## 4 Discussion
### Analysis of Specific Surface Area Changes of Mixed Coal before and after Adsorption
The function for macro pore and micro pore surface areas are expressed by Equations (7) and (8):  
\[x_{1}=In\bigg{(}\frac{B_{10}B_{11}}{\theta_{1}}-\frac{1}{B_{11}}\bigg{)}^{- \frac{1}{B_{12}}} \tag{7}\]  
\[x_{2}=In\bigg{(}\frac{B_{20}B_{21}}{\theta_{1}}-\frac{1}{B_{21}}\bigg{)}^{- \frac{1}{B_{22}}} \tag{8}\]  
By substituting Equations (7) and (8) into Equation (2), we obtain:  
\[W_{1}=A_{10}\left[In\bigg{(}\frac{B_{10}B_{11}}{\theta_{1}}-\frac{1}{B_{11}} \bigg{)}^{-\frac{1}{B_{12}}}\right]^{2}+A_{11}In\bigg{(}\frac{B_{10}B_{11}}{ \theta_{1}}-\frac{1}{B_{11}}\bigg{)}^{-\frac{1}{B_{12}}}+A_{12} \tag{9}\]  
\[W_{2}=A_{20}\left[In\bigg{(}\frac{B_{20}B_{21}}{\theta_{2}}-\frac{1}{B_{21}} \bigg{)}^{-\frac{1}{B_{22}}}\right]^{2}+A_{21}In\bigg{(}\frac{B_{20}B_{21}}{ \theta_{2}}-\frac{1}{B_{21}}\bigg{)}^{-\frac{1}{B_{22}}}+A_{22} \tag{10}\]  
Figure 9: Variation in the difference of mass ratios of coal samples before and after adsorption. (**a**) Difference of macro pores’ specific surface area; (**b**) Difference of micro pores’ specific surface area. Let \(\frac{B_{10}B_{11}}{\theta_{1}}-\frac{1}{B_{11}}=\frac{D_{10}-\theta_{1}}{D_{11} \cdot\theta_{1}},\frac{B_{20}B_{21}}{\theta_{2}}-\frac{1}{B_{21}}=\frac{D_{20}- \theta_{2}}{D_{21}\cdot\theta_{2}}\). Therefore, the total adsorption amount can be expressed as: \(W_{m}=W_{1}+W_{2}\). Merging Equations (9) and (10), we obtain:  
\[W_{m}=F_{i}\sum_{i,j=0}^{2}\left[In\left(\frac{D_{ij}-\theta_{j}}{D_{ij+1}- \theta_{j}}\right)^{-\frac{1}{D_{ij+1}}}\right]^{2}+F_{i+1}\sum_{i,j=0}^{2}In \left(\frac{D_{ij}-\theta_{j}}{D_{ij+1}-\theta_{j}}\right)^{-\frac{1}{D_{ij+1} }}+F_{i+2} \tag{11}\]  
where \(W_{1}\), \(W_{2}\) are the adsorption amount of macro pores and micro pores, mL-g-1. \(B_{ij}\), \(D_{ij}\), \(F_{i+n}\) (\(i=1\), \(2\); \(j=i-1\)) are fitting constants, which are affected by the characteristics of the coal sample. The relationship between the strain after the adsorption of mixed coal and the change of specific surface area before and after adsorption can be obtained by substituting Equation (11) into Equation (6). \[\varepsilon_{m}=K_{m}\left[F_{i}\sum_{i,j=0}^{2}\left[In\left(\frac{D_{ij}- \theta_{j}}{D_{ij+1}-\theta_{j}}\right)^{-\frac{1}{D_{ij+1}}}\right]^{2}+F_{i +1}\sum_{i,j=0}^{2}In\left(\frac{D_{ij}-\theta_{j}}{D_{ij+1}-\theta_{j}} \right)^{-\frac{1}{D_{ij+1}}}+F_{i+2}\right] \tag{12}\]  
Under a certain depth, the rock will put pressure on coal. Thus, the swelling strain of the coal after gas adsorption is limited. Therefore, additional stress will be generated, called adsorption swelling strain. Because of the increase of the swelling stress, the stress of coal will increase and the strength of coal will decrease, which will promote the occurrence of gas outbursts. [PERSON] [52] studied the adsorption swelling when adsorbing CO2 using a rigid dynamometer. It was found that there is a good linear relationship between the adsorption deformation stress and the desorption deformation stress. Because the adsorption of CO2 and CH4 on coal are physical adsorption processes, the linear relationship between the adsorption deformation stress and the desorption deformation stress can be used to express the relationship between the swelling deformation and the expansion stress of coal following the adsorption of gas. Combining the experimental data, the relationship between swelling stress and the change of pore specific surface area before and after adsorption can be obtained using the following equation:  
\[\sigma_{m}=E_{m}K_{m}\left[F_{i}\sum_{i,j=0}^{2}\left[In\left(\frac{D_{ij}- \theta_{j}}{D_{ij+1}-\theta_{j}}\right)^{-\frac{1}{D_{ij+1}}}\right]^{2}+F_{i +1}\sum_{i,j=0}^{2}In\left(\frac{D_{ij}-\theta_{j}}{D_{ij+1}-\theta_{j}}\right) ^{-\frac{1}{D_{ij+1}}}+F_{i+2}\right] \tag{13}\]  
where \(\sigma_{m}\) is the swelling stress of mixed coal after adsorption, MPa. \(E_{m}\) is the proportionality coefficient corresponding to \(\sigma_{m}\), which is equivalent to the elastic modulus of coal, MPa.