Linear and non-linear regression analysis for the adsorption kinetics of SO 2 in a fixed carbon bed reactor – a case study

: Kinetic parameters of SO 2 adsorption on unburned carbons from lignite fly ash and activated carbons based on hard coal dust were determined. The model studies were performed using the linear and non-linear regression method for the following models: pseudo first and second-or-der, intraparticle diffusion, and chemisorption on a heterogeneous surface. The quality of the fitting of a given model to empirical data was assessed based on: R 2 , R , Δq , SSE , ARE , χ 2 , HYBRID , MPSD , EABS , and SNE . It was clearly shown that it is the linear regression that more accurately reflects the behaviour of the adsorption system, which is consistent with the first-order kinetic reaction – for activated carbons (SO 2 +Ar) or chemisorption on a heterogeneous surface – for unburned carbons (SO 2 +Ar and SO 2 +Ar+H 2 O (g) +O 2 ) and activated carbons (SO 2 +Ar+H 2 O (g) +O 2 ). Importantly, usually, each of the approaches (linear/non-linear) indicated a different mechanism of the studied phenomenon. A certain universality of the χ 2 and HYBRID functions has been proved, the minimization of which repeatedly led to the lowest SNE values for the indicated models. Fitting data by any of the non-linear equations based on the R or R 2 functions only, cannot be treated as evidence/prerequisite of the existence of a given adsorption mechanism.


Introduction
The structure of fuel consumption in Poland, based on hard coal and lignite, makes the energy sector one of the main sources of pollutants emitted into the air. According to the information presented in the report of the National Center for Balancing and Emission Management in Warsaw, in 2015-2017 the commercial power industry was responsible for 43-52% of the national SO2 emissions [1]. In the EU countries, on the other hand, the emission of sulfur oxides (total) from the sector of Thermal power plants and other combustion installations, in 2014 accounted for 66.9% of the total emissions from all installations covered by the provisions of the Directive on the Establishment of the European Pollutant Release and Transfer Register (E-PRTR) [2].
Due to the fast and unlimited spread of pollutants and direct impact on the natural environment, a significant tightening of emission standards for air pollutants is observed. Pursuant to EU regulations, emission limits of up to 200 mg SO2·Nm -3 have been in force since 2016, and according to the projections developed in 2019, the national commitment to reduce emissions in the period 2020-2029 and from 2030 was set at 59% and 70%, respectively, compared to the emissions recorded in 2005 [3].
In the light of the information presented in the literature, the least invasive method that does not interfere with the combustion process is the capture of pollutants after the combustion process (i.e. post-combustion capture of pollutants). One of the solutions presented in the literature is an innovative technology for the use of unburned carbon from

Materials
The subjects of the research presented in this paper are selected fractions of unburned carbon recovered from lignite fly ash, resulting from the nominal operation of the pulverized carbon boiler BB-1150 in Bełchatów Power Plant (370 MW unit). Unburned carbon along with fly ash was collected with the use of demonstration installation from the ash hoppers located under the second pas chamber and rotary air heater (more in [29]). The combustible parts have been separated by a mechanical classification system with a capacity of 500 kg·h -1 into three grain classes: ~0.8 mm and 57.3% (marked UnCarb_HAsh), ~1.0 mm and 44.6% (marked UnCarb_MAsh), and ~1.5 mm and 12.8% (marked Un-Carb_LAsh). The commercial activated carbons AKP-5 and AKP-5/A were used as reference materials, manufactured and distributed by GRYFSKAND Sp. z o.o., Hajnówka Branch, Active carbon Production Plant (more in [30]). Both products were developed for the treatment of industrial gases, boiler flue gases in power plants, or waste incineration plants, including sulfur dioxide, nitrogen oxides, hydrogen chloride or dioxins, and furans.

