Modeling and Optimization of Pollutants Removal during Simultaneous Adsorption onto Activated Carbon with Advanced Oxidation in Aqueous Environment

The paper presents the results of studies on the modeling and optimization of organic pollutant removal from an aqueous solution in the course of simultaneous adsorption onto activated carbons with varied physical characteristics and oxidation using H2O2. The methodology for determining the models used for predicting the sorption and catalytic parameters in the process was presented. The analysis of the influence of the sorption and catalytic parameters of activated carbons as well as the oxidizer dose on the removal dynamics of organic dyes-phenol red and crystal violet-was carried out based on the designated empirical models. The obtained results confirm the influence of specific surface area (S) of the activated carbon and oxidizer dose on the values of the reaction rate constants related to the removal of pollutants from the solution in a simultaneous process. It was observed that the lower the specific surface area of carbon (S), the greater the influence of the oxidizer on the removal of pollutants from the solution. The proposed model, used for optimization of parameters in a simultaneous process, enables to analyze the effect of selected sorbents as well as the type and dose of the applied oxidizer on the pollutant removal efficiency. The practical application of models will enable to optimize the selection of a sorbent and oxidizer used simultaneously for a given group of pollutants and thus reduce the process costs.


Introduction
The increasing population, environmental pollution and climate change effects, such as severe droughts, place a heavy burden on water resources. Industry is the sector which exerts the strongest environmental pressure on water quality. Along with its intensive development, increasing amounts of new and complex chemical compounds occur, which together with wastewater, as well as municipal and industrial wastes, infiltrate into the natural environment, usually in the first stage during migration to surface waters [1,2]. Since industrial wastewater is characterized by heterogeneous process are mixed. There is no information on the properties that should characterize the activated carbons used in these reactions and what is the influence of the oxidizing agent on the physicochemical properties of activated carbons. As indicated in previous studies [14], the efficiency of pollution removal in simultaneous adsorption and advanced oxidation processes is governed by a broad range of parameters, the most relevant of which include the sorption and catalytic properties of the adsorbent. These parameters largely determine the process efficiency as well as the possibility of applying a sorbent under specific conditions. The application of simultaneous methods for their optimization enables to analyze the influence of the properties of selected sorbents as well as the type and dose of the applied oxidizer on the efficiency of pollutant removal.
The application of mathematical models for predicting the course of the adsorption process with set operational sorption parameters was investigated in numerous papers [40][41][42][43]. The literature sources present numerous methods and mathematical models based on the kinetics and adsorption isotherms enabling to predict the process. The most commonly used kinetic expressions to explain the solid/liquid adsorption processes are the pseudo-first-order kinetics and pseudo-second-order kinetic models [44][45][46][47]. This approach enables modeling the process dynamics in time. Due to the high costs of conducting experiments, the methods which allow for decreasing the number of measurements, i.e., design of experiments (DoE) are highly useful. Response surface method (RSM) can be mentioned as an example [48]. The DoE methods reduce the number of necessary measurements, which shortens the required time, decreases the material consumption and the costs of experiments [49][50][51]. Due to the limited application of the models designated by means of RSM, the models devised with the data mining methods, for instance artificial neural networks (ANN), are highly useful [52][53][54].
However, in a majority of cases, the models presented describe the sorption of single pollutants on selected carbon adsorbents. Moreover, greater attention was drawn to the sorption process conditions, rather than to the physicochemical properties of the employed activated carbon. As a result, the practical application of such models is limited, because they cannot be used for predicting sorption of another group of adsorbents. No information on the modeling of sorption in simultaneous processes in the presence of oxidizers was found in the literature. Therefore, devising the models for predicting sorption on activated carbons, which account for the diversity of the physicochemical parameters of a sorbent, different adsorbates and the presence of an oxidizing medium, seems to be reasonable.
The paper presented the methodology of creating the mathematical models for optimization of a simultaneous sorption process of activated carbon adsorption with chemical oxidation of pollutants from aqueous solutions via hydrogen peroxide, on the example of crystal violet and phenol red. The article showed the methodology of conducting statistical analyses aimed at determining the influence of selected carbon characteristics (dechlorination half-length (D ehl ), methylene number (MBN), iodine number (IN), S, soluble substance content (Cont.s.sol)) and oxidizer dose (H 2 O 2 ) on the dynamics of the sorption process (Freundlich and Langmuir isotherms) and removal of pollutants from aqueous solutions (reaction rate constants in pseudo-first-and pseudo-second-order equations). The obtained results were used to designate mathematical models (by means of the multiple regression method) for calculating the parameters describing the dynamics of sorption and pollutant removal based on the activated carbon characteristics and oxidizer dose. The calculations were performed using the results of laboratory studies as well as identification of sorption and pollutant removal dynamics in aqueous solutions [36]. The obtained dependencies enable modeling the above-mentioned processes with diversified characteristics and equal oxidizer doses. The obtained models enable optimizing the selected parameters of the simultaneous process of pollutant removal (crystal violet and phenol red) from aqueous solutions by adjusting the oxidizer dose that enables to achieve shortest pollutant reduction time, depending on the activated carbon used.
The practical application of the devised models will enable to optimize the selection of the sorbent and the oxidizer used simultaneously for a given group of pollutants and thus will reduce the process costs of their removal from aqueous environment.

