# Nonlinear Modelling of Kinetic Data Obtained from Photocatalytic Mineralisation of 2,4-Dichlorophenol on  a Titanium Dioxide Membrane

Authors: Ignazio Renato Bellobono; Roberto Scotti; Massimiliano D'Arienzo; Franca Morazzoni; Riccardo Bianchi; Rodica Stanescu; Cristina Costache; Liliana Bobirica; Gabriela Cobzaru; Paola Maria Tozzi; Mauro Rossi; Mauro Luigi Bonardi; Flavia Groppi; Luigi Gini
Journal: International Journal of Photoenergy
 (2009)

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## Abstract

Photomineralisation of 2,4-dichlorophenol (DCP) in aqueous solutions (10.0–100.0 mg/L of C) was systematically studied at318±3 K, in an annular laboratory-scale reactor, by photocatalytic membranes immobilizing titanium dioxide, as a function of substrate concentration, and absorbed power per unit length of membrane. Kinetics of both substrate disappearance, to yield intermediates, and total organic carbon (TOC) disappearance, to yield carbon dioxide, were followed (first series of experiments). At a fixed value of irradiance (1.50 W⋅cm−1), other series of mineralization experiments were repeated (second series of experiments) by carrying out only analyses of chemical oxygen demand (COD), in order to compare modelling results of the two sets of experiments. In both sets of experiments, stoichiometric hydrogen peroxide was used as oxygen donor. For the first series of experiments, a kinetic model was employed, already validated in previous work, from which, by a set of differential equations, four final optimised parameters,
k1 and K1, k2 and K2, were calculated. By these parameters, the whole kinetic profile could be fitted adequately. The influence of irradiance on k1 and k2 could be rationalised very well by this four-parameter kinetic model. Modelling of quantum yields, as a function of irradiance, could also be carried out satisfactorily. As has been found previously for other kinds of substrates, modelling of quantum yields for DCP mineralization is consistent with kinetics of hydroxyl radicals reacting between themselves, leading to hydrogen peroxide, other than with substrate or intermediates leading finally to carbon dioxide, paralleled by a second competition kinetics involving superoxide radical anion. For the second series of experiments, on the contrary, the
Langmuir-Hinshelwood model was employed. Uncertainties of COD analyses, coupled with discrepancies of this model and with its inability to reproduce kinetics up to complete mineralization, are underlined.


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## Body

## 1. Introduction

Among the  advanced oxidation processes, which
were  developed during  the last twenty five-thirty years, photocatalysis [1–3]
in the presence of semiconductors, such as titanium dioxide, with UV radiation,
is certainly one of the most studied. If the UV radiation is from the sun, the
treatment costs are reduced drastically, by transforming photocatalysis in one
of the most appealing and environmentally friendly processes for complete
mineralisation of organic pollutants. Furthermore, photocatalytic processes may
be used as a pre-treatment of toxic chemicals, in order to convert them into
fully biodegradable compounds to be finally processed by the most economical
technology presently available, that of active sludge [4]. Combination of
photocatalysis with active sludge techniques, particularly if solar energy is
directly or indirectly employed, is undoubtedly one of the most promising
technologies to degrade and mineralise, efficiently and inexpensively, wastes
containing toxic compounds.From the point of view of mechanisms
of surface photochemical reactions that take place in heterogeneous systems, a
recent article [5] has amply criticised the a
priori assumed validity of the Langmuir-Hinshelwood kinetic model to
interpret the experimental observations as well as the assertion that the
kinetics of a heterogeneous photoreaction are either first-order or half-order
dependent on photon flow. Such a criticism has already been emphasized in literature also from the point of view of engineering applications and dates
back to earlier pilot plant studies [6]. These studies have shown, for example,
the influence of flow rate on apparent thermodynamic constant of adsorption/desorption, relative to the Langmuir-Hinshelwood model as well as the
inadequacy of this model to fit kinetics of mineralisation, other than in the
segment of the initial rate [7–10]. This has been evidenced particularly when using
immobilised photocatalysts. Consequently, this pseudo-thermodynamic parameter
of the Langmuir-Hinshelwood model, being also linked to reactor geometry and
fluid dynamics, should have a, partly at least, kinetic character.Comparison of efficiency of
different kinds of supported or immobilized photocalysts and objective
characterisation of their photocatalytic power are thus the major problems to
define kinetic reactivity. In most of literature works, this kind of assessment
is often made simply by matching qualitatively concentration profiles of
degraded species. In addition, and unfortunately, in many cases described in
literature, no regard is made to total organic carbon (TOC) mineralisation,
but merely to transformation of substrate into some intermediate compound, for
example, bleaching of a dye, as the first step of the complicated reaction
sequence leading finally to carbon dioxide. Moreover, it is a common practice,
even when quantitative evaluations are made, to consider the photocatalytic process
as a first-order reaction. On the contrary, it has been definitely ascertained
that apparent reaction order varies with concentration of substrate [6, 8], with
an evident transition from a limiting zero-order, at high concentrations, to a
limiting first-order at low concentrations. Modelling of photocatalytic
reactions, oppositely, in a wide range of concentrations, and over all the
kinetic concentration profile of TOC, leading to full mineralisation, has been
approached systematically in previous papers of this series [11, 12], by using
a four-parameter kinetic model. This has been done also from the stand-point
of quantum yields and energy efficiencies, a very important aspect, which is
often neglected in the literature pertinent to photocatalysis. For this latter
aspect, variation of quantum yields with irradiance and concentration of substrate
was taken specifically into consideration [11, 12].Photocatalytic degradation of
2,4-dichlorophenol (DCP) has been widely studied in literature [13–24],
mainly because DCP is a key intermediate in the synthesis of  the herbicide
2,2-dichlorophenoxyacetic acid, the thermal or bio-oxidative decomposition which may give rise to polychloro dioxins. DCP is consequently classified as a
toxic, nonbiodegradable compound [25], which may cause some pathologies and
changes to endocrine systems of humans [26, 27]. Among the photocatalytic
degradation studies of DCP, kinetic models, both of Langmuirian type [15–17, 20], or modified, for example, in the form of
a four-parameter kinetic model [23, 24], have been used and/or developed, in
order to express dependency of rates on concentration of substrate, able to
reproduce, as closely as possible, experimental data in more or less broad
concentration ranges.In the present paper, nonlinear
modelling, previously employed for methane, as a model molecule of aliphatic
hydrocarbons, and for phenol, as a model molecule of aromatics, [11, 12], has
been extended to kinetic runs relative to photocatalytic mineralisation of DCP
in aqueous solution, as a model molecule of polychloro aromatics. A laboratory-scale membrane reactor was employed, with the main purpose of assessing and
comparing results of the four-parameter kinetic modelling with those of the
standard Langmuir model, in the perspective of establishing a reliable
standardization method of the photocatalytic performance of materials. A
particular goal of the work was also to highlight the problems encountered when
trying to compare data obtained from TOC measurements to those obtained from
chemical oxygen demand (COD) measurements. These latter, being easier and less
expensive, are often used to determine the efficacy of wastewater treatments.
Therefore, we deemed useful also to evaluate treatment of data obtained by these
two analytical methods.


