and absence of chloride 
ions (Cl−) (Ghazouani et al. 2016) and also the combination 
of electrocoagulation and anodic oxidation for the simul-
taneous removal of nitrates and phosphates, and the humic 
acids mineralization on the boron-doped diamond anode 
surface (Trellu et al. 2016; Ghazouani et al. 2020).
The main disadvantage of these studies is the huge 
amount of fitting parameters. Furthermore, the rate constants 
defined by such an approach could not be considered 
reliable and independent experiments are required for their 
accurate determination. However, such models allow a 
better understanding of the mechanisms involved and the 
related kinetics. They are more often used as “auxiliary” 
ones and cannot help the researcher to determine the optimal 
parameters of the oxidation process or the influence of 
various factors on the process.
Two‑mode models
Such models consider two different operating regimes 
(current control and mass transfer control); using some 
assumptions, it is easy to obtain an analytical expression 
for the dependence of the concentration of the oxidized 
compound on time, the hydrodynamic parameters of the 
system (using the mass transfer coefficient), the applied 
current density and the electrical current consumption.
The earliest description of a two-mode model was pro-
posed by Simond et al. (1997). In his study, the model takes 
into account the electrochemical oxidation of organic com-
pounds and oxygen evolution reaction at the active anode. 
Two different cases are considered: negligible concentration 
polarization, i.e., current control and significant concentra-
tion polarization, i.e., mass transfer control. The model 
Fig. 8 General representation 
of pseudo first-order kinetic 
model. It assumes that the rate 
of each reaction depends only 
on the concentration of organic 
compound. Material balance 
equation is written for each 
considered component (Ri) of 
the system. Rate constants (ki), 
hydroxyl radicals (·OH), e− 
(electron)
Pseudo first-order kinetic model
products 
Mineralization pathway
[ ]1
1 1
= −
d R
k R
dt
[ ]2
2 2 1 1
= − +
d R
k R k R
dt
[ ]
1 1− −= − +n
n n n n
d R
k R k R
dt
1 2
, ,.., −nk k k fitting parameters....
Material balance equations:
1
k
1
R
/OH e⋅ −− 2
R 2
k
/OH e⋅ −− 3
R nk
/OH e⋅ −−.... nR
1540 Environmental Chemistry Letters (2024) 22:1521–1561
consists of simple equations giving the current efficiency as 
in the pioneering work of Comninellis (1994), but it takes 
into account the surface coverage of higher oxide, its satu-
ration concentration and the mass transfer coefficient in the 
case of significant concentration polarization. This model 
allows obtaining the ratio ofthe rate constants of the organic 
species oxidation and the oxygen evolution reaction, i.e., 
the correlation between the reactivity of the organic to be 
oxidized and the nature of redox couple on the anode. The 
authors proposed the dimensionless parameter, ϕ, expressing 
the ratio between the chemical reaction rate and the mass 
transfer coefficient, km, and the effectiveness factor, ε, evalu-
ating how much the current efficiency decreases as a result 
of concentration polarization. The expression obtained in 
this study shows that the surface coverage of higher oxide 
increases linearly with the applied current and depends on 
the morphology of the anode. This model was experimen-
tally validated by Simond and Comninellis (1997).
Panizza et al. (2001a) proposed a model, which allows 
obtaining the time dependence of chemical oxygen demand 
and instantaneous current efficiency during the electrochem-
ical oxidation of organic pollutants in a batch recirculation 
system. The main assumption of this model is that the rate 
of the electrochemical combustion of the organic compounds 
by generated ·OH radicals and/or direct electron transfer is 
a fast reaction and is controlled by mass transport of the 
organic compounds toward the anode. Using this assump-
tion and some others, mass balance law and Faraday’s law 
analytical expressions for temporal trends of chemical oxy-
gen demand are obtained (Tables 2, 3). Two main modes 
of the electrolysis process under galvanostatic conditions 
were introduced in this work: the first, iappl < ilim, where the 
process is under current control, instantaneous current effi-
ciency is 100% and the chemical oxygen demand decrease 
linearly with time: the second, iappl > ilim, where the pro-
cess is under mass transport control, secondary reactions, 
i.e., oxygen evolution, commence, resulting in instantane-
ous current efficiency < 100% followed by a decrease, and 
the chemical oxygen demand also decreases exponentially 
(Fig. 9). The model shows that an increase in the current 
density at the same initial concentration of organics leads to 
a decrease in the current efficiency due to an increase in the 
fraction of the current consumed for the oxygen evolution 
reaction, while the temperature has a negligible effect on the 
process efficiency.
In the series of works by Gherardini et al. (2001), Rodrigo 
et al. (2001) and Iniesta et al. (2001b, a), this model was 
experimentally validated and was successfully applied with-
out any moderation in the later work of Fierro et al. (2009). 
It should be noted that in the study of Gherardini et al. 
(2001) a new parameter, the normalized current efficiency, 
Table 2 Equations for the calculation of critical values describing the 
transition from the current control to the mass transport control
tcr—critical time (s), α = iappl/ilim0, iappl—applied current density (A 
 m−2), ilim0—initial limiting current density (A m−2), VR—reservoir 
volume (m3), A—electrode area (m2), km—mass transfer coefficient 
(in the electrochemical reactor) (m s−1), Xcr—critical conversion, 
Qcr—critical specific charge (Ah m−3), 4—number of exchanged 
electrons per mol of O2, F—Faraday’s constant (C mol−1), COD0—
initial chemical oxygen demand (mol O2 m−3)
Parameter Equation
Critical time (s) tcr =
1−�
�
VR
Akm
Critical conversion Xcr = 1 − �
Critical specific charge (Ah m−3) Qcr = i0
lim
(1−�)
km3600
=
4FCOD0(1−�)
3600
Table 3 Equations that describe 
parameters evolution during 
organics oxidation at boron-
doped diamond electrode
ICE—instantaneous current efficiency (%), A—electrode area (m2), km—mass transfer coefficient (in the 
electrochemical reactor) (m s−1), VR—reservoir volume (m3), α = iappl/ilim0, iappl—applied current density 
(A m−2), ilim0—initial limiting current density (A m−2), COD—chemical oxygen demand (mol O2 m−3), 
t—time (s), COD0—initial chemical oxygen demand (mol O2 m−3), τ—electrolysis time (s), X—COD 
conversion, V—volume of electrolyte (dm3), tcr—critical time (s), Esp—specific energy consumption 
(kW h kg COD−1), F—Faraday’s constant (C mol−1), 8—equivalent mass of O2, Vd—potential of water 
decomposition, Rc—electrolyte ohmic resistance (Ω), Areq—required electrode area (m2), 4—number of 
exchanged electrons per mol of O2, P—given loading (mol COD s−1). Adapted from Panizza et al. (2008)
Parameter Under current limited control (iappl < ilim) Under mass transport control (iappl > ilim)
ICE ICE = 1 ICE = exp
(
−
Akm
VR
t +
1−�
�
)
COD COD(t) = COD0
(
1 −
�Akm
VR
t
)
COD(t) = �COD0 exp
(
−
Akm
VR
t +
1−�
�
)
τ � =
XV
�Akm
τ = tcr −
V
Akm
[
ln
(
1−X
α
)]
= −
V
Akm
[
ln
(
1−X
α
)
−
1−α
α
]
Esp Esp =
1
3600
F
8
(
Vd + RcA�i
0
lim
)
Esp =
1
3600
F
8
(
Vd + RcA�i
0
lim
) 1−α[1+ln(1−X∕α)]
X
Areq Areq = 4F
XP
�i0
lim
Areq =
4FP
�i0
lim
{
1 − �
[
1 + ln
(
1−X
�
)]}
1541Environmental Chemistry Letters (2024) 22:1521–1561 
φ, is introduced. This parameter can be defined as the ability 
of the anode to promote the electro-oxidation and to reduce 
the side reaction of oxygen evolution. Starting from the 
model described above by Panizza et al. (2001b), a model 
was developed to predict the specific energy consumption 
and the required electrode active area for the electrochemical 
oxidation of organic compounds on boron-doped diamond 
anode. The authors showed that an increase in conversion 
leads to an increase in both required electrode area and 
specific energy consumption, and an optimization problem 
exists, also, the relative importance of these two quantities 
must be taken into account for each situation. Kapałka et al. 
(2008) summarize the research carried out starting from the 
model by Panizza et al. (2001a) on the electrochemical oxi-
dation of organic pollutants for wastewater treatment since 
the end of the 1990s. This paper proposes to use an operat-
ing mode to maximize the efficiency of the process in which 
the applied current density constantly approaches the limit 
value, but does not reach it. Later work of Panizza et al. 
(2008) applies the formulated above model to multiple cur-
rent steps electrolysis and to semi-continuous current control 
electrolysis and shows that this approach allows obtaining 
the 100% process efficiency.