Experimental studies
The model tests were carried out on the results of laboratory tests for SO2 adsorption on a fixed carbon bed, which were the subject of one of the author's earlier works, published in [30]. The experiments were carried out at a temperature of 120 °C, in the presence of gas mixtures flowing linearly through 0.173 dm 3 of the bed and with the following composition: 1. 5% (v/v) of sulfur dioxide and 95% (v/v) of argon (as a carrier gas) and a volumetric flow rate of 2 l•min -1 ; 2. 2.5% (v/v) of sulfur dioxide, 11% (v/v) of water vapor, 20% (v/v) of oxygen and 66.5% (v/v) of argon (as a carrier gas) and a volumetric flow rate of 2.05 l•min -1 . Measurements were made on a fixed-bed reactor (Figure 1), which enabled the assessment of both the degree and dynamics of the adsorption process. The water vapor was generated using Ar from a bubbling container that was bathed in 60.5 ± 0.1 °C water, and the relative humidity was controlled using the Ar flow based on the water vapor Antonio equation. The gas flow line to the reactor was maintained at an elevated temperature (120 °C) to prevent condensation. The final concentration of sulfur in the solid phase was used to assess the effectiveness of sulfur dioxide adsorption, which was carried out in accordance with the PN-EN 04584:2001 standard while correcting this value by the share of the so-called fuel sulfur: m S, t =m S,∞ -m S,0 (1) where: mS,t is the mass of adsorbed sulfur, mg; mS,∞ is the total mass of sulfur in the sample after the adsorption process, mg; mS,0 represents the mass of sulfur in the sample before the adsorption process, mg.
Due to the possibility of adsorption of various forms of sulfur dioxide and the occurrence of indirect chemical reactions, as a consequence of the presence of O2 and H2O(g) in the reaction system, no comparative analyzes were performed for the participation of sulfur dioxide in the solid phase.

Reaction kinetics models
Processes carried out in the environment of SO2+Ar gases (UnCarb_HAsh, Un-Carb_MAsh, UnCarb_LAsh, AKP-5, and AKP-5/A samples) and SO2+O2+H2O(g)+Ar (Un-Carb_LAsh and AKP-5/A samples) were subjected to model tests. For this study, four models were chosen [31][32][33][34][35] chemisorption on a heterogeneous surface called the Elovich or Roginski-Zeldowicz model, which were verified by means of linear regression determined with the use of the least squares method and non-linear regression determined with the use of a numerical algorithm solved by means of the Solver in MS Excel.

Pseudo first-order kinetic model (PFO)
The pseudo-first-order kinetic model, hereinafter referred to as model 1, makes the adsorption rate of sulfur dioxide/oxidized forms of sulfur dioxide (dmS,t·dt -1 , g·kg -1 min -1 ) dependent on the reaction rate constant k1 (min -1 ) and the difference in adsorbate mass after time t (mS,t, g·kg -1 ) and ∞ (mS,∞, g·kg -1 ), according to the relationship: The mS,∞ value was determined experimentally by washing the adsorbent bed with the gas mixture for 1, 5, 15, and 30 minutes. In order to determine the rate constant k1 (min -1 ), the relationship (2) was integrated with the range from 0 to mS,∞, obtaining a linear equation: which was then presented in semi-logarithmic coordinates (t, ln(mS,∞-mS,t)) so that the parameter k1 corresponds to the slope a, according to the relationship a=-k1. Integrating the differential equation (2) with the above boundary conditions also gave a non-linearized function:

Pseudo second-order kinetic model (PSO)
The pseudo second-order kinetic model hereinafter referred to as model 2, assumes that the adsorption rate changes depending on the constant k2 (kg·g -1 ·min -1 ) and the square of the adsorbate mass difference over time t and ∞, according to the equation: the integration of which in the range from 0 (for t=0) to mS,∞ (for t=t), allowed to obtain the relationship: The value of the total adsorbate mass (after time ∞) mS,∞, was not determined experimentally (as was the case for model 1), but it was determined together with the rate constant k2, based on the slope of the line (6) and the intercept in the system coordinates with a linear scale (t, t·mS,t -1 ). Integrating the differential equation (5) with the above boundary conditions also gave a non-linearized function:

Model of intraparticle diffusion
The intraparticle diffusion model, hereinafter referred to as model 3, assumes that the amount of adsorbed sulfur dioxide/oxidized forms of sulfur dioxide at time t can be written by a simple equation: where: the kid coefficient is called the intraparticle diffusion rate constant (g·kg -1 ·min -0.5 ), and C (g·kg -1 ) is the thickness of the layer, called the thickness. If the only factor determining the speed of the process is intramolecular diffusion, then the linear relationship of q(t) to time t 1/2 should be a straight line with a slope coefficient kid and going through the zero intercept, i.e. C=0. However, the deviation from linearity indicates the existence of other factors limiting the rate of the adsorption process, such as: surface diffusion, diffusion of the boundary layer, gradual adsorption in the adsorbent pores, and adsorption on the active sites of the adsorbent [26].

