## 4 Simulation and Analysis of Coal's Adsorption of Gas
### Influence of Coal with Different Pore Diameters on the Diffusion of CH\({}_{4}\)  
Coal with different pore diameters can affect the kinetic behavior of CH\({}_{4}\) molecules [23]. Under the conditions of constant temperature and constant pressure, the diffusion coefficients of methane in the pore models of anthracite with pore diameters ranging from 10 A to 60 A were calculated, respectively. The mean square displacement and its diffusion coefficient D, which can describe the random motion of CH\({}_{4}\) molecules diffusing in different pore diameters, were calculated by selecting the diffusion coefficient (MSD) through the Analysis function of the Forcite module. The calculation formula is as follows [24]:  
\[MSD=\frac{1}{N}\sum_{i=1}^{N}\Big{[}r_{i}(t)-r_{i}(0)]^{2} \tag{2}\]  
\[D=\frac{1}{6N}\textit{lim}\frac{d}{dt}\sum_{i=1}^{N}\Big{[}r_{i}(t)-r_{i}(0)] ^{2} \tag{3}\]  
\[D=\frac{1}{6}\lim_{t\rightarrow\infty}(\frac{MSD}{t})=\frac{1}{6}K_{MSD} \tag{4}\] \begin{table}
\begin{tabular}{c c c c c} \hline
**Pore** & **Surface** & **Free** & **Occupied** & **Porosity\%** \\
**Diameter/Å** & **Area/Å\({}^{2}\)** & **Volume/Å\({}^{3}\)** & **Volume/Å\({}^{3}\)** & **Porosity\%** \\ \hline
10 & 2468.89 & 8355.84 & 8116.71 & 50.72 \\
20 & 2460.12 & 10,631.16 & 8118.79 & 56.69 \\
30 & 2470.55 & 12,920.03 & 8107.32 & 61.44 \\
40 & 2463.97 & 15,189.34 & 8115.41 & 65.17 \\
50 & 2459.37 & 17,461.22 & 8120.93 & 68.25 \\
60 & 2468.89 & 19,744.05 & 8115.49 & 70.86 \\ \hline \end{tabular}
\end{table} 
Table 3: Parameters of coal pore model with different pore sizes. In the formula, \(N\) represents the number of molecules and \(r(t)\) and \(r(0)\) are the position vectors of the center of mass of the diffusing molecules from time \(t\) to \(t\) = 0, respectively. K\({}_{\text{MSD}}\) is the slope of the mean square displacement (MSD) curve, that is, the diffusion coefficient is one-sixth of the slope of the MSD curve. The integrated data of the mean square displacement of CH\({}_{4}\) in anthracite with different pore diameters are shown in Figure 5. The slope of the mean square displacement in the larger pore diameter models is greater than that in the smaller pore diameter models. This is because the larger pore diameters provide a larger area for the diffusion motion, giving CH\({}_{4}\) molecules more diffusion space. By calculating the slope of each point on the curve, the diffusion coefficients of CH\({}_{4}\) molecules in coal samples with different pore structures were finally obtained. The influence of different pore diameters on the diffusion ability of CH\({}_{4}\) is in the order of 60 A \(>\) 50 A \(>\) 40 A \(>\) 30 A \(>\) 20 A \(>\) 10 A. It can be seen from Tables 4 and 5 that among the six different pore diameters of anthracite, the diffusion coefficient of CH\({}_{4}\) is the smallest when the pore diameter is 10 A, with a value of 0.567 A/ps; when the pore diameter is 60 A, the diffusion coefficient is the largest, with a value of 3.892 A/ps. The diffusion coefficients of pore diameters from 10 A to 50 A increase relatively gently at an average rate of 32.25%. However, the increase in the diffusion coefficient from a pore diameter of 50 A to 60 A surges by approximately 126%. This is due to the limited pore volume and the relatively strong binding effect of van der Waals forces on CH\({}_{4}\) in smaller pores, which increases the diffusion resistance inside the coal pores. Most of the CH\({}_{4}\) remains in a free state and moves randomly and freely; so, the diffusion coefficient of CH\({}_{4}\) increases slowly. As the pore diameter increases, its pore volume also becomes larger, and the binding effect on CH\({}_{4}\) gradually weakens. When the migration state of CH\({}_{4}\) begins to change from being mainly in a free state to mainly in an adsorbed state under the action of pressure, the diffusion coefficient in the model will increase rapidly when the pore diameter is 50 A.  
Figure 5: Fitting results of CH\({}_{4}\) rms displacement in different coal pore sizes.