#
Linear and Non-Linear Regression Analysis for the Adsorption Kinetics of SO_{2} in a Fixed Carbon Bed Reactor—A Case Study

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}adsorption on unburned carbons from lignite fly ash and activated carbons based on hard coal dust. The model studies were performed using the linear and non-linear regression method for the following models: pseudo first and second order, 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 the linear regression 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.

## 1. Introduction

_{2}emissions [1]. In 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].

_{2}·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].

_{2}emissions in installations in the energy production and transformation sector, adsorption on the surface of the carbon adsorbent turns out to be one of the most frequently analyzed solutions [10]. Despite the quite extensive variety of methods for removing sulfur dioxide from boiler flue gases [11,12], the practical significance of most of them is limited, and the research does not go beyond laboratory work.

_{2}adsorption kinetics by the method of linear and non-linear regression for the following models: pseudo first-order and pseudo second-order kinetic model, intraparticle diffusion, and chemisorption on a heterogeneous surface. The quality of the fitting of a given model to empirical data was assessed based on the following: determination coefficient (R

^{2}), correlation coefficient (R), 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), the sum of absolute errors (EABS), and the sum of normalized errors (SNE). The subject of research is selected fractions of unburned carbon recovered from lignite fly ash, created as a result of the nominal operation of the pulverized carbon boiler of a Polish power unit. Selected commercial activated carbons dedicated to industrial gas purification processes and traded on the domestic and foreign markets were used as reference materials.

## 2. Experimental Section

#### 2.1. Materials

^{−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 UnCarb_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. (Gryfino, Poland), 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.

#### 2.2. Experimental Studies

_{2}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 1.73 × 10

^{−4}m

^{3}of the bed and with the following composition (in volume concentration):

- 5% of sulfur dioxide and 95% of argon (as carrier gas) and a volumetric flow rate of 2 × 10
^{−3}m^{3}∙min^{−1}; - 2.5% of sulfur dioxide, 11% of water vapor, 20% of oxygen and 66.5% of argon (as carrier gas) and a volumetric flow rate of 2.05 × 10
^{−3}m^{3}∙min^{−1}.

_{S,t}is the mass of adsorbed sulfur, mg; m

_{S,∞}is the total mass of sulfur in the sample after the adsorption process, mg; m

_{S,0}represents the mass of sulfur in the sample before the adsorption process, mg.

_{2}and H

_{2}O

_{(g)}in the reaction system, no comparative analyzes were performed for the participation of sulfur dioxide in the solid phase.

#### 2.3. Modelling Studies

#### 2.3.1. Reaction Kinetics Models

_{2}+ Ar gases (UnCarb_HAsh, UnCarb_MAsh, UnCarb_LAsh, AKP-5, and AKP-5/A samples) and SO

_{2}+ O

_{2}+ H

_{2}O

_{(g)}+ Ar (UnCarb_LAsh and AKP-5/A samples) were subjected to model tests. For this study, four models were chosen [31,32,33,34,35], i.e.,:

- pseudo first-order kinetic model developed by Legergren,
- pseudo second-order kinetic model developed by Ho i McKaya,
- Weber-Morris intraparticle diffusion model, and
- chemisorption on a heterogeneous surface called the Elovich or Roginski-Zeldowicz model,

#### Pseudo First-Order Kinetic Model (PFO)

_{S,t}·dt

^{−1}, g·kg

^{−1}min

^{−1}) dependent on the reaction rate constant k

_{1}(min

^{−1}) and the difference in adsorbate mass after time t (m

_{S,t}, g·kg

^{−1}) and ∞ (m

_{S,∞}, g·kg

^{−1}), according to the relationship:

_{S,∞}value was determined experimentally by washing the adsorbent bed with the gas mixture for 1, 5, 15, and 30 min. In order to determine the rate constant k

_{1}(min

^{−1}), the relationship (2) was integrated with the range from 0 to m

_{S,∞}, obtaining a linear equation:

_{S,∞}−m

_{S,t})) so that the parameter k

_{1}corresponds to the slope a, according to the relationship a = −k

_{1}. Integrating the differential Equation (2) with the above boundary conditions also gave a non-linearized function:

#### Pseudo Second-Order Kinetic Model (PSO)

_{2}(kg·g

^{−1}·min

^{−1}) and the square of the adsorbate mass difference over time t and ∞, according to the equation:

_{S,}

_{∞}(for t = t), allowed to obtain the relationship:

_{S,∞}, was not determined experimentally (as was the case for model 1), but it was determined together with the rate constant k