Lan et al. (2018) have extended the model of Panizza 
et al. (2001a) by taking into account two possible ways of 
oxidation: the direct electron transfer and the oxidation via 
·OH. They assumed that the reaction of ·OH generation 
occurs only when the applied current density, iappl, is higher 
than the limiting current density of direct electron transfer, 
ilim,ne−. They also introduced into consideration the initial 
limiting current density, i·OH, corresponding to the total 
mineralization of organic compounds. This permitted them 
to distinguish three regimes of oxidation: (1) iappl ≤ ilim,ne−; 
(2) ilim,ne− < iappl < i·OH; (3) iappl > i·OH The developed model 
has been implemented for the investigation of the salt effect, 
i.e., the oxidation of organic compounds by electrogenerated 
oxidizing species from the salt. This model allows an 
evaluation of different oxidation pathways: direct electron 
transfer, reaction with ·OH or with strong electrogenerated 
oxidants.
The work of Monteil et al. (2021) can be attributed to 
this group of models. The authors have investigated a new 
4
= appl
cr
m
i
COD
Fk
0( ) 1 = −α 
 
m
R
AkCOD t COD t
V
3
2(mol O m )COD −
0
1( ) exp −α = α − + α 
m
R
AkCOD t COD t
V
1exp −α = − + α 
m
R
AkICE t
V
( )t h
( )t h
crt
(%)ICE
Zone A Zone Ba
b
Fig. 9 a Typical evolution of chemical oxygen demand (COD) and 
b instantaneous current efficiency (ICE) as a function of time. Zone 
A—under the kinetics control COD decreases linearly while the ICE 
value remains constant at 100%. This is because mass transfer is fast 
enough to ensure a high concentration at the electrode surface, hence, 
the CODremoval rate is determined by the oxidation reaction rate 
which is constant at a given applied current; ICE is constant because 
all applied current is consumed by the organic oxidation reaction. 
Zone B—under the mass transfer control both COD and ICE decrease 
exponentially. In these conditions, the rate of concentration decrease 
at the electrode surface is higher than the mass transfer of the sub-
stance from the solution, hence, COD removal rate is determined by 
the mass transfer coefficient, km, which constantly decreases; ICE is 
reduced because not all of the applied current is consumed by the 
organic oxidation reaction. COD0—initial chemical oxygen demand 
(mol O2 m−3), t—time (s), A—electrode area (m2), km—mass transfer 
coefficient (in the electrochemical reactor) (m s−1), VR—reservoir vol-
ume (m3), α = iappl/ilim0, iappl—applied current density (A m−2), ilim0—
initial limiting current density (A m−2), CODcr—critical COD value 
at which the transition from current control to mass transport control 
occurs, tcr -critical time at wich the transition from current control to 
mass transport control occurs. Redrawn with the permission of ACS 
Publications from Panizza and Cerisola (2009)
1542 Environmental Chemistry Letters (2024) 22:1521–1561
continuous flow electrochemical reactor with boron-doped 
diamond anode and carbon felt cathode. They used a simple 
model similar to the one described above, i.e., the authors 
derived the equations representing the anodic oxidation rate 
for two modes of operation, current control and mass transfer 
control, and substituted them into the mass balance law but 
they do not solve this problem analytically. Instead of this, 
they combined the kinetic model with the hydrodynamic 
one. For the modeling of hydrodynamics, two approaches 
were used: the dispersed plug flow reactor model and the 
model of continuous stirred tank reactors in series with dead 
zone. The latter was chosen for the combination with the 
kinetic model. The model has only one fitting parameter—
the mass transfer coefficient and the solution is obtained 
numerically. The experimental data and the theoretical ones 
have a good agreement. However, the authors emphasize that 
the model needs to be improved by considering mediated 
electrochemical oxidation and by improving the description 
of mass transfer phenomena.
To summarize all of the above, it should be clarified again 
that these models provide simple analytical expressions for 
modeling the oxidation of organic pollutants under mass 
transfer limitations. This simplicity is achieved by assuming 
that the oxidation reaction of organic compounds by ·OH is 
much faster than the oxygen evolution reaction. However, 
there is experimental evidence to refute this. Adams et al. 
(2009) showed that the composition of oxygen-evolving 
anodes can influence the kinetics of oxidation of organic 
pollutants even when the applied current density is much 
higher than the limiting current. In the experimental work 
of Fierro et al. (2009), it is shown that at currents below the 
limit current, the oxidation reaction of organics proceeds 
in parallel with the oxygen evolution reaction. These 
observations once again confirm that the assumption of a 
much higher rate of oxidation of organic compounds than 
the rate of the oxygen extraction reaction is not always valid. 
Nevertheless, for boron-doped diamond electrodes such 
models can be applied successfully.
Multi‑zone models
Multi-zone modeling approach divides the system under 
study into two or three zones: one or two electrochemical 
zones close to electrodes where ·OH (or other oxidating 
radicals) exist and where oxidation/reduction takes place 
and bulk zone where there is no radicals and the chemi-
cal reaction there can be neglected. The models are based 
on the material balance equation, which is written for each 
component within each zone separately; it is often assumed 
that concentration within one zone is spatially independent 
and changes only over the time (Fig. 10) (Table 4).
Cañizares et al. (2002, 2003) were the first who presented 
a multi-zone model to describe the oxidation of phenol and 
carboxylic acids on the boron-doped diamond electrode. 
Two zones were considered: the electrochemical zone, i.e., 
the thickness of which is equal to the diffusion layer thick-
ness and was measured experimentally, and the bulk zone. In 
both zones, the concentration of each compound is consid-
ered to be only time dependent and constant at any position 
of the zone. The authors assume that this approximation 
is valid if the residence time in the electrochemical cell is 
small and the concentration profiles in the flow direction are 
negligible. In the bulk zone, the concentration is the same as 
the concentration measured experimentally. In the reaction 
zone, the concentration has a value between the concentra-
tion at the anode surface (which cannot be measured) and 
the concentration in the bulk zone. Mass transport processes 
between both zones were quantified by assuming that the 
local rate of exchange between reaction and bulk zones is 
proportional to the concentration difference in these two 
zones. For modeling the kinetics in the reaction zone, the 
following equation is used:
in which ri is the oxidation of each compound, i, in the 
reaction zone, r·OH is the OH generation rate (assumed to 
be r·OH = i/F, i is the current intensity (A), F is the Faraday’s 
constant), is multiplied by the instantaneous current 
efficiency (ICE) to give the amount of ·OH that oxidizes 
organics and by θi to determine the quantity of ·OH that 
attacks organics. The parameter θi represents the oxidation 
efficiency and depends on the organic composition and the 
operating conditions.
In this model, it is a fitting parameter. In the literature, a 
similar parameter can be found, but its value is related only 
to the electrode properties (Gherardini et al. 2001). The reac-
tions in the bulk zone are neglected. For each compound in 
each zone, the material balance equation is applied.
This model gives a good agreement with experimental 
data, and the only adjustable parameters are the oxidizability 
factors for the different organic compounds. Cañizares et al. 
(2004) presented a modification of the model described 
above, in which three zones are considered and a new 
approach to reactivity description is introduced. The main 
innovation in this article is the estimation of the electrical 
current fraction going to each reaction. It is assumed that 
the difference between the cell potential, ΔVwork, and the 
oxidation or reduction potential, Vi, is the driving force 
occurring due to the distribution of electrons. Thus, the 
fraction of current directed to each reaction can be calculated 
using the following equation:
(36)ri = k
⋅OH(ICE)�i
(37)�i =
�
ΔVwork − ΔVi
�
∑
i
�
ΔVwork − ΔVi
�
1543Environmental Chemistry Letters (2024) 22:1521–1561 
here αi is the proportion of electrons involved in a particular 
electrochemical process corresponds to each process, i, 
ΔVwork is the cell potential and ΔVi is the oxidation potential 
of each process i.
Theoretical data obtained using this model are in good 
agreement with experimental ones, which confirms that the 
proposed assumptions are consistent for the formulation of 
the problem.
Polcaro et al. (2009) present a simple stationary math-
ematical model of Cl−, ClO3
− and Cl2 transport in an elec-
trolysis system. The presence and transport of organic pol-
lutants were not considered. All three zones are represented 
as perfectly stirred reactors. The main feature of this model 
is its simplicity (the resulting equation system consists of 
six algebraic equations). This model takes into account the 
convective transfer of system components and allows us to 
calculate the faradaic yield as well as the concentration of 
 Cl2 derivatives in a permeate solution.
The multi-zone models are easy to use and allowpredict-
ing the concentration of system components at the outlet of 
the electrolyzer and determining of optimal parameters of 
the system. The main disadvantage of such models is the 
oversimplification of the reaction mechanisms and mass 
transport in diffusion layers.
Diffusion‑kinetic models
Diffusion-kinetic models are also based on the material bal-
ance equation that takes into account the diffusion of organic 
pollutants inside the reaction zone and the kinetics of sev-
eral chemical or electrochemical reactions (Table 5). These 
models allow to calculate the reaction zone thickness and 
describe the mass transfer limitations (Fig. 11).
