Effect of Equation of States on High Pressure Volumetric

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Effect of Equation of States on High Pressure Volumetric Measurements of Methane-Coal Sorption Isotherms - Part 1: Volumes of Free Space and Methane Adsorption Isotherms Jamiu Ekundayo, and Reza Rezaee Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b04016 • Publication Date (Web): 24 Jan 2019 Downloaded from http://pubs.acs.org on January 25, 2019

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Effect of Equation of States on High Pressure Volumetric Measurements of Methane-Coal Sorption Isotherms - Part 1: Volumes of Free Space and Methane Adsorption Isotherms Jamiu M. Ekundayo* and Reza Rezaee Discipline of Petroleum Engineering, Western Australian School of Mines: Minerals, Energy and Chemical Engineering

Abstract High pressure volumetric analysis (HPVA) experiments were carried out using Particulate System’s HPVA-II 200® at temperatures of 25 oC & 40 oC, and pressures up to 7 MPa to quantify the methane adsorption capacities of a coal sample. Six well-known equations of state (EOSs) namely PengRobinson’s (PR), Soave-Redlich-Wonk’s (SRK) and their volume translated forms (PR-Peneloux and SRK-Peneloux), Soave’s modified Benedict-Webb-Rubin’s (SBWR) and Lee-Kesler’s (LK) were used for data interpretation and the results were compared with those calculated by the equipment using zfactors from NIST-refprop® software’s implementation of McCarty and Arp’s EOS for helium and Setzmann & Wagner’s EOS for methane. Due to the variations in the z-factors of helium & supercritical methane obtained from the different EOSs, large variations were observed in the calculated isotherms and Langmuir parameters, with the PR-Peneloux and PR-EOS family showing the largest relative deviations and the SRK-EOS & SRK-Peneloux gave more negative adsorption for each temperature. All the EOSs gave significantly lower density for the adsorbed phase at 40 oC compared to the value at 25 o

C. This is consistent with the negative effect of temperature on adsorption.

Keywords: High pressure volumetric analysis, adsorption capacities, equation of states, NIST Refprop, Z-factor, gas compressibility factor, shale negative adsorption 1. Introduction Adsorbed gas represents a significant proportion of the total gas in place in coal seams and gas shale 1 due to their characteristic pore structures and mineral make-ups 2-3. Hence, investigations, mostly experimental, of adsorption capacities of coal samples to methane have been widely explored

4-7

. Menon

8

discussed

different approaches for laboratory measurements of high-pressure gas-solid sorption isotherms. These include the glass-piezometric method used for methane-coal sorption studies up to 50 MPa, Mitchel’s modified glass-piezometric method for gas-solid sorption and general PVT studies up to 300 MPa, modified gas-chromatographic method for both pure and multicomponent gas-solid sorption studies up to about 14 MPa and the modified gravimetric method used for methane-silica gel up also up to 14 MPa

8

. More

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recently, Keller and Staudt 9 also discussed in details, several experimental approaches for measuring gassolid sorption isotherms. These include the volumetric method, the manometric method, the gravimetric method, the rotational oscillometric method, the impedance spectroscopic or dielectric method and some combinations thereof 9. Based primarily on its simplicity 9, the volumetric method seems to have gained a wider acceptance among researchers. Zhang and Liu 7 used the volumetric method to measure isotherms for different coal samples using subcritical CO2 gas & supercritical methane with focus on sorption hysteresis 7. Busch, et al. 10 also used the volumetric approach for CO2 and CH4 adsorption isotherms on coal up to 23 MPa and 45 oC 10. Gasparik, et al. 11 measured shale gas adsorption up to 25 MPa using volumetric method at 65 oC. Simply put, volumetric method operates primarily on gas expansion and equilibration processes resulting in a pressure profile that is converted to amounts using the ideal gas equation corrected for non-ideality by real gas compressibility (z-) factors calculated with a chosen equation of states (EOS) 12. Volumetric balance at each equilibrium pressure step allows for the calculation of adsorbed gas volume. Despite its wide use, there is no known standard for reporting measured data 12. This lack of reporting standard for high-pressure volumetrically measured gas/solid sorption isotherms has made it nearly impossible to compare or reproduce, with any degree of certainty, reported data

12-14

. One obvious issue identified from published

reports is the varied (and sometimes inconsistent) use of EOSs. For example, Tang, et al. 15 measured the adsorption of methane on coal by volumetric approach up to 9 MPa and calculated the z-factors of methane using Redlich-Kwong EOS

15

. Zhang, et al.