Materials
The object of research involved the two dyes: crystal violet (C 25 H 30 CIN 3 ) and phenol red (C 19 H 14 O 5 S), commonly used for dyeing fabrics, paper products and printing ink, thus found in the textile and dye industry wastewater. They are also used as analytic reagents. These substances are characterized by low efficiency of removal from aqueous solutions and wastewater using physical and biological methods.
In the research, H 2 O 2 , which is found in the wastewater from the textile, dye, and printing industries, was employed as the oxidizing agent [55].
In order to determine the correlations between the activated carbon properties and the removal efficiency of the analyzed organic substances, the following organic carbons were selected (see Table S1 Supplementary Materials): • WDex and WG-12-fresh commercial granulated activated carbons produced in Poland by Gryfskand (Gryfino), used in the treatment of water and wastewater, removal of organic compounds, including pesticides, as well as dechlorination of water and improvement of its taste; • F-200S-spent F-300 carbon following several years of use in a water treatment station in Goczałkowice, to be disposed of; • F-200R-F-200S activated carbon following several years of use in a water treatment station in Goczałowice, subsequently regenerated with the Fenton's agent.

Methods
The methodology for the designation of models predicting the kinetic parameters in the models describing the sorption isotherms and removal of pollutants in aqueous solutions using activated carbons, was proposed ( Figure 1).

Materials
The object of research involved the two dyes: crystal violet (C25H30CIN3) and phenol red (C19H14O5S), commonly used for dyeing fabrics, paper products and printing ink, thus found in the textile and dye industry wastewater. They are also used as analytic reagents. These substances are characterized by low efficiency of removal from aqueous solutions and wastewater using physical and biological methods.
In the research, H2O2, which is found in the wastewater from the textile, dye, and printing industries, was employed as the oxidizing agent [55].
In order to determine the correlations between the activated carbon properties and the removal efficiency of the analyzed organic substances, the following organic carbons were selected (see Table  S1 Supplementary Materials): • WDex and WG-12-fresh commercial granulated activated carbons produced in Poland by Gryfskand (Gryfino), used in the treatment of water and wastewater, removal of organic compounds, including pesticides, as well as dechlorination of water and improvement of its taste; • F-200S-spent F-300 carbon following several years of use in a water treatment station in Goczałkowice, to be disposed of; • F-200R-F-200S activated carbon following several years of use in a water treatment station in Goczałowice, subsequently regenerated with the Fenton's agent.

Methods
The methodology for the designation of models predicting the kinetic parameters in the models describing the sorption isotherms and removal of pollutants in aqueous solutions using activated carbons, was proposed ( Figure 1). Identification of the pollutants removal: -pseudo-first-order reaction -pseudo-second-order reaction Analysis of correlations between the kinetic parameters, the activated carbon characteristics, and the applied oxidizers Designation of empirical models for forecasting the reaction rate constants using multiple regression   The analyses in the assumed methodology are based on the results of laboratory tests related to sorption and pollutant removal performed on the activated carbons with diversified physical characteristics (specific surface area-S, pore volume-V p , dechlorination half-length-De hl , iodine number-IN, detergent number-DetN., methylene number-MBN, ash content-Cont.ash, soluble substance content-Cont.s.sol., pH of the aqueous extract-pH) with hydrogen perodixe (H 2 O 2 ) as the oxidizer.

Methodology of Laboratory Tests
The measurement experiments can be divided into several stages which involve analytic determination of the activated carbon characteristics, removing crystal violet and phenol from a solution through: sorption, oxidation via H 2 O 2 and a simultaneous process (sorption + oxidation with H 2 O 2 ) as well as regeneration of spent activated carbon. The subsequent measurement steps and their methodology are discussed in detail in Figure 2. The analyses in the assumed methodology are based on the results of laboratory tests related to sorption and pollutant removal performed on the activated carbons with diversified physical characteristics (specific surface area-S, pore volume-Vp, dechlorination half-length-Dehl, iodine number-IN, detergent number-DetN., methylene number-MBN, ash content-Cont.ash, soluble substance content-Cont.s.sol., pH of the aqueous extract-pH) with hydrogen perodixe (H2O2) as the oxidizer.

Methodology of Laboratory Tests
The measurement experiments can be divided into several stages which involve analytic determination of the activated carbon characteristics, removing crystal violet and phenol from a solution through: sorption, oxidation via H2O2 and a simultaneous process (sorption + oxidation with H2O2) as well as regeneration of spent activated carbon. The subsequent measurement steps and their methodology are discussed in detail in Figure 2.  [36,56]. , pH of the aqueous extract (pH) were determined. The methodology followed in the determination of physico-chemical properties of activated carbons was discussed in details in the study by Dąbek et al. [36,56]. The porous structure was determined on the basis of studies of the low-temperature nitrogen adsorption isotherm (77K) performed with the Sorptomatic 1900 apparatus (Carlo Erba, Italy, Austria).

