Pure methane, carbon dioxide, and nitrogen adsorption on anthracite from China over a wide range of pressures and temperatures: experiments and modeling

Abstract

The adsorption isotherms and kinetics characteristics were investigated at 294 K, 311 K, 333 K, and 353 K with pressures up to 18 MPa for CH₄ and N₂ and 5.5 MPa for CO₂ on anthracite from China using a volumetric method. The excess adsorption for N₂ belongs to the type I isotherm, while the adsorption capacity for CH₄ and CO₂ initially increased, followed by a sequence decrease with increasing pressure. The preferential ratio of the maximum in the excess adsorption for N₂ : CH₄ : CO₂ are 1 : 1.1 : 1.6, 1 : 1.3 : 1.8, 1 : 1.4 : 1.9, and 1 : 1.2 : 1.6 at 294 K, 311 K, 333 K, and 353 K, respectively. In addition, the excess adsorption capacity was predicted using the Langmuir + k and simplified Ono–Kondo lattice models. The Langmuir + k monolayer model has higher accuracy in modeling pure gas adsorption on coal, especially for CH₄ and N₂. Moreover, the pressure decay method was used to analyse the adsorption kinetics of gases on coal. It is observed from the kinetics data that the adsorption rate and the effective diffusivity increase with the increasing pressure for CH₄ at 0.55–6.97 MPa and N₂ at 0.63–10.05 MPa. However, for CO₂, an increase in pressure reduces the adsorption rate and the effective diffusivity at 0.11–3.96 MPa due to intensive gas molecule–molecule collisions and the strong coal matrix adsorption swelling. The adsorption rate and the effective diffusivity with temperature are similar for the three gases, which increase with increasing temperature. Through verification of the experimental data, the diffusion model can be used to model the kinetics data of gases on coal under low and medium pressure conditions.

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1 Introduction

Facing the double crisis of environmental concerns and the energy crisis in the world, it is necessary to reduce CO₂ emissions and to exploit unconventional natural gas resources. CO₂-ECBM (enhanced coal bed methane recovery) technology is a possible approach to the geological storage of CO₂, and can recover coal bed methane by injecting CO₂ into the deep coal seam.^1,2 In addition, N₂-ECBM has been applied as an alternative alone or together with CO₂ to solve the problem of coal swelling during pure CO₂ injection.^3–6 CO₂ storage and CH₄ production are controlled by the relative affinity of the two gases to the adsorption sites and their adsorption rates and relative transport, which are related to the thermodynamic and kinetic processes. It is important to estimate the storage capacity of coal and to describe the adsorption/desorption kinetic characteristics in deep coal seams. Therefore, measurements of the adsorption isotherms involve wide temperature and pressure ranges.

More and more experimental data about CH₄, CO₂ and N₂ adsorption on coal have been published relating to high pressures and reservoir temperatures in recent years.^7–17 For CH₄, Krooss et al.¹⁰ found that the excess adsorption increased monotonously with the increasing pressure at 60 and 80 °C and with the pressure up to 20 MPa. Also, the same conclusion was also proposed by Busch et al.¹⁸ at 22 °C and for a pressure up to 11 MPa. However, others presented an initial increase, followed by a decreasing trend at higher pressures.^11,14,19,20 So it is necessary to perform experiments on CH₄ adsorption on coal. The Langmuir + k monolayer adsorption model and the Ono–Kondo lattice model have been used to describe the adsorption of supercritical fluids.^{8,15,16,21–25} In particular, the Ono–Kondo model is able to describe the peak of excess adsorption isotherms, and also has the potential to describe the adsorption behavior based on the physical properties of the adsorbates and the accessible characterization of the adsorbent. However, most works to date have been performed on adsorbents such as activated carbons, zeolite and molecular sieves, only a few studies have focused on the adsorption on coal.

Recently, studies were reported about the adsorption kinetics of CH₄ and CO₂ on coal.^12,26–44 For all the studies, it was observed that CO₂ diffusivity in coal is higher than that of CH₄ at the same temperature and pressure. However, the kinetic parameters of N₂ in coal are less discussed compared with CH₄ and CO₂. In addition, some contradictory results were found about the effect of the reservoir temperature and pressure on the effective diffusivity. Based on the experimental results, it is proposed that the adsorption and diffusion rates reduce at lower temperatures.^26–28,43 However, Li et al.¹² found no dependencies with temperature for the diffusivity of CH₄ and CO₂ in coal. Moreover, their conclusions about the influence of pressure on diffusion are different. Their diffusion parameters increased with an increased pressure,^28,33,37,45 while others reported the opposite.^{26,27,30,38–41}

As a result, the main content of this research work is focused on isothermal adsorption and kinetics characteristics for CH₄, CO₂, and N₂ adsorption on anthracite from China in a wide temperature and pressure range. First, the pore structures for coal were measured by using a low temperature N₂ adsorption method. Then, adsorption isotherms were determined in the temperature range of 294 to 353 K and the pressure range up to 18 MPa for CH₄ and N₂, and up to 5.5 MPa for CO₂, respectively. To describe the excess adsorption curves, two types of thermodynamic model were used. Finally, the diffusion characteristics of the three gases in coal were investigated by modeling the adsorption rate.

