## 2 Model  
### Desorption Model for Breakage Gas Bearing Coal  
For simplicity, if coal is assumed to be homogeneous and spherical in shape and its diffusion coefficient is constant, then its diffusion model with boundary conditions being constants is [31]:  
\[\begin{cases}\frac{\partial C}{\partial t}=D\Big{(}\frac{\partial^{2}C}{ \partial\pi^{2}}+\frac{t}{2}\frac{\partial C}{\partial\tau}\Big{)}&(0<r<R_{0}, t>0)\\ \frac{\partial C}{\partial\tau}\Big{|}_{r=0}=0,\,C|_{r=R_{0}}=C_{a}&(t>0)\\ C|_{t=0}=C_{0}&(r\geq 0)\end{cases} \tag{1}\]  
where \(C\) is the gas concentration within coal, \(\mathrm{g/cm^{3}}\); \(C_{a}\) is the gas concentration at the boundary, \(\mathrm{g/cm^{3}}\); \(C_{0}\) is the initial gas concentration in the coal particle, \(\mathrm{g/cm^{3}}\); \(D\) is the gas diffusion coefficient for coal particles, \(\mathrm{cm^{2}/s}\); \(D_{e}=D/R_{0}^{2}\) is the gas effective diffusion coefficient for coal particles, \(/s\); \(t\) denotes time, \(s\); \(R_{0}\) is the radius of coal particles, cm.  
Solving Equation (1) leads to the ratio of the accumulative gas diffusion quantity to the limiting accumulative diffusion quantity from the coal particle, so-called gas diffusivity for a coal particle.  
\[\frac{M_{t}}{M_{\infty}}=g(D_{e}(R_{0}),t) \tag{2}\]  
where \(M_{\infty}\) is the corresponding quantity after infinite time for a certain mass of coal; \(M_{t}\) is the accumulative amount of gas desorbing after time \(t\). Equation (2) proves that the gas diffusivity is determined by the effective gas diffusion coefficient \(D_{e}(R_{0})=D/R_{0}\)2. When the effective gas diffusion coefficient stays constant, the gas diffusivity increases over time.  
Several models, including the Gaudin-Schumann distribution, the Rosin-Rammler distribution, the Gaussian distribution, and the Fractal distribution, have been proposed to describe the particle distribution of broken coal [46; 47]. The size of constituent coal grains ranges between \([R_{\text{min}},R_{\text{max}}]\), where \(R_{\text{min}}\) and \(R_{\text{max}}\) are a definite number.  
Then, the volume of coal grains with a size of \(R\) occurs can be written as:  
\[\text{d}V(R)=\varphi(R)V_{0}\text{d}R \tag{3}\]  
where \(V_{0}\) is the total coal volume; \(\varphi(R)\) is the function of grain size distribution; Then, the size distribution function for sieved coal volume could be expressed as \(\int_{0}^{R}\varphi(R)\text{d}R\).  
The ultimate gas amount released from a certain mass of coal sample is constant and unchangeable if given long enough time to desorb. Then, the total amount of gas desorption after infinite time can be calculated as follows:  
\[M_{\infty}=(C_{0}-C_{a})V_{0} \tag{4}\]  
Meanwhile, the ultimate gas desorption amount remains constant and unaffected by coal particle size distribution. This is because most of the gas in coal is adsorbed onto the micropores of the coal matrix, which are not influenced by changes in particle size that mainly impact mesopore characteristics [48; 49].  
Further, if we assume the gas diffusivity for coal grains with a size of \(R\) is \(g(D_{e}(R),t)\), we can derive the gas desorption volume \(M_{t}\) as  
\[M_{t}=M_{\infty}\int_{R_{\text{min}}}^{R_{\text{max}}}g(D_{e}(R),t)\varphi(R) \text{d}R \tag{5}\]  
Then, gas diffusivity for broken coal is:  
\[g(D_{e}(R),t)=\int_{R_{\text{min}}}^{R_{\text{max}}}g(D_{e}(R),t)\varphi(R) \text{d}R \tag{6}\]