# Nonlinear and Linear Equation of Gas Diffusion in Coal--Theory and Applications
## 3 Gas Diffusion Equation in Sorbent Grains
In order to establish it, we shall calculate a density integral over a grain radius [38]:  
\[\gamma=\frac{3}{r_{0}^{3}}\int_{0}^{r_{0}}r^{2}\cdot\xi^{\prime}(r,t)dr \tag{12}\]  
where:  
\(\gamma\) is the average gas density inside a grain. Following the integration of Function (11) according to formula (12), we shall receive:  
\[\gamma=1-\frac{6}{\pi^{2}}\cdot\sum_{n=1}^{\infty}\frac{1}{n^{2}}\cdot\ \exp\left(-\frac{n^{2}\cdot\pi^{2}\cdot D^{\prime}\cdot t}{r_{0}^{2}}\right) \tag{13}\]Function (13) can be approximated with its asymptotic expansions for large and small time values. After a sufficiently long amount of time passes in the final desorption stage, function \(\gamma\) can be approximated by means of the following formula:  
\[\gamma_{\infty}=1-\frac{6}{\pi^{2}}\cdot\ \exp\left(-\frac{\pi^{2}\cdot D^{\prime} \cdot\ t}{r_{0}^{2}}\right) \tag{14}\]  
where:  
\(\gamma_{\infty}\) is the amount of gas released at the final desorption stage. For the initial moments of the desorption process, in turn, the function approximation is expressed by the following formula:  
\[\gamma_{m}=\frac{6}{r_{0}}\sqrt{\frac{D^{\prime}\cdot\ t}{\pi}} \tag{15}\]  
where:  
\(\gamma_{m}\) is the amount of gas released at the initial desorption stage. Figure 1 presents an exact, radius-averaged solution to Equation (7) (formula (13)) and its asymptotic approximations (14) and (15). As can be seen, when the relative amount of the gas released from a sorbent grain reaches the value \(\gamma=0.8\), the approximate solution of the diffusion coefficient is identical with the exact solution. Let us denote as \(t_{0.8}\), the time after which the amount of gas being released will come to 0.8 of the maximum value. Using formula (14), we can calculate the effective diffusion coefficient \(D^{\prime}_{\infty}\):  
\[D^{\prime}_{\infty}=-\frac{r_{0}^{2}}{\pi^{2}\cdot t_{0.8}}\cdot\ \ln\frac{0.2\cdot\pi^{2}}{6}=0.113\frac{r_{0}^{2}}{t_{0.8}} \tag{16}\]  
Figure 1: Comparison of approximate solutions (Equations (14) and (15)) of the linear diffusion equation with the exact solution (Equation (13)). Similarly, comparing the exact solution with the approximation for the initial desorption stadium, we can see that the former corresponds to the latter up to the value \(\gamma=0.2\). Therefore, on the basis of formula (15), we can determine the following:  
\[D^{\prime}_{m}=\left(\frac{0.2}{6}\right)^{2}\cdot\frac{r_{0}^{2}\cdot\pi}{t_{0.2}}=0.21\frac{r_{0}^{2}}{t_{0.2}} \tag{17}\]  
where:  
\(t_{0.2}\) is the time after which the amount of gas being released reaches 0.2 of the maximum value (s). Analyzing the exact solution (13), one can demonstrate that, for \(\gamma=0.5\), the value of the effective diffusion coefficient will be inversely proportional to the half-time, \(t_{0.5}\), according to the following relationship:  
\[\frac{\partial\xi^{\prime\prime}}{\partial t}=2\cdot D\cdot\xi_{0}\cdot\
abla ^{2}\xi^{\prime} \tag{10}\]0  
where:  
\(t_{0.5}\)--is the time after which the amount of gas being released reaches 0.5 of the maximum value [s].