# Optimization Method to Determine the Kinetic Rate Constants for the Removal of Benzo[a]pyrene and Anthracene in Water through the Fenton Process

^{1}

^{2}

^{*}

## Abstract

**:**

^{®}software for representing AN and BaP elimination by the Fenton process under an experimental domain. These algorithms were derived from the first-, second- and third-order kinetic models, as well as from the double exponential and the Behnajady-Modirshahla-Ghanbery (BMG) kinetic models. Regarding the AN and BaP removal kinetics, the double exponential and the BMG models were found to exhibit the highest correlation coefficients (>0.98 and >0.95, respectively) in comparison with those ones obtained from the first-, second- and third-order kinetic models (>0.80, >0.85 and >0.88, respectively). It was found that the algorithms can be used to optimize and fit the rate constants by creating an objective function that fits and represents the experimental data obtained concerning the removal of the compounds of interest through the Fenton advanced oxidation process.

## 1. Introduction

_{2}), water (H

_{2}O), and mineral salts is achieved; i.e., the total mineralization of the parent pollutant is found. Among all the AOP, the Fenton process, employing ferrous ion (Fe

^{2+}) and hydrogen peroxide (H

_{2}O

_{2}) as the catalyst and the oxidant, respectively, has been considered as one of the most attractive and powerful advanced oxidation technologies [3]. The Fenton process is able to degrade recalcitrant pollutants that cannot be efficiently removed by conventional treatment processes to a high degree [4,5,6]. Additionally, inherent toxic and dangerous substances are not generated by the Fenton oxidation process when applied to water. The Fenton process is also easy and safe to operate, and the reagents used are widely available [3,6].

## 2. Materials and Methods

#### 2.1. Chemicals and Reagents

_{4}7H

_{2}O) and H

_{2}O

_{2}30% w/w, which were obtained from Sigma-Aldrich (St. Louis, MA, USA) and JT Baker (Mexico City, Mexico), respectively. Additionally, sulfuric acid (H

_{2}SO

_{4}, 95–97%), obtained from Merck (Darmstadt, Germany), was used for reducing the solution pH up to 2.8.

#### 2.2. Experimental Setup

_{2}O

_{2}concentrations of 0.44 mg L

^{−1}and 10.50 mg L

^{−1}, respectively, were achieved. Detailed information on the procedure followed is established in [12].

#### 2.3. Analytical Methods

^{−1}and 4.26 ng L

^{−1}, respectively. The detailed information on the analytical procedure used, including the validation studies, are reported elsewhere [13,14].

#### 2.4. Kinetic Model Representing AN and BaP Removal

^{2+}and H

_{2}O

_{2}. In turn, the reaction where Fe

^{3+}and H

_{2}O

_{2}were involved was related to the second stage of the reaction [20,21]. Both stages have pseudo-first-order kinetics. In turn, the behavior of Orange G dye in water when subjected to the Fenton process was studied by Suna and coworkers. In this case, the dye degradation was found to occur in one step. In addition, it was observed that the reaction kinetics followed a pseudo-second order [22].

_{i}and n refer to the reaction rate constant and the reaction order, respectively. Finally, t is the time. For a first-order reaction, n is equal to 1. After integration, Equation (1) becomes in Equation (2), where C

_{o}and C

_{t}are the initial concentration and the concentration at the reaction time t of AN or BaP, respectively.

_{t}and 1/C

_{t}

^{2}vs. time should be linear with a slope equal to $-{k}_{1}$, ${k}_{2}$ and ${k}_{3}$, and an intercept equal to $\mathrm{ln}[{C}_{o}]$, 1/C

_{o}and 1/C

_{t}

^{2}, respectively. Linear least-squares analysis can be used to evaluate the slopes and intercepts.

_{1}is expressed in min, while A

_{2}is a dimensionless parameter. Several authors determined that 1/A

_{1}stands for the initial pollutant removal rate within the bulk (k

_{4}). Consequently, a higher 1/A

_{1}value results in a faster pollutant initial decay rate. Moreover, 1/A

_{2}was the theoretical maximal removal fraction of the pollutant of interest (k

_{5}). This value can be equal to the maximal oxidation capacity of the process at the end of the Fenton reaction. To solve these constants, a number of authors linearized Equation (6) to obtain Equation (7).