_{2}, based on the slope of the line (6) and the intercept in the system coordinates with a linear scale (t, t·m

_{S,t}

^{−1}). Integrating the differential Equation (5) with the above boundary conditions also gave a non-linearized function:

#### Model of Intraparticle Diffusion

_{id}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 k

_{id}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

_{S,∞}(for t = t) allows for the obtainment of the relationship:

^{−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), m

_{S,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:

#### 2.3.2. Linear vs. Non-Linear Approach

^{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):

_{S,t,mod}is the model amount of adsorbate adsorbed by the adsorbent mass as a function of time (g·kg

^{−1}), m

_{S,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, and EABS, and the highest possible values for the criteria, R

^{2}and R (Figure 2 and Table 2).

## 3. Results and Discussion

_{2}was presented in the previous work by one of the authors [30]. This work also includes the characterization of the porous structure and the quantitative and qualitative analysis of surface oxygen functional groups, the key to the efficiency of the sulphur dioxide binding process. Therefore, this paper focuses on the mathematical description, which enables a deeper understanding of the mechanism of the observed reactions and to identify the optimum way to predict the behaviour of unburned carbons.

#### 3.1. Linear Regression

^{−1}(AKP-5/A, SO

_{2}+ Ar + H

_{2}O

_{(g)}+ O

_{2}) to 0.423 min

^{−1}(UnCarb_HAsh, SO

_{2}+ Ar) for model 1 and from 0.0156 kg·g

^{−1}·min

^{−1}(UnCarb_HAsh, SO

_{2}+ Ar) up to 0.114 kg·g

^{−1}·min

^{−1}(UnCarb_LAsh, SO

_{2}+ 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 k

_{1}and k

_{2}has not been confirmed. Interestingly, the calculations made for model 1 show a reduction in the rate of the adsorption process in the presence of H

_{2}O

_{(g)}and O

_{2}(0.123–0.155 min

^{−1}under SO

_{2}+ Ar + H

_{2}O

_{(g)}+ O

_{2}vs. 0.214–0.423 min

^{−1}under SO

_{2}+ Ar).

_{S,∞}(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 min), and the discrepancies (averaged for all analyzes) do not exceed 3.5% (Figure 3).

_{id}coefficient range from 2.81 AKP-5/A, SO

_{2}+ Ar) to 7.77 g·kg

^{−1}·min

^{−0.5}(UnCarb_LAsh, SO

_{2}+ O

_{2}+ H

_{2}O

_{(g)}+ Ar), while parameter C varies from 1.76 (AKP-5/A, SO

_{2}+ Ar) to 12.1 g·kg

^{−1}(UnCarb_LAsh, SO

_{2}+ O

_{2}+ H

_{2}O

_{(g)}+ Ar) (Table 2). In view of the information from [40], high C values and the low k

_{id}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. However, it is interesting that the addition of H

_{2}O

_{(g)}and O

_{2}to the gas mixture significantly increased the C value (4.13 and 12.1 g∙kg

^{−1}vs. 1.76 and 9.56 g∙kg

^{−1}, respectively, for the AKP-5/A and UnCarb_LAsh tests). Considering the information presented above, it should be assumed that under the conditions of the SO

_{2}+ Ar + H

_{2}O

_{(g)}+ O

_{2}mixture, the boundary layer effect in the SO

_{2}adsorption process will be greater.

_{2}adsorption on unburned carbons, which would make it possible to compare the obtained results. Interestingly, the registered change in kinetic parameters for the addition of H

_{2}O

_{(g)}and O

_{2}to the gas mixture would indicate a change in the kinetics of SO

_{2}adsorption. In the case of the UnCarb_LAsh test, a decrease was noted in both the value of the reaction rate constant α and the degree of surface coverage with the β adsorbate (0.515 g∙kg

^{−1}min

^{−1}and 0.160 kg∙g

^{−1}vs. 5917 g∙kg

^{−1}min

^{−1}and 0.370 kg∙g

^{−1}); for the AKP-5/A sample, the intensification of each of them (62.6 g·kg

^{−1}min

^{−1}and 0.285 kg·g

^{−1}against 8.24 g·kg

^{−1}min

^{−1}and 0.206 kg·g

^{−1}) (Table 2).

_{2}+ Ar mixture, for commercial samples of activated carbons, regardless of the statistical error function, the quality of the results suggests that SO

_{2}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.

^{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).

_{2}+ O

_{2}+ H

_{2}O

_{(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.