Probably, Mascia et al. (2007) were the first to present the 
time-dependent diffusion-kinetic model of anodic oxidation. 
Fick’s second law in differential form with chemical reaction 
Fig. 10 Three zones model. 
The system is divided into three 
zones according to the presence/
absence of the oxidizing radical 
and, thus, depending on the 
presence/absence of chemical 
reaction. Each zone (cathodic 
and anodic reaction zones and 
chemical reaction zone) is 
presented as continuous stirred-
tank reactor. All reactors are 
interconnected by mass transfer 
equation, km—mass transfer 
coefficient
stirrer stirrer
stirrer
km km
Cathodic
reaction zone
Chemical reaction zone
Anodic reaction 
zone
Feed 
solution
Treated 
solution
Table 4 Three-zone models
Ci—concentration of the ith species (mol m−3), Ri—reactive term (mol m−3  s−1), t—time (s), dreac—reaction zone thickness (m), δ (exp)—
diffusion layer thickness obtained from the experiment (m), ·OH—hydroxyl radicals, r·OH –·OH generation rate, i—current intensity (A), F—
Faraday’s constant (C mol−1), ri—oxidation of each compound i, in the reaction zone, ICE—instantaneous current efficiency (%), θi—parameter 
represents the oxidation efficiency, ii—current density spent on ith electrochemical reaction, αi—proportion of electrons involved in a particular 
electrochemical process corresponds to each process i, ΔVwork—cell potential, ΔVi—oxidation potential of each process i, εi—faradaic yield of 
each process i
Cañizares et al. (2002, 2003) Cañizares et al. (2004) Polcaro et al. (2009)
Equation on which model is based Mass balance law:�Ci
�t
= Ri
Relation between zones Mass transfer equation
Reaction zone thickness dreac = �(exp)
Zones, considered as stirred-tank reactors All zones
Electrode reactions r
⋅OH =
i
F
ii = �i
i
F
,
�i =
(ΔVwork−ΔVi)
∑
i
(ΔVwork−ΔVi)
ii = �i
i
F
,
Chemical reactions ri = k
⋅OHICE�i, Second-order rate First-order rate
1544 Environmental Chemistry Letters (2024) 22:1521–1561
Ta
bl
e 
5 
 C
om
pa
ris
on
 o
f d
iff
us
io
n-
ki
ne
tic
 m
od
el
s
C
—
co
nc
en
tra
tio
n 
(m
ol
  m
−
2 ), 
t—
tim
e 
(s
), 
J—
flu
x 
(m
ol
 m
−
2   s−
1 ), 
R—
re
ac
tiv
e 
te
rm
 (m
ol
  m
−
3   s−
1 ), 
O
ER
—
O
xy
ge
n 
ev
ol
ut
io
n 
re
ac
tio
n,
 C
lO
3·—
ch
lo
ra
te
, C
lO
4−
—
pe
rc
hl
or
at
e,
 x
—
di
st
an
ce
 (m
), 
δ—
di
ffu
si
on
 la
ye
r t
hi
ck
ne
ss
 (m
), 
∞
—
fa
r d
ist
an
ce
, α
·O
H
—
th
e 
te
rm
 a
cc
ou
nt
s f
or
 th
e 
fr
ac
tio
n 
of
 c
ur
re
nt
 d
ire
ct
ed
 to
w
ar
d 
·O
H
 p
ro
du
ct
io
n,
 i a
pp
l—
ap
pl
ie
d 
cu
rr
en
t d
en
si
ty
 (A
  m
−
2 ), 
F—
Fa
ra
da
y’
s c
on
st
an
t 
(C
  m
ol
−
1 ), 
D
—
di
ffu
si
on
 c
oe
ffi
ci
en
t (
m
2   s
−
1 ), 
A—
el
ec
tro
de
 a
re
a 
 (m
2 ), 
ε C
l−
—
fa
ra
di
c 
yi
el
d 
as
 a
 f
un
ct
io
n 
of
 c
hl
or
id
e 
 (C
l−
) 
co
nc
en
tra
tio
n;
 in
de
x 
i r
ef
er
s 
to
 o
rg
an
ic
 c
om
po
un
d,
 ·O
H
—
hy
dr
ox
yl
 
ra
di
ca
ls
, D
L—
di
ffu
si
on
 la
ye
r, 
B
—
bu
lk
 so
lu
tio
n,
 O
X
—
ac
tiv
e 
ch
lo
rin
e 
 (C
l 2)
 sp
ec
ie
s, 
0—
in
iti
al
 st
at
e
K
ap
ał
ka
 e
t a
l. 
(2
00
9)
G
ro
en
en
-S
er
ra
no
 e
t a
l. 
(2
01
3)
D
on
ag
hu
e 
an
d 
C
ha
pl
in
 
(2
01
3)
M
as
ci
a 
et
 a
l. 
(2
00
7)
M
as
ci
a 
et
 a
l. 
(2
01
0)
Th
e 
eq
ua
tio
n 
on
 w
hi
ch
 th
e 
m
od
el
 is
 b
as
ed
M
as
s b
al
an
ce
 la
w
:�
C
i
�
t
=
−
d
iv
J
i
+
R
i
Th
e 
co
ns
id
er
in
g 
re
ac
tio
ns
 in
 
w
hi
ch
 th
e 
·O
H
 a
re
 sp
en
t
O
xy
ge
n 
ev
ol
ut
io
n 
re
ac
tio
n
or or
ga
ni
c 
ox
id
at
io
n
O
xy
ge
n 
ev
ol
ut
io
n 
re
ac
tio
n 
or
/
an
d
ox
id
at
io
n 
of
 o
ne
 o
r t
w
o 
or
ga
ni
c 
sp
ec
ie
s
O
xy
ge
n 
ev
ol
ut
io
n 
re
ac
tio
n 
an
d
or
ga
ni
c 
ox
id
at
io
n
an
d
th
e 
re
ac
tio
n 
of
 C
lO
4−
 
fo
rm
at
io
n 
fro
m
 C
lO
3·
D
ea
ct
iv
at
in
g 
pr
oc
es
se
s
an
d
or
ga
ni
c 
ox
id
at
io
n
In
iti
al
 c
on
di
tio
ns
–
C
i(
∀
x
,t
=
0
)
=
C
0 i
–
C
D
L
⋅
O
H
=
0
C
D
L
i
=
C
B i
=
C
i0
,∀
x
C
D
L
⋅
O
H
=
C
D
L
O
X
=
C
B O
X
=
0
C
D
L
i
=
C
B i
=
C
i0
,∀
x
B
ou
nd
ar
y 
co
nd
iti
on
s
C
⋅
O
H
=
0
at
x
=
∞
C
⋅
O
H
(x
=
0
) o
bt
ai
ne
d 
fro
m
 th
e 
as
su
m
pt
io
n 
th
at
 a
ll 
cu
rr
en
t 
is
 d
ire
ct
ed
 to
 ·O
H
 fo
rm
at
io
n 
(i.
e.
, J
⋅
O
H
(x
=
0
)
=
i a
p
p
l/
F
)
C
⋅
O
H
=
0
at
x
=
�
C
⋅
O
H
(x
=
0
) o
bt
ai
ne
d 
fro
m
 th
e 
as
su
m
pt
io
n 
th
at
 a
ll 
cu
rr
en
t 
is
 d
ire
ct
ed
 to
 ·O
H
 fo
rm
at
io
n 
(i.
e.
, J
⋅
O
H
(x
=
0
)
=
i a
p
p
l/
F
)
— J
⋅
O
H
(x
=
0
)
=
�
⋅
O
H
i a
p
p
l/
F
,
α 
ob
ta
in
ed
 fr
om
 th
e 
fit
tin
g 
th
e 
ad
di
tio
na
l e
xp
er
im
en
ta
l 
da
ta
D
⋅
O
H
�
C
⋅
O
H
�
x
=
−
i a
p
p
l
A
F
,x
=
0
D
i
�
C
D
L
i
�
x
=
0
,
x
=
0
C
D
L
i
=
C
B i
,x
=
�
C
⋅
O
H
=
0
,
x
→
∞
D
⋅
O
H
�
C
⋅
O
H
�
x
=
−
(
1
−
�
C
l−
)
i a
p
p
l
A
F
,x
=
0
D
i
�
C
D
L
i
�
x
=
0
,
x
=
0
C
D
L
i
=
C
B i
,x
=
�
C
⋅
O
H
=
0
,
x
→
∞
Th
e 
re
ac
tio
n 
zo
ne
 th
ic
kn
es
s
1 
nm
–1
 μ
m
 <
 20
 n
m
 <
 1 
μm
 <
 10
 n
m
–
Th
e 
m
ax
im
um
 su
rfa
ce
 H
O
• 
co
nc
en
tra
tio
n
 <
 0.