2 Experiment section

2.1 Materials

In this study, the fresh coal was taken from the Datong coal mine located in the Shanxi province of China, and for which the ultimate, proximate, and petrographic analyses are given in Table 1, which indicate that the raw coal is a typical anthracite. The coal was crushed into particles with a size fraction of 0.25–0.38 mm, preheated for about 8 hours in a drying oven at 378.15 K and then stored in a sealed plastic container.

Table 1 Physical properties and pore data of the raw coal used in the experiments^a

Proximate analysis (wt% dry basis)		Ultimate analysis (wt% dry basis)		Petrographic analysis (vol%, mmf)		Pore data
a Remarks: SA is the BET surface area (m^2 g^−1); PV is the BJH cumulative adsorption pore volume (cm^3 g^−1); PD is the average pore diameter for BJH adsorption branch (Å).
Ash	37.48	C	50.73	Vitrinite	73.0	SA	80.688
Moisture	0.86	H	2.169	Liptinite	0.0	PV	0.113
Volatile	10.48	N	7.653	Inertinite	16.2	PD	39.721
Fixed carbon	51.18	O	35.63	Mineral	10.8
		S	0.927	Ro'max	3.72

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The pore structure of the coal was measured using a Micromeritics 3Flex Surface Characterization Analyzer, which used a static volumetric method to determine the amount of adsorbed N₂. Prior to the measurement, the crushed coal was degassed under vacuum at 423 K for at least 12 hours.^46,47 After degassing, the coal sample was exposed to N₂ at 77.15 K in a relative pressure (P/P′) range from 0.010 to 0.993, where P is the equilibrium pressure and P′ is the saturation pressure of N₂ under laboratory conditions. Based on a low temperature N₂ adsorption isotherm, the surface area, pore volume, and pore distribution were calculated using adsorption theories, i.e., Brunauer–Emmett–Teller (BET) and Barret–Joyner–Halenda (BJH) in this study.

The CH₄, CO₂, N₂, and helium used in this study were obtained from Dalian Da-te Gas Co., Ltd. with a purity of 99.99%, 99.999%, 99.999% and 99.999%, respectively.

2.2 Apparatus and procedure

The experiments of gas adsorption on the crushed coal sample were performed using a high-pressure volumetric analyzer (HPVAII-200) designed by Micromeritics Instrument Corporation, shown schematically in Fig. 1a. HPVA was used to measure the adsorption in the temperature range from 77 to 773 K and pressures up to 22 MPa. The sample temperature was controlled by a Julabo-ME refrigerated/heating circulator whose accuracy was ±0.01 K. The pressure was measured by a high pressure transducer (HP, accuracy ±0.04%, stability ±0.1%) and a 1000 torr pressure transducer (LP, accuracy ±0.15%).

Fig. 1 (a) Experimental set-up of the volumetric system for gas sorption on a coal sample. HP, high pressure transducer; LP, 1000 torr pressure transducer; T, temperature probe; (1) analysis port valve; (2) vent valve; (3) manifold valve; (4) full vent valve; (5) full vacuum valve; (6) CH₄/CO₂/N₂ gas valve; (7) helium gas valve; (8) 1000 torr isolation valve; (9) degas port valve. (b) The temperature zones of the area below valve 1.

The adsorption measurement was performed using the following procedures:

(1) Degassing of the coal sample in the furnace at 378.15 K. The coal sample was placed in a cylinder and degassed for approximately 12 hours until the sample pressure decreased to 1 × 10⁻⁶ MPa, and was then cooled to ambient temperature.

(2) Measuring the volume of free space at ambient temperature. First, the manifold and sample tube were vacuumed; second, about 0.08 MPa of helium was supplied to the dosing manifold, allowed to equilibrate, and then dosed into the sample tube. During this process, two equations for the mass balance of helium are utilizes for the cases in the manifold and the sample tube, as shown in eqn (1) and (2), respectively.

Then, the volume of free space at ambient temperature V_AFS was solved using eqn (3). For this analysis, it was assumed that V_S and V_xL had the same temperature.

V_AFS = V_xU + V_xL + V_S = V_xU + V_SxL (3)

As shown in Fig. 1b, V_xU is the upper stem volume, and is approximately 3.5 cm³, V_xL is the lower stem volume, and V_S is the volume of free space at the analysis temperature.

