# Modified Dual-Site Langmuir Adsorption Equilibrium Models from A GCMC Molecular Simulation
## 4 Results and Discussion
### Modification of the TDSL Model That Reflects Temperature Changes  
In the model parameter regression, we found that the parameters that were obtained by the DSL model and the M-DSL model were quite different at different temperatures and that the adsorption energy, which was found to have a linear relationship with the pressure obtained by a GCMC simulation, can modify the DSL model at different temperatures and had a good effect. In order to further verify this conclusion, we chose the DSL model that reflects temperature changes (TDSL), which is used in the adsorption module of Aspenplus and has been used in previous research [16; 17; 18], and attempted to improve its accuracy and universality. The TDSL model is a two-parameter adsorption equilibrium model that is based on the DSL model and takes temperature and pressure as variables together  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Adsoorbent**} & \multirow{2}{*}{**Adaorbate**} & \multirow{2}{*}{**T (K)**} & \multirow{2}{*}{**\(\mathbf{d_{\text{eff}}}\)**} & \multirow{2}{*}{**\(\mathbf{d_{\text{eff}}}\)**-1**} & \multirow{2}{*}{**\(\mathbf{k_{\text{eff}}}\)**(Br\({}^{-1}\))**} & \multirow{2}{*}{**\(\mathbf{k_{\text{eff}}}\)**(Br\({}^{-1}\))**} & \multirow{2}{*}{**\(\mathbf{k_{\text{eff}}}\)**(Br\({}^{-1}\))**} & \multirow{2}{*}{**\(\mathbf{d_{\text{eff}}}\)**(Br\({}^{-1}\))**} & \multirow{2}{*}{**\(\mathbf{d_{\text{eff}}}\)**(Br\({}^{-1}\))**} & \multirow{2}{*}{**R\({}^{2}\)**} & \multirow{2}{*}{**SSE**} \\ \cline{5-11} \multirow{6}{*}{HHPAC} & & 223.15 & 3.487 & 3.553 & 3.441\(\times 10^{-3}\) & 0.3387 & 1.131\(\times 10^{-2}\) & 1.006\(\times  
It is also based on the three assumptions of the Langmuir model. With this model, the adsorption equilibrium data obtained at different temperatures in the same system can be correlated at the same time. The form of the TDSL model is shown in Equation (12):  
\[q=\frac{(k_{1}+k_{2}T)\exp(\frac{k_{3}}{RT})P}{1+k_{4}\exp(\frac{k_{3}}{RT})P}+ \frac{(k_{5}+k_{6}T)\exp(\frac{k_{7}}{RT})P}{1+k_{8}\exp(\frac{k_{7}}{RT})P}, \tag{12}\]  
where \(P\) is the adsorption equilibrium pressure, \(q\) is the adsorption capacity at the equilibrium pressure, R is the gas constant, T is the adsorption temperature, \(k_{1}\), \(k_{2}\), \(k_{5}\), and \(k_{6}\) are the coefficients of the linear relationship between the saturated adsorption capacity and temperature in the original DSL model, \(k_{3}\) and \(k_{7}\) are the adsorption energies corresponding to different adsorption sites, and \(k_{4}\) and \(k_{6}\) are pre-exponential factors. Based on the conclusion that adsorption energy has a linear relationship with pressure, the TDSL model can be modified to obtain the M-TDSL model by substituting Equation (5) into Equation (12). The form of the modified TDSL (M-TDSL) model is shown in Equation (13):  
\[q=\frac{(k_{1}+k_{2}T)\exp(\frac{c_{1}+a_{1}P}{RT})P}{1+k_{4}\exp(\frac{c_{  
on CO\({}_{2}\)-HHPAC well. Its R\({}^{2}\) was greater than 0.99, and its SSE was in a reasonable range. From the physical meaning of the parameters, it can be seen that the values of \(k_{3}\) and \(k_{7}\) were the adsorption energies at the two adsorption sites, respectively, and the results also showed that the adsorption energies were \(-\)19.98 and \(-\)28.99 kJ\(\cdot\)mol\({}^{-1}\), respectively, which are in the general range of physical adsorption energies. This indicated that the regression results were reliable and accurate and that the physical meanings of the corresponding parameters were clear and definite. The regression effect of the M-TDSL model on the adsorption equilibrium data was slightly improved as compared to that of the original model, and the accuracy of the regression was increased by 14%. For the CH\({}_{4}\)-BPL and CO\({}_{2}\)-H-Mordenite systems, the TDSL model had a poor correlation, which cannot produce the expected effect. In fact, the M-TDSL model was better than the TDSL model with respect to the fitting correlation effects of these systems.