07
 m
M
(in
 th
e 
ab
se
nc
e 
of
 o
rg
an
ic
 
sp
ec
ie
s)
 <
 0.
1 
m
M
(in
 th
e 
ab
se
nc
e 
of
 o
rg
an
ic
 
sp
ec
ie
s)
 <
 0.
02
 m
M
(in
 th
e 
pr
es
en
ce
 o
f o
rg
an
ic
 
sp
ec
ie
s)
 <
 0.
5 
μM
(in
 th
e 
pr
es
en
ce
 o
f 
or
ga
ni
c 
sp
ec
ie
s)
–
1545Environmental Chemistry Letters (2024) 22:1521–1561 
term was used to describe the processes in the diffusion layer 
near the electrode surface. The bulk solution is considered 
to be ideally mixed. The model takes into account the oxida-
tion of all intermediate products as a second-order reaction. 
It is believed that the entire electrical current of the system 
is spent on the formation of ·OH; side reactions with ·OH 
are modeled using a first-order reaction with a lumped con-
stant. This model makes it possible to calculate both the time 
dependences of the concentration of all components of the 
systems in bulk solution and their distribution in the entire 
diffusion layer. Later Mascia et al. (2010) presented a model, 
based on ones, which were previously published by Mascia 
et al. (2007) and Polcaro et al. (2009). The proposed model 
combines all the advantages of multi-zone and diffusion-
kinetic models.
The work of Kapałka et al. (2009) proposes a station-
ary one-dimensional model describing the formation of a 
·OH concentration profile in proximity to the boron-doped 
diamond anode surface. The spatial ·OH concentration dis-
tribution is described by analytical expression, which is a 
solution of Fick’s second law with a chemical reaction term. 
Two limiting cases are considered: the absence of organic 
compounds, when only ·OH recombination reaction occurs, 
and the presence of organic compounds, when there is no 
recombination reaction and only organic oxidation reac-
tion takes place. The surface concentration of ·OH is found 
by assuming that all current is directed to the ·OH forma-
tion and that the concentration of the organic compound is 
constant and spatially independent.Using this model, the 
authors estimated the reaction zone thickness: It is equal 
to 1 μm in the absence of organic substances and is of the 
order of nanometers or tens of nanometers in the case of 
the presence of organic substances. The work of Skolotneva 
et al. (2020) expanded the above model, an analytical expres-
sion was obtained for the distribution of the concentration 
of ·OH, taking into account both parallel reactions, but with 
the same assumptions. The results of this work show that in 
the current problem formulation the impact of the recombi-
nation reaction on the thickness of the reaction zone in the 
presence of organic compounds is insignificant.
In the study of Donaghue and Chaplin (2013), a one-
dimensional steady-state model was developed to under-
stand ClO4
− formation as a function of organic compound 
concentration and current density. This model allows one 
to describe the transport of compounds in a diffusion layer 
adjacent to the anode surface, as well as theoretically deter-
mine the inhibition of ClO4
− formation in the presence of 
organic substances. Several fitting parameters are used, i.e., 
the diffusion coefficients of organic substances and the rate 
constant of the reaction of ClO3· with ·OH. The surface con-
centration of ClO3
− radicals and the fraction of the current 
directed to ·OH generation are used as the boundary con-
ditions and obtained by fitting the model and the data of 
additional experiments. The problem is solved numerically. 
The discrepancy between the fitted values of the diffusion 
coefficients of organic substances and those calculated by 
the Wilke and Change method is explained by the fact that 
organic substances may be involved in some physical or 
chemical processes that are not taken into account by the 
model. Calculations using this model show that the inhibi-
tion of ClO4
− formation linearly depends on the thickness 
of the ClO4
− formation reaction zone, which confirms the 
assumption that free ·OH exist in the volume of the reaction 
zone, and are not only adsorbed on the anode surface.
The one-dimensional nonstationary diffusion-kinetic 
model of Groenen-Serrano et al. (2013) allows one to cal-
culate the time and spatial dependencies of ·OH and organic 
substances concentration near the surface of a boron-doped 
diamond film anode during competitive oxidation, i.e., in the 
presence of two organic substances. The model does not use 
any fitting parameters, and it takes into account that ·OH are 
simultaneously consumed in two parallel reactions: recom-
bination and oxidation of organic substances. The problem 
is solved numerically. The study shows two main points: 
(1) Substance with a higher rate constant is predominantly 
oxidized, and (2) substance begins to be noticeably removed 
only when the substance which is oxidized more favorably 
reaches a sufficiently low concentration. Authors claim that 
the model could be developed to describe systems with more 
than two compounds.
Ma et al. (2023a, b) using a kinetic-diffusion model based 
on Fick's second law and the Butler-Volmer equation stud-
ied the mechanism of paracetamol oxidation on plate TiOx 
and boron-doped diamond anodes. This is the first work 
that takes into account both the parallel competitive course 
C, mM
x, nm
reaction zone
diffusion layer
CHO
CR
H2O
Ri
R'i+1
·OH
Ri
Ri+1
COH
Distance from electrode surface
·OH and R
concentrations
Fig. 11 Typical concentration profiles of hydroxyl radicals (·OH) 
and organic compounds (R) during the anodic oxidation process. Ini-
tial organic compound (Ri), organic compound oxidized by hydroxyl 
radical (Ri+1), organic compound oxidized by direct electron transfer 
(R′i+1). The dependence of ·OH and R concentrations (C·OH and CR, 
respectively) on the distance (x) from the electrode surface is pre-
sented. The reaction zone thickness is usually tens of times less than a 
diffusion layer thickness (δ)
1546 Environmental Chemistry Letters (2024) 22:1521–1561
of electrochemical reactions (the reaction of the formation 
of ·OH and the oxidation of paracetamol by direct electron 
transfer), and homogeneous reactions in the volume of the 
solution (recombination of ·OH, degradation of the target 
component and mineralization of by-products). It was shown 
that the oxidation of paracetamol on the surface of boron-
doped diamond and Ti4O7 electrodes is due to ·OH, but in 
the presence of scavengers of these radicals, such as ethanol, 
direct electron transfer becomes the main mechanism. It was 
also found no significant competition between the mother 
molecule and degradation by-products under mass transport 
limitation.
Marshall and Herritsch (2018) proposed a model of 
the oxidation of an organic compound on an active anode, 
which most fully describes the kinetics of oxygen evolution 
reaction. The model takes into account the competitiveness 
of the organic oxidation reaction and oxygen evolution 
reaction, describes the first two steps of oxygen evolution 
reaction using the Butler-Volmer equation, and describes 
mass transfer using Fick's second law. All oxygen evolution 
reaction stages are considered reversible, organic oxidation 
is not. The kinetic parameters for oxygen evolution reaction 
are adjusted according to the experiment without organic 
compounds, only the processes in diffusion layer are 
modeled. It is assumed that diffusion layer is of constant 
thickness, and the concentration in the volume of the 
solution does not change. The model opens a new pathway 
for the oxidation of organic compounds, i.e., molecular 
oxygen forms a higher oxide with an active site, which then 
oxidizes the organic compound molecule, which makes it 
possible to exceed the efficiency of 100%. Also, this model 
allows to determine the number of active sites per unit 
electrode area and the surface coverage of adsorbed oxygen 
and ·OH.
All the works presented in this section (except for the 
last one) theoretically confirm that during oxidation on a 
non-active electrode in the reaction zone a homogeneous-
like reaction occurs between organic substances and ·OH. 
Thereby, ·OH is not adsorbed but can diffuse from the anode 
surface, forming a thin reaction zone, the thickness of which 
varies from 1 nm in the presence of organic substances to 
1 μm in the absence of it. It means that the rate constants of 
purely homogeneous reactions between organic compounds 
and ·OH can be used in calculations. Also, each of these 
works reveals the parameters that affect the reaction zone 
thickness and allows to separately quantify their effects: 
the applied current density, the nature and the initial 
concentration of organic substances.
All the presented models are one-dimensional and only 
indirectly take into account the two-dimensionality of the 
system. Therefore, the roughness of the electrode surface 
and the lateral concentration distribution along the solution 
flow are not quantitatively included in the existing models.
Modeling of anodic oxidation with porous 
3D electrodes
The features of porous electrodes
As it has been said above, the implementation of porous 
electrodes in flow-through configuration is the most 
promising way to solve mass transport limitation problem 
existing in anodic oxidation (Chaplin 2014; Trellu et al. 
2018a).
The use of porous electrodes in electrochemistry is 
no longer a novelty. Paul Léon Hulin developed the first 
patent for a flow-through porous electrode in 1893 (Hulin 
1897). Since then, porous electrodes are widely used in 
electrochemistry, and nowadays, many advanced areas 
of electrochemistry are inconceivable without porous 
electrodes.
They have found their application for energy storage: 
Numerous porous electrode materials are used in lithium-
ion batteries, and various carbon-based nanocomposites are 
currently pursued as supercapacitor electrodes (Vu et al. 
