# Modified Dual-Site Langmuir Adsorption Equilibrium Models from A GCMC Molecular Simulation
## 4 Results and Discussion
### Regression of Adsorption Equilibrium Data with the DSL Model and the M-DSL Model  
The DSL model and the M-DSL model were used to regress the adsorption equilibrium data by the nonlinear regression method, and the corresponding model parameters were obtained. The nonlinear regression was realized by the lsqnonlin function in MATLAB, and we used the Levenberg-Marquardt algorithm. The results were calculated with Equation (9), which was written as:  
\[\text{min}\sum\limits_{i=1}^{N}\left(q_{i,exp}-q_{i,pre}\right)^{2}, \tag{9}\]  
where \(i\) is the number of experimental data points, \(N\) is the total number of experimental data points, \(q_{i,exp}\) is the experimental value, and \(q_{i,pre}\) is the regression value. The lsqnonlin function in MATLAB is based on the nonlinear least squares method. In the process of regression using this method, the selection of an initial value is very important, different initial values may obtain different results. The specific operation process is as follows: (1) an appropriate initial value is selected according to experience of regression calculation; (2) the results of the first regression are taken as the initial values of regression; and (3) regression is carried out, until the obtained model parameter value no longer changes. After the operation process is completed, the R\({}^{2}\) of the model reaches the maximum, and the SSE has the minimum value, and the results are the final model parameter values. Therefore, using this method to regress the same system could get the same results, which showed that the results had good repeatability, and the model parameters obtained by this method were considered to be accurate and reasonable. The accuracy of the regression was evaluated by the square sum of the error between the original experimental data and the fitting data, that is, the SSE, which was shown in Equation (10):  
\[SSE=\sum\limits_{i=1}^{N}\left(y_{i,exp}-y_{i,pre}\right)^{2}, \tag{10}\]  
where \(i\) is the number of experimental data points, \(N\) is the total number of experimental data points, \(y_{i,exp}\) is the experimental value, and \(y_{i,pre}\) is the regression value. The closer the SSE is to 0, the closer the predicted values of the model are to the experimental values, and the better the model describes the experimental data. The square of the correlation coefficient (R\({}^{2}\)) can also be used as an evaluation index of data fitting. It is defined as the ratio of the sum of squares of the fitting deviation to the sum of squares of the actual deviation and can be calculated with Equation (11):  
\[R^{2}=\frac{\sum\limits_{i=1}^{N}\left(y_{i,pre}-\overline{y}\right)^{2}}{ \sum\limits_{i=1}^{N}\left(y_{i,exp}-\overline{y}\right)^{2}}, \tag{11}\]  
where \(i\) is the number of experimental data points, \(N\) is the total number of experimental data points, \(y_{i,exp}\) is the experimental value, \(y_{i,pre}\) is the regression value, and \(\overline{y}\) is the average number of experimental values used to determine the adsorption capacity. R\({}^{2}\) ranges from 0 to 1. The smaller the value is, the worse the fit between the model and the data is; the closer the value is to 1, the better the fit of the model is. In this study, SSE and R\({}^{2}\) were combined in order to analyze the regression results. We used the adsorption equilibrium data on C\({}_{2}\)H\({}_{4}\)-HHPAC, CO\({}_{2}\)-MIL-101 (Cr), CH\({}_{4}\)-BPL active carbon, and a CO\({}_{2}\)-H-Mordenite molecular sieve [31; 32], as determined by experiments, in the DSL model and the M-DSL model for the nonlinear regression. The reason why we selected these data is that they have a wide range of adsorption temperatures and pressures and are commonly used to test adsorption equilibrium models. In addition, the adsorbents used in these systems are commonly used in industry, among which BPL is cylindrical, H-Mordenite is spherical and HHPAC is powder. Therefore, the shapes of adsorbents selected in this study almost contained the shapes of adsorbents commonly used in industry.