## 2. Experimental

### 2.1. Materials

2,4-Dichlorophenol (DCP) was
obtained from Aldrich (purity greater than99.5%). It was used as received with
no further purification. Ultrapure water (maximum contents of Na+ and heavy metal ions 0.02 and 0.004 mg Kg-1, resp.) was used
for preparation of its solutions. Concentrations of DCP, expressed as carbon,
were varied in the range 10–100 mg⋅L-1. No buffer system was
added to solutions, either initially or during photodegradation. Initial pH was
6.2–6.7, and it dropped to no lower than 2.5 when HCl content, produced from
chlorinated phenol, was greater than 95% with respect to equivalent initial
concentration of substrate, at the highest concentration examined.


### 2.2. Photocatalytic Membranes

The photocatalytic membranes (PHOTOPERMBIT/313), kindly
supplied by B.I.T. srl, Milan,
Italy, were standard photocatalytic membranes, immobilising 30±3 wt.% of titanium
dioxide (P25 by Degussa, Germany). These membranes were
prepared (patents pending) by grafting and graft-polymerising, onto a nonwoven microporous polypropylene tissue, 250 ± 9 g⋅m-2 of a
prepolymeric blend containing 30±3 wt.% of titanium dioxide, under rheological
control, in the presence of proprietary photoinitiating and photosensitising
systems.


### 2.3. Apparatus and Procedures

The laboratory-scale photoreactor was
substantially the same as that already described [28], with some differences in
the operating procedures specified later on. Stoichiometric hydrogen peroxide
was used as the oxygen donor. As it has been found in all preceding works of
the present series [6–12], at low values of absorbed power density, hydrogen
peroxide allowed to attain maximum
quantum yields, thus suppressing any contribution of charge carriers
recombination. Overall volume of solution being treated was 24.0±0.08L; the
ratio between the overall reacting volume and the length of the lamp,
corresponding to the length of the irradiated side of the membrane, was 160±6 cm3/cm.
A high pressure mercury arc lamp with a nominal power of 1.00 kW was employed,
kindly supplied by Chimia Prodotti e Processi (Muggiò, Milan, Italy);
it was further provided with a step-by-step power regulation device, so that
power absorbed by the membrane, as determined actinometrically, could be varied
between 0.10 and 4.0 W⋅cm-1. Radiating power
per unit surface of membrane was preferably expressed by radiating power per
unit length, in W⋅cm-1 units, as has been done in all preceding
works of this series, instead of  using W⋅cm-2 units, because the membrane was concentric to the lamp. Consequently, the
transformation of the latter into the former units involves the membrane
circumference. Furthermore, also for engineering purposes, the nominal
irradiance of lamps is given, for a cylindrical geometry, in W⋅cm-1 units.
This allows to obtain directly the fraction of power density, which is
effectively absorbed by the membrane, on which quantum yields may be based. Several
sets of experiments, each one at a certain value of absorbed power per unit
length of membrane, were thus carried out, every one by varying the
concentrations of DCP within each set, in the range 10–100 mg⋅L-1, as  stated above.  The
membrane being concentric to the lamp in the photoreactor used, absorbed power
per unit length of lamp was also the same, if referred to unit length of
membrane. At 1.50 W⋅cm-1 of absorbed power per unit length of the
membrane, the overall power absorbed by the membrane, within the absorption
range of immobilized semiconductor, in the experimental conditions of the
present work, was 150 W, corresponding to 1.26 × 10-2 Einstein⋅min-1.
As was done in all preceding papers of this series, concerning laboratory-scale experiments carried out in the presence of hydrogen peroxide as oxygen
donor, the radiating flow was filtered by a suitable borosilicate glass. Thus, the
radiation wavelengths ranged from the band gap of TiO2 downwards to
315 nm. Consequently, direct photolysis of hydrogen peroxide to give hydroxyl
radicals was inhibited. Owing to the refrigeration system of the lamp, by means
of water circulating in the lamp sheath, made of quartz, mean temperature,
during the runs, was 318 ± 3 K.In the present as well as in
previous papers of this series, membranes were placed and fixed coaxially with
the lamp, in such a way that the flow was perpendicular to the membrane. Owing
to the microporous structure of the membrane, the pressure drop was very
modest; it corresponded to 0.9–1.6 mm Hg, when the flow rate of the liquid
system was 4 m3/h in the laboratory-scale photoreactor.The disappearance of DCP, as such,
was followed by a high performance liquid chromatograph (HPLC Finnigan P4000)
equipped with a UV detector, by using a reverse-phase separation column
(Restek Pinnacle II, d-C18 4.6 mm × 250 mm, 5μm) and acetonitrile:water:acetic acid (68:30:2) as the mobile phase.Besides
the rate of DCP disappearance, as such, the overall rate of organic carbon disappearance
was also followed, in this series of experiments, by total organic carbon (TOC)
analysis. To this purpose, a
Shimadzu TOC-V instrument was used, by which the contribution of inorganic
carbon present was always evaluated, in order to obtain TOC by difference from
total carbon and inorganic carbon determinations. Determinations were precise
within 20 ppb C.A second series of experiments was
also carried out, in which the rate of DCP disappearance, as such, was not
measured. In this series, the overall decrease of organic compounds, either
substrate or intermediates, was followed by determination of chemical oxygen
demand (COD), in order to compare these sets of data with those relative to the
direct measurement of TOC disappearance. Furthermore, contrarily to the first series
of experiments cited above, in which absorbed power was varied in each set of
experiments, in this second series absorbed power was fixed and kept constant
at 150 W (1.5 W/cm). In all experiments of this kind, the overall reacting
volume was4.00±0.06 L.During all the duration of the runs,
no significant decrease of activity of the immobilized photocatalyst was
observed, as checked both by random repetition of experiments at various
initial concentrations of DCP and by reiteration of whole series of tests at
the beginning and at the end of the experimentation period.