Sorption + oxidation
In conical flasks with a capacity of 300 cm 3  For a conical flask with a capacity of 300 cm 3 , 200 cm 3 of a color solution with a concentration of 20 mg/L and a quantity of H2O2 solution in an amount to give a concentration of 1500 mg H2O2/L were added. The reaction mixture was vigorously stirred. At specified time intervals, the dye concentration in the reaction mixture was determined by spectrophotometric method using a Marcel Media UV/VIS spectrophotometer. The test was repeated for the oxidant concentration at the level of 3750 and 7500 mg H2O2/L. The changes in the dye concentration were analyzed in the period from 20 to 160 min.
200 cm 3 of a color solution with a concentration of about 20 mg/ L was added to the conical flasks with a capacity of 300 cm 3 , 0.5 g of tested activated carbon and H2O2 were also added in the amount ensuring the concentration of 1500 mg H2O2/L, 3750 mg H2O2/L and 7500 mg H2O2/L. The reaction mixture was vigorously stirred. The changes in the dye concentration were analyzed in the period from 20 to 160 min.
Regeneration of spent activated carbon 5 g of activated carbon intended for regeneration was weighed into conical flasks with a capacity of 300 cm 3 , and then 200 cm 3 of distilled water acidified with sulfuric acid (VI) was added to pH = 3 ÷ 4.1, and then FeSO4·7H2O and H2O2 solutions were added simultaneously in the amount of providing a 1:5 weight ratio of Fe 2+ : H2O2, and mixed for 30 min. Then the solution with Fenton's reagent was decanted, and the activated carbon was washed with 5 × 100 cm 3 distilled water and dried at 378 K until constant weight. Figure 2. Methodology of laboratory studies [36,56].

Identification of the Pollutant Removal
On the basis of the conducted experiments (sorption of pollutants onto activated carbon, removal of pollutants from the solution), the empirical models were fitted to the theoretical data. For the sorption models, the process kinetics parameters were determined based on the Langmuir and Freundlich isotherms. While removing pollutants from the solution (activated carbon as the oxidizer), the process kinetics was identified based on the pseudo-first-and second-order equations. Thus, the values of reaction rate constants were determined.
The influence of carbon characteristics and the oxidizer dose on their dynamics was analyzed on the basis of the process kinetics calculations (sorption, removal of pollutants from the solution). For this purpose, the correlation matrices were calculated. This enabled to determine which parameters of activated carbons and oxidizers affect the sorption and pollutant removal processes. Moreover, the correlation between the analyzed characteristics of activated carbons, having a significant influence on the selection of optimal independent variables for the kinetic models, was determined. This served as the basis for designating the models for predicting the values of process kinetics parameters. The characteristics of activated carbons were optimized in order to achieve high pollutant removal efficiency in shortest time possible. The proposed solution, in contrast to those presented in other papers [57][58][59] is universal, because it enables to analyze the influence of the analyzed sorbent (here: activated carbon) and employed oxidizer as well as their properties. The previous models usually did not account for these aspects and considered only a single activated carbon, which led to their numerous limitations.
The analysis of sorption onto activated carbons was performed assuming the following equations: • Freundlich: • Langmuir: where q e -amount of adsorbate adsorbed per unit mass of activated carbon (mg/g), c e -equilibrium concentration of the adsorbate (mg/L), K F -coefficient in the Freundlich isotherm model (mg/g), n-coefficient in the Freundlich isotherm model, K L -coefficient in the Langmuir isotherm model (L/mg), a m -sorption capacity value in the Langmuir isotherm model (mg/g). On the basis of the studies on the changes in the concentration of crystal violet and phenol red in the solution in the presence of activated carbon and oxidizer [33] the following kinetic equations were assumed: • Pseudo-first-order: • Pseudo-second-order: where c 0 -initial substrate concentration (mg/L), c t -substrate concentrations after a given time (t) (mg/L), t-reaction time (s), k 1 -first order reaction rate constant (s −1 ), k 2 -second order reaction rate constant (M −1 ·s −1 ) In the case of sorption and pollutant removal from the solution, the correlation between the results and calculations was measured using the coefficient of determination (R 2 ). On the basis of the obtained value of this parameters, the identified sorption mechanisms (process physics) were discussed for the analyzed variants of the experiments (activated carbon parameters, oxidized dose) in terms of the designated R 2 values and differences between them. For instance, when the calculated R 2 value for the data interpolated using a pseudo-second order equation was greater than the data modeled with a pseudo-first ordered equation, it was assumed that the kinetics of pollutant removal was in line with Equation (4). In turn, when the R 2 value obtained for the data interpolated with the Freundlich isotherm equation was greater than the results of calculations using the Langmuir isotherm, it was assumed that the sorption mechanism was in accordance with relation Equation (1).