2012; Jiang et al. 2013). Optimized for salt storage, ion 
and electron transportporous electrodes have significant 
potential for capacitive deionization (Porada et al. 2013). 
3D electrodes are also exploited as sensors and for 
heterogeneous catalysis (Sun et al. 2012; Walcarius 2012; 
Zhu et al. 2017).
The wide application of porous electrodes is due to their 
valuable advantages:
• The pores ensure good entry of the electrolyte to the 
electrode surface.
• The surface area of the porous material is relatively large, 
which facilitates charge transfer across the electrode or 
electrolyte interface.
• The walls of active material surrounding the pores can be 
very thin (micrometers to tens of micrometers), reducing 
path lengths for molecule diffusion.
• The small feature sizes permit increased utilization of 
active material.
• The walls and pores in a porous electrode can be 
bicontinuous, thereby providing continuous electron 
transport paths through the active phase (walls) and the 
electrolyte phase (pores).
Porous electrodes used for anodic oxidation are also 
called reactive electrochemical membranes, as they com-
bine separation and electrooxidation processes. A timeline 
of reactive electrochemical membranes development and 
investigation is summarized in Wei et al. (2020), which was 
updated and expanded by Andersson et al. (1957), Hayfield 
(1983), Smith et al. (1998), Qi et al. (2022) and Yin et al. 
1547Environmental Chemistry Letters (2024) 22:1521–1561 
(2023) (Fig. 12). The results obtained over the past ten years 
have shown that such a solution is a revolutionary technol-
ogy for the electrooxidation of organic pollutants for water 
purification systems (Trellu et al. 2016; Gayen et al. 2018; 
Fu et al. 2019). Research in this direction is carried out by 
the world's leading laboratories in the field of electrochem-
istry. Recent research has been focused on the development 
of porous TinO2n−1 electrodes in order to improve (i) the 
electroactive surface area and (ii) mass transport conditions, 
particularly in flow-through configuration (Radjenovic et al. 
2020; Mousset 2022). Therefore, it is important to be able 
to control the porous structure of the material (Trellu et al. 
2018a).
In Trellu et al. (2018b), Gayen et al. (2018) and Fu et al. 
(2019), it was shown that a high degree of purification can 
be achieved with certain system parameters, e.g., pumping 
rate, solution concentration, current strength, though the 
energy consumption increases. For some pollutants, a local 
maximum is observed on the dependence curve of energy 
consumption on the flux density of organic substances. Thus, 
the formation of insoluble fouling in the dead zone occurs 
and, in addition, the degradation of the anodes.
However, there are a number of problems that are espe-
cially noticeable when working with porous electrodes. 
During electrolysis, gas bubbles formed in the pores of the 
electrode can act as fouling substances, they partially or 
completely block the pores, which leads to a decrease in the 
hydrodynamic permeability of the system and a decrease in 
the mass transfer coefficient. There is one more problem—
heterogeneity of properties such as conductivity, reaction 
rate and diffusivity across the electrode. As a result, the 
experimental characterization of a porous electrode is much 
more complicated than that of a plate electrode.
Modeling of anodic oxidation in the systems 
with reactive electrochemical membranes
Compared to plate electrodes, the mathematical description 
of systems with porous electrodes is difficult because it is 
necessary to describe the transport of particles within their 
volume. So, in pores, in addition to normal diffusion (diffu-
sion along the x-axis, Fig. 13), there is also axial diffusion: 
from the center of the pore to its walls. A similar problem 
exists for the distribution of electric current: Its streamlines 
can be bent not only due to the inhomogeneity of the system 
conductivity but also because the electrochemical reactions 
that cause the flow of current proceed unevenly over the 
volume of the electrode. In addition, the surface areas of 
the pores are not equally accessible, that is, the path that 
the specie needs to overcome from the center of the pore to 
its wall at each point along the x-axis is different (Fig. 13).
Polcaro and Palmas (1997) presented a simulation of 
the oxidation of 2-chlorophenol and 2,6-dichlorophenol on 
porous carbon felt in a fixed bed mode. This model can be 
attributed to the kinetic group. It makes possible to predict 
the dependence of the concentration of the initial compo-
nent and intermediate reaction products on time, as well 
as the effect of the applied current density on the process 
efficiency. The model takes into account the adsorption 
of organic compounds on carbon as a pseudo-first-order 
reaction. It is believed that the entire current goes to the 
generation of ·OH. As in works with kinetic models for 
Ti oxides were first
synthesized and
characterized,1957
Commercially available
material Ebonex ® – TiOx
was patented, 1983
Electrochemical cell
including an electrode
comprising TinO2n−1
disclosed for use with
 redox reactions, 1988
Focused on Ti4O7
material, 1998
First REM was
fabricated from a
commercially
available Ebonex®
electrode, 2013
Ti4O7 REM with high purity 
was synthesized by 
mechanical pressing of 
TiO2 powders, 2022
IrO2
REM,
2015
Ti4O7
REM,
2014
RuO2-Sb2O5-
SnO2 REM, 2017
Stainless steel
mesh/polymeric
REM, 2015
Electrocatalytic 
membrane 
reactor, 2010
Seepage electrode 
reactor, primary 
 REM, 2009
Nano-MnO2
REM, 2013 
Actual wastewater 
Treatment / Pilot 
study, 2016-2017 
Study of mechanism 
of anti-fouling and 
regeneration, 2016
TiO2 
mesoflower 
interlayer REM, 
2016 
Nanostructure
macroporous
PbO2 REM, 2017
Bi-doped
SnO2-TinO2n−1
REM, 2018
TiO2-REM doped 
with Pd-Based 
catalyst , 2018
Ti sub-oxide
REM, 2018 
Ceramic-REM
TiO2-SnO2-Sb 
anode, 2018 
Effect study of
pore structure
of REM, 2018
Coal-
based
carbon
REM,
2018
Carbotherma
l reduction of 
TiO2 REM, 
2018
Carbon-Ti4O7
REM, 2019
Ozonation 
and REM 
coupled 
process 
with Ti4O7
electrode, 
2020
Moving-bed 
REM, 2019
Pd-Cu/Ti4O7
REM , 2020
Manganese 
oxide-coated 
graphite felt 
REM, 2020
Model 
study of 
REM, 
2020 
EO of bio-treated landfill 
leachate using a novel 
dynamic reactive 
electrochemical 
membrane (DREM), 2023
SnO2-Sb
REM,
2016
3-D printed
electrodes,
2023
1950s 1980s 1990s 2000s 2010s 2020s
is
Fig. 12 Development of reactive electrochemical membranes (REMs) 
during 1957–2023. The explosive development of electrochemical 
membrane technology began in the 2010s. The figure is redrawn from 
Wei et  al. (2020) with modifications from Andersson et  al. (1957), 
Hayfield (1983), Smith et  al. (1998), Qi et  al. (2022) and Yin et  al. 
(2023)
1548 Environmental Chemistry Letters (2024) 22:1521–1561
plate electrodes, the model formulation consisted of four 
(according to the number of compounds, the change in 
the concentration of which is modeled) material balance 
equations with a reaction term. The problem was solved 
analytically, the reaction rate constants were found by pro-
cessing the experimental data, and only one was a fitting 
parameter.
Mascia et al. (2012) presented a simple two-dimensional 
stationary convection–diffusion model of active Cl2 gen-
eration for water disinfection in fixed bed reactors with 
3D electrodes (titanium coated with Ru/Ir oxides) in con-
tinuous mode. This model takes into account only the direct 
electron transfer reaction, and pseudo-first-order kinetics is 
used to describe chemical and electrochemical reactions. As 
a continuation of Mascia's work on modeling plate elec-
trodes, the new model assumes that the reactor is divided 
into several zones: two reaction zones (cathode and anode) 
and three flow zones (inlet, outlet and between the reac-
tion zones) (Mascia et al. 2007, 2010). Fluid dynamics are 
modeled using residence time distribution. The hydrody-
namic was interpretedby a simple plug flow model, in which 
axial dispersion accounts for the non-ideal flow behavior 
of the system. The common limiting current technic has 
been adopted for mass transport characterization. Mascia 
et al. (2016) apply the same model to describe the genera-
tion of various oxidizing radicals in a fixed bed reactor with 
three-dimensional conductive diamond grid electrodes. The 
main differences are (1) one-dimensional approximation is 
applied and (2) electrochemical reactions are modeled not as 
pseudo-first-order kinetics but in accordance with Faraday’s 
law. These models allow to simulate the performances of 
such reactor.
In classical electrochemistry, a great contribution to the 
modeling of porous electrodes was made by John Newman 
group’s (Newman and Tiedemann 1975; Trainham and New-
man 1977). In these works, a one-dimensional model of 
flow-through porous electrodes operating above and below 
the limiting current was developed. The model takes into 
account the possibility of multiple reactions occurring and 
it shows a nonuniform distribution of reaction rates due to 
ohmic, mass transfer, and heterogeneous kinetic limitations. 
