## 2 Model
### Desorption Rate Ratio
#### 2.2.1 Relationship between Effective Diffusion Coefficient and Particle Size  
The diffusivity obtained from the solution of the diffusion Equation by the method of separate variables and the method of Laplace transformation are, respectively. The unipore diffusion model is:  
\[g(D_{\epsilon}(R_{0}),t)=1-\frac{6}{\pi^{2}}\sum_{n=1}^{\infty}\frac{1}{n^{2}} \exp\left(-n^{2}\pi^{2}D_{\epsilon}t\right) \tag{10}\]  
\[g(D_{\epsilon},t)=\frac{M_{t}}{M_{\infty}}=6\cdot\sqrt{D_{\epsilon}t}\left( \frac{1}{\sqrt{\pi}}+2\sum_{n=1}^{\infty}\mathrm{ierfc}\frac{n}{\sqrt{D_{ \epsilon}t}}\right)-3D_{\epsilon}t \tag{11}\]  
The effective diffusion coefficient in the dynamic diffusion coefficient model is expressed as:  
\[D_{\epsilon}=D_{\epsilon 0}e^{-\beta t} \tag{12}\]  
where \(D_{\epsilon 0}\) is the initial effective diffusion coefficient for gas diffusion in coal; \(\beta\) is the decay coefficient. The specific form of diffusivity is obtained by solving the diffusion Equation using the separated variable method and the Laplace transform method when the diffusion coefficient varies with time. The dynamic diffusion model is:  
\[g(D_{\omega}(R),\beta(R),t)=\frac{M_{t}}{M_{\infty}}=1-\frac{6}{\pi^{2}}\sum_{ n=1}^{\infty}\frac{1}{n^{2}}\exp\left(-n^{2}\pi^{2}\frac{D_{\epsilon 0}\left(1-e^{-\beta t} \right)}{\beta}\right) \tag{13}\]  
\[g(D_{\omega}(R),\beta(R),t)=\frac{M_{t}}{M_{\infty}}\] \[=6\cdot\sqrt{\frac{D_{\omega}\left(1-e^{-\beta t}\right)}{\beta} }\left(\frac{1}{\sqrt{\pi}}+2\sum_{n=1}^{\infty}\mathrm{ierfc}\frac{n}{\sqrt {\frac{D_{\omega}\left(1-e^{-\beta t}\right)}{\beta}}}\right)-3\frac{D_{ \omega}\left(1-e^{-\beta t}\right)}{\beta} \tag{14}\]  
In a specific case, the coal particles are of the same size before and after breakage, with coal being broken down into particles of a single size corresponding to a specific distribution. Taking the most simplified case as an example, the radius of coal particles before breaking is \(R_{1}\) and after crushing is \(R_{2}\).The coal particle size distribution function before and after coal is broken can be expressed as:  
\[\psi(R)=\delta(R-R_{1})= \begin{cases}0&(R\
eq R_{1})\\ \infty&(R=R_{1})^{\prime}\\ \end{cases}  \tag{15}\] \[\varphi(R)=\delta(R-R_{2})= \begin{cases}0&(R\
eq R_{2})\\ \infty&(R=R_{2})\end{cases} \]  
The substitution of Equations (10) and (15) into Equation (8) leads to the variation of the accumulative gas diffusion before and after coal is broken. \[\eta(t)=\frac{g(D_{\epsilon}(R_{2}),t)}{g(D_{\epsilon}(R_{1}),t)}=\frac{1- \frac{6}{\pi^{2}}\sum\limits_{n=1}^{\infty}\frac{1}{n^{2}}\exp\left(-n^{2}\pi^ {2}D_{\epsilon}(R_{2})t\right)}{1-\frac{6}{\pi^{2}}\sum\limits_{n=1}^{\infty} \frac{1}{n^{2}}\exp\left(-n^{2}\pi^{2}D_{\epsilon}(R_{1})t\right)} \tag{16}\]Substituting Equations (10) and (15) into Equation (9) produces changes in the desorbing rate before and after coal breakage as:  
\[\xi(t)=\frac{g^{\prime}(D_{\varepsilon}(R_{2}),t)}{g^{\prime}(D_{\varepsilon}(R_{ 1}),t)}=\frac{D_{\varepsilon}(R_{2})\sum\limits_{n=1}^{\infty}\exp\bigl{(}-n^{2 }\pi^{2}D_{\varepsilon}(R_{2})t\bigr{)}}{D_{\varepsilon}(R_{1})\sum\limits_{n =1}^{\infty}\exp(-n^{2}\pi^{2}D_{\varepsilon}(R_{1})t)} \tag{17}\]  
which is the radius of coal with two sizes, \(R_{1},R_{2}\) respectively. \(M^{\prime}_{t1}\) and \(M^{\prime}_{t2}\) represent the gas desorption rates of coal at different times before and after coal crushing, respectively, per unit mass of coal. The gas desorption rate ratio of the two coal sizes at the initial time, i.e., \(t=0\), is  
\[\xi(0)=\frac{g^{\prime}(D_{\varepsilon}(R_{2}),0)}{g^{\prime}(D_{\varepsilon} (R_{1}),0)}=\frac{D_{\varepsilon}(R_{2})}{D_{\varepsilon}(R_{1})} \tag{18}\]  
Equation (18) illustrates that the change in the gas desorbing rate before and after coal breakage can be expressed as the ratio of the effective gas diffusion coefficient. Methane flows through multiscale pores via a stepwise mechanism from interior to surface pores.