Multiple Linear Regression
Multiple linear regression (MLR) constitutes one of the simplest and most commonly employed methods for creating linear models. In this method, the forecast relationship has the following form [60]: where a-model parameters determined with the least squares method, x-input variables (independent), y-model output, i-number of independent variables (i = 1, 2, 3, . . . , M). In order to optimally assign the independent variables to the empirical models, the correlation coefficients were calculated between the characteristics of carbons as well as the kinetics parameters in the sorption models (Freundlich and Langmuir isotherms) and pollutant removal (dyes-crystal violet, phenol red). These calculations enabled to eliminate the dependent variables from the models. By applying the multiple regression method, the empirical models for calculating the coefficients pertaining to sorption (K L , a m , K F , and n) as well as removal of pollutants from the solution (k) were determined.

Optimization of Pollutant Removal from the Solution
During the process design, the costs of reagents, its dynamics and degree of pollutant reduction in wastewater are of great importance. In order to design a process optimally, the costs of reagent consumption have to be decreased while simultaneously achieving the highest reduction of pollutants in shortest time possible. On the basis of Equations (3) and (4), the subsequent conditions for process optimization were described using the following relationships: • Pseudo-first-order: • Pseudo-second-order: where β-degree of pollutant removal expressed as c t c 0 −1 ; c 0 -in line with the research methodology (mg/L), the value c 0 = 20 mg/L was assumed in the calculations for phenol red and crystal violet; c t -substrate concentrations after a given time (t) (mg/L).
In the presented analyses, the main criterion for the optimization of pollutant removal from wastewater was time t, which enabled to achieve high reduction of dye concentration in the aqueous solution. Therefore, the combination of carbon characteristic and oxidizer dose, for which time t enabling high reduction of pollutants (η = 90%) in the solution was the shortest, was sought. On the basis of Equations (6) and (7), the influence of activated carbons characteristics and oxidizer doses on shortening the time of pollutant removal from the solution (crystal violet and phenol red) was analyzed assuming the required reduction.

Visualization of Data with Multidimensional Scaling Method
The multidimensional scaling (MS) method was used in order to visualize the similarity level for the component of matrix containing the data connected with the properties of employed activated carbons along with the H 2 O 2 dose and the kinetic parameters of dye removal. The MS method [61,62] made it possible to obtain sets of points distributed in a 2-dimensional space in such a way that in the analyzed cases (tested activated carbons in different versions of the experiment), the similar points were closest to each other, while different ones were spread far apart [63]. Its aim was to show in one figure the general similarity between the processes carried out with the use of four activated carbons (WDex, WG-12, F-200R, F-200S) characterized by different properties (S, IN, MBN, De hl ), using different doses of the oxidizer (0, 1500, 3750, 7500 mg H 2 O 2 /L), which were associated with obtaining different kinetic constants k 1 and k 2 with respect to the removal of both analyzed dyes from the water solution (crystal violet, phenol red).

Identification of the Sorption Process
On the basis of the conducted research (Figure 1), the theoretical curves were matched to the empirical data. The calculation results for sorption were presented in Table 1, whereas for the removal of pollutants in the presence of crystal violet and phenol red-in Table 2. Taking into account the data presented in Table 1, the sorption mechanism of the analyzed dyes cannot be unequivocally determined for all activated carbons. In the case of fresh carbons (WDex, WG-12), the sorption process dynamics of the crystal violet solution is in line with the Langmuir isotherm. However, for phenol red, the situation is not as clear-in the case of WDex, the coefficient of determination R 2 , similarly as for crystal violet, is higher for the Langmuir model, whereas for WG-12, the R 2 coefficients are at a similar level. In turn, for the spent activated carbons (F-200R and F-200S) it can be stated that in the case of the regenerated spent carbon F-200S, the sorption dynamics is clearly in accordance with the Freundlich isotherm for both considered dyes. However, the match is not as obvious for spent carbon F-200R, the coefficients of determination R 2 are at a similar level in relation to both models; however, for phenol red, R 2 is greater for the Langmuir model. Due to slight differences between the coefficients of determination R 2 , in majority of cases it was difficult to unequivocally determine the sorption mechanism, which was discussed in detail in the paper by Dąbek et al. [36]. While analyzing the literature data, it was stated that most authors, in relation to the sorption of dyes onto activated carbon, pointed at the monolayer sorption mechanism [64][65][66][67].  The obtained sorption results indicate that in the case of the fresh activated carbons, this process can be employed as a method for the removal of the analyzed dyes; however, this process requires longer time and is difficult to implement in an industrial installation, due to the presence of other adsorbing substances and suspensions.

Removal of Pollutants from the Solution (Cystal Violet and Phenol Red)
As indicated by the abovementioned results, the sorption process efficiency can be improved by introducing an oxidizing agent into the system. The literature review showed that although the authors unequivocally indicate the purposefulness of integrating sorption and AOPs in order to improve to efficiency of pollutant removal, the mechanism of the simultaneous process is not fully explained; therefore, in order to ensure the optimal conditions, it is necessary to correlate the parameters related to the sorbent parameters as well as the type and dose of the oxidizer. On the basis of Equations (3) and (4), the reaction rate constants in the dye decomposition processes (crystal violet and phenol red) in the presence of an oxidizer (H 2 O 2 ), were determined. The calculation results were presented in Table 2.