### 2.4. Kinetic Modelling

Modelling kinetic  data for most of
experiments, that is those for which the rate of DCP disappearance was measured
together with the rate of TOC disappearance, the same procedure described in
preceding papers ([23, 24] and particularly [11]) was closely followed, by using
the four-parameter model illustrated with all details in these works.In this model, mineralisation of
substrateS to CO2 is supposed to occur through one single
intermediate I, following(1)S→I→CO2,and if both S and I show a
competitive apparent adsorption onto the immobilized semiconductor surface, as
expressed by apparent adsorption constants K1 and K2,
respectively,
regression equations for experimental data are not expressed in a closed form
equation:(2)y(calc)=f(x),but as the following system  of first-order ordinary differential equations in the
dependent variables CS,  CI, and CCO2:(3)dCSdt=−k1⋅K1⋅CS1+K1⋅CS+K2⋅CI,dCIdt=k1⋅K1⋅CS−k2⋅K2⋅CI1+K1⋅CS+K2⋅CI,dCCO2dt=k1⋅K1⋅CI1+K1⋅CS+K2⋅CI,where C denotes
concentrations of species to which the pertinent suffix refers, t time; k1 and k2 the kinetic constants relative to
degradation of S and I, respectively. Intermediate I is supposed to simulate the behaviour
of all the often uncountable, numerous intermediates, which are on the route
from the substrate S to CO2.
Experimentally speaking, only kinetic profiles of S and CO2 have to be followed. By solving these
equations, together with considerations concerning the overall carbon balance
[11], the four parameters, k1 and K1, k2 and K2, may be obtained, by optimisation of all experimental
data, as described in previous work [11, 12].Two examples of experimental data, relative to
the series of experiments carried out at the value of 150 W of overall absorbed
power, corresponding to 1.26 × 10-2 Einstein/min, are collected in Figures
1 and 2. Figure 1 regards the variation of DCP concentration, expressed as
carbon, with time, for all the initial concentrations of DCP examined in the
present paper. Figure 2 regards the variation of TOC with time, for all the
corresponding kinetics. All the experimental points of the corresponding
couples of single curves reported in Figures 1 and 2 (at equal initial
concentrations) were subjected to the optimisation procedure described above.
This optimisation gave rise to four parameters, k1 and K1, k2 and K2, for each
couple of curves. These parameters differed for each couple only within the
uncertainty of the treatment, as a proof that the optimisation method was
satisfactory. By extending this treatment to all the couples of Figures 1 and 2,
the mean values of the four optimised parameters could be calculated, able to
yield the continuous calculated curves. These values are reported inside Figures 1 and 2 themselves. For each value of the optimised parameters, the
probable error is equally indicated, between parentheses, in the same figures,
shown as uncertainty of the last digit.Figure 1
Kinetics of disappearance of
2.4-dichlorophenol (DCP), from its aqueous solutions, in a laboratory-scale
photoreactor (see Experimental), with 1.50 W⋅cm-1 of absorbed
radiation power per unit membrane length, and standard photocatalytic
membranes, immobilising 30 wt.% of P-25 titanium dioxide. Concentrations, C in mg/L (ppm; mass/volume), for DCP,
are expressed as carbon, which was not transformed into intermediates, as a
function of time t, expressed in minutes.
In the figure, the four optimised parameters, k1 and K1, k2 and K2, obtained by
kinetic modelling, are also reported, together with the probable error, shown
between parentheses, as uncertainty in the last digit.Figure 2
Kinetics of disappearance of total
organic carbon (TOC) from 2,4-dichlorophenol (DCP) aqueous solutions. TOC is
expressed as concentration,CTOC in mg/L (ppm; mass/volume) of carbon, which was not mineralised, as a function
of time t, expressed in min. Absorbed
radiation power per unit membrane length was 1.50 W⋅cm-1 during these
runs. In the laboratory-scale photoreactor (see Experimental) standard
photocatalytic membranes, immobilising 30 wt.% of P-15 titanium dioxide, were
employed. In the Figure the four optimised parameters, k1 and K1, k2 and K2, obtained by
kinetic modelling, are also reported for this series of runs, together with the
probable error, shown between parentheses, as uncertainty in the last digit.On
the contrary, for the series of experiments in which only COD was determined,
COD data were first transformed into TOC data, by supposing that the ratio
between total organic compound, as expressed by COD, and carbon itself was the
same as that which was present in DCP. This is already an uncertainty factor,
since intermediates do not surely have the same C/H ratio as that of the
substrate. An example of this kind of kinetics is shown in Figure3, relatively to the initial concentration of
DCP equal to 53 ppm (mass/volume, i.e., mg⋅L-1). From this sort of graphs, the initial rate r0 of TOC disappearance was
measured and plotted, for the various runs carried out at different initial
concentrations C0 of DCP,
as the reciprocal of r0 as
a function of the reciprocal of C0,
following the well-known Langmuir-Hinshelwood linear form of the adsorption and
reaction mechanism:Figure 3
TOC concentrations,CTOC, in mg⋅L-1,
obtained by calculation from COD measurements, as a function of time t, in minutes, for a kinetic run, starting
from a DCP concentration of 84 ppm, expressed as carbon, during which
measurements of the decrease of organic substances, either substrate or
intermediates, versus time t, expressed in minutes, was measured by the
COD method. Absorbed radiation power per unit membrane length was 1.50 W⋅cm-1 and overall reacting volume 4.0 L. In the laboratory-scale photoreactor (see Experimental), standard
photocatalytic membranes, immobilising 30 wt.% of P-15 titanium dioxide, were
employed. The curve reported in this figure is that simulated by the four-parameter
kinetic model, used in the present work, at the same initial concentration of
DCP, and at the same radiating power (see Table 1).(4)1/r0=(1/k)+(1/kKC0),where k is the rate constant at infinite concentration of substrate, and K the apparent thermodynamic constant
for adsorption/desorption equilibrium of substrate between photocalytic
membrane and solution.An
example of the graph resulting from classical equation (4) is reported in Figure
4. From this graph, both k and K could be easily obtained by linear
regression analysis, for this series of experiments.Figure 4
The Langmuir-Hinshelwood plot (4) reporting (1/r0) (in min⋅L⋅mg-1 of mineralised carbon) as a function of (1/C0)
(in L⋅mg-1 of carbon), for all kinetic runs in which analysis has
been carried out by COD measurements, these latter being transformed by
calculation into total organic carbon. Absorbed radiation power per unit
membrane length was 1.50 W⋅cm-1 and overall reacting volume 4.0 L, in all the runs of
this kind. In the laboratory-scale photoreactor (see Experimental), standard
photocatalytic membranes, immobilising 30 wt.% of P-25 titanium dioxide, were employed.