#### 3.2. Non-Linear Regression

^{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 analyses were performed to 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 (SO

_{2}+ Ar + H

_{2}O

_{(g)}+ O

_{2}), the noted values of χ

^{2}were in the range 4.27–8.09—the lowest for models 2, 3, and 4.

_{2}+ 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.

_{2}+ 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 (SO

_{2}+ Ar and SO

_{2}+ Ar + H

_{2}O

_{(g)}+ O

_{2}) and AKP-5/A (SO

_{2}+ Ar + H

_{2}O

_{(g)}+ O

_{2}) samples, diffusion in boundary layers or inside the pores of adsorbents (model 3) could have been the stage limiting the adsorption rate. Considering 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 external diffusion should rather be noted for the UnCarb_HAsh sample (C = 0) (Table 5), similar to the case [48].

_{2}O

_{(g)}and O

_{2}to the gas mixture, the kinetic parameters of SO

_{2}adsorption have changed (Table 5). Analogously to the linear regression method (Table 2), for both samples (UnCarb_LAsh and AKP-5/A) an increase in the boundary layer effect was noted (in accordance with C). Moreover, for the AKP-5/A test, the rate of SO

_{2}adsorption under the SO

_{2}+ Ar + H

_{2}O

_{(g)}+ O

_{2}mixture was intensified (in accordance k

_{1}, k

_{2}and α).

#### 3.3. Comparative Analysis of Linear and Non-Linear Regression

_{2}+ Ar + H

_{2}O

_{(g)}+ O

_{2}) 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).

_{id}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%).

## 4. Conclusions

^{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 was the most reliable for linear and non-linear regression, respectively, leading 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 SO
_{2}+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 SO_{2}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, UnCarb_LAsh), regardless of the process conditions, and the AKP-5/A (SO
_{2}+ Ar + H_{2}O_{(g)}+ O_{2}) 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).

_{2}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.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Summary of model and experimental curves for linear regression for the following mixtures: (

**a**–

**e**) SO

_{2}+ Ar, (

**f**,

**g**) SO

_{2}+ O

_{2}+ H

_{2}O

_{(g)}+ Ar.

**Figure 4.**SNE error analysis for kinetic models solved by non-linear regression method for the following mixtures: (

**a**–

**e**) SO

_{2}+ Ar, (

**f**,

**g**) SO

_{2}+ O

_{2}+ H

_{2}O

_{(g)}+ Ar.

**Figure 5.**Summary of model and experimental curves for linear and non-linear regression for the following mixtures: (

**a**–

**e**) SO

_{2}+ Ar, (

**f**,

**g**) SO

_{2}+ O

_{2}+ H

_{2}O

_{(g)}+ Ar.

Function | Equation | |
---|---|---|

Determination coefficient (R^{2}) | ${\mathrm{R}}^{2}=\frac{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{{(\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{mod}}-\overline{{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}})}^{2}}{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{{(\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{mod}}-\overline{{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}})}^{2}+{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{{(\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{mod}}-{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}})}^{2}}$ | (12) |

Correlation coefficient (R) | $\sqrt{{\mathrm{R}}^{2}}=\mathrm{R}$ | (13) |

Relative standard deviation (Δq) | $\Delta \mathrm{q}=\sqrt{\frac{{{\displaystyle \sum}}_{\mathrm{i}=1}^{\mathrm{n}}{\left(\frac{{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}-{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{mod}}}{{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}}\right)}^{2}}{\mathrm{N}-1}}$ | (14) |

Sum of squared deviations (SSE) | $\mathrm{SSE}\text{}=\text{}{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}{\left({\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}-{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{mod}}\right)}^{2}$ | (15) |

Average Relative Error (ARE) | $\mathrm{ARE}=\frac{100}{\mathrm{N}}{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}|\frac{{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}-{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{mod}}}{{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}}|$ | (16) |

Chi-square test (χ^{2}) | ${\mathsf{\chi}}^{2}={\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}\frac{{{(\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}-{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{mod}})}^{2}}{{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}}$ | (17) |

Hybrid fractional error function (HYBRID) | $\mathrm{HYBRID}=\frac{100}{\mathrm{N}-\mathrm{p}}{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}\frac{{{(\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}-{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{mod}})}^{2}}{{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}}$ | (18) |

Marquardt’s percent standard deviation (MPSD) | $\mathrm{MPSD}=100\sqrt{\frac{1}{\mathrm{N}-\mathrm{p}}{\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}{\left(\frac{{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}-{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{mod}}}{{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}}\right)}^{2}}$ | (19) |