Analysis of Correlation between the Activated Carbon Characteristics
In order to comprehensively evaluate the relationships between the particular activated carbon parameters, a Spearman's rank correlation coefficient was prepared (Table 3), which indicates that between the dechlorination half-length iodine number, ash content and pH of the aqueous extract occurs at the confidence interval of p = 0.05. In Table S2 in Supplementary Materials determined values of the test probability are given for the correlation between the characteristics of the activated carbons.
It should be noted that an increase in the pH of the aqueous extract and the iodine number decreases the dechlorination half-length, i.e., the dechloration ability increases, which indirectly indicates enhanced catalytic properties of carbon in the reaction involving decomposition of hydrogen peroxide with the formation of hydroxyl radicals [14]. A strong correlation between the iodine number as well as the methylene number and specific surface area is observed as well. A similar correlation was indicated by Nunes et al. [68], who stated that the iodine number and methylene blue number are important for the surface area estimation of the activated carbons, since the materials possess different pore sizes which can be accessed by the different molecules according to the pore geometry. Moreover, while analyzing the obtained correlation matrix (Table 3), correlations between specific surface, pore volume, pH of the aqueous extract and ash content, were noted.

Adsorption onto Activated Carbons
In order to complete the analyses, the possibility of determining the values of the model parameters (K F and n in the Freundlich isotherm model, a m and K L in the Langmuir isotherm model) using empirical models was considered. This aspect is very important from the point of generalizing the calculation results and creating universal models. Taking into account the correlation of selected characteristics of activated carbons (Table 3) and the different sorption process course (crystal violet, phenol red), Table 4 presents examples of relationships K F = f(x i ) and n = f(x i ) as well as a m = f(x i ) and K L = f(x i ). Taking into account the results of calculations (Table 3), the investigations were narrowed down to the analysis of independent variables. Table 4 presents the dependencies for which the coefficients of correlation and determination are high.
The results presented in Table 4 indicate a strong correlation between the specific surface area (S), and K F and n, as well as between the content of dissolved substance (Cont.s.spl.) and n. The obtained relations between the parameters in the Freudlich and Langmuir models confirm in most cases that it is possible to develop universal models in which the kinetics of the sorption process is estimated based on the characteristics of the carbons.

Designation of Empirical Models for Determining the Reaction Rate Constant k
Correlation matrices were created first as part of the conducted analyses, in order to evaluate the influence of selected activated carbon characteristics (WDex, WG12, F200R, F200S) on the concentration of selected pollutants. The results of analyses were presented in Tables 5 and 6. The conducted analyses (Table 5) in the confidence interval of p = 0.05 indicated a strong correlation between the mean dye concentration and the dechloration number, iodine number, as well as specific surface area. In Table S3 in Supplementary Materials determined values of the test probability are given for the correlation between the selected characteristics of the activated carbons and average dye concentration (c av ).  The correlation matrix also indicates that the greater the dechloration number (lower catalytic capacity of the activated carbon), the greater the mean dye concentration in the solution following the process.
Due to the high costs of the studies and the time of conducting the supplementary determinations, an attempt was made to designate the empirical models for predicting the reaction rate constants (k) in the pseudo-first (or second) order equations. For this purpose, the correlation matrices reflecting the relations between the determined kinetic parameters and the characteristics of carbons and the oxidizer dose, were designated. Then, depending on the obtained values of correlation coefficients, the empirical models were determined. The multiple regression method was employed for this purpose.
The identification of independent variables for the model was performed using the Fischer-Snedecor distribution model. Statistically significant variables, in the confidence interval of 0.05 were assumed for analyses. Due to the limited amount of measurement data, the multiple cross validation method was used for the numerical experiment. The values of correlation coefficients for crystal violet and phenol red were given in Tables 7 and 8, respectively. In Table S4 in Supplementary Materials determined values of the test probability are given for the correlation between the selected characteristics of the activated carbons and the reaction rate constants in the pseudo-first and pseudo-second order equations.

The conducted analyses (Tables 7 and 8) indicated that:
• There is a correlation (r > 0.5) between the pseudo-first (second) order kinetic parameters and specific surface area (S), methylene number (MBN), dechlorination half-length (De hl ) and oxidizer dose (H 2 O 2 ).

•
There is almost complete correlation (r > 0.9) between the pseudo-first (second) order kinetic parameters and specific surface area (S), methylene number (MBN) and dechlorination half-length (De hl ).