Model of chemisorption on a heterogeneous surface
The last of the applied models (model 4) was developed to describe the chemisorption on a heterogeneous surface. According to the Elovich equation, the adsorption rate of sulfur dioxide/oxidized forms of sulfur dioxide is described by the relationship: the integration of which in the range from 0 (for t=0) to mS,∞ (for t=t) allows to obtain the relationship: where: α is the initial adsorption rate (g·kg -1 min -1 ), and β is the Elovich constant, reflecting the degree of surface coverage and activation energy for chemisorption (kg·g -1 ). Presenting it in the system of semi-logarithmic coordinates (ln(t), mS,t) makes it possible to determine the parameters α and β based on the slope of the straight line and the intercept.
Integrating the differential equation (9) with the above boundary conditions also gave a non-linearized function:

Linear vs. non-linear approach
In order to determine the linear kinetic parameters, the equations presented in Chapter 2.3.1 were used, i.e. eq. (3) for model 1, eq. (6) for model 2, eq. (8) for model 3, and eq. (10) for model 4. The determined kinetic parameters made it possible to determine the curve which shows the course of the reaction as a function of time. On the basis of these curves, a model amount of adsorbed component was determined and compared with the values measured experimentally. The discrepancies between the model and experimental data were analyzed by comparing 9 statistical criteria (summarized in Table 1), i.e. the determination coefficient (R 2 ), the correlation coefficient (R), the relative standard deviation (Δq), sum squared error (SSE), average relative error (ARE), chi-square test (χ 2 ), hybrid fractional error function (HYBRID), Marquardt's percent standard deviation (MPSD), and the sum of absolute errors (EABS). Table 1. Statistic error functions [36,37].

Function Equation
Hybrid fractional error function (HYBRID) where: mS,t,mod is the model amount of adsorbate adsorbed by the adsorbent mass as a function of time (g·kg -1 ), mS,t,exp is the experimental amount of adsorbate adsorbed by the adsorbent mass as a function of time (g·kg -1 ), N is the number of experimental points and p is the number of parameters in a given mathematical model. The high data convergence is evidenced by the lowest possible value of the criteria: Δq, SSE, ARE, χ 2 , HYBRID, MPSD, EABS, and the highest possible values for the criteria: R 2 and R. The text continues here ( Figure 2 and Table 2). In order to determine the kinetic parameters via the linear method, the equations presented in chapter 2.3.1 were used, i.e. eq. (4) for model 1, eq. (7) for model 2, eq. (8) for model 3, and eq. (11) for model 4. For each data series, these equations were solved in 9 different variants, assuming the minimization of individual statistical criteria (collected in Table 1). To select the optimal variant for the best convergence of the model and experimental results, the criterion of the sum of normalized errors (SNE) was applied, which took into account the values of each statistical error, in accordance with the method described in [38,39]. The variant with minimal SNE error was considered to be the optimal non-linear variant. In order to compare the effectiveness of the linear and non-linear approach, Chapter 3.3 compares the values of 9 statistical error functions and model curves for the best linear variant with the selected, optimal non-linear variant (determined on the basis of the lowest SNE value).

Results and discussion
A detailed analysis of the adsorption capacity of unburned carbon from lignite fly ash and activated carbons based on hard coal dust in relation to SO2 was presented in the previous work by one of the authors [30]. Therefore, this paper focuses on the mathematical description, which enables a deeper understanding of the mechanism of the observed reactions and the selection of optimal conditions for the SO2 adsorption process.