(3) Measuring the volume of free space at the analysis temperature. For this analysis, V_S and V_xL had different temperatures. A new mass balance of helium was set up in the sample tube as eqn (4), which was used to solve V_S.

(4) Performing the experiment. First, the manifold and sample tubes were vacuumed; second, the adsorbate was supplied to the dosing manifold, allowed to reach equilibrium, and then dosed into the sample tube. The sample pressure and temperature data logging interval was 2 min or 5 × 10⁻⁴ MPa. The equilibrium criteria were set as 3 × 10⁻⁴ MPa min⁻¹ at each pressure step. After the adsorption reached equilibrium, the pressure was increased to the next pressure in an incremental way up to the maximum pressure. Similarly, the content of the desorbed gas was measured in a degressive pressure way. During this process, two equations of the mass balance for adsorbate were obtained in the cases of the manifold and the sample tube, as shown in eqn (5) and (6), respectively.

The adsorption capacity could be calculated using eqn (7).

n_ads = n_dosed − n_Nads (7)

It is worth noting that n_D is the moles of helium dosed into the sample tube, and is applied in calculating the free space volume at ambient and at the analysis temperature, whereas n_dosed is the moles of adsorbate (CH₄, CO₂ or N₂) dosed into the sample tube.

3 Modeling

To predict the adsorption capacity and the kinetics characteristics, thermodynamic and kinetic models were employed to model the adsorption isotherm and diffusivity of gas in coal. The Langmuir + k and simplified Ono–Kondo lattice approaches were utilized in this work, due to their applicability for high pressure and accuracy in engineering calculations.

3.1 Langmuir + k and Ono–Kondo lattice models

In this work, a 32-parameter modified Benedict–Webb–Rubin (MBWR) equation of state (EOS) was used in the data acquisition system of HPVA, which gave the compressibility factor at different temperatures and pressures. Therefore, the bulk phase density was calculated by EOS for the actual gases, and also it could be used directly in the following thermodynamic models. The Langmuir + k model is based on the concept of the equilibrium between the adsorbed gas molecules at the adsorption site and the free gas molecules. The form of the Langmuir + k isotherm in eqn (8) was adapted by Sakurovs et al.,²⁵ and it consists of the Langmuir adsorption component (gas density instead of pressure) and the ‘Henry’ absorption component k, which is a term proportional to pressure, following Henry's law. If the absorption of gas on coal is accompanied by swelling, and the swelling contribution is represented by the volume of condensed gas inside the coal, the swelling is associated with the direct proportion of the amount absorbed. In this study, the adsorbed phase densities for CH₄, N₂, and CO₂ were 0.808, 0.421, and 1.027 g cm⁻³, which were taken from the suggestion of Arri et al.⁴⁸ and Radovic et al.⁴⁹

n_ex = n₀(1 − ρ_b/ρ_a)[ρ_b/(ρ_b + ρ_L)] + kρ_b(1 − ρ_b/ρ_a) (8)

According to the research of Ottiger et al.,⁵⁰ the lattice model can only be used in the calculation of adsorption in coal pores without considering the swelling of the coal matrix. So the Ono–Kondo lattice model was used to estimate the adsorption capacity from a microscopic perspective in this work. The coal structure was simplified to a graphite-like crystalline structure, then the adsorption process could be directly mapped between the graphite planes.²² The thermodynamic equilibrium equation in Ono–Kondo lattice model is expressed as eqn (9).

where i = 2,3,…, with the boundary condition i = 1:

Compared with |ε_is|, |ε_ii| is much smaller and can be neglected. Thus, eqn (10) is simplified as:

Based on the Ono–Kondo lattice theory, the excess adsorption can be expressed as follows:

Aranovich and Donohue²¹ suggested that the adsorption of a supercritical fluid exhibits two-layer characteristics. Considering the boundary condition x₁ = x_m, eqn (12) can be written as:

n_ex = 2n′₀(x₁ − x_b) (13)

where x_b = ρ_b/ρ_mc. Combined with eqn (11) and (13), eqn (14) is deduced as:

If order:

Then, we have

Y = aX + b (16)

In this way, ρ_mc can be obtained by the experimental points (X, Y) to optimize the correlation coefficient of linear regression. The parameters of n′₀ and ε_is/k′ are then obtained by the slope and intercept of the linear fitting.

3.2 Adsorption rate analysis

As a normalized parameter, M_t/M_∞, which is the ratio of the cumulated amount of gas adsorbed at time t and at equilibrium, represents the adsorption rate of gas on coal. In this work, the adsorption kinetic data could be recorded by monitoring the change of pressure with time during the adsorption process. The method of pressure decay was used to analyze the adsorption rate of the three gases.²⁷ Considering the non-ideality of the gas, the pressure values were corrected using the compressibility factor as follows:⁵¹

As some researchers mentioned earlier, the pressure increases in a few seconds to attain thermal equilibrium after opening valve 1 and allowing the gas into the sample tube. The time to reach thermal equilibrium is shorter at low pressure than at high pressure. As a result, the pressure P₀ was considered as the maximum pressure valve in the sample cell at each pressure step.