## 2.1. Materials

2,4-Dichlorophenol (DCP) was
obtained from Aldrich (purity greater than99.5%). It was used as received with
no further purification. Ultrapure water (maximum contents of Na+ and heavy metal ions 0.02 and 0.004 mg Kg-1, resp.) was used
for preparation of its solutions. Concentrations of DCP, expressed as carbon,
were varied in the range 10–100 mg⋅L-1. No buffer system was
added to solutions, either initially or during photodegradation. Initial pH was
6.2–6.7, and it dropped to no lower than 2.5 when HCl content, produced from
chlorinated phenol, was greater than 95% with respect to equivalent initial
concentration of substrate, at the highest concentration examined.


## 2.2. Photocatalytic Membranes

The photocatalytic membranes (PHOTOPERMBIT/313), kindly
supplied by B.I.T. srl, Milan,
Italy, were standard photocatalytic membranes, immobilising 30±3 wt.% of titanium
dioxide (P25 by Degussa, Germany). These membranes were
prepared (patents pending) by grafting and graft-polymerising, onto a nonwoven microporous polypropylene tissue, 250 ± 9 g⋅m-2 of a
prepolymeric blend containing 30±3 wt.% of titanium dioxide, under rheological
control, in the presence of proprietary photoinitiating and photosensitising
systems.


## 2.3. Apparatus and Procedures

The laboratory-scale photoreactor was
substantially the same as that already described [28], with some differences in
the operating procedures specified later on. Stoichiometric hydrogen peroxide
was used as the oxygen donor. As it has been found in all preceding works of
the present series [6–12], at low values of absorbed power density, hydrogen
peroxide allowed to attain maximum
quantum yields, thus suppressing any contribution of charge carriers
recombination. Overall volume of solution being treated was 24.0±0.08L; the
ratio between the overall reacting volume and the length of the lamp,
corresponding to the length of the irradiated side of the membrane, was 160±6 cm3/cm.
A high pressure mercury arc lamp with a nominal power of 1.00 kW was employed,
kindly supplied by Chimia Prodotti e Processi (Muggiò, Milan, Italy);
it was further provided with a step-by-step power regulation device, so that
power absorbed by the membrane, as determined actinometrically, could be varied
between 0.10 and 4.0 W⋅cm-1. Radiating power
per unit surface of membrane was preferably expressed by radiating power per
unit length, in W⋅cm-1 units, as has been done in all preceding
works of this series, instead of  using W⋅cm-2 units, because the membrane was concentric to the lamp. Consequently, the
transformation of the latter into the former units involves the membrane
circumference. Furthermore, also for engineering purposes, the nominal
irradiance of lamps is given, for a cylindrical geometry, in W⋅cm-1 units.
This allows to obtain directly the fraction of power density, which is
effectively absorbed by the membrane, on which quantum yields may be based. Several
sets of experiments, each one at a certain value of absorbed power per unit
length of membrane, were thus carried out, every one by varying the
concentrations of DCP within each set, in the range 10–100 mg⋅L-1, as  stated above.  The
membrane being concentric to the lamp in the photoreactor used, absorbed power
per unit length of lamp was also the same, if referred to unit length of
membrane. At 1.50 W⋅cm-1 of absorbed power per unit length of the
membrane, the overall power absorbed by the membrane, within the absorption
range of immobilized semiconductor, in the experimental conditions of the
present work, was 150 W, corresponding to 1.26 × 10-2 Einstein⋅min-1.
As was done in all preceding papers of this series, concerning laboratory-scale experiments carried out in the presence of hydrogen peroxide as oxygen
donor, the radiating flow was filtered by a suitable borosilicate glass. Thus, the
radiation wavelengths ranged from the band gap of TiO2 downwards to
315 nm. Consequently, direct photolysis of hydrogen peroxide to give hydroxyl
radicals was inhibited. Owing to the refrigeration system of the lamp, by means
of water circulating in the lamp sheath, made of quartz, mean temperature,
during the runs, was 318 ± 3 K.In the present as well as in
previous papers of this series, membranes were placed and fixed coaxially with
the lamp, in such a way that the flow was perpendicular to the membrane. Owing
to the microporous structure of the membrane, the pressure drop was very
modest; it corresponded to 0.9–1.6 mm Hg, when the flow rate of the liquid
system was 4 m3/h in the laboratory-scale photoreactor.The disappearance of DCP, as such,
was followed by a high performance liquid chromatograph (HPLC Finnigan P4000)
equipped with a UV detector, by using a reverse-phase separation column
(Restek Pinnacle II, d-C18 4.6 mm × 250 mm, 5μm) and acetonitrile:water:acetic acid (68:30:2) as the mobile phase.Besides
the rate of DCP disappearance, as such, the overall rate of organic carbon disappearance
was also followed, in this series of experiments, by total organic carbon (TOC)
analysis. To this purpose, a
Shimadzu TOC-V instrument was used, by which the contribution of inorganic
carbon present was always evaluated, in order to obtain TOC by difference from
total carbon and inorganic carbon determinations. Determinations were precise
within 20 ppb C.A second series of experiments was
also carried out, in which the rate of DCP disappearance, as such, was not
measured. In this series, the overall decrease of organic compounds, either
substrate or intermediates, was followed by determination of chemical oxygen
demand (COD), in order to compare these sets of data with those relative to the
direct measurement of TOC disappearance. Furthermore, contrarily to the first series
of experiments cited above, in which absorbed power was varied in each set of
experiments, in this second series absorbed power was fixed and kept constant
at 150 W (1.5 W/cm). In all experiments of this kind, the overall reacting
volume was4.00±0.06 L.During all the duration of the runs,
no significant decrease of activity of the immobilized photocatalyst was
observed, as checked both by random repetition of experiments at various
initial concentrations of DCP and by reiteration of whole series of tests at
the beginning and at the end of the experimentation period.


