Prediction of methane adsorption capacity in different rank coal at low temperature by the Polanyi‐based isotherm model

To investigate the adsorption properties of methane in coal under low temperatures, the isothermal adsorption tests of the three coal samples with different metamorphic degrees were conducted at the ambient temperatures of 253.15−293.15 K, and the low‐temperature nitrogen adsorption (LNA) tests of coal were also performed. Then a relational expression of equilibrium pressure, temperature, and methane adsorption capacity (T−P model) was deduced to predict the adsorption isotherm at any other temperature based on the Polanyi adsorption theory. The results show that the gas adsorption capacity of coal can be significantly increased at low temperatures (below 273.15 K), and the adsorbed methane in anthracite is obviously more than that in lean coal and coking coal by the virtue of possessing a larger micropore/transition pore volume and specific surface area. The relations between adsorption potential (ε) and adsorption volume (ω) at different temperatures can be drawn on one single logarithmic curve, and a suitable pseudo‐saturation pressure can be obtained by the improved Amankwah's method. The predicted adsorbed capacities via the T−P model are in line with the measured results at other equilibrium conditions, indicating that the model can contribute to the deep coalbed methane resources estimation and the gas disaster prevention and control in coal mines.

reservoirs due to coalbed as hydrocarbon source rocks, which results in that there are mainly two types of gas states in coal formation: free gas and adsorbed gas. [4][5][6] The adsorbed gas on the internal surface of coal micropores makes up a sizeable share rather than the free gas in the fracture space. Thus, it is significant to understand the adsorption characteristics of methane on coal not only for CBM resource evaluation and productivity prediction but also for evaluating the potential severity of gas disasters in new mines or in unmined areas of existing mines. 7 The isothermal adsorption experiments targeted at the organic-rich coal from different formations are commonly used to investigate the methane adsorption behavior on coal, which reveals the influences of various factors on the methane adsorption capacity, including gas pressure, 8,9 temperature, 10 moisture, 11,12 organic matter, 13 clay minerals, 8 pore structure, 14,15 and fracturing fluid, 16,17 and so forth. The pressure and temperature of CBM reservoirs are both increased with the increased depth. The methane adsorption on coal reservoirs is all above the critical temperature of methane (190.6 K). When the reservoir pressure is higher than 4.59 MPa, it belongs to a type of supercritical adsorption. Therefore, a series of isothermal adsorption tests at different temperatures are needed to estimate the adsorption capacity of methane on the relevant reservoir occurrence conditions. For the methane adsorption modeling, many scholars have made valuable contributions, in which the Langmuir equation (single site) is the most widely used by the virtue of simplicity and good fitting effect of adsorbed quantity under low-pressure conditions. 18,19 However, the monolayer assumption of the Langmuir model and the saturation vapor pressure of pore-filling model [20][21][22] are not valid for describing the supercritical methane adsorption in coal, as these models are proposed for describing the subcritical gas adsorption behavior, and there are distinguishable differences between the measured and the actual adsorption values of the reservoir at high pressures. 23 Besides this, the Langmuir model can not predict the adsorption isotherms beyond test conditions, impeding its application in the deep CBM resources estimation. 24 For physical adsorption, the interaction between adsorbate and adsorbent mainly depends on the dispersion force, as well as called van der Waals force, which is independent of temperature. 25 Thus, the Polanyi adsorption potential theory 26 provides a framework for such predictions, relying on the characteristic curve of an adsorbate−adsorbent system. In the theory, any adsorbate molecule near the adsorbent surface has some adsorption potential, which is a function of its proximity to the surface and the nature of the adsorbate. Based on the relationship between the adsorption potential and the adsorbate volume, the adsorption isotherms at different temperatures can be transformed into a single characteristic curve. Dubinin et al. 27 successfully applied the theory to vapor adsorption equilibrium on activated carbon and zeolite for nonpolar and weakly polar substances. Manes et al. 28 extended the theory to the adsorption in an aqueous solution. Subsequently, the Dubinin−Astakhov (D−A) equation 29 was used to describe the characteristic curves of CH 4 and CO 2 adsorption on activated carbons. Du et al. 30 investigated adsorption isotherms of hydrogen on four kinds of zeolites above the critical temperature at different pressures. Giraldo et al. 31 applied the D−A equation to describe the characteristic curves for asphaltene on a rock surface. Liu et al. 32 deduced a calculation model of partition coefficient of volatile organic compounds to study the relations between air emission and building materials. Askalany and Saha 33 proposed a new thermodynamic equation on adsorbed phase volume to calculate the isosteric heat of adsorption. Then, an exponential equation 34 was proposed in the prediction of CO 2 and CH 4 adsorption on carbon molecular sieves.
The reduction of gas desorption capacity and kinetics by cooling coal samples below 273.15 K is probably a feasible way to improve the accuracy of CBM content estimation. 35 Although the D−A equation 36 and the polynomial 37 were also applied to discuss the adsorption and thermodynamic characteristics of methane on coals, very few studies on the methane adsorption on coal at low temperature were conducted via the adsorption potential theory by far. In the current work, we tried to establish the relations between gas pressure, temperature, and methane adsorption capacity on coal. By choosing the different rank coal, the isothermal adsorption tests and the LNA tests were conducted to explore the coal affinity on methane at 253.15−293.15 K. Based on the Polanyi adsorption theory, a relational expression (T−P model) of adsorption capacity, pressure, and temperature was then deduced and verified to predict methane adsorption isotherms at other equilibrium conditions, which can contribute to the CBM storage historical course, resource estimation, and gas disaster prevention.