Sum of absolute errors (EABS) | $\mathrm{EABS}={\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}|{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{exp}}-{\mathrm{m}}_{\mathrm{S},\mathrm{t},\mathrm{mod}}|$ | (20) |

Sample | Model 1 | Model 2 | Model 3 | Model 4 | |||
---|---|---|---|---|---|---|---|

k_{1} | k_{2} | m_{S,∞} | k_{id} | C | α | β | |

min^{−1} | kg·g^{−1}·min^{−1} | g·kg^{−1} | g·kg^{−1}·min^{−0.5} | g·kg^{−1} | g·kg^{−1}min^{−1} | kg·g^{−1} | |

SO_{2} + Ar | |||||||

UnCarb_HAsh | 0.423 | 0.0156 | 27.5 | 5.01 | 2.33 | 17.2 | 0.156 |

UnCarb_MAsh | 0.247 | 0.0527 | 29.5 | 5.03 | 6.56 | 107 | 0.210 |

UnCarb_LAsh | 0.214 | 0.114 | 29.2 | 4.47 | 9.56 | 5917 | 0.370 |

AKP-5 | 0.286 | 0.0449 | 15.6 | 2.81 | 1.94 | 7.92 | 0.244 |

AKP-5/A | 0.222 | 0.0273 | 18.2 | 3.30 | 1.76 | 8.24 | 0.206 |

SO_{2} + O_{2} + H_{2}O_{(g)} + Ar | |||||||

UnCarb_LAsh | 0.155 | 0.0293 | 48.3 | 7.77 | 12.1 | 515 | 0.160 |

AKP-5/A | 0.123 | 0.0363 | 22.5 | 3.73 | 4.13 | 62.6 | 0.285 |

Sample | R^{2} | R | Δq | SSE | ARE | χ^{2} | HYBRID | MPSD | EABS |
---|---|---|---|---|---|---|---|---|---|