3.3 Diffusion model

The kinetics of the adsorption process for CH₄, CO₂, and N₂ in coal are presented by means of the solutions of Fick's II law using the diffusion model by the expression:where y is the fractional uptake, and 0 ≤ y ≤ 1.

The above-mentioned equation is based on the assumption that the adsorbate concentration remains constant following the initial step change. The value of the diffusion parameter C, which is a fundamental property of a coal-gas system, was calculated on the basis of the analytical solution of eqn (18). However, it is difficult to estimate the parameter because the solution would be different depending on the method used for the estimation. Terzyk and Gauden⁵² compared the experiment to fitting data by using a simplified algorithm to estimate the parameter C with a minimum error from the expressions:

The C value was determined for which both f₁(y)/t and f₂(y)/t yielded the same results, and the result used as an input to eqn (18) for modeling the kinetics data considering the first ten terms in the infinite series. It is worth noting that the parameter C estimated by the above-mentioned method is an average value in diffusion processing for each pressure step. In addition, the kinetics data obtained from the different values of n were compared with the experimental data for each pressure in this study. The results presented that n = 10 was the best value for the estimation with satisfying the accuracy with the experimental data in eqn (18).

4 Results and discussion

4.1 Pore structure and reproducibility tests

The N₂ isotherms adsorption/desorption method is a known quantitative detection method to study adsorption pores at 77.15 K. It has been suggested that the adsorption pore structure of the coal can be described by the shape of the N₂ adsorption/desorption isotherms.^53,54 The adsorption and desorption curves of the crushed coal sample in this research are shown in Fig. 2a. A hysteresis exists in desorption, which reflects that the adsorption pores are mostly open pores. This type of isotherm is commonly dominated by adsorption pores with a large maximum adsorption volume, as well as a high BET pore surface area and BJH cumulative pore volume values. At the same time, the adsorption pore structure has the best adsorption capacity and connectivity, and is the most favourable for CBM adsorption, desorption and diffusion.⁵⁵

Fig. 2 Isotherms and pore size distribution results for a coal sample following N₂ adsorption/desorption test at 77.15 K.

The BET pore surface area is 80.688 m² g⁻¹. For the determination of pore size distribution, the adsorption branch of the isotherm was used for the BJH method, as shown in Fig. 2b. The cumulative pore volume is 0.113 cm³ g⁻¹ when the pore diameter is between 17 Å and 2637 Å. The average pore diameter is 39.721 Å which makes the main coal sample in this study mesoporous following the pore size classification of the International Union of Pure and Applied Chemistry (IUPAC).

Fig. 3 presents the results of the reproducibility tests for CH₄, CO₂ and N₂ excess adsorption at 311 K. These measurements show good agreement between the replicate runs. The error analysis indicates that the average uncertainties for the CH₄, N₂ and CO₂ adsorption measurements are approximately 0.40% (0.004 mmol g⁻¹), 1.23% (0.010 mmol g⁻¹) and 0.43% (0.006 mmol g⁻¹), respectively.

Fig. 3 Isotherm reproducibility of gas adsorption on a coal sample at 311 K. The solid and empty symbols refer to the first and second data, respectively. ◆◇, CO₂; ▲△, CH₄; ●○, N₂.

4.2 Adsorption/desorption isotherms

Gas adsorption/desorption measurements for CH₄ and N₂ were conducted at pressures of up to 18 MPa on the coal sample at 294 K, 311 K, 333 K and 353 K. For CO₂, the experiments were performed at up to 5.5 MPa because of the restriction of HPVA. Fig. 4 presents the experimental and predictive adsorption/desorption isotherms for CH₄, CO₂ and N₂, respectively. The adsorption and desorption curves almost coincide for a single gas, and the excess adsorption decreases with the increasing temperature for all three gases due to the exothermic physical adsorption.

Fig. 4 Experimental and predictive isotherms for gas adsorption/desorption on a coal sample. (a), CH₄; (b), CO₂; (c), N₂. The solid and empty symbols refer to adsorption and desorption, respectively. ◆◇, 294 K; ▲△, 311 K; ●○, 333 K; ■□, 353 K.