| Adsorbent preparation
The three fresh coal samples with different ranks as adsorbents were respectively collected from Xingwu coal mine (Liulin, Shanxi province of China), Liulong coalmine (Liuzhi, Guizhou province of China), and Jiulishan coal mine (Jiaozuo, Henan province of China), as shown in Figure 1. In the reservoir conditions, the obtained samples are more similar to the molded coal than coal powder by the compaction action. Therefore, the three kinds of coal samples were firstly ground and sieved using 0.17−0.25 mm metal sifters, and evenly mixed with an appropriate amount of distilled water. Then they were placed into a mold and loaded with 60 kN pressure for 30 min. After molding and dehydration for 3 h at 378.15 K, the prepared samples were stored in a dehydrator for later use. The physical parameters of coal samples were determined by Chinese national standards (GB/T 212-2008, 217-2008, 6949-1998) as shown in Table 1: ash content (A ad ), volatile matter (V ad ), moisture (M ad ), true relative density (TRD 20 20 ), apparent relative density (ARD 20 20 ) , firmness (f) and porosity. The mass of dry ash-free basis in coal (m r ) is evaluated as follows: where m t is the total mass of coal sample, kg.

| Isothermal adsorption experiments
The isothermal adsorption tests of methane (at 253. 15, 263.15, 273.15, 283.15, and 293.15 K) on the coal samples with different ranks were separately conducted with the self-developed low-temperature isothermal apparatus ( Figure 2). It consists of an airtight sample canister, a low-temperature incubator, vacuum degassing devices, quantitative gas charging system, a data acquisition system, and pipelines. The operating temperature range of the incubator is 223.15−373.15 K. Based on the volumetric method, 12 the general procedures of isothermal adsorption are shown as follows: (1) Place the molded coal samples into the coal canister and double-check the tightness of the whole test system with soap water.
(2) Vacuum the coal canister below 10 Pa, and then charge a certain amount of methane into the canister via the reference cell. The adsorbed gas content is calculated by the following equation 38 : where m g is the total mass of charged gas, kg; m a is the mass of adsorbed gas; m f is the mass of free gas; ΔP is the pressure difference of the reference cell when the sorption system reaches equilibrium state at the setting pressure, MPa; V cell reference is the volume of reference cell (here the gas reduction in reference cell is actually the total amount of charged gas into coal canister), cm 3 ; P 1 is the gas pressure in coal canister; V f is the volume of free gas in coal canister; T is the ambient temperature, K; Z is the compressibility coefficient of methane, derived by the Redlich−Kwong equation; M is the molar mass of methane, 16 g/mol; and R is the universal gas constant, 8.314 J/(mol K). (3) Monitor the pressure variation in the coal canister to determine the point of the adsorption equilibrium state or suspend the adsorption time. Only when the gas pressure remains unchanged at the set temperature for 12 h, can the equilibrium point be reached and the adsorption capacity be obtained. (4) Repeat Steps 2−3 until all the scheduled adsorption pressures are reached, and eventually, the gas adsorption isotherms can be drawn at different temperatures.

