# Nonlinear and Linear Equation of Gas Diffusion in Coal--Theory and Applications
## 2 Assumptions of the Gas Diffusion Equation in a Granular Sorbent  
In order to provide a theoretical description of the emission of a gaseous sorbate from a grain of a sorbent, we shall look into the equation of the diffusion process with its source in desorption of a sorbate bound with a sorbent. We shall assume that the sorbent grains have a shape of a sphere, whose radius is \(r_{0}\). Inside a grain, a gaseous sorbate in two states occurs: pore-located free gas and sorbed gas [32; 33; 34].  
Let us denote the mass of free gas as \(m_{g}\), the mass of sorbed gas as \(m_{z}\), and the grain volume as \(V_{w}\). The porosity of the grain, understood as the relationship of the pores' volume to the total grain volume, equals \(\psi\). For these denotations, the densities of free gas, \(\rho_{g}\), and sorbed gas, \(\rho_{z}\), shall be expressed with the following formulas:  
\[\rho_{g}=\frac{m_{g}}{\psi.V_{w}},\ \rho_{z}=\frac{m_{z}}{V_{w}} \tag{1}\]  
where:  
\(\rho_{g}\) is the density of free gas (kg/m\({}^{3}\));  
\(\rho_{z}\) is the density of sorbed gas (kg/m\({}^{3}\));  
\(m_{g}\) is the mass of free gas (kg);  
\(m_{z}\) is the mass of sorbed gas (kg);  
\(V_{w}\) is the grain volume (m\({}^{3}\));  
\(\psi\) is the porosity (-).  
Generally, in the case of a non-stationary process, densities \(\rho_{g}\) and \(\rho_{z}\) are functions of time and a radius, \(\rho(r,\ t)\). Both the theoretical and experimental analyses [5] demonstrate that the process of sorption proper, understood as the transition of gas molecules from the bonded state to the free state, is almost instantaneous. Therefore, let us assume that sorption has no impact on the speed with which gas penetrates the pore structure of a sorbent. It also needs to be assumed that the process of gas emission from a grain is isothermal in its nature. Free gas is treated as ideal gas, and thus the relationship between pressure and gas density is expressed as follows:  
\[p=C\cdot\rho_{g} \tag{2}\]  
where:  
\(p\) is the gas pressure (Pa);  
\(C=\frac{R\cdot T}{M}\) (m\({}^{2}/s^{2}\));  
\(R\) is the universal gas constant (J/(mol\(\cdot\)K));  
\(T\) is the temperature (K);  
\(M\) is the molecular mass of gas (kg/mol).  
The next assumption concerns the sorption isotherm. In the case of the processes occurring under low sorbate pressure, it might be assumed that the sorption isotherm is linear and that it can be described by means of [PERSON]'s law [36]:  
\[\rho_{z}=H\cdot p=H\cdot C\cdot\rho_{g} \tag{3}\]  
where:  
\(H\) is [PERSON]'s coefficient (s\({}^{2}\)/m\({}^{2}\)).  
The relationship between the density of sorbed gas, \(\rho_{z}\), and the amount of sorbed gas, \(a_{m}\), expressed in cubic meters of gas per one kilogram of sorbent and adapted via calculation to the standard conditions of pressure and temperature, is demonstrated by the following formula:  
\[\rho_{z}=\frac{m_{z}}{V_{w}}=\frac{m_{w}\cdot M\cdot p_{N}}{R\cdot T_{N}\cdot V _{w}}a_{m}=\frac{\rho_{w}\cdot M\cdot p_{N}}{R\cdot T_{N}}a_{m}=S\cdot a_{m} \tag{4}\]  
where:  
\(m_{w}\) is the grain mass (kg);  
\(T_{N}\) is the temperature under standard conditions (K);  
\(p_{N}\) is the pressure under standard conditions (Pa);  
\(\rho_{w}\) is the coal density \(\frac{m_{w}}{V_{w}}\) (kg/m\({}^{3}\));  
\(a_{m}\) is the amount of sorbed gas (m\({}^{3}\)/kg);  
\(S=\frac{\rho_{w}\cdot M\cdot p_{N}}{R\cdot T_{N}}\) is the constant value (kg\({}^{2}\)/m\({}^{6}\)).