Model 1 ^{1} | |||||||||

UnCarb_HAsh | 0.955 | 0.977 | 54.9 | 24.6 | 23.9 | 5.16 | 172 | 63.4 | 6.55 |

UnCarb_MAsh | 0.906 | 0.952 | 27.6 | 73.6 | 14.5 | 4.51 | 150 | 31.8 | 12.2 |

UnCarb_LAsh | 0.745 | 0.863 | 38.9 | 268 | 20.8 | 12.6 | 421 | 45.0 | 22.9 |

AKP-5 | 0.998 | 0.999 | 3.72 | 0.346 | 2.10 | 0.0383 | 1.28 | 4.29 | 0.776 |

AKP-5/A | 0.979 | 0.989 | 8.39 | 5.05 | 3.98 | 0.373 | 12.4 | 9.68 | 2.38 |

Model 2 ^{1} | |||||||||

UnCarb_HAsh | 0.958 | 0.979 | 49.9 | 20.7 | 22.6 | 4.27 | 142 | 57.6 | 7.15 |

UnCarb_MAsh | 0.958 | 0.979 | 19.9 | 26.2 | 8.06 | 2.04 | 67.9 | 23.0 | 5.23 |

UnCarb_LAsh | 0.987 | 0.994 | 7.04 | 7.77 | 3.28 | 0.392 | 13.1 | 8.13 | 3.42 |

AKP-5 | 0.961 | 0.980 | 31.1 | 6.52 | 14.0 | 1.62 | 54.1 | 37.0 | 3.27 |

AKP-5/A | 0.962 | 0.981 | 42.4 | 8.24 | 18.4 | 2.38 | 79.5 | 48.9 | 3.84 |

Model 3 ^{1} | |||||||||

UnCarb_HAsh | 0.840 | 0.917 | 43.2 | 92.0 | 28.3 | 5.91 | 197 | 49.9 | 20.1 |

UnCarb_MAsh | 0.771 | 0.878 | 19.3 | 144.8 | 13.5 | 3.85 | 128 | 22.2 | 23.5 |

UnCarb_LAsh | 0.647 | 0.804 | 22.1 | 211 | 15.9 | 4.77 | 159 | 25.5 | 29.4 |

AKP-5 | 0.851 | 0.922 | 21.3 | 26.8 | 16.2 | 1.91 | 63.8 | 24.6 | 10.7 |

AKP-5/A | 0.847 | 0.920 | 33.3 | 38.1 | 23.1 | 3.14 | 105 | 38.5 | 12.7 |

Model 4 ^{1} | |||||||||

UnCarb_HAsh | 0.952 | 0.976 | 28.3 | 27.8 | 17.4 | 2.17 | 72.5 | 32.7 | 10.1 |

UnCarb_MAsh | 0.966 | 0.983 | 11.0 | 21.5 | 7.70 | 0.961 | 32.0 | 12.7 | 8.24 |

UnCarb_LAsh | 0.988 | 0.994 | 5.20 | 6.99 | 3.50 | 0.272 | 9.07 | 6.01 | 4.40 |

AKP-5 | 0.945 | 0.972 | 20.0 | 11.8 | 14.0 | 1.18 | 39.3 | 23.1 | 6.36 |

AKP-5/A | 0.944 | 0.972 | 17.4 | 16.0 | 12.4 | 1.22 | 40.6 | 20.1 | 6.74 |

Model 1 ^{2} | |||||||||

UnCarb_LAsh | 0.776 | 0.881 | 41.1 | 587.4 | 22.7 | 19.6 | 653 | 47.5 | 35.6 |

AKP-5/A | 0.824 | 0.908 | 40.8 | 86.3 | 22.8 | 7.42 | 247 | 47.1 | 14.1 |

Model 2 ^{2} | |||||||||

UnCarb_LAsh | 0.992 | 0.996 | 5.17 | 12.4 | 3.69 | 0.355 | 11.8 | 5.97 | 6.59 |

AKP-5/A | 0.982 | 0.991 | 7.33 | 6.14 | 4.46 | 0.361 | 12.0 | 8.46 | 3.90 |

Model 3 ^{2} | |||||||||

UnCarb_LAsh | 0.978 | 0.876 | 19.5 | 354 | 14.0 | 5.48 | 183 | 22.6 | 38.2 |

AKP-5/A | 0.867 | 0.931 | 16.6 | 41.1 | 11.9 | 1.57 | 52.3 | 19.2 | 13.2 |

Model 4 ^{2} | |||||||||

UnCarb_LAsh | 0.994 | 0.997 | 4.14 | 9.15 | 2.75 | 0.245 | 8.17 | 4.78 | 5.03 |

AKP-5/A | 1.00 | 1.00 | 0.898 | 0.0787 | 0.614 | 0.00494 | 0.165 | 1.04 | 0.471 |

^{1}SO

_{2}+ Ar;

^{2}SO

_{2}+ O

_{2}+ H

_{2}O

_{(g)}+ Ar.

**Table 4.**SNE error analysis for kinetic models solved by non-linear regression method—the most appropriate values.

Sample | Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|---|

SO_{2} + Ar | ||||

UnCarb_HAsh | HYBRID | HYBRID | EABS | χ^{2} |

5.40 | 4.84 | 4.39 | 5.06 | |

UnCarb_MAsh | SSE | χ^{2} | χ^{2} | Δq |

8.16 | 7.13 | 5.37 | 5.00 | |

UnCarb_LAsh | R^{2} | EABS | MPSD | EABS |

8.97 | 8.43 | 3.16 | 5.59 | |

AKP-5 | HYBRID | HYBRID | EABS | HYBRID |

6.08 | 6.28 | 6.94 | 5.60 | |

AKP-5/A | HYBRID | χ^{2} | χ^{2} | MPSD |

5.42 | 4.97 | 6.77 | 4.84 | |

SO_{2} + Ar + H_{2}O_{(g)} + O_{2} | ||||

UnCarb_LAsh | SSE | R | χ^{2} | EABS |

8.92 | 7.55 | 4.04 | 5.52 | |

AKP-5/A | SSE | χ^{2} | χ^{2} | χ^{2} |

7.91 | 8.09 | 4.27 | 6.18 |

Sample | Model 1 | Model 2 | Model 3 | Model 4 | |||
---|---|---|---|---|---|---|---|

k_{1} | k_{2} | m_{S,∞} | k_{id} | C | α | β | |

min^{−1} | kg·g^{−1}·min^{−1} | g·kg^{−1} | g·kg^{−1}·min^{−0.5} | g·kg^{−1} | g·kg^{−1}min^{−1} | kg·g^{−1} | |