_{t}/C

_{o}) vs. t results in straight lines with an intercept (${A}_{1}$) and a slope (${A}_{2}$). This model is known in the literature as the BMG model [25,26,27].

_{3}and A

_{4}are fractions of the initial concentration of the pollutant.

#### 2.5. Optimization Method to Determine the Pollutant Removal Rate Constants

^{®}, Minitab, Maple, etc., fitting models can be carried out through a number of functions, among which the function fminsearch has a highlighted position. The fitting of non-linear models relies on non-trivial hypothesis, unlike the linear regression. Therefore, users are required to carefully ensure and validate the entire modeling. By utilizing some variant of the least squares’ criterion, the estimation of parameters can be conducted through an iteration process. Optimal factors are thus ideally calculated. For example, the fminsearch is an implementation of the Nelder-Meda simplex algorithm that allows minimization of a non-linear function of several variables [28,29].

- The algorithm reads the vector of experimental data (C
_{data}) and the vector that records the time when the measurements of the species concentration were performed (t_{data}). The fitting function (g) should be defined. This function depends on one or several parameters to be fitted. Fitting the curve via optimization means finding these parameters that minimize the sum of squared errors.

- clear all
- close all
- clc
- fig = figure(3)
- ExpData = xlsread(‘Fenton’,’hoja1’, ’A1.B21’); % Read experimental data from an excel file
- tdata = ExpData(:,1);
- Cdata = ExpData(:,2);

- Then, the experimental data related to concentration vs. time are plotted using the function plot of MATLAB
^{®}.

- plot(tdata,Cdata,’ro’); % Plot the experimental data
- hold on;
- h = plot(tdata,Cdata,’b’);
- hold off;

- In the algorithm, it is necessary to define the objective function to be minimized that accepts the parameters to be optimized, in this particular case. When Cdatai (tdatai) represents the experimental AN or BaP concentration values measured throughout the time of treatment and ${g}_{i}\left({t}_{dat{a}_{i}},v\right)$ are the simulated data, the function (F) can be rewritten as Equation (9).

_{1}and the experimental data vectors t

_{data}and C

_{data}, and returns the sum of squared errors for the model is required. Additionally, all the variables to be optimized (k

_{1}) must be put in a single vector variable (x).

- function E = Order1(x,tdata,Cdata)
- k1 = x(1);
- E = sum((Cdata - exp(-k1*tdata)).^2);

^{®}path.

_{data}, C

_{data}), the objective function allows calculation of the error in the fitting for this equation with respect to the experimental data. Additionally, the objective function allows updating of the line (h) of the algorithm. All the procedures providing the estimation of the non-linear parameters require initial values. An initial estimation of x is then carried out, and the function fminsearch is invoked. A random positive set of parameters (x

_{0}) is taken and, afterwards, fminsearch allows finding of the parameters that minimize the objective function. By adjusting x, Fminsearch minimizes the error resultant from the objective function. It returns the final value of x. Finally, an output function to plot intermediate fits is used. The initial values selection will affect the estimation algorithm convergence, leading to no convergence and to convergence after a few iterations for the worst and the best cases, respectively. For the first-order kinetic model, the following steps of the algorithm are conducted:

- fun = @(x)Order1(x,tdata,ydata);
- x0=rand(1,1);
- outputFcn = @(x,optimvalues,state) fitoutputfun(x,optimvalues,state,tdata,ydata,h);
- options = optimset(‘OutputFcn’,outputFcn,’TolX’,1e-80,’MaxFunEvals’, 10,000,000);
- bestx=fminsearch(fun,x0,options)

_{data}and C

_{data}are not variables to be optimized; however, they are data to be used for the optimization. Therefore, the objective function is required to be defined for fminsearch as a function of x alone. On the other hand, employing an output function is required so that the optimization function is called during each iteration. By employing this function, recording the history of the data that the algorithm generates and producing a graphical output, or halting the algorithm based on the data at the current iteration, is feasible. In MATLAB

^{®}, it is possible to use the function OutputFcn with the optimization functions fminsearch. In the function OutpuntFcn, the variable x is the point computed by the algorithm at the current iteration, the variable optimValues is a structure containing data from the current iteration, and the variable state is the current state of the algorithm.

- For checking the quality of the fit, the resulting fitted response curve and the data are plotted. The response curve is created from the returned parameters of the model. Finally, the coefficient of determination (R
^{2}) was used to choose the best model that describes the removal of the compounds of interest.