•
An increase in S, De hl , and H 2 O 2 dose leads to an increase in the values of kinetic parameters (k 1 , k 2 ) of pseudo-first(second) order.
On the basis of the above-mentioned analyses, independent variables for the models used to calculate the k 1 and k 2 values were determined with the Fischer-Snedecor distribution method. This served as a basis for creating the empirical models using the MLR method, having the following form: • crystal violet (pseudo-first order): • crystal violet (pseudo-second order): • phenol red (pseudo-first order): • phenol red (pseudo-second order):

Optimization of Pollutant Removal from the Solution
The influence of the activated carbon characteristics as well as the activated carbon characteristics and the oxidizer dose on the dynamics of pollutant removal from the aqueous solution was investigated. In the case of the analyzed dyes, higher coefficients of correlation were obtained using the pseudo-second order reactions (see Table 2). Therefore, it was assumed that the removal of phenol red and crystal violet from aqueous solutions are described by the pseudo-second order reaction. Therefore, while determining the pollutant removal time, only Equation (7) were used, whereas the empirical models for indicating the reaction rate constants were based on Equations (9)-crystal violet and (11)-phenol red. The influence of carbon characteristics and carbon dose on the pollutant removal dynamics was investigated using the following relationships: where S 0 -minimal surface of activated carbon assumed for calculations; here, it amounts to S 0 = 750 m 2 , ∆S-calculation step assumed in the analysis pertaining to the impact of changes in the specific surface of activated carbon on the reduction of pollutants; in this work, it was assumed as ∆S = 25 m 2 , H 2 O 2 -oxidizer dose, assumed as 0-3500 mg; β-degree of pollutant reduction in the solution, assumed as 90%. Equation (12) enables one to analyze the influence of increasing specific surface of activated carbon for different assumed oxidizer doses on the changes in the pollutant removal time for the required reduction degree. In turn, Equation (13) (14) were used to designate the η t(1) , η t(2) and η t(3) curves describing the influence of activated carbon characteristics and oxidizer dose on the removal of the investigated pollutants from the solution (Figures 3 and 4). The obtained results indicate a significant influence of specific surface of carbon on the efficiency of dye removal. For instance, increasing the S value from 750 m 2 to 850 m 2 assuming the H 2 O 2 dose of 800 mg, shortens the pollutant removal time by 29.10% in the case of phenol red (90% reduction) (Figure 3a). Increasing the oxidizer dose to H 2 O 2 = 3200 mg indicates that the increase in S within the range of 750-800 m 2 shortens the minimum time required to achieve the assumed dye reduction level by 24.01%. Identical relations were between the analyzed variables η t(1) = f(S, H 2 O 2 ) were determined for crystal violet (Figure 4a). For example, an increase in the carbon surface from S = 750 m 2 to S = 850 m 2 at the oxidizer dose of H 2 O 2 = 800 mg shortens the time required to achieve 90% pollutant reduction by 34.13%. An identical change of S for the H 2 O 2 dose of 3200 mg shortens that time by 17.01%.
On the basis of the obtained η t(2) curves, it was observed that introducing an oxidizer into the system leads to an increase in the reaction rate constants. For De hl = 4.5 and S = 850 m 2 the reaction rate constant amounts to k 2 = 0.0105 (phenol red)- Figure 3b and k 2 = 0.011 (crystal violet)- Figure 4b. An oxidizer addition in the amount of H 2 O 2 = 1600 mg increases their values to k 2 = 0.012 and k 2 = 0.023, respectively.
This translates into a shortening of the pollutant removal time required to achieve the assumed reduction degree. For instance, inclusion of 800 mg H 2 O 2 in the reaction system (S = 850 m 2 and De hl = 4.5) shortens the pollutant removal time (phenol red) by 7% in comparison to the system without an oxidizer (Figure 3c). Assuming S = 850 m 2 , introduction of H 2 O 2 = 1600 mg into the system shortens the dye removal time (90% elimination) by 33.5% compared to the non-modified process. This is in line with the results obtained by Santos et al. [37], who indicated that a synergistic effect occurs in the case of simultaneous sorption onto activated carbon and oxidation via H 2 O 2 , which enhances the solution decolorization.
While analyzing the obtained η t(2) curves, it was observed that the lower the specific surface of activated carbon (S), the greater the influence of oxidizer on the pollutant removal process. For example, in the case of S = 800 m 2 and H 2 O 2 = 1600 mg, 90% crystal violet removal can be achieved 57.0% faster than in the system without this compound. In turn, for S = 900 m 2 , the same pollutant removal effect can be achieved 44.53% quicker. While analyzing the removal of phenol red from the solution, it was observed that for S = 800 m 2 and H 2 O 2 = 1600 mg, the 90% removal can be obtained 15.24% faster than in the system without an oxidizer. In turn, for S = 900 m 2 and H 2 O 2 = 1600 mg, 90% phenol red removal is 11.60% quicker than without the oxidizing compound.
On the basis of the performed analyses, the η t(3) = f(S, De hl , H 2 O 2 ) curve was prepared for crystal violet (Figure 4d) and phenol red (Figure 3d). The curve obtained in this way reflects the relative extension of the pollutant removal time enabling to achieve the assumed removal degree. Thus, it can be stated that an increase in De hl decreases the dynamics of pollutant removal. For example, for S = 850 m 2 the increase in De hl by 4.5-4.7 extends the time required for 90% crystal violet removal by 17.58%. In turn, for S = 850 m 2 the increase in De hl by 4.5-4.7 extends the 90% phenol red removal time by 7.96%. The obtained curves (Figures 3d and 4d) also confirm that specific surface of activated carbon and oxidizer dose constitute key factors affecting the removal of pollutants from a solution in a simultaneous process. This is in line with the findings of Faria et al. [26], who indicated that activated carbon is characterized not only by high adsorption capacity, but also the catalytic properties, owing to its high specific surface area.     This translates into a shortening of the pollutant removal time required to achieve the assumed reduction degree. For instance, inclusion of 800 mg H2O2 in the reaction system (S = 850 m 2 and Dehl = 4.5) shortens the pollutant removal time (phenol red) by 7% in comparison to the system without an oxidizer (Figure 3c). Assuming S = 850 m 2 , introduction of H2O2 = 1600 mg into the system shortens the dye removal time (90% elimination) by 33.5% compared to the non-modified process. This is in line with the results obtained by Santos et al. [37], who indicated that a synergistic effect occurs in the case of simultaneous sorption onto activated carbon and oxidation via H2O2, which enhances the solution decolorization.
While analyzing the obtained ηt(2) curves, it was observed that the lower the specific surface of activated carbon (S), the greater the influence of oxidizer on the pollutant removal process. For example, in the case of S = 800 m 2 and H2O2 = 1600 mg, 90% crystal violet removal can be achieved 57.0% faster than in the system without this compound. In turn, for S = 900 m 2 , the same pollutant removal effect can be achieved 44.53% quicker. While analyzing the removal of phenol red from the solution, it was observed that for S = 800 m 2 and H2O2 = 1600 mg, the 90% removal can be obtained 15.24% faster than in the system without an oxidizer. In turn, for S = 900 m 2 and H2O2 = 1600 mg, 90% phenol red removal is 11.60% quicker than without the oxidizing compound.
On the basis of the performed analyses, the ηt(3) = f(S, Dehl, H2O2) curve was prepared for crystal violet (Figure 4d) and phenol red (Figure 3d). The curve obtained in this way reflects the relative extension of the pollutant removal time enabling to achieve the assumed removal degree. Thus, it can be stated that an increase in Dehl decreases the dynamics of pollutant removal. For example, for S = 850 m 2 the increase in Dehl by 4.5-4.7 extends the time required for 90% crystal violet removal by 17.58%. In turn, for S = 850 m 2 the increase in Dehl by 4.5-4.7 extends the 90% phenol red removal time by 7.96%. The obtained curves (Figures 3d and 4d) also confirm that specific surface of activated carbon and oxidizer dose constitute key factors affecting the removal of pollutants from a solution in a simultaneous process. This is in line with the findings of Faria et al. [26], who indicated that

