# Nonlinear and Linear Equation of Gas Diffusion in Coal--Theory and Applications
## 3 Gas Diffusion Equation in Sorbent Grains  
The differential form of the law of mass conservation, applied to a grain of a sorbent in which gas in a free and sorbed form occurs, is expressed as follows [29]:  
\[\frac{\partial\big{(}\rho_{z}+\psi\rho_{g}\big{)}}{\partial t}+\text{div} \Big{(}\rho_{g}\cdot\overrightarrow{v}\Big{)}=0 \tag{5}\]  
where:  
\(\overrightarrow{v}\) is the diffusion process rate (m/s). The equation of gas motion is expressed by [PERSON]'s law [17]:  
\[\overrightarrow{v}=-\frac{k}{\mu}\cdot\text{grad}p \tag{6}\]  
where:  
\(k\) is the permeability coefficient (m\({}^{2}\));  
\(\mu\) is the gas dynamic viscosity (N\(\cdot\)s/m\({}^{2}\)). It can be assumed that the permeability coefficient, \(k\), is constant and independent of both the radius and time. Once the motion Equation (6) is substituted into the continuity equation--while taking into account the gas state Equation (2) and linear sorption isotherm Equation (3) and introducing the dimensionless gas density, \(\xi(r,t)=\frac{\rho_{g}(r,t)}{\rho_{max}}\) (\(\rho_{max}\)--the maximum, initial gas density)--we obtain the following formula:  
\[\frac{\partial\xi}{\partial t}=D\cdot\
abla^{2}\xi^{2} \tag{7}\]  
where:  
\(D=\frac{k\cdot C\cdot\rho_{max}}{2\cdot(H\cdot C+\psi)}\) is the diffusion coefficient (m\({}^{2}\)/s). Equation (7) is analyzed in a spherical coordinate system. According to the conditions of the experiment, the equation should be solved for the following boundary-initial conditions:  
\[\left\{ \begin{array}{ccc}\rho_{S}(r,0)=\rho_{max,}&\xi(r,0)=1,&0\leq r\leq r_{0 },\\ \rho_{S}(r_{0},t)=0,&\xi(r_{0},t)=0,&t\geq 0,\\ \frac{\partial\xi}{\partial r}=0,&\frac{\partial\xi}{\partial r}=0,&t\geq 0,r=0.\end{array} \right. \tag{8}\]  
The first condition means that, at the initial moment, the density of free gas inside a grain equals \(\rho_{max}=\frac{p_{0}}{C}\), where \(p_{0}\) is the pressure of a gaseous sorbate saturating a porous sorbent. The second condition means that, outside a grain, gas density equals zero. The third condition concerns the symmetry of the distribution of free gas density in relation to the center of a grain of a sorbent. Equation (7), with its boundary-initial conditions (8), can be solved in a numerical way by applying the finite element method (FEM). A condition for solving the equation is knowledge of the diffusion coefficient, \(D\). The density of gas in a grain of a sorbent may change to a significant degree, which is why the analyzed equation is nonlinear. However, if we investigate the final or the initial stadium of the process kinetics, we shall see that the gas density within a grain changes to a much lesser extent. Therefore, in these particular stages of the process, the diffusion equation can assume the linear form:  
\[\xi=\xi_{0}+\xi^{\prime} \tag{9}\]  
where:  
\(\xi_{0}\) is the density of gas after the time period in which 80% of the sorbed gas was released. After substituting Equation (9) into Equation (7), while neglecting the small quantities of the second order, \(\left(\xi^{\prime}\right)^{2}\), we shall obtain the following formula:  
\[\frac{\partial\xi^{\prime\prime}}{\partial t}=2\cdot D\cdot\xi_{0}\cdot\
abla ^{2}\xi^{\prime} \tag{10}\]  
Equation (10) is the linear diffusion equation, whose solution, under condition (8), is expressed by the series given below [37]:  
\[\xi^{\prime}=1-\frac{2\cdot r_{0}}{\pi\cdot r}\cdot\sum_{n=1}^{\infty}\frac{ \left(-1\right)^{n+1}}{n}\cdot\ sin\left(\frac{n\cdot\pi\cdot r}{r_{0}}\right) \cdot exp\left(-\frac{n^{2}\cdot\pi^{2}\cdot D^{\prime}\cdot t}{r_{0}^{2}} \right), \tag{11}\]  
where:  
\(D^{\prime}=2\cdot D\cdot\xi_{0}\left(\text{m}^{2}/\text{s}\right)\). In the course of the experiment, it is possible to determine the amount of gas released from a grain--thus, we are interested in the average gas density inside a grain.