The model makes it possible to calculate the distribution 
of the potential, currents, and concentration of the target 
component inside the porous electrode.
a
Transition to 3D unit cell 3D unit cell 2D unit cell
Diffusion
layer
Electrode
Electrode
Electrode
Diffusion layer
1D unit cell (Newman-Misal-Chaplin model)
CR
jn
Cw
i
ik
( )n m W Rj k C C= −
jx
Electrode 
Diffusion
layer
Electrode 
j(CR)
с
b
d e
Fig. 13 a Transition from real electrode to b 1D,e 2D and d 3D 
model unit cells (Skolotneva et al. 2020; Misal et al. 2020). b In the 
transition to one-dimensional geometry, the cross-section of the elec-
trode is considered, the porous structure is modeled using porosity. 
c In the transition to a three-dimensional structure, a uniform pore 
distribution is assumed and then d an individual pore is modeled. e 
As the 3D pore has axial symmetry, a transition to 2D geometry is 
possible. ik—current density in solution (A m−2), i—current density 
in electrode material (A m−2), jx—flux density of reactive species in 
the solution flow direction (mol m−2  s−1), jn—flux density of reactive 
species to the pore wall (mol m−2  s−1), CR—concentration of reac-
tive species in the pore bulk (mol m−3), Cw—concentration of reac-
tive species at the pore wall (mol m−3), km—mass transfer coefficient 
(m s.−1)
1549Environmental Chemistry Letters (2024) 22:1521–1561 
Based on the classical works of Newman, Misal et al. 
(2020) have developed a stationary reactive transport model 
for the study of electrochemical oxidation (and reduction) 
of sulfamethoxazole using reactive electrochemical mem-
branes based on Ti4O7 and Pd-Cu/Ti4O7. Two phases are 
considered: the solution phase and the electrode material 
phase. The current in the solution is due to the occurrence of 
(electro) chemical reactions (this model considers only one 
reaction—the oxidation or reduction of sulfamethoxazole 
by direct electron transfer, the rate of which is described by 
the Butler-Volmer equation). The local electrical neutral-
ity assumption is used, while the charge that has left the 
phase of the electrode material automatically passes into the 
solution phase and vice versa. The effects of axial diffusion, 
dispersion and convection are included. Some parameters 
(specific surface area, exchange current density and formal 
potentials) were optimized according to the experimental 
data. The simulations allowed for an analysis of the effect of 
the applied potential and flow rate on the concentration, cur-
rent, and potential distribution within the porous electrode 
under both anodic and cathodic polarizations. Under anodic 
conditions, the entire volume of the reactive electrochemical 
membrane was assumed to be electroactive. It is shown that 
under kinetically limited conditions the reactive area was 
approximately uniformly distributed in the bulk of the reac-
tive electrochemical membrane but shifted to the inlet of the 
electrode under mass transport-limited conditions. In their 
further article, authors applied the model to simulate the 
anodic oxidation of perfluorooctanoic acid and perfluorooc-
tanesulfonic acid on a porous Ti4O7 anode disk and showed 
that increasing the reactive electrochemical membrane phase 
conductivity above a certain threshold value did not improve 
the conversion of organics as the solution phase resistance 
limited the performance of anodic oxidation (Khalid et al. 
2022). The authors also developed the reactors-in-series 
model and found that increasing the specific surface area 
of reactive electrochemical membranes and operating under 
conditions that minimize the total number of reactors is the 
efficient approach for anodic oxidation of target organic 
compounds.
This model was applied by Skolotneva et al. (2021) to 
describe the oxalic acid oxidation by direct electron transfer 
simultaneously with the oxygen evolution reaction on Mag-
néli phase reactive electrochemical membrane. The addition 
of the second chemical reaction leads to a sharp increase 
in the number of adjustable parameters. It is shown that at 
low oxalic acid fluxes the oxygen evolution reaction domi-
nates in the system, but the concentration of oxygen just 
slightly surpasses the solubility limit. The reaction rates rise 
from the center of reactive electrochemical membrane bulk 
toward the inlet and outlet if the kinetic limit is not reached. 
This behavior is due to the similar values of electrode and 
solution phase conductivities. At conditions close to the 
kinetic limit the rate of direct electron transfer of oxalic 
acid increases from the outlet to the inlet of reactive elec-
trochemical membrane. Using a brief theoretical analysis, it 
was found that even at a high oxalic acid flux (70 mgC  L−1) 
99.9% removal and 50% current efficiency may be achieved 
at high current densities (− 300 A  m−2).
Mareev et al. (2021) have modified the model of Newman 
for investigation of the influence of gas bubble formation 
on the efficiency of anodic oxidation of paracetamol in the 
tubular electrolyzer with Magnéli phase reactive electro-
chemical membrane as an anode. Two electrode reactions 
(·OH generation and oxygen evolution) were considered. 
The special balance equation was deduced to simulate the 
transition of solved oxygen into the gas phase. The authors 
also used Darcy’s law to describe the hydrodynamics with 
Helmholtz–Smoluchowski equation to take into account the 
electroosmotic flow. The results confirm that the considera-
tion of bubble formation is necessary to describe with high 
accuracy the permeate flux in such systems; the oxygen bub-
bles form during the first 15 min of the experiment and, after 
that, their size remains constant (under applied conditions); 
the zeta potential of the reactive electrochemical membrane 
pore surface changes with time. Nevertheless, this model 
contains a large amount of fitting parameters and the oxygen 
evolution reaction is modeled in a way that is inconvenient 
in the literature. It should be noted that some works are exist 
in the literature, where the oxygen evolution into the porous 
electrode phase is modulated, but they do not consider other 
reactions (such as oxidation of organic pollutants or recom-
bination of ·OH) (Saleh et al. 2006; Saleh 2007, 2009).
To the best of our knowledge, only one paper is presented 
a two-dimensional micrometer scale model of the transport 
of organic species during the anodic oxidation in the system 
with reactive electrochemical membrane operated in flow-
through mode (Skolotneva et al. 2020). The pore shape was 
considered cylindrical throughout the reactive electrochemi-
cal membrane depth and the cylindrical symmetry assump-
tion was applied to present the system in two coordinates. 
The organic oxidation by direct electron transfer andoxygen 
evolution reaction was not taken into account, and the con-
ductivity of the electrode phase was considered to be several 
times higher than that of the solution phase. The model takes 
into account the convection using the Navier–Stokes equa-
tion. The assumptions decrease the number of adjustable 
parameters to only one—the rate constant of by-products 
mineralization reaction. In contrast to the results obtained 
using the Misal model, this work shows that the electrical 
current streamlines thicken at the entrance to the pore and 
become less dense in its depth, which means that the elec-
troactive portion of the electrode is located at the entrance 
of the pore. However, the calculated reaction zone thick-
ness is in good agreement with previous studies (see the 
reaction–diffusion models) and the effect of two crucial 
1550 Environmental Chemistry Letters (2024) 22:1521–1561
geometrical parameters of reactive electrochemical mem-
brane, porosity and pore radius, is as expected in the litera-
ture: The degradation rate decreases with increasing pore 
radius or decreasing porosity (Trellu et al. 2018a).
There is a single paper in the literature in which the tran-
sition line model is developed for the simulation of an elec-
trochemical impedance spectrum to study the fouling in the 
reactive electrochemical membrane (Jing and Chaplin 2016). 
Although earlier, the model of the impedance of porous 
film electrode by Bisquert (2000) was applied to extract 
the electroactive surface area of the reactive electrochemi-
cal membrane (Zaky and Chaplin 2013). Jing and Chaplin 
were the first, whose work was deduced especially to the 
simulation of reactive electrochemical membrane fouling. 
The main advantage of this work is that it can accurately 
detect changes in the impedances at the three physical inter-
faces (outer membrane surface, active and support layers) 
and therefore is capable of detecting the dominant fouling 
mechanism (e. g., adsorption at outer, active and support 
layers, and pore blockage at the outer membrane surface). 
To apply the electrochemical impedance spectroscopy 
model to the reactive electrochemical membrane, authors 
have transformed the three-dimensional porous geometry 
into one dimension by assuming the reactive electrochemical 
membrane contains a collection of cylindrical homogeneous 
pores of uniform radius. This model was validated experi-
mentally in a separate study (Jing et al. 2016).
Wei et  al. (2017) modeled the flow dynamics in the 
tubular electrolyzer (one of the ends of the cylinder was 
sealed, and the reactor seemed to be a dead end) with a 
tubular porous Ti membrane electrode by computational 
fluid dynamics. The mass and momentum transport inside 
the reactor is described by the Navier–Stokes equation with 
a source term for porous media in the momentum term. The 
authors have investigated the influence of reactor length and 
diameter on its performance. They found that the distribution 
of permeate velocity along the tubular reactor was uniform 
and the short length and large diameter of the reactor provide 
an enhanced mass transfer.