## 2.4. Kinetic Modelling

Modelling kinetic  data for most of
experiments, that is those for which the rate of DCP disappearance was measured
together with the rate of TOC disappearance, the same procedure described in
preceding papers ([23, 24] and particularly [11]) was closely followed, by using
the four-parameter model illustrated with all details in these works.In this model, mineralisation of
substrateS to CO2 is supposed to occur through one single
intermediate I, following(1)S→I→CO2,and if both S and I show a
competitive apparent adsorption onto the immobilized semiconductor surface, as
expressed by apparent adsorption constants K1 and K2,
respectively,
regression equations for experimental data are not expressed in a closed form
equation:(2)y(calc)=f(x),but as the following system  of first-order ordinary differential equations in the
dependent variables CS,  CI, and CCO2:(3)dCSdt=−k1⋅K1⋅CS1+K1⋅CS+K2⋅CI,dCIdt=k1⋅K1⋅CS−k2⋅K2⋅CI1+K1⋅CS+K2⋅CI,dCCO2dt=k1⋅K1⋅CI1+K1⋅CS+K2⋅CI,where C denotes
concentrations of species to which the pertinent suffix refers, t time; k1 and k2 the kinetic constants relative to
degradation of S and I, respectively. Intermediate I is supposed to simulate the behaviour
of all the often uncountable, numerous intermediates, which are on the route
from the substrate S to CO2.
Experimentally speaking, only kinetic profiles of S and CO2 have to be followed. By solving these
equations, together with considerations concerning the overall carbon balance
[11], the four parameters, k1 and K1, k2 and K2, may be obtained, by optimisation of all experimental
data, as described in previous work [11, 12].Two examples of experimental data, relative to
the series of experiments carried out at the value of 150 W of overall absorbed
power, corresponding to 1.26 × 10-2 Einstein/min, are collected in Figures
1 and 2. Figure 1 regards the variation of DCP concentration, expressed as
carbon, with time, for all the initial concentrations of DCP examined in the
present paper. Figure 2 regards the variation of TOC with time, for all the
corresponding kinetics. All the experimental points of the corresponding
couples of single curves reported in Figures 1 and 2 (at equal initial
concentrations) were subjected to the optimisation procedure described above.
This optimisation gave rise to four parameters, k1 and K1, k2 and K2, for each
couple of curves. These parameters differed for each couple only within the
uncertainty of the treatment, as a proof that the optimisation method was
satisfactory. By extending this treatment to all the couples of Figures 1 and 2,
the mean values of the four optimised parameters could be calculated, able to
yield the continuous calculated curves. These values are reported inside Figures 1 and 2 themselves. For each value of the optimised parameters, the
probable error is equally indicated, between parentheses, in the same figures,
shown as uncertainty of the last digit.Figure 1
Kinetics of disappearance of
2.4-dichlorophenol (DCP), from its aqueous solutions, in a laboratory-scale
photoreactor (see Experimental), with 1.50 W⋅cm-1 of absorbed
radiation power per unit membrane length, and standard photocatalytic
membranes, immobilising 30 wt.% of P-25 titanium dioxide. Concentrations, C in mg/L (ppm; mass/volume), for DCP,
are expressed as carbon, which was not transformed into intermediates, as a
function of time t, expressed in minutes.
In the figure, the four optimised parameters, k1 and K1, k2 and K2, obtained by
kinetic modelling, are also reported, together with the probable error, shown
between parentheses, as uncertainty in the last digit.Figure 2
Kinetics of disappearance of total
organic carbon (TOC) from 2,4-dichlorophenol (DCP) aqueous solutions. TOC is
expressed as concentration,CTOC in mg/L (ppm; mass/volume) of carbon, which was not mineralised, as a function
of time t, expressed in min. Absorbed
radiation power per unit membrane length was 1.50 W⋅cm-1 during these
runs. In the laboratory-scale photoreactor (see Experimental) standard
photocatalytic membranes, immobilising 30 wt.% of P-15 titanium dioxide, were
employed. In the Figure the four optimised parameters, k1 and K1, k2 and K2, obtained by
kinetic modelling, are also reported for this series of runs, together with the
probable error, shown between parentheses, as uncertainty in the last digit.On
the contrary, for the series of experiments in which only COD was determined,
COD data were first transformed into TOC data, by supposing that the ratio
between total organic compound, as expressed by COD, and carbon itself was the
same as that which was present in DCP. This is already an uncertainty factor,
since intermediates do not surely have the same C/H ratio as that of the
substrate. An example of this kind of kinetics is shown in Figure3, relatively to the initial concentration of
DCP equal to 53 ppm (mass/volume, i.e., mg⋅L-1). From this sort of graphs, the initial rate r0 of TOC disappearance was
measured and plotted, for the various runs carried out at different initial
concentrations C0 of DCP,
as the reciprocal of r0 as
a function of the reciprocal of C0,
following the well-known Langmuir-Hinshelwood linear form of the adsorption and
reaction mechanism:Figure 3
TOC concentrations,CTOC, in mg⋅L-1,
obtained by calculation from COD measurements, as a function of time t, in minutes, for a kinetic run, starting
from a DCP concentration of 84 ppm, expressed as carbon, during which
measurements of the decrease of organic substances, either substrate or
intermediates, versus time t, expressed in minutes, was measured by the
COD method. Absorbed radiation power per unit membrane length was 1.50 W⋅cm-1 and overall reacting volume 4.0 L. In the laboratory-scale photoreactor (see Experimental), standard
photocatalytic membranes, immobilising 30 wt.% of P-15 titanium dioxide, were
employed. The curve reported in this figure is that simulated by the four-parameter
kinetic model, used in the present work, at the same initial concentration of
DCP, and at the same radiating power (see Table 1).(4)1/r0=(1/k)+(1/kKC0),where k is the rate constant at infinite concentration of substrate, and K the apparent thermodynamic constant
for adsorption/desorption equilibrium of substrate between photocalytic
membrane and solution.An
example of the graph resulting from classical equation (4) is reported in Figure
4. From this graph, both k and K could be easily obtained by linear
regression analysis, for this series of experiments.Figure 4
The Langmuir-Hinshelwood plot (4) reporting (1/r0) (in min⋅L⋅mg-1 of mineralised carbon) as a function of (1/C0)
(in L⋅mg-1 of carbon), for all kinetic runs in which analysis has
been carried out by COD measurements, these latter being transformed by
calculation into total organic carbon. Absorbed radiation power per unit
membrane length was 1.50 W⋅cm-1 and overall reacting volume 4.0 L, in all the runs of
this kind. In the laboratory-scale photoreactor (see Experimental), standard
photocatalytic membranes, immobilising 30 wt.% of P-25 titanium dioxide, were employed.