Linear regression
The results of the model tests for linear regression are shown in Figure 2. As shown by the test results, the highest sorption capacity against sulfur dioxide is shown by unburned carbons UnCarb_MAsh and UnCarb_LAsh (Figures 2b, 2c). By mass, these materials adsorbed 28.90 and 28.95 g of S per kg of adsorbent, respectively. Among the selected materials, the lowest concentration of the active agent is characteristic of commercial activated carbons formed on the basis of hard coal dust. The mass of adsorbed sulfur dioxide for the AKP-5 and AKP-5/A samples is 41 and 32% lower than the least adsorbing unburned carbon (UnCarb_HAsh), for which 25.15 g S per kg of adsorbent was demonstrated. Additionally, due to the presence of oxygen and water vapor in the measurement system, the sorption capacity of the samples increased. The percentage of sulfur in the solid phase after the process increased 1.6 times for the UnCarb_LAsh material, while for commercial materials this value did not exceed 1.3 (Figures 2f, 2g). As can be observed, the reaction rate constants determined during the tests range from 0.214 min -1 (UnCarb_LAsh, SO2+Ar) to 0.423 min -1 (UnCarb_HAsh, SO2+Ar) for model 1 and from 0.0156 kg·g -1 ·min -1 (UnCarb_HAsh, SO2+Ar) up to 0.114 kg·g -1 ·min -1 (Un-Carb_LAsh, SO2+Ar) for model 2 ( Table 2). According to the theory, for both models, materials that quickly bind the adsorbate should be characterized by high reaction rates. However, in practice, the correlation between the values of k1 and k2 has not been confirmed. A model parameter of great practical importance is the amount of adsorbate related to the equilibrium conditions mS,∞ (for unlimited contact time). It is interesting that this coefficient, determined on the basis of model 2, reaches a value similar to that obtained experimentally (for a contact time of 30 minutes), and the discrepancies (averaged for all analyzes) do not exceed 3.5% (Figure 3). The calculations made for model 3 show that the values of the kid coefficient range from 2.81 AKP-5/A, SO2+Ar) to 7.77 g·kg -1 ·min -0.5 (UnCarb_LAsh, SO2+O2+H2O(g)+Ar), while parameter C varies from 1.76 (AKP-5/A, SO2+Ar) to 12.1 g·kg -1 (UnCarb_LAsh, SO2+O2+H2O(g)+Ar). In view of the information from [40], high C values and the low kid would indicate a role that the diffusion-controlled boundary layer could play. The reverse configuration of the discussed parameters would prove that the speed-limiting stage of the process was diffusion inside the pores of the solid phase surface. Nevertheless, as shown in Figure 2, the described model does not faithfully reflect the course of the reaction, which to some extent confirms the kinetic nature of the experiments performed.
The kinetic parameters determined for model 4 are theoretical and physicochemical interpretation is difficult. Moreover, as far as the author is aware, the literature lacks studies on the kinetics of SO2 adsorption on unburned carbons, which would make it possible to compare the obtained results. Table 3 presents the analysis of statistical errors in kinetic models solved by the linear regression method. The highlighted data (in colors and bold) indicate the most appropriate values for a given sample out of the four analyzed models.
In the case of the SO2+Ar mixture, for commercial samples of activated carbons, regardless of the statistical error function, the quality of the results suggests that SO2 adsorption is a first-order kinetic reaction. However, bearing in mind the considerations of Płaziński and Rudziński in [41,42], we should be cautious to hypothesize about a specific physical model of adsorption in the case of equation (3). There is a belief that the indicated equation is not able to reflect changes in the mechanism controlling the adsorption kinetics, and the adjustment of the model data to the experimental data, especially in the case of systems close to the equilibrium state, results rather from mathematical foundations.
In the case of the UnCarb_HAsh trial, inconsistency in the indication of error values was obtained. It is highly likely related to the heterogeneity of the sample (ash content 57.3% for UnCarb_HAsh, 44.6% for UnCarb_MAsh, 12.8% for the UnCarb_LAsh [30]). Nevertheless, as evidenced in Table 3, 5 (Δq, ARE, χ 2 , HYBRID, MPSD) out of 9 functions indicate that model 4 reflects the empirical data most accurately. The determination (R 2 ) and correlation (R) coefficients, as well as the sum squared error (SSE) indicate model 2; and the sum of absolute errors (EABS) -model 1. However, bearing in mind the information that in the case of the first and second-order models (models 1 and 2), the ability to fit data may result only from the mathematical properties of equations (3) and (6), and not from specific physical assumptions, the compliance of adsorption with the kinetic mechanism of chemisorption on a heterogeneous surface was adopted for further comparative analyzes (according to model 4).
In the case of the UnCarb_MAsh and UnCarb_LAsh trials, greater consistency of the statistical error values was obtained, and their quality indicates the importance of the chemisorption phenomenon. This confirms the observations described in [30] that even in the absence of molecular oxygen in the gas mixture, the interaction between the adsorbate molecules and the carbon material occurs both due to relatively weak intermolecular van der Waals forces (corresponding to physical adsorption), as well as the chemical binding of sulfur dioxide.
The change of the atmosphere into SO2+O2+H2O(g)+Ar indicates that the reliability of the analyzed models changes towards model 1 < model 3 < model 2 < model 4. These data, in line with the results of experimental research [30], also prove the formation of strong chemical bonds between the adsorbent and the adsorbate in the presence of oxygen and water vapor, thus indicating a strong inhomogeneity of the adsorbent surface.