- A = 1;
- k1 = bestx(1);
- yfit = A*exp(-k1 *tdata);
- FS=10;
- plot(tdata,ydata,’*’);
- hold on
- plot(tdata,yfit,’r’);
- hold on
- axis ([ 0 90 0 1.0 ])
- title(‘Experimental Data and Best Fitting Curve’)
- xlabel (‘Time (min)’)
- ylabel (‘[AN]/[AN]_o’)
- g=legend(‘Experimental data Fenton-NW’,’Fitting curve’,’location’,’best’);
- set(g,’Box’,’on’,’EdgeColor’,[1 1 1])
- set(gcf, ‘color’,’white’)
- set(gca,’FontSize’,FS,’yticklabel’,num2str(get(gca,’ytick’)’,’%.1f’));
- box off
- grid on
- a=corr(ydata,yfit)^2

## 3. Results and Discussion

^{2}values for the removal of a mixture of AN and BaP by the Fenton process under several operating conditions were calculated by applying the algorithm developed for the first-, second-, and third-order kinetic models, as well as for the double exponential and the BMG kinetic models. In order to compare the kinetic models constructed, the kinetic model providing the best fit was determined by the highest R

^{2}associated, which was obtained among empirical and theoretical data [31].

^{−1}BaP after only 2 min of reaction under 3.75 mg L

^{−1}Fe(II) and H

_{2}O

_{2}in the range from 20 to 150 mg L

^{−1}at pH 3.5 was observed [33]. This fast pollutant conversion was also observed for 2 mg L

^{−1}NA using 8 mM Fe(II) at pH 4 [30].

_{2}O

_{2}, resulting in a degradation of AN and BaP of 17.53% and 11.23%, respectively. The second stage occurring by the catalytic reaction of Fe(III) with H

_{2}O

_{2}, obtaining a final degradation extent of 20.44% and 13.29% for AN and BaP, respectively. Therefore, an increase in only 2.91% and 2.06% for AN and BaP was observed when moving from the first to the second stage. Note also that due to the dependence on the Fenton reaction on the H

_{2}O

_{2}and Fe(II) content within the bulk, and the results obtained from the Fenton experiments, lower degradation values of AN and BaP were attained for the same treatment time. This double behavior concerning both reaction stages was also found by Mitsika and coworkers for acetamiprid [31], a persistent toxic substance.

^{2}, as mentioned above.

^{2}for the five models are different.

^{2}values are approached to 1. The use of this correlation coefficient for determining the fit of a model has been also reported in the literature by [31] during the removal of 5 mg L

^{−1}acetamiprid (0.023 mmol L

^{−1}) at different concentration of H

_{2}O

_{2}and the catalyst in an aqueous solution at a pH value of 2.9. The referred authors found R

^{2}values higher than 0.80, in general terms, although R

^{2}values lower than 0.20 were also achieved, which demonstrated that, under certain operating conditions, the experimental data are not fitted by the models tested.

^{2}values associated are >0.80, >0.85 and >0.88, respectively. The double exponential and the BMG kinetic models have the highest R

^{2}values in comparison with the first- and second-order kinetic models (R

^{2}values ranging from 0.95 to ~1.0 were found), indicating that the AN and BaP decay kinetics were well described by the double exponential and the BMG models using the Fenton oxidation system.

## 4. Conclusions

^{2}values in comparison with the first-, second- and third-order models.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Removal of anthracene (AN) in (

**a**) natural water (NW) and (

**b**) deionized water (DW), and benzo[a]pyrene (BaP) in (

**c**) NW and (

**d**) DW by the Fenton oxidation process. [AN]

_{0}= 3 µg L

^{−1}, [BaP]

_{0}= 3 µg L

^{−1}; [Fe(II)]

_{0}= 0.44 mg L

^{−1}; [H

_{2}O

_{2}]

_{0}= 10.50 mg L

^{−1}.