The isotherms for the excess adsorption of supercritical CH₄ increase to a maximum at first and then decrease afterwards with the increasing pressure. The peak can be analysed by eqn (21) as follows. At lower pressure, the bulk gas density is so small that the absolute adsorption and the excess adsorption are nearly equal. With the pressure increasing, n_ab increases up to saturation. Afterwards, the increase rate of n_ab decreases gradually. However, ρ_b increases with the pressure rising, which results in n_ex decreasing, so that the excess adsorption isotherm shows a peak. This phenomenon was also found by Cui et al.⁸ and Pini et al.⁴ for gas adsorption on coal. The CO₂ isotherms also exhibit a maximum in the experimental pressure range, whereas it belongs to the type I isotherm for N₂ and the peak is not visible in this work. The excess adsorption isotherm of CO₂ is easier to show the maximum for than that of CH₄, which is easier than N₂. As can be seen from Fig. 4a and b, the maximum of the isotherms move to higher pressure with the increasing temperature for CH₄ and CO₂, which is in accordance with other studies.^7,8,56 In addition, the maximum excess adsorption for CO₂ is in the range of 5–7% weight per unit mass of dry coal and 1.3–1.8% for CH₄ at the four temperatures. These values provide estimates for the capacity of CO₂ storage and for the maximum theoretical desorption capacity of coalbed methane.

n_ab = n_ex/(1 − ρ_b/ρ_a) (21)

The ratios between the maximum in the excess adsorption for N₂ : CH₄ : CO₂ are 1 : 1.1 : 1.6, 1 : 1.3 : 1.8, 1 : 1.4 : 1.9, and 1 : 1.2 : 1.6 at 294 K, 311 K, 333 K, and 352 K, respectively. This indicates the preferential adsorption of CO₂ for coal. For anthracite, the ratios of CO₂/CH₄ are about 1.5, which is close to the results of Busch et al.¹⁸ for higher rank coal.

4.3 Comparison of the thermodynamic model with the experimental results

Eqn (8) and (11) were used with the free parameters: n₀, ρ_L, and k for the Langmuir + k model, and n′₀, ε_is/k′, and ρ_mc for the simplified Ono–Kondo lattice theory. Tables 2 and 3 give the models' parameters and the error analysis, expressed in terms of the average relative error (ARE), of which it can be observed that the ARE of the Langmuir + k and simplified Ono–Kondo model are less than 2%, 6%, 1% and 6%, 8%, 8% for CH₄, CO₂ and N₂, respectively. The accuracy of the Langmuir + k model is higher than that of the simplified Ono–Kondo model for each gas. As a whole, both of the models apply to pure gas adsorption on coal in this work. In addition, the fitted isotherms, as shown as drawn lines in Fig. 4, can also well describe the adsorption behaviors of pure gas on coal under the experimental temperature and pressure conditions, especially for CH₄ and N₂. For CO₂, the data are not adequate to represent the excess adsorption isotherm. The possible reason for this is that the adsorbed phase density of 1.027 g cm⁻³ of CO₂ is chosen for the model when it approaches bulk density at high pressures, but the adsorption experiments are actually performed for CO₂ at low pressures in this study.

Table 2 Parameters and error analysis of the Langmuir + k model for CH₄, CO₂ and N₂ adsorption^a

	T (K)	NDTS	ρa (g cm^−3)	n0 (mmol g^−1)	ρL (g cm^−3)	k (cm^3 g^−1)	ARE
a NDTS: number of data points estimated; , average relative error.
CH4	293.81	14	0.421	1.467	0.005	−1.371	1.9%
	311.14	14	0.421	1.379	0.006	−1.432	1.8%
	333.14	14	0.421	1.251	0.007	−0.528	1.6%
	352.88	14	0.421	1.253	0.010	−2.480	2.0%
CO2	293.73	11	1.027	1.652	0.003	1.102	5.7%
	311.17	11	1.027	1.616	0.005	−0.064	5.8%
	333.06	11	1.027	1.514	0.007	−0.396	5.2%
	352.74	11	1.027	1.583	0.012	−2.789	5.6%
N2	293.71	13	0.808	0.887	0.017	2.400	0.9%
	311.08	13	0.808	0.915	0.023	1.162	0.9%
	333.07	13	0.808	0.838	0.029	0.846	0.6%
	352.78	13	0.808	0.741	0.034	1.546	0.5%

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Table 3 Parameters and error analysis of the simplified Ono–Kondo lattice model for CH₄, CO₂, and N₂ adsorption

