## 2 Model
### Desorption Rate Ratio
#### 2.2.2 Approximation of the Desorption Rate Ratio  
The distribution pattern of pulverized coal particles can be analyzed using the Gates-Gaudin-Schuhmann (GGS) model, which is expressed as:  
\[F=\left(\frac{d}{d_{\max}}\right)^{\alpha} \tag{21}\]  
where \(F\) is the volume percentage of particles with a diameter smaller than \(d\); \(d\) is the particle diameter; \(d_{\max}\) is the maximum particle diameter; \(\alpha\) is the distribution index of granularity, which means the degree of coal particle fragmentation.  
Equation (21) can also be written as:\[F(R)=\left(\frac{R}{R_{\max}}\right)^{\alpha} \tag{22}\]  
The derivative of Equation (22) with respect to particle size yields the particle size distribution, which is given by:  
\[\varphi(R)=\alpha\frac{R^{\alpha-1}}{R_{\max}^{\alpha}} \tag{23}\]  
By Equations (9) and (14), the gas desorption rate ratio of coal before and after crushing can be approximately obtained in a short period of time as follows:  
\[\xi(t)=\frac{\int_{R_{\min}}^{R_{\max}}\sqrt{\frac{D_{\omega}(R)}{\pi}}\sqrt{ \frac{\beta(R)}{1-e^{-\beta(R)t}}}e^{-\beta(R)t}\varphi(R)\mathrm{d}R}{\int_{R _{\min}}^{R_{\max}}\sqrt{\frac{D_{\omega}(R)}{\pi}}\sqrt{\frac{\beta(R)}{1-e^{- \beta(R)t}}}e^{-\beta(R)t}\psi(R)\mathrm{d}R} \tag{24}\]  
Assuming that coal exists in a fixed particle size composition before crushing, which changes into different particle sizes after crushing, the particle size distribution of coal before fragmentation and crushing can be expressed as:  
\[\psi(R)=\delta(R-R_{\max}) \tag{25}\]  
which \(R_{\max}\) is the radius of particles before coal breakage. The upper limit of the integral in Equation (24) is also \(R_{\max}\). Substituting Equations (23) and (25) into (24) yields that the desorption rate ratio is:  
\[\xi(t) =\frac{\int_{R_{\min}}^{R_{\max}}3\cdot\sqrt{\frac{D_{\omega}(R)}{ \pi}}\sqrt{\frac{\beta(R)}{1-e^{-\beta(R)t}}}e^{-\beta(R)t}\xi\frac{R^{\alpha -1}}{R_{\max}^{\alpha}}\mathrm{d}R}{\int_{R_{\min}}^{R_{\max}}3\cdot\sqrt{ \frac{D_{\omega}(R)}{\pi}}\sqrt{\frac{\beta(R)}{1-e^{-\beta(R)t}}}e^{-\beta(R )t}\xi(R-R_{\max})\mathrm{d}R} \tag{26}\] \[=\frac{\int_{R_{\min}}^{R_{\max}}\sqrt{\left(\frac{R_{\max}}{R} \right)^{\beta}}\frac{\beta(R)}{1-e^{-\beta(R)t}}e^{-\beta(R)t}\xi\frac{R^{ \alpha-1}}{R_{\max}^{\alpha}}\mathrm{d}r}{\sqrt{\frac{\beta(R_{\max})}{1-e^{ -\beta(R_{\max})t}}}e^{-\beta(R_{\max})t}}\]  
When the decay coefficient \(\beta\) changes little with the decrease in particle size, Equation (26) can be simplified as  
\[\xi(t)\approx\int_{R_{\min}}^{R_{\max}}\sqrt{\frac{D_{\omega}(R)}{D_{\omega}( R)_{\max}}}\alpha\frac{R^{\alpha-1}}{R_{\max}^{\alpha}}\mathrm{d}R \tag{27}\] \[=\int_{R_{\min}}^{R_{\max}}\sqrt{\left(\frac{R_{\max}}{R}\right)^ {\gamma}}\alpha\frac{R^{\alpha-1}}{R_{\max}^{\alpha}}\mathrm{d}R\] \[=\begin{cases}\alpha\ln\frac{R_{\max}}{R_{\min}}&\alpha=\gamma/2 \\ \frac{\alpha}{\alpha-\gamma/2}\left(1-\left(\frac{R_{\min}}{R_{\max}}\right)^{ \alpha-\gamma/2}\right)&\alpha\
eq\gamma/2\end{cases}\]  
Equation (27) demonstrates that the change in gas desorption rate before and after coal crushing can be determined by applying the rules of particle crushing distribution and effective diffusion coefficient.