SO_{2} + Ar | |||||||

UnCarb_HAsh | 0.224 | 6.57E-03 | 31.8 | 4.59 | 0 | 13.6 | 0.123 |

UnCarb_MAsh | 0.581 | 2.41E-02 | 31.0 | 5.38 | 7.48 | 54.4 | 0.175 |

UnCarb_LAsh | 1.19 | 6.85E-02 | 29.4 | 1.97 | 18.2 | 3955 | 0.353 |

AKP-5 | 0.312 | 1.97E-02 | 17.1 | 3.45 | 0.460 | 11.8 | 0.280 |

AKP-5/A | 0.252 | 1.19E-02 | 20.6 | 3.52 | 0.775 | 9.44 | 0.213 |

SO_{2} + O_{2} + H_{2}O_{(g)} + Ar | |||||||

UnCarb_LAsh | 0.865 | 2.59E-02 | 47.5 | 4.80 | 24.3 | 390 | 0.153 |

AKP-5/A | 0.578 | 3.71E-02 | 21.5 | 2.74 | 8.17 | 61.7 | 0.284 |

Model | R^{2} | R | Δq | SSE | ARE | χ^{2} | HYBRID | MPSD | EABS |
---|---|---|---|---|---|---|---|---|---|

UnCarb_HAsh ^{1} | |||||||||

Model 4 L | 0.952 | 0.976 | 28.3 | 27.8 | 17.4 | 2.2 | 72.5 | 32.7 | 10.1 |

Model 3 NL | 0.752 | 0.867 | 29.1 | 152.9 | 17.8 | 7.1 | 235.9 | 33.6 | 17.7 |

UnCarb_Mash ^{1} | |||||||||

Model 4L | 0.966 | 0.983 | 11.0 | 21.5 | 7.7 | 0.96 | 32.0 | 12.7 | 8.24 |

Model 4NL | 0.961 | 0.980 | 9.6 | 27.4 | 5.4 | 1.01 | 33.6 | 11.1 | 7.4 |

UnCarb_Lash ^{1} | |||||||||

Model 4L | 0.988 | 0.994 | 5.2 | 7.0 | 3.5 | 0.3 | 9.1 | 6.0 | 4.4 |

Model 3NL | 0.220 | 0.469 | 9.7 | 358.9 | 5.7 | 1.0 | 34.4 | 11.2 | 25.8 |

AKP-5 ^{1} | |||||||||

Model 1L | 0.998 | 0.999 | 3.7 | 0.3 | 2.1 | 0.04 | 1.3 | 4.3 | 0.8 |

Model 4NL | 0.967 | 0.983 | 10.6 | 6.4 | 7.9 | 0.5 | 17.0 | 12.2 | 4.4 |

AKP-5/A ^{1} | |||||||||

Model 1L | 0.979 | 0.989 | 8.4 | 5.1 | 4.0 | 0.4 | 12.4 | 9.7 | 2.4 |

Model 4NL | 0.953 | 0.976 | 12.3 | 13.1 | 7.3 | 0.9 | 29.6 | 14.2 | 5.5 |

UnCarb_Lash ^{2} | |||||||||

Model 4L | 0.994 | 0.997 | 4.1 | 9.2 | 2.8 | 0.2 | 8.2 | 4.8 | 5.0 |

Model 3 NL | 0.465 | 0.682 | 9.0 | 633.3 | 6.5 | 1.2 | 38.9 | 10.4 | 36.5 |

AKP-5/A ^{2} | |||||||||

Model 4L | 1.00 | 1.00 | 0.9 | 0.08 | 0.6 | 0.005 | 0.2 | 1.0 | 0.5 |

Model 3NL | 0.686 | 0.828 | 7.5 | 72.1 | 5.6 | 0.3 | 11.2 | 8.7 | 12.5 |

^{1}SO

_{2}+ Ar;

^{2}SO

_{2}+ O

_{2}+ H

_{2}O

_{(g)}+ Ar.

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**MDPI and ACS Style**

Kisiela-Czajka, A.M.; Dziejarski, B. Linear and Non-Linear Regression Analysis for the Adsorption Kinetics of SO_{2} in a Fixed Carbon Bed Reactor—A Case Study. *Energies* **2022**, *15*, 633.
https://doi.org/10.3390/en15020633

**AMA Style**

Kisiela-Czajka AM, Dziejarski B. Linear and Non-Linear Regression Analysis for the Adsorption Kinetics of SO_{2} in a Fixed Carbon Bed Reactor—A Case Study. *Energies*. 2022; 15(2):633.
https://doi.org/10.3390/en15020633

**Chicago/Turabian Style**

Kisiela-Czajka, Anna M., and Bartosz Dziejarski. 2022. "Linear and Non-Linear Regression Analysis for the Adsorption Kinetics of SO_{2} in a Fixed Carbon Bed Reactor—A Case Study" *Energies* 15, no. 2: 633.
https://doi.org/10.3390/en15020633