| Low-temperature nitrogen adsorption (LNA) tests
Affected by various coalification processes, the pore structure in coal has different shapes and sizes, not only rich in micropores and transition pores but also macropores, visible pores, and fractures, which provide a place and channel for gas adsorption, storage, and migration. 39 LNA method is conducive to understanding the nanopore (<100 nm) structure of coal. Therefore, the LNA experiments were conducted with an ASAP2020 specific surface area analyzer (Micromeritics). Firstly, the three coal samples were ground and sieved to 0.5 mm, and then dehydrated at 353.15 K for 6 h. After the coal samples were cooled, they were respectively tested by pressurized nitrogen injection under the low-temperature condition of 77 K, and then the adsorption isotherms can be drawn according to the corresponding N 2 adsorption capacities in different relative pressures. The specific surface area and pore volume of coal were calculated by the BET multilayer adsorption model and BJH model, respectively.

| Polanyi adsorption potential theory
The adsorption potential (ε) is a function of its proximity to the surface and the nature of the adsorbate, which can be quantitatively described as follows 40 : where P s is the saturated vapor pressure of methane, MPa; and P 1 is the adsorption equilibrium pressure, MPa.
When the temperature is below the critical temperature, the saturation vapor pressure of methane can be obtained via the National Institute of Standards and Technology (NIST). However, the methane adsorption on coal surface at high pressure is a type of supercritical adsorption, and the vapor pressure would lose physical meaning above the critical temperature (190.6 K). Thus, the concept of pseudo-saturation vapor pressure was proposed, and the saturated vapor pressure P s can be derived from the Amankwah's improved equation as follows 41 : where P c is the critical pressure of methane, 4.62 MPa; T c is the critical temperature of methane, 190.6 K; m is a specific parameter related to the adsorption system. The adsorbed phase volume of the absolute adsorption capacity is defined as the volume at the boundary where the density is practically equal to the bulk density. Hence, the volume of adsorbed phase (ω) can be evaluated as follows 42 : where n ad is the absolute adsorbed amount of methane at the equilibrium pressure, mol; ρ ad is the density of the adsorbed phase, g/cm 3 .
Here, we adopt the empirical equation proposed by Ozawa et al. 42 to determinate the density of the adsorbed phase as follows: where ρ b is the methane density at the normal boiling point, 0.424 g/cm 3 ; and T b is the boiling point of methane, 111.5 K.

| Adsorption isotherms
On the assumptions of monolayer adsorption and uniformity on the adsorbent surface, the Langmuir model (Equation 3) is widely used to describe the adsorption characteristics of methane owing to simplicity and functionality. The adsorption isotherms of different coal samples are shown in Figure 3 at the ambient temperatures of 253.15−293.15 K.
where Q is the adsorbed methane quantity by the dry ash-free basis in coal under specific pressure and temperature, cm 3 /g; P is the equilibrium pressure, MPa; a and b are the adsorption constants. Figure 3 shows that the adsorbed methane capacities of different coal samples all increase with the ambient temperature drop, and the adsorbed quantities below 273.15 K are significantly larger than these above 273.15 K. At the same temperature, the adsorbed methane quantity in coal increases with the equilibrium pressure rising. When the pressure reaches a threshold value, the adsorbed methane tends to be stable and saturated. Furthermore, the adsorbed quantity in Jiulishan anthracite is obviously larger than that in Xingwu coking coal and Liulong lean coal.
The effect of temperature on the methane adsorption capacity can be reflected by the Langmuir adsorption constants a and b. Adsorption constants a represents the saturated adsorption capacity on the total coal surface, and b is affected by the physical properties of adsorbent, the adsorbate−adsorbent interaction, and the state of the adsorption system. The variations of adsorption constants a and b with temperature are listed in Table 2.
As shown in Figure 4, the adsorption constants a and b of the three coal samples all decrease with the temperature rising. The saturated adsorption capacity a at the low temperature (below 273.15 K) is obviously higher than that at 293.15 K, which is mainly related to the changes in the kinetic energy and the free path of methane molecules. Cooling weakens the kinetic energy of gas molecules, and more gas molecules would be adsorbed on the surface of coal when the kinetic energy is lower than the adsorption potential well. Additionally, the average free path of a gas molecule is decreased by cooling, 43 and therefore, the where λ is the average free path of methane molecule; d 0 is the effective diameter of the methane molecule, nm; K is the Boltzmann constant, 1.38 × 10 −23 J/K.