	T (K)	NDTS	ρmc (g cm^−3)	Y = aX + b	R^2	εis/k′ (K)	n′0 (mmol g^−1)	ARE
CH4	293.81	14	0.304	Y = 0.1987X + 0.0052	0.9986	−1070	0.786	5.7%
	311.14	14	0.344	Y = 0.2684X + 0.0038	0.9998	−1325	0.649	2.4%
	333.14	14	0.377	Y = 0.3109X + 0.0058	0.9994	−1326	0.619	1.5%
	352.88	14	0.317	Y = 0.3023X + 0.0061	0.9991	−1377	0.535	4.3%
CO2	293.73	11	0.843	Y = 0.4644X + 0.0022	0.9993	−1572	0.912	7.8%
	311.17	11	0.526	Y = 0.2931X + 0.0035	0.9983	−1378	0.909	7.7%
	333.06	11	0.567	Y = 0.3606X + 0.0048	0.9978	−1439	0.796	5.6%
	352.74	11	0.615	Y = 0.4713X + 0.0053	0.9924	−1583	0.660	5.3%
N2	293.71	13	0.902	Y = 0.6585X + 0.0245	0.9929	−956	0.713	7.3%
	311.08	13	0.730	Y = 0.6202X + 0.0298	0.9973	−944	0.618	4.6%
	333.07	13	0.682	Y = 0.6575X + 0.0387	0.9975	−943	0.551	3.6%
	352.78	13	0.937	Y = 0.9208X + 0.0476	0.9974	−1045	0.536	2.8%

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In addition, the sequence of the interaction energy between the adsorbate and coal in the simplified Ono–Kondo model is as follows: CO₂ > CH₄ > N₂, which means that the adsorption strength of CO₂ on coal is stronger than that of CH₄ and N₂. This also implies the feasibility of the CO₂-ECBM process. Fig. 5 shows that the monolayer adsorption capacity decreases slightly with the increasing temperature. One possible reason for this is that the adsorbate molecules can be adsorbed only in spaces with a diameter smaller than the characteristic pore, which decreases with the increasing temperature. Dubinin et al.,⁵⁷ Sakurovs et al.⁵⁸ and Zhang et al.¹⁶ also observed this phenomenon in their studies. Moreover, the monolayer adsorption capacities for CO₂ on coal are higher than for CH₄ and N₂ in Fig. 5, which is in good agreement with the excess adsorption in Fig. 4a–c.

Fig. 5 Temperature dependence of the monolayer adsorption capacity obtained from the thermodynamic models.

4.4 Adsorption kinetics

4.4.1 Adsorption rate. Using 311 K as an example, the adsorption rates of gas on the coal sample are presented in Fig. 6. It can be seen that the adsorption rates increase with the increasing pressure for CH₄ and N₂ at the three different pressure steps. Furthermore, CH₄ adsorption rates are steeper than the corresponding N₂ adsorption rates at each pressure step, and CH₄ adsorption attains equilibrium much quicker than N₂. In contrast to CH₄ and N₂, although it is not so obvious, an increase in pressure reduces the adsorption rates for CO₂ when the adsorption time is less than 250 s. However, the time to reach adsorption equilibrium for CO₂ is longer at lower pressures for the whole adsorption process. Similarly, CO₂ adsorption rates are steeper than CH₄, which is attributed to the characteristics of CO₂, including kinetic diameter, polarity and elongated shape.

Fig. 6 Results of the adsorption kinetic experiment and model fitting for CH₄, CO₂, and N₂ on a coal sample at 311 K.

A comparison of adsorption rates for gases on coal are shown in Fig. 7 at 294 K, 311 K, 333 K and 353 K. It can be observed from the graphs, for the three gases, that the adsorption rates increase with increasing temperature at each pressure step. Compared with CH₄ and N₂, a slight increase in CO₂ adsorption rates with temperature is not obvious within the temperature range considered here. The dependence of the adsorption rate is temperature is in agreement with literature data.^26–28,43

Fig. 7 Effect of temperature on adsorption kinetics for CH₄, CO₂, and N₂ on coal sample.

4.4.2 Diffusivity. The adsorption kinetic data was fitted using the diffusion model mentioned in section 3.3. The values of the kinetic parameter C were estimated using eqn (19) and (20) and put in eqn (18) for modeling the adsorption rate. The fitting results of the diffusion model are shown in Fig. 6 as drawn lines, where it can be seen that the accuracy of CH₄ and N₂ are better than that of CO₂. A high deviation for CO₂ adsorption on coal has also been presented in other diffusion models because CO₂ changes the physical structure of coal, including by coal swelling and permeability changes. The diffusion model cannot predict the diffusion at high pressures very well in the initial time, because of the temperature variation caused by the Joule–Thomson effect after high pressure CO₂ is injected into the sample tube. This has also been proven by researchers for the experimental studies of CH₄ and CO₂ adsorption on coal.^12,26,29

The parameter C is converted into the effective diffusion coefficient D_e, as shown in Tables 4–6 for CH₄, N₂ and CO₂, respectively. It can be seen from the tables that the D_e values increase with increasing temperature for the three gases, and increase with increasing pressure for CH₄ at 0.55–6.97 MPa and for N₂ at 0.63–10.05 MPa. Clarkson and Bustin²⁹ proposed that analytical model diffusivities increase with increasing pressure for CH₄ in coal at 1–5 MPa due to nonlinearity of the isotherms. In contrast, the diffusivities of CO₂ through coal decrease with increasing pressure of 0.11–3.96 MPa due to intensive gas molecule–molecule collisions and the strong coal matrix adsorption swelling. This may indicate that molecular diffusion is the dominant transport mechanism for the mesopore structure in this study. This conclusion appears to be in agreement with Cui et al.³⁰ On the other hand, it is important to note the temperature dependence of D_e for all three gases.