Wang et al. (2015) presented a 3D model through which 
a novel tubular electrochemical reactor with a mesh-plate 
electrode perpendicular to fluid flow and a traditional 
concentric tubular reactor were compared. Fluid dynamics 
is described by the Navier–Stokes equations and the 
re-normalization group k-epsilon turbulence model, and the 
kinetics of organic oxidation is described by the Comninellis 
model for reactions limited by mass transfer with some 
simplifications (it is assumed that all organic compounds 
have the same diffusion coefficient). The results of this 
work show that the orthogonal flow through the mesh-plate 
electrodes clearly enhanced the mass transfer coefficient 
and improved the removal rate of pollutants in tubular 
electrochemical reactors. Earlier Ibrahim et al. (2013) have 
used computational fluid dynamics and residence time 
distribution to investigate the flow dynamics in the tubular 
electrochemical reactor with a cylindrical mesh anode. The 
obtained results show that the application of mesh electrodes 
positively affects the performance of the reactor and in such 
systems the presence of dead zones and short-circuiting in 
the reactor decreased with an increase in the flow rate.
Modeling anodic oxidation 
in the FM01‑LC electrochemical reactor
The FM01-LC is a laboratory-scale, electrochemical filter 
press cell with a projected electrode area of 64 cm2 and a 
rectangular electrolyte flow channel which was originally 
based on the larger FM21-SP electrolyzer of 2100  cm2 
projected electrode area designed for the chlorine-alkali 
industry then diversified to other applications. Currently, 
this flow reactor is used in many areas of electrochem-
istry (chloralkali synthesis, electrosysntesis, electrowin-
ning, metal ion removal and recycling, electrooxidation of 
organic pollutants, flow batteries and fuel cells for energy 
conversion and storage). Such a wide range of applications 
is due to two main advantages of FM01-LC: (1) flexible 
cell design, which may accommodate different types of 
electrodes (textured, coated, profiled or porous), polymer 
mesh turbulence promoters and microporous separators 
or ion-exchange membranes and (2) well-studied fluid 
dynamics that make this cell a flow cell with controlled 
hydrodynamics (Rivera et al. 2015a) (Fig. 14).
There is a wide range of works on the modeling of 
hydrodynamics in FM01-LC with different configurations 
(Trinidad and Walsh 1996; Trinidad et al. 2006; Rivero 
et al. 2012; Cruz-Díaz et al. 2014). Let us consider the 
ones that describe the FM01-LC with mesh electrodes 
because these reactors are used in the system of anodic 
oxidation of organic compounds. Bengoa et al. (2000) 
modeled the flow pattern in the electrochemical cell using 
a coupled model: a dispersed plug flow model for reaction 
zone and a continuous stirred-tank reactors in series for 
inlet–outlet. They reported the influence of the inlet geom-
etry of the cell on flow establishment in the reaction area. 
In the paper of Rivera et al. (2010), the liquid phase mix-
ing flow pattern is studied at low and intermediate Reyn-
olds numbers by means of the residence time distribution 
model combined with the “axial dispersion model” and 
“plug dispersion model” and using ‘closed-closed vessel” 
boundary conditions. Under these conditions, the effects of 
canalization and stagnant zones are important, and devia-
tions from the ideal flow pattern should be considered.
Cruz-Díaz et  al. (2012) presented a parametric flow 
dispersion model with an electrochemical reaction rate 
limited by mass transfer expression coupled with Poisson 
and continuous stirred tank equations for describing the 
1551Environmental Chemistry Letters (2024) 22:1521–1561 
electrooxidation of thiourea in FM01-LC reactor coupled 
to recirculation continuous stirred tank. The stagnant zones 
through the reactor are assumed negligible, and the electrical 
conductivity of the liquid bulk phase and electrode meshes 
is assumed to be the same. The model formulation consists 
of three equations: the material balance equation in differ-
ential form with a reaction term for the reactor, Poisson's 
equation and the material balance equation in integral form 
without a reaction term for recirculating continuous stirred 
tank. Since the authors assume that the electrochemical 
reaction is carried out under limiting current density condi-
tions, the reaction rate depends only on the mass transfer 
coefficient. The boundary conditions were set considering a 
closed-closed vessel system and using the tertiary potential 
model discussed by Fedkiw (1981). The problem was solved 
numerically. This work is aimed at obtaining the concentra-
tion dependence of the target component on time and the 
potential distribution inside the FM01-LC. In addition, this 
model does not contain fittingparameters. Cruz-Díaz et al. 
(2018) introduced another work in which they modified the 
above model to describe the electrochemical oxidation of 
dyeing wastewater. The main difference of the new model 
is that it also considers indirect electrochemical oxidation. 
It is assumed that only the formation of oxidizing species 
at the anode (modeled as pseudo-first-order reactions) and 
their reduction at the cathode (modeled as reactions limited 
by mass transfer, and their rate is expressed only through 
the mass transfer coefficient) occur in the reactor. In turn, 
the oxidation of organics and dye proceeds in recirculation 
continuous stirred tank (modeled as homogeneous second-
order reactions). Also, Poisson’s equation is removed from 
the model. The new model contains four fitting parameters 
(reaction rate constants) and describes well the evolution of 
different chemical species.
Density functional theory
Determining the kinetics of the oxidation reaction of an 
organic compound during anodic oxidation is an extremely 
complicated task. The main problem is that the oxidation 
reaction rate of by-products is high and often they cannot 
even be detected. As a result, experimental methods for 
determining reaction pathways and rate constants of 
oxidation reactions are not applicable. Density functional 
theory modeling can be used to gain insights into probable 
reaction pathways for the electrochemical oxidation of aimed 
organic compounds and subsequent by-product formation.
Density functional theory is a successful theory to calcu-
late the electronic structure of atoms, molecules, and solids. 
Its goal is the quantitative understanding of material proper-
ties from the fundamental laws of quantum mechanics. Den-
sity functional theory is today the most widely used method 
to study interacting electrons, and its applicability ranges 
from atoms to solid systems, from nuclei to quantum fluids. 
Knowledge of the electronic structure allows one to calculate 
the adsorption energies, the reaction energies and activation 
barriers, which in turn helps to determine a thermodynami-
cally favorable reaction pathway.
The choice of approximating functions and solution 
methods plays a key role in modeling the electronic structure 
using density functional theory. In existing studies on density 
functional theory modeling of chemical reactions occurring 
during the anodic oxidation of organic compounds frequency 
and geometry optimization as well as energy calculations are 
performed using built-in basis sets of the Gaussian software 
package. Exchange and correlation are mostly modeled with 
gradient-corrected Becke, three-parameter, Lee–Yang–Parr 
functionals. Also in all models the implicit water solvation 
is taking into account. The activation energy is calculated 
mostly according to Marcus theory (Jing and Chaplin 2017; 
Gayen et al. 2018; Lin et al. 2020). Also Anderson and Kang 
Fig. 14 The FM01-LC electro-
chemical reactor. This is one of 
the most used and studied filter 
press reactors. It has rectan-
gular electrolyte flow channel. 
Redrawn with the permission 
of Elsevier from Rivera et al. 
(2015a) Electrolyte outlet
Electrolyte inlet
Turbulence promoter 
Channel distributor 
Electrode attachment 
1552 Environmental Chemistry Letters (2024) 22:1521–1561
method can be applied (Azizi et al. 2011; Zaky and Chaplin 
2014). The following is a brief review of the results obtained 
by density functional theory modeling in the field of anodic 
oxidation of organic pollutants.
In the paper of Zaky and Chaplin (2014), density 
functional theory simulations were performed to elucidate 
possible reaction mechanisms of p-substituted phenols. 
The authors were looking for a reason why p-nitrophenol 
and p-methoxyphenol react differently. The results of the 
study showed that the 2 e− oxidation mechanism is most 
likely for p-methoxyphenol, and the 1 e− polymerization 
mechanism is for p-nitrophenol. It was shown that 
benzoquinone is not oxidized by direct electron transfer. It 
was also calculated which carbon atoms are most likely to 
be hit by the ·OH radical for different forms of p-nitrophenol 
and p-methoxyphenol. Electron-donating substituents, i.e., 
–OCH3 (methoxy) groups, increase the electron density of 
the phenolic ring and allow direct electron transfer reactions 
to proceed at lower anodic potentials relative to p-substituted 
phenolic compounds with electron withdrawing substituents, 
i.e., –NO2 (nitrogen dioxide). Therefore, the anodic 
potential at which the mechanism for p-substituted phenolic 
compound removal switches from the 1e− polymerization 
mechanism to the 2e− oxidation mechanism, is determined 
by the electronegativity of the substituent.