| Pore structure characteristics
The LNA curves of the three samples slightly differ in shape ( Figure 5), and they all accord with the characteristics of Ⅱ-type adsorption isotherm based on the IUPAC classification criteria. In the low-pressure range (0 < P/P 0 < 0.2), the N 2 adsorption quantity increases slowly with the relative pressure rising; the adsorption curve is almost flat with no increment of adsorption ZHAO ET AL.
| 1389 quantity in the middle-pressure section (0.2 < P/ P 0 < 0.8), which indicates that N 2 adsorption on the coal pore surface has reached saturated state; and the adsorption curve increases sharply with the relative pressure further rising (0.8 < P/P 0 < 1), which is related to the multilayer adsorption and capillary condensation existing in pores. The desorption curve of each coal sample is associated with the pore morphology. There is an obvious adsorption hysteresis loop in the high-pressure section of each coal. It can be roughly estimated that a certain amount of open-type pores exist in each sample. When the relative pressure approaches 0.5, there is a sharp downward inflection point in the desorption curves of coking coal and lean coal, indicating that the semiopen flask pores account for a certain proportion. When the relative pressure is below 0.5 (pore size < 4 nm), the desorption curves of coking coal and lean coal almost coincide with the adsorption curves, indicating that these micropores are sealed at one end and have poor connectivity. However, there is no sudden drop point near the relative pressure of 0.5 in the anthracite desorption curve, indicating that open-type pores account for the main part of micropores in anthracite.
To directly compare the distribution of the BETspecific surface area of different rank coal, the results of LNA tests were drawn into a histogram, as shown in Figure 6. It can be seen that the cumulative specific surface area of the three samples gradually increases with the increase of coal metamorphic degree; and the total pore area of anthracite is 2.77 m 2 /g, which is significantly larger than that of coking coal and lean coal. The micropores of the samples account for the largest proportion of the total surface area, in which the micropore proportion of anthracite reaches 70.36%, so the adsorption capacity of anthracite is the strongest among them. Additionally, the transition holes of coal samples also account for a high proportion, ranging from 26.07% to 54.35%. The mesopore proportion of samples can be almost ignored, which is mainly related to the measurement range of pore size of 1.9−300 nm by the LNA tests.
The distribution of BJH pore volume of coal samples is shown in Figure 7. The total pore volume of the three samples also gradually increases with the increase of coal metamorphic degree. But unlike the distribution of BET special surface area, welldeveloped mesopores make a major contribution to BJH pore volume for coking coal and lean coal, accounting for 71.43% and 66.23%, respectively. The total pore volume of the anthracite sample is 0.011 ml/g; the mesopore proportion is 36.36%, and the proportions of transition pores and micropores cannot be ignored, which reach 45.46% and 18.18%, respectively. Therefore, anthracite coal should be of stronger gas adsorption and reservoir capacities.

