## 4 Results
### Relationship between Desorption Rate Ratio and Coal Particle Breakage Ratio  
When \(D_{e}t\) is small, meaning the effective diffusion coefficient is a constant, and time is not long, Equation (11) can be reduced to:  
\[g(D_{e}t)=\frac{M_{t}}{M_{\infty}}=6\cdot\sqrt{\frac{D_{e}t}{\pi}} \tag{28}\]  
In the initial time period, combining Equation (28) and Equation (16) can be reduced to:  
\[\eta(t)=\sqrt{\frac{D_{e}(R_{2})}{D_{e}(R_{1})}} \tag{29}\]  
In combination with Equations (20) and (29), the ratio of gas amount diffusing from coal before being broken to that from crushed coal during a time period \(t\) could be expressed by the ratio of coal particle size before and after coal breakage as:  
\[\eta(t)=\left(\frac{R_{1}}{R_{2}}\right)^{\gamma/2}=x^{\gamma/2} \tag{30}\]  
Equations (16) and (29) can be used to calculate the maximum and initial approximate ratio of accumulative gas desorption for coal particles of varying sizes relative to an initial diameter of 7.5 mm in Yang and 2 mm in Guo. The ratio of the initial period of cumulative desorption is approximately equal to the ratio of the maximum cumulative gas desorption.  
In the initial period, the differential of the diffusivity is:  
\[g^{\prime}(D_{e},t)=3\cdot\sqrt{\frac{D_{e}}{\pi t}} \tag{31}\]  
The desorption rate ratio for the initial period is easily obtained as follows:  
\[\xi(t)=\sqrt{\frac{D_{e}(R_{2})}{D_{e}(R_{1})}} \tag{32}\]  
Equations (29) and (32) demonstrate that the ratio of gas desorption volume from fragmented coal to that from intact coal equals the ratio of gas desorption rate during a given time period. Both ratios can be estimated by taking the square root of the ratio of the effective gas diffusion coefficient for broken coal particles to that for original coal particles.  
By combining Equations (20) and (32), the ratio of the gas desorption rate in crushed coal to that in undisturbed coal during a specific time period can be expressed as the ratio of the coal particle size before and after fragmentation:  
Figure 6: The ratio of gas desorption rate after and before coal is broken for a long time (Yang).  
\[\xi(t)=\left(\frac{R_{1}}{R_{2}}\right)^{\gamma/2}=x^{\gamma/2} \tag{33}\]  
Equation (33) demonstrates that the ratio of desorption amount is equivalent to the ratio of gas desorption rate during the initial period.  
The Equation (33) was used to fit the relationship between the coal particle breakage ratio and the gas desorption rate ratio after and before coal fragmentation. The results are depicted in Figure 7.  
The data fit parameters \(\gamma\) for Yang and Guo were 0.5829 and 0.5149, respectively, with coefficients of determination of 0.9303 and 0.9822, respectively. Figure 7 illustrates the approximate value for the ratio of the gas desorption rate from broken coal to that from intact coal after desorption lasts for a time period \(t\).  
The ratio of the gas diffusing rate from broken coal particles at the initial time to that from intact coal can be related to the coal particle breakage ratio \(x=R_{2}/R_{1}\) from Equation (18):  
\[\xi(0)=x^{\gamma} \tag{34}\]  
This \(\xi(0)\) is the initial value for the ratio of the gas release rate from broken coal to that from coal before being broken, which is the largest value for the ratio. The initial gas desorption rate ratio is proportional to the coal particle breakage ratio.