4.1. Dynamic Characteristics of Methane Adsorption
Figure 2,
Figure 3 and
Figure 4 show the plots of pressure response and adsorption versus time under different pressure step at 30, 40 or 50 °C, respectively. Obviously, it is shown that the pressure drops fast at first, then slowly in the middle and reaches and maintains stability at last. Meanwhile, the absolute amount of methane adsorption increases quickly initially, then slowly in the middle and reaches a constant at last. These two curves, which describe the adsorption dynamic of methane, have similar characteristics to the other studies [
8,
24], indicating that the methane molecules are mainly adsorbed in the initial stage.
To be more specific, at the beginning, most of the active surface sites are vacant and favorable for methane molecules adsorption on shale powders because the adsorption rate is positively correlated with the available vacancies [
25]. Furthermore, at first, the high concentration driving force between the methane molecules spurs the mass free phase gas to the adsorbed phase gas. Additionally, in the middle, the repulsion of methane molecules gradually becomes the major force to determine the adsorption dynamic of methane molecules [
35]. Therefore, the tendency of the adsorption dynamic curves demonstrates the synthetic influence of the high concentration driving force and repulsion of methane molecules on the adsorption dynamic of methane molecules.
4.2. Dynamic Model Fitting
Figure 5 shows the continuous change of
versus ln(
t) under seven pressure steps using Equation (3) based on the Bangham model at 30 °C. It is clearly shown that
increases linearly with the increasing ln(
t) because the correlation coefficients (R
2), respectively, are 0.9253, 0.9418, 0.9435, 0.9745, 0.9706, 0.9655 and 0.8585, as listed in
Table 1, with the average of 0.9400. Therefore, q
t can be well fitted with t by using the Bangham model at each pressure step.
The fitting equations, the Bangham model, the adsorption rate constant and the constant z are listed in
Table 1. At different pressure steps, the adsorption rate constants are 2.2632, 2.7286, 2.9761, 3.1221, 3.3848, 3.5758 and 3.7014, respectively, indicating that the adsorption rate constant increases with the equilibrium pressure increasing. The constant z, respectively, is 0.1457, 0.1295, 0.1259, 0.1076, 0.1145, 0.1244 and 0.1379, revealing that z (a constant of the Bangham model) first decreases and then increases with the equilibrium pressure increasing.
Figure 6 shows the relationship between
and ln(
t) under six pressure steps by using Equation (3) at 40 °C. It can be seen that
increases linearly with ln(
t) increasing because the correlation coefficients at each pressure step, respectively, are 0.9938, 0.9430, 0.9411, 0.9600, 0.9512 and 0.9355, as listed in
Table 2, with the average of 0.9541. Therefore, the Bangham model can be well fitted in the relationship between
qt and t for different pressure steps.
The fitting results, including the fitting equations, the Bangham model, the adsorption rate constant and the constant z are listed in
Table 2. At six pressure steps, the adsorption rate constants are 1.7074, 2.4665, 2.8255, 3.1015, 3.4404 and 3.4411, respectively, indicating that k (adsorption rate constant) increases with the equilibrium pressure increasing. The constant z, respectively, is 0.1993, 0.1795, 0.1469, 0.1240, 0.1544 and 0.2012, revealing that z (a constant of the Bangham model) first decreases and then increases with the equilibrium pressure increasing.
Figure 7 shows the plots of
and ln(
t) under six pressure steps using Equation (3) at 50 °C. It is clearly shown that
increases linearly with ln(
t) increasing because the correlation coefficients at each pressure step, respectively, are 0.9170, 0.9743, 0.9864, 0.9610, 0.9411 and 0.9964, as listed in
Table 3, with the average of 0.9627. Thus,
qt can be well fitted with t by using the Bangham model for six pressure steps.
Table 3 lists the fitting results, including the fitting equations, the Bangham model, the adsorption rate constant and the constant z. K (adsorption rate constant) at six pressure steps, respectively, is 1.3814, 1.9866, 2.5544, 2.8499, 3.1000 and 3.2091, indicating that k increases with the equilibrium pressure increasing. The constant z (a constant of the Bangham model), respectively, is 0.2086, 0.1791, 0.1508, 0.1415, 0.1514 and 0.1932, revealing that z first decreases and then increases with the equilibrium pressure increasing.