| Polanyi-based isotherm model
To explore the methane adsorption characteristics on coal via the Langmuir model, it is inevitable to carry out isothermal adsorption experiments at multiple temperature points, which brings difficulties to the prediction of CBM content under extreme conditions. The advantage of the adsorption potential theory is that the adsorption isotherm at any temperature can be evaluated based on the relationship between the adsorption potential (ε) and the adsorption volume (ω) at one temperature. It is before establishing the characteristic curve of adsorption potential. Here the specific parameter m is temporarily taken as 3, and then the adsorption characteristic curves of ε−ω are fitted based on the adsorption results of the  Figure 8. It can be seen from Figure 8 that the ε−ω characteristic curves of coal samples at different temperatures are almost on the same curve, which means the ε−ω curve is temperature-independent. Consequently, the interaction between coal and methane molecules is mainly on dispersion forces, and the process belongs to physical adsorption. The relation between ε and ω conforms to the logarithmic equation (ε A ω B = − ln + ), where A and B are the fitting constants.

| Derivation of the T−P model
Since the relation between the adsorption potential (ε) and the volume of adsorbed phase (ω) conforms to the logarithmic equation (ε A ω B = − ln + ), and there is a linear relation between ω and the adsorbed quantity (Q), Equation 9 can be obtained as follows: where f and g are also the fitting constants. By taking exponents on both elements of the above equation, we can get: The above equation can be also expressed as a logarithmic form: Then, the following equation can be written by taking where M and N are defined as the characteristic constants of adsorption isotherm.
Here, the experimental adsorption results of coal samples at 283.15 K are adopted to calculate the characteristic constants M and N ( Figure 9). Hence, M XW = 0.361, N XW = 2.478; M LL = 0.367, N LL = 2.663; and M JLS = 0.237, N JLS = 3.30. By substituting these constants into Equation 11, the adsorption prediction relations of the coal samples are given as follows:

| Prediction of adsorption isotherms
To validate the prediction performance of the T−P model in adsorption isotherms, the predicted adsorption capacities of coal samples at 263.15 and 253.15 K are calculated to compare with the measured values, as shown in Figure 10. The results show the predicted adsorption capacity of methane by the T−P model is very close to that by the isothermal tests under the corresponding pressures. Overall, the predicted adsorption capacity is slightly smaller than the measured at the equilibrium pressures of 0−7 MPa, and if taking the difference ratio between predicted and measured value as the relative error, its ranges are within 6%. Despite the same value of specific parameter m being adopted at each temperature point, the relative error is not large enough to affect the predicted results of the T−P model. The predicted adsorption capacity also increases with the equilibrium pressure rising, and the adsorption capacity of anthracite is larger than these of coking coal and lean coal at the same equilibrium conditions.
The predicted adsorption isotherms of the coal samples by the T−P model are drawn in Figure 11 at the temperatures of 243.15−333.15 K. It is noted that the temperature and pressure values applied in prediction should meet as close to the practical conditions of the reservoir as possible. Consequently, it can be concluded that the T−P model proposed here is applicable in the estimation of methane adsorption capacity of coking coal at any temperature and pressure, which can provide a reliable reference for the estimation of deep CBM resources and the design of coal mine gas prevention and control.

| CONCLUSIONS
The isothermal adsorption tests of different rank coal were conducted at the temperatures of 253.15−293.15 K, and the LNA tests of coal were performed to analyze the pore structure characteristics of samples. Based on the Polanyi adsorption theory, a relational expression of equilibrium pressure, temperature, and adsorption capacity (T−P model) was deduced and verified to predict the methane adsorption isotherm at any other temperature. The following conclusions can be drawn: (1) The methane adsorption capacity of coal samples can be increased at low temperatures (below 273.15 K); furthermore, the adsorbed quantity of anthracite is significantly larger than that of coking coal and lean coal by the virtue of possessing a larger micropore/transition pore volume and specific surface area. (2) In the Polanyi isotherm model, the relations between adsorption potential (ε) and adsorption volume (ω) at different temperatures are drawn on a single logarithmic curve, and then a suitable pseudo-saturation pressure is obtained by the improved Amankwah's method; consequently, a relational expression of adsorption capacity, pressure, and temperature (T−P model) is proposed.