# Nonlinear and Linear Equation of Gas Diffusion in Coal--Theory and Applications
## 5 Comparison of the Measurement Results with the Solution of the Nonlinear Diffusion Coefficient  
For the sake of the experimental research, the grain samples of hard coal of four different fractions were used [23]. The coal came from the \"Thorez\" coal mine in the Lower-Silesian Coal Basin, Poland [39]. The research was carried out by means of the volumetric method [4; 40], at a stabilized temperature of 30 \({}^{\circ}\)C. The sorbate was carbon dioxide (CO\({}_{2}\)), with which the coal sorbent was saturated for 24 h at a stabilized pressure of 0.25 MPa. The saturation process was preceded by outgassing the coal for 24 h by means of a vacuum pump, under a temperature of 80 \({}^{\circ}\)C. For the chosen grain fraction, these time values proved sufficient to obtain sorption equilibriums. Table 1 shows the results of CO\({}_{2}\) desorption measurements from coal samples of different grain sizes after initial saturation of these samples to a pressure of 0.25 MPa and emission of the gas to the vacuum. The amount of released gas was determined by the volumetric method, based on the measurement of pressure changes in the container to which the gas was released from the sample. Time dependencies of sorption changes (\(a(t)\)) were determined from the Clapeyron equation. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Grain Size** & **Equivalent Grain Size \(r_{0}\)** & **Sorption Capacity \(a_{m}\)** & \(t_{0.2}\) & \(t_{0.8}\) & \(D_{m}^{{}^{\prime}}\cdot 10^{-13}\) & \(D_{\infty}^{{}^{\prime}}\cdot 10^{-13}\) \\
**(mm)** & **(mm)** & **(cm\({}^{3}\)/g)** & **(s)** & **(s)** & **(m\({}^{2}\)/s)** & **(m\({}^{2}\)/s)** \\ \hline
25–32 & 14.1 & 9.81 & 0.33 & 65 & 16.5 & 2.7 \\
63–72 & 33.6 & 9.95 & 1.08 & 340 & 32.1 & 2.9 \\
90–100 & 47.4 & 9.90 & 3.03 & 964 & 23.3 & 2.4 \\
200–250 & 111.6 & 9.78 & 27.7 & 4520 & 12.6 & 2.5 \\ \hline \end{tabular}
\end{table} 
Table 1: Measurement results of CO\({}_{2}\) desorption from coal of different fractions obtained after initial saturation at a pressure of 0.25 MPa and emission to the vacuum. Figure 2: Scheme of the spherical grain division adopted for the calculation of gas density changes within its individual parts during gas emission. For the model calculations, for each grain size range, an equivalent grain size \((r_{0})\) was assumed, which was calculated from the formula:  
\[r_{0}=\frac{1}{2}\sqrt[3]{\frac{2\cdot d_{1}{}^{2}\cdot d_{2}{}^{2}}{d_{1}+d_{2}}} \tag{23}\]  
where: \(d_{1}\) and \(d_{2}\) are the grain size limits of the size of the analyzed grain fraction. By determining the times, \(t_{0,2}\) and \(t_{0,8}\), from the registered courses, the diffusion coefficients, \(D^{\prime}_{m}\) and \(D^{\prime}_{\infty}\), were calculated from Equations (16) and (17). Comparing the values of these coefficients, it can be seen that, in the initial stage, the process of gas emission from the grains of coal samples is much faster, and the \(D^{\prime}_{m}\) values are about an order of magnitude higher than \(D^{\prime}_{\infty}\). Therefore, courses that are described by nonlinear diffusion, Equation (10), are difficult to characterize by one parameter that determines the gas emission rate. The results of the experiment were compared with the model based on the exact solution of the linear diffusion equation and the numerical solution of the nonlinear diffusion equation (Figure 3). According to expectations, linearization of the diffusion equation results in a poorer representation of the experimental course, which is confirmed by a substantially bigger number of deviation squares. The values of the diffusion coefficients differ by more than two times. Figure 4 shows a comparison of the experimental results with the model results described by the nonlinear diffusion equation, obtained for four different grain classes. The initial CO\({}_{2}\) saturation pressures were 0.25 MP, and then the gas emission into the vacuum was recorded for 100 min.