Non-linear regression
In the case of describing sulfur dioxide adsorption by non-linear regression, the socalled sum of normalized errors (SNE) method was applied, allowing to select of the most appropriate error function used to optimize kinetic parameters. This method makes it possible to estimate the values that are not burdened with the error resulting from the use of only one type of function and enables the selection of the model that best describes the adsorption process. Figure 4 shows the distribution of the parameter of the sum of normalized errors for all tested samples. As can be seen, the SNE value determined for one data series varies greatly. Within a given model, it may even decrease twofold (e.g. for the UnCarb_HAsh trial and model 3: 8.81 in the case of minimizing the R 2 criterion and 4.39 in the case of minimizing the EABS criterion). Especially in the case of models 3 and 4, there is a correlation that minimization of the determination coefficient (R 2 ) and correlation (R) leads to high SNE values. This observation does not confirm the commonly used assumption that the models with R 2 > 0.7 describe the studied phenomena reliably [43,44]. It is therefore clear that fitting data by any of the non-linear equations based on the R or R 2 functions only, cannot be treated as evidence or prerequisite of the existence of a mechanism that determines the kinetics or dynamics of adsorption in a given system. Notwithstanding the fact that it is quite common in the literature to use them as a basis for the assessment of the quality of fitting kinetic data to experimental data [45][46][47]. Interestingly, the analyzes performed prove a certain universality of the χ 2 and HYBRID functions. As noted, in 15 out of 28 cases, the minimization of these functions led to the lowest SNE values for individual models (Table 4). For example, for the AKP-5 sample, HYBRID values in the range 5.60-6.28 were recorded -the lowest for models 1, 2, and 4; in the case of the AKP-5/A sample (SO2+Ar+H2O(g)+O2), the noted values of χ 2 were in the range 4.27-8.09 -the lowest for models 2, 3 and 4. Figure 4. SNE error analysis for kinetic models solved by non-linear regression method. Table 4 distinguishes the error functions used for non-linear regression (out of 9), for which the most appropriate values of the SNE function were obtained. These values served as a criterion for selecting an appropriate mathematical model for the discussed adsorption case. As can be seen, regardless of the tested sample and process conditions, in the case of models 1 and 2, the lowest SNE values were obtained by minimizing the complex fractional error function (HYBRID), and for models 3 and 4, by Marquardt's percentage standard deviation (MPSD). Interestingly, all the indicated values correspond to the SO2+Ar mixture. As a result of wetting and oxygenating the gas mixture, the functions of 9 statistical errors for each model generated higher SNE values.  (Tables 4, 5), at the level of the tested samples and process conditions, clearly indicates that under the conditions of the SO2+Ar mixture, in the case of commercial activated carbons and the unburned activated carbon UnCarb_MAsh sample, permanent bonding of sulfur dioxide could have occurred. Compatibility of adsorption with the Elovich equation (model 4) shows that the adsorption sites increased exponentially with the course of the process, which resulted in multilayer adsorption. Interestingly, for the UnCarb_HAsh and UnCarb_LAsh (SO2+Ar and SO2+Ar+H2O(g)+O2) and AKP-5/A (SO2+Ar+H2O(g)+O2) samples, diffusion in boundary layers or inside the pores of adsorbents (model 3) could have been the stage limiting the adsorption rate. Taking into account the high values of parameter C (od 8.17 do 24.3 g·kg -1 ) ( Table 5), it can be indicated that in the case of the UnCarb_LAsh and AKP-5/A samples, internal diffusion of sulfur dioxide dominated over the general adsorption kinetics. The phenomenon of diffusion in boundary layers should rather be noted for the Un-Carb_HAsh sample (C=0) ( Table 5), similar to the case [48].