## 3. Results and Discussion

Photomineralisation of DCP in
aqueous solutions, and in a concentration range corresponding to 10.0–100.0 ppm (mass/volume, i.e., mg⋅L-1) of carbon, was studied at 318 ± 3 K, in a
laboratory-scale reactor. Stoichiometric hydrogen peroxide was used as oxygen
donor, and photocatalytic membranes immobilising 30 ± 3% of titanium dioxide were
employed in the photoreactor. Kinetics of both DCP disappearance, to yield
intermediates, and total organic carbon (TOC) disappearance, to yield carbon
dioxide, were followed. These studies were repeated at various values of
overall absorbed power. An example of experimental results of these runs,
relative to the set of experiments carried out at an overall value of absorbed
power equal to 150 W (1.50 W/cm), is collected in Figures 1 and 2,
respectively.For
treating these data, a kinetic model was employed [11], from which, by a set of
differential equations (see Experimental), four final optimised parameters, k1 and K1, k2 and K2,
were calculated. These parameters, by the use of differential equations above,
were able to fit the whole kinetic profile very satisfactorily, as shown in
the example of Figures 1 and 2. In these Figures comparison is made between
experimental data and the continuous curves calculated by the model employed
from the values of the four optimised parameters reported in the same figures.Results
relative to all the sets of experiments, carried out as a function of absorbed
power per unit length of membrane, are collected in Table1, in which the four
parameters above are reported at the different values of absorbed power per
unit length of membrane employed in the variety of experiments.Table 1
Parameters♥, k1 and K1, k2 and K2, able to fit the whole
photomineralisation curve, following Scheme (1), at various values of absorbed
power per unit length of membrane expressed in W⋅cm-1. They were
obtained by optimisation of kinetic curves, by using the set of differential
equations corresponding to kinetic model of Scheme (1). All the parameters are
expressed in ppm of carbon (mass of C/volume, mg/L) and refer to the
photodegradation of DCP in aqueous solutions, in the presence of stoichiometric
hydrogen peroxide as oxygen donor (treated volume of the aqueous solutions 24.0 L ; membrane length
100.0±0.5 cm ;
ratio between overall reaction volume and membrane length 160 ± 6 cm3/cm2; PHOTOPERMBIT/313 membranes immobilising 30 ± 3 wt.% TiO2, in the absence of any
photopromoter).
Absorbed power per unit length of membrane (W⋅cm-1)k1 (ppm C/min)k2 (ppm C/min)K1 (ppm C)-1K2 (ppm C)-10.150.821(3)0.159(3)0.081(5)0.076(3)0.301.64(4)0.321(3)0.079(3)0.078(3)0.703.61(5)0.671(4)0.080(4)0.081(3)0.904.32(6)0.789(4)0.076(4)0.075(3)1.505.86(8)1.06(4)0.077(3)0.077(4)2.006.54(9)1.13(3)0.078(5)0.079(3)3.006.49(8)1.17(6)0.077(3)0.078(3)4.006.51(7)1.15(7)0.080(4)0.079(3)♥Uncertainties are indicated between parentheses and expressed, relatively to the last digits, as probable errors of the means of optimised values from nine sets of runs in the range of concentrations tested.As
it has been remarked and underlined in preceding papers (see particularly [11, 12]), K1 and K2
parameters have very similar values, and for many substrates, as for DCP in the
present work, show to coincide, within the limits of experimental uncertainty.
On the contrary, they show a marked dependence on flow rate, on reactor
geometry, and fluid dynamics [6, 11], besides the chemical nature of substrate.
Their kinetic behaviour, particularly from the point of view of reactor
engineering, stems out undoubtedly. At least, this behaviour is surely shown by
TiO2 immobilized in photocatalytic membranes tested. The dependency
of K values on chemical character of
the corresponding substrates is probably the only remaining aspect of their
thermodynamic nature.Furthermore, Table1
points out that, while k1 and k2 appear to be a clear
function of irradiance, showing a saturation behaviour at absorbed powers per
unit membrane length greater than about 2 W⋅cm-1, K1 and K2 parameters show an evident independency on
irradiance. The kinetic role of substrate and intermediates is adequately and
quantitatively highlighted by k1 and k2 values. For
alkanes and alkane-like molecules [11, 29, 30] k2 values are higher than k1, and thus the
rate is controlled by k1,
that is, by attack and functionalisation of the starting aliphatic substrate. The
contrary occurs for aromatic substrates [6, 8, 12, 23, 24], the attack of the
aromatic ring by hydroxyl radicals being faster than radical reactions
involving intermediates, particularly when aliphatic fragments are formed in
the way between the starting substrate molecule, and carbon dioxide, as the
final product. In the latter cases, consequently, as for DCP in the present work, the rate
limiting step is given by transformation of intermediates into carbon dioxide.An additional and
outstandingly important aspect branching fromk1 and k2 values, as well as from K1~K2 values of Table 1, is
that by using these optimised parameters [11, 12], the rates of photomineralisation may be
evaluated as a function of initial concentration of substrate, C0, and consequently, by operating at a certain value of the photon
flow, the corresponding quantum
yields Φ0.As has been found in all previous papers
of this series, also for DCP quantum yieldsΦ0,
calculated from rates, followed an apparently Langmuirian function of initial
concentration of substrate C0,
by which Φ∞ values at “infinite” concentration could be
obtained (from the physical point of view, Φ0
values approach Φ∞ values at a concentration of substrate equal
or slightly more than 10-3 M)(5)1Φ0=1Φ∞+1Φ∞C0K,where the K value for DCP substantially coincided with K1~K2
(see Table 1). Finally, by considering all sets of experiments of Table 1, at
different values of the absorbed radiant power per unit length of
photocatalytic membrane, which was varied in the range 0.10–4.0 W⋅cm-1,
and by applying (5) to each set of these experiments, the Φ∞
values as a function of the absorbed radiant
power, per unit length of photocatalytic membrane, were obtained, as reported
in Figure 5.Figure 5
Quantum efficienciesΦ∞ (mol/Einstein)
for photocatalytic mineralisation of DCP in aqueous solution, by photocatalytic
membranes immobilising titanium dioxide, extrapolated by (5), at
“infinite” concentration of substrate, evaluated as mean values (black dots),
relative to measurements of kinetic parameters reported in Table 1, as a
function of absorbed power per unit length of irradiated photocatalytic
membrane, expressed as W⋅cm-1.This
curve of Figure5 looks very much like those observed in preceding papers of this
series (see particularly [11, 12]). It first shows an initial plateau, up to
about 0.4 W/cm, matching the maximum allowable Φ∞ value (0.0417 mol/Einstein = 1/24); this value corresponds to a quantitative
transformation of one molecule, in the case of DCP, by 24 photons yielding 24
hydroxyl radicals able to mineralise the substrate into chloride ions, carbon
dioxide, and water, following exactly the stoichiometric equation. In other
words, experimental quantum yields at infinite concentration appear to result
in the maximum allowable ones when irradiance is low. Then, quantum yields
decrease sharply by increasing irradiance values from about 0.4 W/cm to about
2.–2.5 W/cm, and much more gently from about 2.5 W/cm on. In the case of phenol
and methane [11, 12], this part of the curve had a very low slope, so that
experimental points at values higher than about 2.5 W/cm could simulate an
apparent plateau.To explain the behaviour of sigmoid
curves, such as that of Figure5, a competition kinetics has been envisaged
[11, 12]. A reaction of hydroxyl radicals with themselves to give hydrogen