4.4. Effect of Temperature on Adsorption Rate Constant
The plot of the adsorption rate constant versus equilibrium pressure at 30, 40 and 50 °C is shown in
Figure 9 to illustrate the temperature effect on the adsorption dynamic of methane molecules on shale powders. Obviously, it is shown that the adsorption rate constants all slowly drop with the equilibrium pressure increasing under different temperature conditions, revealing that it is much easier for methane molecules to adsorb on gas shale powders at a higher pressure. This is mainly because at a higher pressure condition, the high concentration driving force is the main controlling force that can promote the adsorption rate of methane molecules. Moreover, at the same pressure point, a smaller Bangham adsorption rate constant is attained due to a higher temperature, which indicates low temperatures are favorable for methane adsorption on shale powders. This is mainly because the methane adsorption dynamic on shale powders is exothermic.
Furthermore, to quantitatively analyze the relationship between the adsorption rate constant and the equilibrium pressure, the linear correlation relationships between
k (adsorption rate constant) and ln(
P) at 30, 40 and 50 °C are plotted in a semi-logarithmic coordinate system, as shown in
Figure 10. The fitted results are, respectively, expressed as follows:
The correlation coefficients at 30, 40 and 50 °C, respectively, are 0.9932, 0.9924 and 0.9937, indicating that the adsorption rate constant is linearly positively correlated with the natural log of the equilibrium pressure.
4.5. Effect of Temperature on Adsorption Isotherm
To compare the difference in the adsorption amount at different temperatures, the adsorption amounts under different equilibrium pressures at 30, 40 and 50 °C are plotted versus the equilibrium pressure in
Figure 11. It is clearly shown that the adsorption amount under the different equilibrium pressure at 30 °C is the biggest, followed by that of 40 °C and 50 °C, which indicates low temperatures are favorable for methane adsorption on shale powders.
Figure 12 shows the plot of adsorption amount versus equilibrium pressure in a logarithmic coordinate system at each stable temperature. Obviously, it is shown that the adsorption amount is linearly positively correlated with the equilibrium pressure. The fitted results, including the fitted equation, the correlation coefficient (R
2), the Freundlich model, the Freundlich constant K and the constant
n, are listed in
Table 4. The correlation coefficients at 30, 40 or 50 °C, respectively, are 0.9945, 0.9987 and 0.9925, indicating that the relationship between the adsorption amount and the equilibrium pressure can be well fitted by the Freundlich model. K (Freundlich constant) at 30, 40 and 50 °C, respectively, is 5.1487, 2.1062 and 1.7857, indicating that K decreases with the temperature increasing. The constant
n, respectively, is 0.2182, 0.2120 and 0.1967, revealing that the constant
n decreases with the temperature increasing. Therefore, low temperatures are favorable for methane adsorption on shale powders, and high temperatures can obviously reduce constant K and
n of the Freundlich model.
4.6. Effect of Temperature on Isostatic Enthalpy
The isostatic enthalpy of methane adsorption is derived from the Van’t Hoff equation, and it is expressed as follows [
16]:
where
P is the pressure in kPa,
T is the temperature in K,
n is the absolute adsorption amount, R is the ideal gas constant in kJ/mol, and
is the enthalpy of adsorption in kJ/mol.
Equation (10) can be integrated and rearranged, and the linear form of this model can be expressed as
where
,
. The plot of ln
P versus 1/
T should be fitted as a linear equation, and then,
can be calculated according to the slope of a.
Figure 13 shows the plot of ln
P versus
n (adsorption amount) at 30, 40 or 50 °C. The fitted results, including the temperature, the fitting equation, the correlation coefficient (R
2) and the parameters of the fitted equation, are listed in
Table 5. It is clearly shown that there exists a well-linear relationship between ln
P and
n because the correlation coefficients, respectively, are 0.9471, 0.9142 and 0.9205, with the average of 0.9273. The slopes of the fitted equation increase with the temperature increasing. Moreover, by using the fitted equation listed in
Table 5, the values of ln
P at different temperatures are calculated and shown in
Table 6.
The relationship of ln
P and 1/
T under different given adsorption amounts is shown in
Figure 14, and the fitted results, including the adsorption amount, the fitted equation, the correlation coefficient (R
2) and the parameters of the fitted equation, are listed in
Table 7. It can be seen that ln
P is linearly positively correlated with 1/
T because R
2 is distributed between 0.8781 and 0.9974, with the average of 0.9705. Furthermore, isostatic enthalpy can be obtained, and the plot of isostatic enthalpy versus adsorption amount is shown in
Figure 15. Obviously, it is shown that there exists a good linear relationship between isostatic enthalpy and the adsorption amount, indicating that isostatic enthalpy is linearly positively correlated with the adsorption amount.