Comparative analysis of linear and non-linear regression
In order to assess the validity of the description of the kinetics and dynamics of adsorption by means of linear or nonlinear regression, the values of statistical errors and model curves were compared for the models for which the smallest deviations from empirical data were recorded ( Figure 5, Table 6). As can be seen, for 6 out of 7 tested trials, the research clearly proves that it is the linear regression that more accurately reflects the behaviour of the adsorption system (regardless of the process conditions). What is particularly interesting, only for the UnCarb_MAsh sample, the method of linear and nonlinear fitting indicates the same mechanism of the studied phenomenon (model 4). Depending on the applied statistical error, the linear and nonlinear approaches may differ even several dozen times. For example, for the AKP-5/A (SO2+Ar+H2O(g)+O2) sample it was noted that the HYBRID error reached the value of 0.2 with linear regression and as much as 56 times more with non-linear regression (11.2).  What is also noteworthy, comparing the kinetic parameters from Table 2 for the linear regression method with the parameters from Table 5 for the non-linear regression method, it can be seen that the differences between them can be over 100%. As can be seen, the kid rate constant for the UnCarb_LAsh trial for the linear fit is 1.97 g·kg -1 ·min -0.5 , and for the non-linear fit it is as much as 4.47 g·kg -1 ·min -0.5 (the difference is 227%).

Conclusion
The aim of this article was to determine the parameters of the kinetics and dynamics of adsorption by linear and non-linear regression for the following models: the pseudo first-order (model 1) and pseudo second-order (model 2) models, intraparticle diffusion (model 3), and chemisorption on a heterogeneous surface (model 4). The quality of fitting the model data to the experimental data was analyzed based on 9 statistical error functions (R, R 2 , Δq, SSE, ARE, χ 2 , HYBRID, MPSD, EABS) and, in the case of non-linear regression, the normalized error sum (SNE) method. The performed measurements and analyzes lead to the conclusion that: -confronting 9 statistical error functions for the models indicated as the most reliable, for linear and non-linear regression, respectively, leads to an unequivocal conclusion that it is the linear regression that more accurately reflects the behaviour of the adsorption system (regardless of the process conditions); -in the case of the SO2+Ar mixture, for commercial samples of activated carbons AKP-5 and AKP-5/A, regardless of the statistical error function, the quality of the results suggests that SO2 adsorption is a first-order kinetic reaction (model 1). However, it should be noted that fitting model data to experimental data for the systems close to the equilibrium state can only result from the mathematical foundations of model 1; -in the case of unburned carbons samples (UnCarb_HAsh, UnCarb_MAsh, Un-Carb_LAsh), regardless of the process conditions, and the AKP-5/A (SO2+Ar+H2O(g)+O2) sample, the quality of the results shows that the adsorption is compatible with the kinetic mechanism of chemisorption on the heterogeneous surface (according to model 4); -the sum of normalized errors, regardless of the tested sample and process conditions, reaches the lowest values for models 1 and 2 by minimizing the hybrid fractional error function (HYBRID), and for models 3 and 4 by the Marquardt's percentage standard deviation (MPSD); -minimization of the determination coefficient (R 2 ) and correlation (R) leads to high SNE values. Fitting data by any of the non-linear equations based on the R or R 2 functions only cannot be treated as evidence or a prerequisite of the existence of a given mechanism determining the kinetics or dynamics of adsorption in a given system.
-only in 1 case (UnCarb_MAsh) out of 7 possible, both linear and non-linear regression indicate the same mechanism of the adsorption phenomenon -identical to chemisorption on a heterogeneous surface (according to model 4).
The analysis presented above proves that linear methods generally enable the determination of kinetic parameters that reflect the character of adsorption more reliably than non-linear methods, although it is puzzling that usually each of the approaches indicates a different mechanism of the phenomenon. Hence, in order to determine the optimal set of kinetic pairs as faithfully reproducing the course of the analyzed processes as possible, it is recommended to perform both linear and non-linear regression, in accordance with the methodology presented in this paper. Moreover, the assessment of the mechanism of the adsorption reaction based solely on the accuracy of the kinetic model may be misleading and, in the opinion of the authors, requires additional discussion supported by experimental studies, as in the case of [30]. Taking into account the limited amount of data in the literature on SO2 adsorption on unburned carbon from lignite fly ash, the indicated work may be the first attempt at a thorough analysis of the chemical kinetics of this process, constituting the basis for considering the industrial application of the adsorption reaction.