peroxide, accompanied by the reaction of these radicals, as well as of
superoxide anion radical and/or its conjugate acid, with the substrate and
intermediates, leads to mineralisation. These reactions are given by reactions
(a) on one side and reactions and (b) and (b') on the other.Reaction (a) may correspond, as it has been shown [11, 12], to the formation
of hydrogen peroxide by recombination of hydroxyl radicals:(a)O•H+O•H→H2O2.Reaction
(b) [11, 12] consists in the
traditional attack of the substrate by hydroxyl radicals, and may be expressed
by(b)OH•+substrate(or intermediates)→mineraliztion.Reaction
(b') [11, 12], involving the O2•−/HO2• superoxide radical couple, may be given by(b')O2•−HO2•+substrate (or intermediates)→mineralisation.This model ignores the possible
contribution of charge carriers or surface recombinations. At low irradiance
values, when maximumΦ∞ values are attained, as it has been found in the present and in preceding
works, in the presence of hydrogen peroxide as oxygen donor, this contribution
is surely negligible. This may not apply equally to high irradiance values.
Anyway, modelling above is substantiated, on one side, by quantitative
detection of hydrogen peroxide [31] formed onto photocatalytic membranes, when
it is not added initially, and on the other side by the compatibility of
kinetic data with absolute kinetic constants of reaction (a) [11, 12]. Therefore, it is not unrealistic to think that onto
photocatalytic membranes immobilising titanium dioxide, especially when
cocatalysts and hydrogen peroxide are used, scavenging of conduction band
electrons is practically quantitative.To sum up, as regards theΦ∞ values, also for DCP, the same behaviour that
was observed previously, for methane, both in gaseous phase and in aqueous
solution, and for phenol in aqueous solution [11, 12], appears to be operative.
Accordingly, the four-parameter kinetic model, used in the present work as
well as in previous papers, performs very well not only from the engineering
point of view, which is the first goal to be reached, for practical
applications, but also in throwing light onto photocatalytic mechanisms.Another way is usually employed in
literature to reach a reliable characterisation of photocatalytic
processes, also in view of its simplicity, notwithstanding the strong
criticism, which emerged [5]. This way consists in the application of the  two-parameter Langmuir-Hinshelwood model. Consequently, we deemed
useful a comparison of results between our model and the classical Langmuir-Hinshelwood equation in its linearised form
(4). By operating at the fixed overall adsorbed power of 150 W, corresponding
to 1.5 W/cm per unit length of membrane, we examined kinetics of
photomineralisation, by using the COD method to determine the transformation of
organic matter into carbon dioxide (see example of Figure 3). Then, by repeating
similar experiments as a function of DCP concentration, the reciprocal of the
initial rate of degradation, as given by the slope of the first two or three
points in curves such as that of Figure 3, the graph of (4) was plotted
in Figure 4, from which the values of 12.5±0.8 ppm⋅min-1 for k, and of 9.9±0.9 for K were obtained.First of all, one should remark in
Figure3 the great uncertainty of data, as resulting from the means of repeated
experiments, which may be attributed to the COD method itself used for
following the kinetics. During the runs, the chemical composition of the
solution continued to vary, as a consequence of the production of intermediate
species. Consequently, the transformation of COD data into total organic carbon
values, which is necessary to compare the experiments (based on the assumption
that the same ratio between COD and TOC, that is, the ratio corresponding to the
starting substrate, was preserved in the runs notwithstanding the changing of
composition), is a rough approximation. This approximation is partly
responsible for the lack of confidence in the experimental data. Other reasons
for this bearing are surely given by the weariness of the time consuming
analytical method, and by its intrinsic difficulties as well as by the great
doubt given by the choice of the experimental points to calculate the
initial rate. To show the great uncertainty of this latter calculation, in Figure
3 the curve simulated by the four-parameter kinetic model described above is
reported, at the same initial concentration of this kinetic run. It may be
easily observed that the general trend of the experimental points determined by
the COD method is very well represented
by the four-parameter model. At the same time, the great difference in initial
rate clearly appears, either if the slope of the first experimental points is
considered to evaluate the initial rate, or if the kinetic criterion should be
that of the half transformation time. In both cases an about 30%–40%
uncertainty results, with respect to the slope of the simulating curve, if the
other two criteria of measurements of the photocatalytic activity are used.
This, on one side, speaks again in favour of the superior reliability given by
the four-parameter model employed in our work; on the other side, this stresses the
fact that some experimental points of Figure 3 may lead to deceptive conclusions.
Research people should bear in mind these deductions, when reporting, and
particularly when comparing, single experiments for assessment of
photocatalytic activity. Anyway, by
taking into account a congruous number of kinetic runs, and by repeating
experimental measurements to estimate the probable error, the correlation
factor of the Langmuir-Hinshelwood plot of Figure 4 (R2 =
0.92), almost at the limit of acceptability, allows to assess most conveniently
the values of parameters k and K reported above.Even
if parametersk and K of (4) are obtained by a
different model with respect to the four-parameter model discussed above, it
is interesting to compare their ability and their limits to interpret the two
kinds of experiments tested in the present work, those relative to substrate
disappearance and TOC measurements, as a function of absorbed power per unit
membrane length, and those relative to COD measurements, by (4). As
the latter has been carried out at 1.5 W/cm, only the parameters of the former
at the same irradiance value (see Table 1) may be comparable. Furthermore, when
comparing k value of the second kind
of experiments with k2 value
of the first kind of experiments, being this step the rate determining one, the k and k2 values, expressed in ppm C/min, should be converted
in mg C/min, by taking into consideration the different reacting volumes in
the two kinds of experiments. A value of 58 mg C/min is thus obtained from k,
and of 25 from k2 at an
irradiance corresponding to 1.5 W/cm, both at an infinite concentration value.
However, it should be first underlined that consideration on rates, based
exclusively on k values, following a
Langmuir-Hinshelwood model, should not be regarded as fully trustworthy. This
notwithstanding, rough indication based on k values can be considered, at high concentrations of substrate, as
complementary to the use of the rate equations (3). This latter is anyway the
most consistent method to treat experimental data, as has been done, for
example, in kinetic modelling of data reported in Table 1, in which the kK factor shown by (3) may
play the most relevant role [11, 12]. This factor is fully taken into account
only when modelling is carried out by (3). Different K and K1, K2
values are consequently obtained in
Langmuir-Kinshelwood traditional modelling, on one
side, and in the four-parameter kinetic modelling of (3), on the
other. Only the latter is able to reproduce kinetic data overall the run and
not only for initial rates, as the Langmuir-Hinshelwood traditional model does.


