## 4 Simulation and Analysis of Coal's Adsorption of Gas
### Influence of Coal with Different Pore Diameters on the Isothermal Adsorption of CH4  
Based on the fact that methane in actual coal seams exists in the pores of anthracite with different pore diameters, under different pressures, methane gas molecules diffuse towards the surface of the coal seam. When they diffuse to the adsorption sites, they will be adsorbed at these sites. For coal samples with different pore diameters, isothermal adsorption simulations of methane gas under the same temperature and different pressure conditions were carried out for coal pore models with pore diameter sizes of 10 A, 20 A, 30 A, 40 A, 50 A, and 60 A, respectively. The linearized formula of the Langmuir model (Formula (1)) was used to process the adsorption equilibrium experimental data under different pressures for the description of the adsorption equilibrium process of CH4 on the coal surface [18; 19; 20; 21; 22]:  
\[Q=\frac{{\it abp}}{{1+bp}} \tag{1}\]  
In the formula, \(Q\) represents the adsorption amount of methane at the equilibrium state of the coal sample, in mmol/g; \(a\) represents the absolute adsorption amount of methane, in mmol/g; \(b\) represents the Langmuir adsorption equilibrium constant, in MPa\({}^{-1}\); and \(p\) represents the adsorption pressure, in MPa. It can be seen from Figure 4 that the isothermal adsorption curves of CH4 gas in the pore models of anthracite with different pore diameters conform to the characteristics of the isothermal lines of the Langmuir model. Therefore, the isothermal adsorption curves obtained from the simulation were fitted according to the linearized formula equation, and the volume constant a and pressure constant b in the Langmuir adsorption constants were obtained. The parameters are shown in Table 3.  
The absolute adsorption amount of CH4 continuously increases with the increase in adsorption pressure. When it approaches the critical pressure of CH4, the absolute adsorption amount increases slowly and tends to stabilize. In the range of 1-5 MPa, the adsorption amount of methane rises rapidly with the increase in adsorption pressure, which is in the rapid adsorption stage. When the adsorption pressure reaches 5-10 MPa, the adsorption amount of CH4 increases slowly with the increase in adsorption pressure, which is in the slow adsorption stage. The results indicate that pressure can promote the  
Figure 4: Isothermal adsorption curves of coal with different pore size pores and CH4. ability of coal to adsorb CH\({}_{4}\), but the effect is not as obvious as that in the range of 1-5 MPa, and the influence gradually decreases in the slow adsorption stage. During the adsorption process, with the increase in pressure, the influence of pore diameter on the adsorption of CH\({}_{4}\) is relatively obvious. When the pore diameter increases from 10 A to 60 A, the absolute adsorption amount of CH\({}_{4}\) increases from 51.07 mmol/g to 108.97 mmol/g. The growth rates in the pores are basically the same. When the pressure is 10 MPa, with every 10 A increase in pore diameter, the absolute adsorption amount increases by an average of 16.43%. After Langmuir nonlinear fitting, the R\({}^{2}\) values are all greater than 0.95, indicating a good fitting effect. As can be seen from Table 3, under the pore diameters ranging from 10 A to 60 A, the Langmuir adsorption constants a of the coal samples are 51.55 mmol/g, 64.00 mmol/g, 78.97 mmol/g, 92.87 mmol/g, 106.80 mmol/g, and 124.60 mmol/g, respectively. The value of the adsorption constant a reflects the absolute adsorption amount of CH4 of the coal sample. It can be observed that the adsorption constant a shows an obvious upward trend with the increase in pore diameter, with an overall increase as high as 58.26%. The change in the adsorption constant b from 50 A to 60 A is only 0.27 MPa\({}^{-1}\), which is negligible compared to the change in the adsorption constant b from 10 A to 20 A.