Table 4 Fitting results of the effective diffusion coefficient for CH₄ (×10⁻³ s⁻¹)

293.81 K			311.14 K			333.14 K			352.88 K
P (MPa)			P (MPa)			P (MPa)			P (MPa)
From	To	De	From	To	De	From	to	De	From	To	De
0.18	0.55	4.41	0.20	0.57	4.84	0.23	0.60	5.41	0.24	0.64	6.34
0.55	1.03	4.37	0.57	1.06	4.98	0.61	1.08	5.81	0.64	1.11	6.52
1.03	1.98	5.21	1.06	1.99	5.76	1.08	1.98	6.97	1.11	1.99	7.60
1.98	3.98	6.47	1.99	3.97	7.19	1.98	3.97	7.86	1.99	3.98	8.33
4.83	5.93	7.92	4.80	5.97	8.00	4.79	5.98	8.17	3.98	5.96	8.56
5.93	6.96	9.00	5.97	6.97	9.21	5.98	6.97	9.69	5.96	6.97	9.07

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Table 5 Fitting results of the effective diffusion coefficient for N₂ (×10⁻⁴ s⁻¹)

293.71 K			311.08 K			333.07 K			352.78 K
P (MPa)			P (MPa)			P (MPa)			P (MPa)
From	To	De	From	To	De	From	To	De	From	To	De
0.24	0.63	4.08	0.26	0.64	4.84	0.27	0.68	6.71	0.28	0.68	7.73
0.63	1.10	4.85	0.64	1.11	5.32	0.68	1.14	7.27	0.68	1.15	8.74
1.10	1.99	5.79	1.11	1.99	6.36	1.14	1.99	7.72	1.15	2.00	9.47
3.39	4.01	6.25	3.40	4.03	8.05	3.40	4.02	7.61	3.40	4.01	9.47
4.01	6.00	6.54	4.03	6.01	8.99	4.02	6.00	9.23	4.01	6.00	9.47
6.00	7.92	8.26	6.01	7.94	9.22	6.00	8.00	9.45	6.00	8.00	10.9
7.92	10.0	9.53	7.94	10.0	9.67	8.00	10.0	9.75	8.01	10.0	11.2

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Table 6 Fitting results of the effective diffusion coefficient for CO₂ (×10⁻² s⁻¹)

293.71 K			311.08 K			333.07 K			352.78 K
P (MPa)			P (MPa)			P (MPa)			P (MPa)
From	To	De	From	To	De	From	To	De	From	To	De
0.02	0.11	1.02	0.02	0.15	1.09	0.04	0.19	1.12	0.05	0.22	1.37
0.11	0.47	1.02	0.15	0.51	1.07	0.20	0.56	1.11	0.22	0.59	1.36
0.47	1.43	0.83	0.51	1.44	0.94	0.56	1.45	1.02	0.59	1.46	1.07
1.44	1.97	0.65	1.44	1.98	0.83	1.45	1.98	0.92	1.46	1.98	1.00
2.46	2.97	0.50	2.48	2.97	0.72	2.47	2.97	0.81	2.49	2.97	0.90
2.97	3.46	0.45	2.97	3.47	0.60	2.97	3.47	0.67
3.46	3.95	0.41	3.47	3.96	0.41

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In addition, the D_e values for gases on coal are in this order: CO₂ > N₂ > CH₄ at the same temperature and at less than 2 MPa. CO₂ has the smallest half width of an exclusive pore and the largest adsorption energy between CH₄, N₂ and CO₂. Consequently CO₂ can diffuse into the coal matrix more easily.^29,30,59,60

5 Conclusions

The adsorption isotherm and kinetics of CO₂, CH₄ and N₂ on coal particles were investigated using an HPVA system at four different temperatures. Simultaneously, the adsorption capacity and the diffusivity of gas on coal were analyzed with thermodynamics and diffusion models. The major conclusions are as follows:

(1) The excess adsorption isotherms for CH₄ and CO₂ exhibit a maximum because of the value changes for the adsorbed phase and the bulk phase. The ratios of the maximum regarding the excess adsorption are 1 : 1.1 : 1.6, 1 : 1.3 : 1.8, 1 : 1.4 : 1.9, and 1 : 1.2 : 1.6 for N₂ : CH₄ : CO₂ at 294 K, 311 K, 333 K, and 353 K, respectively.