## 4. Conclusions

A kinetic model was employed to
study photocatalytic mineralisation of DCP in aqueous solution. An annular
laboratory-scale membrane reactor, fitted with photocatalytic membranes
immobilising titanium dioxide, was used. By a set of differential equations,
four final optimised parameters,k1 and K1, k2 and K2, were
calculated, able to fit the whole kinetic profile satisfactorily. The influence
of irradiance on k1 and k2 may be thus rationalised.Quantum yields of mineralisation of
DCP in annular photoreactors immobilising the photocatalyst in a membrane
structure show a Langmuirian-type dependency on initial concentration of
substrate itself. By this relation limiting quantum yields, corresponding to
“infinite” concentration may be calculated, together with an apparent
thermodynamic constant, coinciding with the lowest betweenK1 and K2 values, very often being K1~K2.These
limiting quantum yields depend on radiant power absorbed per unit membrane
length, in the cylindrical geometry of photoreactors. In the low radiant power
range, up to about 0.4 W/cm they take the shape of a plateau corresponding to
the maximum allowable quantum yields, evaluated on the basis of a single
hydroxyl radical produced per each absorbed photon, thus showing the excellent
performance of the photocatalytic membrane reactors. On the contrary, at high
radiant power values, the limiting quantum yields decrease sharply by
increasing irradiance values from about 0.4 W/cm to about 2.–2.5 W/cm, and much
more gently from about 2.5 W/cm on. In the case of phenol and methane [11, 12],
this part of the curve had a very low slope, so that experimental points at
values higher than about 2.5 W/cm could simulate an apparent plateau.Kinetic
parameters of model above have been compared with parameters of the traditional
Langmuir-Hinshelwood model. The remarkable uncertainties of the latter
model and the fairly greater adequacy of the former were shown in the ability
of the four-parameter model to reproduce the whole kinetic profile, up to
complete mineralisation of the substrate, and not only in the segment of
initial rates, as the Langmuir-Hinshelwood model does.



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*Source: 631768-2009-04-14.xml*