Multidimensional Scaling (MDS)
The visualization of similarity levels for the matrix containing the data connected with the properties of the applied activated carbons with H 2 O 2 and the kinetic parameters for the removal of dyes based on the multidimensional scaling (MDS) method was presented in Figure 5.
There is a visible similarity between the fresh activated carbons WG-12 and WDex and between the spent carbons F-200S and F-200R. Taking into account the amount of added H 2 O 2 oxidizer, it can be seen that the points in the graph for particular carbons move closer to each other as the dose increases, and the increasing similarity of the spent carbon to the general characteristics describing the fresh carbon is much more evident in the case of F-200R. Another similarity presented graphically using Multidimensional Scaling, is that the highest applied oxidizer dose (7500 mg H 2 O 2 /L) causes that the similarities between the analyzed results of dye removal by all employed carbons (WDex, WG12, F-200S, F-200R) are much greater (the points are spread apart) than in the case of the similarity between the employed carbons with a high oxidizer dose and without oxidizer. This is most evident on the example of relatively close proximity of the points corresponding to WDex 7500 and F 200S 7500 compared to the long distance between WDex 7500 and WDex 0. The distance between F-200R 7500 and F-200R 0 is even greater, whereas the point marked as F-200R 7500 lies closer to the points reflecting the process realized using fresh carbons with the greatest oxidizer dose. There is a visible similarity between the fresh activated carbons WG-12 and WDex and between the spent carbons F-200S and F-200R. Taking into account the amount of added H2O2 oxidizer, it can be seen that the points in the graph for particular carbons move closer to each other as the dose increases, and the increasing similarity of the spent carbon to the general characteristics describing the fresh carbon is much more evident in the case of F-200R. Another similarity presented graphically using Multidimensional Scaling, is that the highest applied oxidizer dose (7500 mg H2O2/L) causes that the similarities between the analyzed results of dye removal by all employed carbons (WDex, WG12, F-200S, F-200R) are much greater (the points are spread apart) than in the case of the similarity between the employed carbons with a high oxidizer dose and without oxidizer. This is most evident on the example of relatively close proximity of the points corresponding to WDex 7500 and F 200S 7500 compared to the long distance between WDex 7500 and WDex 0. The distance between F-200R 7500 and F-200R 0 is even greater, whereas the point marked as F-200R 7500 lies closer to the points reflecting the process realized using fresh carbons with the greatest oxidizer dose. This is in line with the results of the conducted analytic studies [36], which showed that-similarly as it the presence of fresh activated carbons-following the introduction of the activated carbon F-200S (with the lowest sorption capacity) to the system comprising dye-hydrogen peroxide, the solution decolorization is accelerated, compared to the results obtained through sorption. However, these changes are not as significant as those obtained in the presence of regenerated activated carbon F-200R, where complete solution decolorization was achieved already after 120 min, at the hydrogen peroxide concentration of 3740 mgH2O2/L, similarly as in the case of WDex. This is in line with the results of the conducted analytic studies [36], which showed that-similarly as it the presence of fresh activated carbons-following the introduction of the activated carbon F-200S (with the lowest sorption capacity) to the system comprising dye-hydrogen peroxide, the solution decolorization is accelerated, compared to the results obtained through sorption. However, these changes are not as significant as those obtained in the presence of regenerated activated carbon F-200R, where complete solution decolorization was achieved already after 120 min, at the hydrogen peroxide concentration of 3740 mgH 2 O 2 /L, similarly as in the case of WDex.