(2) The excess adsorption isotherms can be well simulated by the Langmuir + k and simplified Ono–Kondo lattice monolayer models. The consistency between the experimental data and modeling indicates good applicability of the two models for simulation of the adsorption behavior of CH₄ and N₂. Also, the monolayer adsorption capacity is linearly dependent on temperature for the three gases in the two thermodynamic models.

(3) There are variations based on the pressure dependency of the adsorption rate and D_e for CH₄, N₂ and CO₂ on coal. The adsorption rate and D_e increase with the increasing pressure for CH₄ and N₂. However, for CO₂, an increase in pressure reduces the adsorption rate and D_e. The adsorption rate and D_e increase with increasing temperature for all three gases.

The diffusion model satisfactorily fits the kinetics data for all three gases, but cannot predict the diffusion well in the initial injection time at high pressures due to the Joule–Thomson effect.

Nomenclature

C	Diffusion parameter (s^−1)
De	Effective diffusion coefficient (s^−1)
k	Parameter of the Langmuir + k model (cm^3 g^−1)
k′	Boltzmann's constant, 1.38 × 10^−23 (J mol^−1 K^−1)
Mt/M∞	Fractional uptake
nab	Amount of absolute adsorption (mmol g^−1)
nads	Moles of gas adsorbed by coal (mol)
nD	Moles of helium dosed into the sample tube (mol)
ndosed	Moles of CH4, CO2 or N2 dosed into the sample tube (mol)
nex	Amount of excess adsorption (mmol g^−1)
nNads	Moles of gas not adsorbed by the coal sample (mol)
n0, n′0	Monolayer adsorption capacity for Langmuir + k model and simplified Ono–Kondo model, respectively (mmol g^−1)
PA, P′A	Pressure of helium and adsorbate before dosing in the manifold (bar)
PB, P′B	Pressure of helium and adsorbate after dosing in the manifold (bar)
PS	Sample pressure (bar)
PS1	Pressure of helium after dosing in the sample tube (bar)
Pt	Pressure in the sample tube at time t (bar)
P0	Maximum pressure after dosing in the sample tube (bar)
P∞	Pressure in the sample cell at adsorption equilibrium (bar)
R	Gas constant (cm^3 bar K^−1 mol^−1)
TA, T′A	Temperature of the manifold before dosing (K)
TAM	Ambient temperature, approximated as 23 °C (296.15 K) (K)
TB, T′B	Temperature of the manifold after dosing (K)
TS	Sample temperature (K)
TS1	Temperature in the sample tube after dosing, approximately ambient (K)
TxL	Ambient temperature (K)
TxU	Manifold temperature (K)
Va	Adsorbed phase volume (cm^3)
VAFS	Volume of free space at ambient temperature (cm^3)
Vm	LP manifold volume, 46.7791 cm^3 (cm^3)
V′m	HP manifold volume, 27.0903 cm^3 (cm^3)
VS	Volume of free space at analysis temperature (cm^3)
VSxL	Volume outside of the temperature controlled zone in the manifold (cm^3)
VxL	Lower stem volume (cm^3)
VxU	Upper stem volume, approximately 3.5 cm^3 (cm^3)
xi	Density or fraction of sites occupied by adsorbed molecules in layer i
xb	Fraction of sites occupied by fluid molecules in the bulk phase
zA, z′A	Gas compressibility at PA and TA, P′A and T′A
zB, z′B	Gas compressibility at PB and TB, P′B and T′B
zS	Gas compressibility at PS and TS
zS1	Gas compressibility at PS1 and TS1
zxL	Gas compressibility at PS and TxL
zxL1	Gas compressibility at PS1 and TAM
zxU	Gas compressibility at PS and TxU
zxU1	Gas compressibility at PS1 and TB
z0	Volume coordination number
z1	Monolayer coordination number
z2	Interplane coordination number, z2 = (z0 − z1)/2
Z0	Gas compressibility at P0 and T0
Zt	Gas compressibility at Pt and Tt
Z∞	Gas compressibility at P∞ and T∞
ρa	Adsorbed phase density (g cm^−3)
ρb	Bulk density (g cm^−3)
ρL	Gas density at which the adsorption is half the maximum (g cm^−3)
ρmc	Adsorbed phase density at maximum capacity (g cm^−3)
εii	Interaction energy between the adsorbate molecules, which is limited to nearest neighboring sites of the lattice (kJ mol^−1)
εis	Interaction energy between the adsorbate molecules and the adsorbent surface (kJ mol^−1)

Acknowledgements

This paper was supported by National Science and Technology Major Project of China (no. 2011ZX05026-004-07), National Program on Key Basic Research Project (no. 2011CB707304) and the Fundamental Research Funds for the Central Universities (DUT15LAB22).

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