Kinetic Model | Function |
---|---|

First-order model | function E = Order1(x,tdata,Cdata) k1 = x(1); E = sum((Cdata - exp(-k1*tdata)).^2); |

Second-order model | function E = Order2(x,tdata,Cdata, A) C0 = A; k2 = x(1); E = sum((Cdata -1./(1+C0*k2*tdata)).^2); |

Third-order model | function E = Order3(x,tdata,Cdata, A) C0 = A; k3 = x(1); E = sum((Cdata –sqrt(1./(1+C0^2*k3*tdata))).^2); |

Behnajady-Modirshahla-Ghanbery (BMG) model | function E = Order3(x,tdata,Cdata) A1 = x(1); A2 = x(2); E = sum((Cdata -(1-tdata./(A1+A2*tdata))).^2); |

Double exponential model | function E = Order3(x,tdata,Cdata) A3 = x(1); k6 = x(2); A4 = x(3); k7 = x(4); E = sum((Cdata - (A3*exp(-k6*tdata)+A4*exp(-k7*tdata))).^2); |

**Table 2.**Kinetic parameters of different models and their correlation coefficients (R

^{2}) for the removal of a mixture of AN and BaP in deionized (DW) and natural (NW) water by the Fenton process.

Pollutant | Matrix | First-Order Model | Second-Order Model | Third-Order Model | BMG Model | Double Exponential Model | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

k_{1} (min^{−1}) | R^{2} | k_{2} (µg^{−1} min^{−1}) | R^{2} | k_{3}(µg ^{−2} min^{−1}) | R^{2} | A_{1} (min) | k_{3} (min^{−1}) | A_{2} | k_{4} | R^{2} | A_{3} | k_{6} (min^{−1}) | A_{4} | k_{7} (min^{−1}) | R^{2} | ||

AN | NW | 0.0085 | 0.8094 | 0.0038 | 0.8718 | 0.0034 | 0.9108 | 45.7838 | 0.0218 | 1.8278 | 0.5471 | 0.9526 | 0.6146 | 0.0344 | 0.4061 | −0.0041 | 0.9806 |

DW | 0.1563 | 0.9607 | 0.0945 | 0.9691 | 0.1515 | 0.9819 | 2.1473 | 0.4657 | 1.1161 | 0.8960 | 0.9903 | 0.8515 | 0.2528 | 0.1495 | 0.0003 | 0.9996 | |

BaP | NW | 0.0046 | 0.8148 | 0.0019 | 0.8549 | 0.0015 | 0.8885 | 67.9842 | 0.0147 | 2.9935 | 0.3341 | 0.9868 | 0.2984 | 0.0435 | 0.7074 | −0.0005 | 0.9970 |

DW | 0.0952 | 0.8867 | 0.0500 | 0.9242 | 0.0651 | 0.9652 | 3.1959 | 0.3129 | 1.2282 | 0.8142 | 0.9893 | 0.7633 | 0.2104 | 0.2366 | 0.0002 | 0.9987 |

_{1}, k

_{2}and k

_{3}: kinetic reaction rate constants representing the depletion of the target pollutant according to the first-, second and third-order kinetic models. A

_{1}and A

_{2}: constants related to the reaction kinetics and oxidation capacities of the treatment system, respectively. k

_{4}and k

_{5}: constants concerning 1/A

_{1}and 1/A

_{2}that refer to the initial pollutant removal rate within the bulk and the theoretical maximal removal fraction of the pollutant of interest, respectively. A

_{3}and A

_{4}: fractions of the target pollutant initial concentration. k

_{6}and k

_{7}: kinetic reaction rate constants representing the abattement of the target pollutant for the first and second stages, respectively.

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**MDPI and ACS Style**

Rubio-Clemente, A.; Chica, E.; Peñuela, G.A. Optimization Method to Determine the Kinetic Rate Constants for the Removal of Benzo[a]pyrene and Anthracene in Water through the Fenton Process. *Water* **2022**, *14*, 3381.
https://doi.org/10.3390/w14213381

**AMA Style**

Rubio-Clemente A, Chica E, Peñuela GA. Optimization Method to Determine the Kinetic Rate Constants for the Removal of Benzo[a]pyrene and Anthracene in Water through the Fenton Process. *Water*. 2022; 14(21):3381.
https://doi.org/10.3390/w14213381

**Chicago/Turabian Style**

Rubio-Clemente, Ainhoa, Edwin Chica, and Gustavo A. Peñuela. 2022. "Optimization Method to Determine the Kinetic Rate Constants for the Removal of Benzo[a]pyrene and Anthracene in Water through the Fenton Process" *Water* 14, no. 21: 3381.
https://doi.org/10.3390/w14213381