Jing and Chaplin (2017) published a study in which den-
sity functional theory simulations were performed to inves-
tigate the possibility of direct electron transfer oxidation of 
·OH probes (coumarin, p-benzoquinone, terephthalic acid, 
p-chlorobenzoic acid). Results of these simulations indicate 
that oxidation of coumarin proceeds at potentials much less 
than that for ·OH formation, the oxidation of p-chloroben-
zoic acid occurred via direct electron transfer at potentials 
less than 2.3 V and both reactions pathways (direct elec-
tron transfer and via ·OH oxidation) take place at potentials 
more than 2.3 V. Both terephthalic acid and p-benzoquinone 
were found to be unreactive to direct electron transfer reac-
tions. Density functional theory simulations were also used 
to investigate the possibility of these four probe molecules 
undergoing the Forrester-Hepburn mechanism. The results 
indicated that the nucleophilic addition or substitution of 
·OH to coumarin, terephthalic acid, and p-chlorobenzoic 
acid was unlikely to be significant at room temperature. 
Results from this study indicated that terephthalic acid is 
the most appropriate ·OH probe compound for the charac-
terization of electrochemical and catalytic systems.
Gayen et al. (2018) studied the mineralization of model 
agricultural contaminants—atrazine and clothianidin. 
Density functional theory simulations provided potential-
dependent activation energy profiles for atrazine, 
clothianidin, and various oxidation products (desethyl 
desisopropy atrazine, desisopropyl atrazine, desethyl 
atrazine, cyanuric acid). Results from the density functional 
theory simulations allow concluding that the mechanism of 
oxidation of atrazine and clothianidin involves the direct 
electron transfer and oxidation via ·OH radicals which 
causes the rapid and complete mineralization of atrazine 
and clothianidin at a very short residence time.
Lin et al. (2020) studied the formation of chlorinated 
by-products during the oxidation of model compound—
resorcinol. Several pathways of resorcinol electrooxidation 
have been proposed based on density functional theory 
simulations. Based on the results of liquid chromatography-
mass spectrometry analysis and some assumptions, the 
authors proposed several possible structures for the 
chlorinated products and then also performed density 
functional theory simulations to determine the potential-
dependent activation energy for proposed chlorinated 
products via direct electron transfer.
It should be noted that, to the best of our knowledge, 
there is only one paper in which density functional theory is 
applied to model the activation barriers for reactions occur-
ring on the plate anode during the electrochemical oxidation. 
Azizi et al. (2011) have investigated the possible mechanism 
of ClO4
− formation from ClO3
− on the boron-doped diamond 
anode. The model predicts the reaction rate of ClO4
− for-
mation as a function of electrode potential and temperature, 
and two approaches are used: calculation of the direct elec-
tron transfer coefficient in Butler-Volmer equation (kinetic 
model, Sect. 2.1) and quantum mechanical modeling (density 
functional theory). The authors used the Accelrys Materials 
Studiosoftware package for density functional theory simula-
tions. Using this model, it is possible to study the mechanism 
of the oxidation reaction of ClO3
− ions to ClO4
− on boron-
doped diamond. In other words, the model allows to deter-
mine which of the parallel oxidation reactions (through direct 
electron transfer or through ·OH) is predominant. Such an 
approach allows to develop a mechanistic understanding of 
 ClO4
− formation on boron-doped diamond electrode. Results 
of density functional theory simulations show that direct 
electron transfer of one electron from the ClO3
− molecule 
is an activationless reaction at potentials more than 0.76 V, 
the obtained ClO3
− radical is chemosorbed and it reacts with 
physisorbed ·OH to form the ClO4
−.
Perspective
Recent achievements in the field of anodic oxidation 
modeling enrich an understanding of this process and 
facilitate the design of new more effective systems. 
However, there are some weaknesses that require further 
research. Here the possible ideas for future developments 
are addressed:
1553Environmental Chemistry Letters (2024) 22:1521–1561 
• Most of the models are one-dimensional, in which the 
inhomogeneity of the surface and volume of the elec-
trode is taken into account only indirectly through the 
dispersion coefficient (3D electrodes) or the mass transfer 
coefficient (plate electrodes). Few 2D models are based 
on rough approximations (homogeneity of the electrode 
surface or cylindrical reactive electrochemical membrane 
pores). The possible appearance of 3D models taking into 
account the hydrodynamic characteristics of the liquid 
flow in different geometries will make it possible to more 
accurately estimate the contribution of the diffusion layer 
thickness and the roughness of the electrode surface to 
the characteristics of anode systems.
• The simultaneous presence of the hydrodynamics and 
properties of the anode surface is rare, and therefore 
additional fitting parameters are introduced into 
the models, which reduces their predictive ability. 
Verification of the model is possible only with strict 
control of these two components.
• To simulate electrochemical reactions, most researchers 
either assume that the entire applied current of the system 
is spent on only one useful reaction, or assume that the 
reactions occur in series. Simulation of electrochemical 
reactions running simultaneously will allow a more 
accurate determination of system efficiency. Application 
of the Butler-Volmer equation to implement this feature 
is most desirable.
• Most studies apply simplifying assumptions to 
model chemical reactions. In particular, many use 
lumped constant, as often the reaction pathway of the 
mineralization of the target component and thus the 
reaction by-products are unknown. Mechanisms of 
reactions of organic compounds oxidation by direct 
electron transfer and formation of some radicals also 
require more detailed investigation. In this regard, 
a wider application of density functional theory for a 
mechanistic study of kinetics could be very fruitful.
Conclusion
The effectiveness of anodic oxidation for the removal of 
most known organic pollutants has been proven in many 
studies. Thus, anodic oxidation is a technology that responds 
to the world's demand for clean drinking water and reduction 
of natural water pollution. Nevertheless, further optimization 
of the process is required for its widespread application. This 
can be achieved through mathematical modeling, an essential 
tool for the design of anodic oxidation systems, which 
reflects the researchers' knowledge of system operation, the 
essential interrelationships between system components, and 
the influence of various parameters on system performance. 
With the help of mathematical modeling it is possible to 
develop a mechanistic understanding of anodic oxidation, 
to optimize various trades-off and competitive phenomena 
and to obtain a complete picture of the different system 
characteristics (such as concentration, voltage, current, 
flow velocity) as a function of position and time for different 
reactor configurations and operating conditions. The high-
quality model also has predictive power, which can be used 
to source insights to optimise performance and cell design. 
To the best of our knowledge, the first comprehensive 
systematical overview of existing mathematical models of 
the anodic oxidation of organic compounds is presented in 
this paper. All main approaches are described, equations and 
boundary conditions are given for the simplest models, the 
discussion on advantages and limitations of each group of 
models is provided. The basic principles and equations used 
for mathematical modeling of anodic oxidation are dissected 
and described in detail. Short overviews of historical 
pathway, of reactor’s design and of works applying density 
functional theory are also provided. The current review 
paper may be a starting point for beginning researchers. 
It is shown that simulation can successfully determine the 
mechanisms of the anodic oxidation process, identify its 
limiting stages, and predict the behavior of experimental 
systems. In recent years, a breakthrough has been made in 
the field of describing the dissolved components' transport in 
porous anodes. Based on the results of this literature review 
and considering the main advances in modeling of anodic 
oxidation and the challenges facing researchers to further 
apply this process in practice, future developments have 
been addressed.
Authors' contribution ES, DC, MP and SM contributed to 
conceptualization and writing—review and editing; ES and SM 
contributed to methodology; ES, AK and AK contributed to software, 
project administration, funding, visualization and investigation; SM 
contributed to resources; ES, AK, AK and SM contributed to writing—
original draft preparation; and SM, DC and MP supervised the study. 
All authors have read and agreed to the published version of the 
manuscript.
Funding This research was funded by Russian Science Foundation, 
project No. 22-79-10177.
Availability of data and material Not applicable.
Code availability Not applicable.
Declarations 
Conflict of interest The authors declare no conflict of interest.
Ethical approval Not applicable.
Consent to participate Not applicable.
Consent for publication Not applicable.
1554 Environmental Chemistry Letters (2024) 22:1521–1561
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	Mathematical modeling of the anodic oxidation of organic pollutants: a review
	Abstract
	Introduction
	Basics of the anodic oxidation process
	Competition phenomena in real wastewater treatment
	Implementation of anodic oxidation devices
	Batch cells
	Flow cells
	Flow cells with plate electrodes
	Flow cells with mesh electrodes
	Flow cells with porous electrodes
	Flow cells with particle electrodes
	Historical aspects
	General equations used for anodic oxidation modeling
	Material balance law
	Flux density equations
	Electrochemical and chemical reactions
	Simulation of the flow pattern
	Modeling of anodic oxidation with plate electrodes
	Kinetic models
	Two-mode models
	Multi-zone models
	Diffusion-kinetic models
	Modeling of anodic oxidation with porous 3D electrodes
	The features of porous electrodes
	Modeling of anodic oxidation in the systems with reactiveelectrochemical membranes
	Modeling anodic oxidation in the FM01-LC electrochemical reactor
	Density functional theory
	Perspective
	Conclusion
	References