Conclusions
The obtained relations between the parameters in the Freudlich and Langmuir models confirm in most cases that it is possible to develop universal models in which the kinetics of the sorption process is estimated based on the characteristics of the carbons. The conducted analyses show that there is a strong correlation between the pseudo-first and second-order kinetic parameters, and the dose of the oxidizer (H 2 O 2 ) as well as the characteristics of the carbons: specific surface area (S), methylene number (MBN), dechlorination half-length (De hl ). The conducted laboratory studies showed the key influence of the surface area (S) of active carbon and the dose of oxidizer (H 2 O 2 ) on the course of the pollutant removal process in the aqueous environment. For example, an increase in the S value from 750 m 2 to 850 m 2 (phenol red) assuming a dose of H 2 O 2 = 800 mg leads to a decrease in the removal time (required removal ratio of 90%) by 29.10%. Increasing the oxidizer dose to the value of H 2 O 2 = 3200 mg indicates that an increase in S in the range of 750-800 m 2 leads to a decrease of the minimum time required to removal the dye by 24.01% (removal ratio 90%). The addition of an oxidizer into the system leads to an increase in the value of the constants rate. For De hl = 4.5 and S = 850 m 2 , the constant rate is k 2 = 0.0105 (phenol red) k 2 = 0.011 (crystal violet). The addition of the oxidizer in the amount of H 2 O 2 = 1600 mg leads to an increase in their values to k 2 = 0.012 and k 2 = 0.023. This leads to a required degree of removal of pollutants within a shortened time. It was also found that the smaller the surface of the carbon (S), the greater the influence of the oxidizer on the dynamics of the process of dye removal from the aqueous solution. For example, in the case of S = 800 m 2 and H 2 O 2 = 1600 mg, 90% dye removal (crystal violet) can be achieved with time shortened by 57% (compared to the system without oxidizer). In turn, for S = 900 m 2 , the same effect of dye removal can be obtained in time shortened by 44.53%. Dechlorination half-length (De hl ) also has a significant impact on the dynamics of the process. The increase in the De hl value leads to a decrease in the rate of pollution removal. For example, for S = 850 m 2 , the increase in De hl from 4.5 to 4.7 leads to an extension of the removal time of crystal violet from aqueous solution by 17.58% (for removal ratio 90%). For S = 850 m 2 , the increase in De hl from 4.5 to 4.7 results in an extension of the removal time of phenol red from solution by 7.96% (removal ratio 90%). The aforementioned correlations give the assumption for further modelling and process optimization.
The visualization of the similarity levels for the matrix containing the data related to the properties of the applied activated carbons together with oxidizer as well as the kinetic parameters of dye removal based on the multidimensional scaling method (MDS), indicates that the similarity of the spend carbons to the general characteristics describing fresh carbons increases with the oxidizer dose. The highest dose of the oxidizer used (7500 mg H 2 O 2 /L cause that the similarities between the analyzed results of dye removal by all the carbons used (WDex, WG12, F-200S, F-200R) are much greater than in the case of the similarity between the same applied carbons with a high dose oxidizer and no oxidizer.
The developed models allow for analyzing the influence of the specific surface of activated carbon and applied different doses of the oxidizer on the change of the time of pollutants removal from aqueous solution at the required degree. The above-mentioned models also allow for analyzing the influence of the oxidizer dose increase for given surfaces of activated carbons on the decrease of the time of pollutants removal from the aqueous solution until the assumed degree achieved. The models can be also applied for determination of the impact of the change in the De hl value on the time of pollutants removal for the assumed S values and the oxidizer dose.
The models developed on the basis of the analysis of the correlation between the kinetic parameters of the pollutant removal from the aqueous solution and the characteristics of the carbons together with the doses of the oxidizer used, are universal in nature. They allow for the prediction of simultaneous sorption and oxidation processes, taking into account the diversity of physicochemical properties of the sorbent, various adsorbates as well as the type and dose of the oxidizer. Thus, it will enable to optimize the selection of a sorbent and oxidizer used simultaneously for a given group of pollutants which allows to avoid the costly and time-consuming laboratory tests in order to adapt the process to the changing technological parameters.