16

applied a 32-parameter modified Benedict-Webb-Rubin

(MBWR) EOS for their data interpretation in a similar study with coal. In a more recent study with shale, Yang, et al. 17 used McCarty and Arp EOS to calculate the z-factors of helium for void space calculations and Setzmann and Wagner EOS for z-factors of methane 17. While various explanations have been attempted to explain some of the commonly observed abnormalities in high-pressure sorption isotherms (for example, observations of hysteresis loops in high-pressure supercritical methane isotherms

4-7, 18

, observations of reducing amounts adsorbed at medium to high

pressure especially in coals and shale rocks

16, 19

, etc.), little has been reported on equation of state as a

potential (or primary) reason for some (or all) of these observations. To demonstrate the effect of z-factor on CO2 excess adsorption, Goodman, et al. 12 compared the ideal isotherm (one obtained with Z = 1) with those calculated using z-factor from 2 other sources (Span & Wagner’s and Gas Encyclopedia) and found that CO2 excess adsorption is much lower for the ideal case 12. Lutynski, et al. 19 investigated the effect of equations of state on the adsorption of supercritical CO2 by comparing the performances of PR-EOS and SRK-EOS with Span and Wagner’s EOS (SW-EOS). The authors found that both PR-EOS and SRK-EOS gave unrealistically higher results above the critical pressure (7.39 MPa) following lower results at 2|Page ACS Paragon Plus Environment

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subcritical pressures and attributed these discrepancies to errors in density calculations by these EOS relative to SW-EOS 19. Al-Fatlawi, et al. 20 has also recently reinstated the need for accurate predictions of z-factors for gas reserves estimations, which is one of the direct applications of gas adsorption 20. This paper, therefore, represents the first comparison of the performances of different equations of state relative to a combination (subsequently referred to as NIST-refprop) of McCarty & Arp EOS (for helium) 21

and Setzmann & Wagner’s EOS 22 (for methane) in high-pressure volumetric measurements of methane

adsorption on a coal sample. It focuses on establishing that the choice of EOS can significantly affect not only the volume of free space but also the calculated isotherms and their associated model parameters. 2. Equations of State The thermodynamic state of a system is often defined in terms of the relationship between its pressure (P), temperature (T) and molar volume (vm) such that: ݂ሺܲǡ ܶǡ ‫ݒ‬௠ ሻ ൌ ͲʹǤͲ This relationship is referred to as the equation of state (EOS) 23. By solving equation 2.0 for vm for a given system over a range of pressure and temperature conditions, other physical properties such as the compressibility factor, enthalpy, entropy, fugacity, etc. as well as the phase equilibrium of the system can be calculated 24. Equations of state are empirically classified as cubic and non-cubic based on the degree of the polynomic function 23 represented by equation 2.0. In this paper, only the cubic EOS of Peng-Robison’s (PR) and the Soave-Redlich-Wonk (SRK) and their volume translated forms

25-26

are discussed. Also, the

Soave’s modified Benedict-Webb-Rubin’s (SBWR) 27 and Lee-Kesler’s 28 equations of state are the only non-cubic equations of states discussed in this paper. The details of these equations of state are included in appendix A. 3. Sample and Methods 3.1 Sample A commercial coal sample was used for this study. Pulverized coal sample was sieved to particle sizes in the range 45 – 75μm corresponding to 325 – 200 mesh 29. The coal has a mean vitrinite reflectance of 1.43% 29

, which indicates a coal rank of medium volatile bituminous

30

. The organic composition of this coal

sample is given in table 1. 3.2 Low-Pressure Nitrogen Adsorption Micromeritics® Tristar II 3020 apparatus was used to measure the low-pressure nitrogen adsorption isotherm of the coal sample at a temperature of 77.3 K (≈ -196 oC). The measurements were performed at


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equilibrium relative pressure (P/Po, where P = equilibrium pressure) range of about 0.01 to 1 using a saturation pressure, Po of 0.1 MPa. The pulverized sample was degassed at a temperature of 110 oC for at least 8 hours prior to the measurements. The experimental isotherms were theoretically described by the Brunauer-Emmett-Teller (BET) model 31 and this, combined with the non-local density-functional theory (NLDFT) available in equipment’s in-built software, was used to invert the data for the PSD. 3.3 HPVA Measurements & Data Interpretation One equipment that has been reportedly used for this type of measurements is the Particulate System’s HPVA-II® 200 unit

16, 29, 32

. The fully automated HPVA has the capacity to measure up to 500 oC

temperature and 20 MPa pressure 33 and it uses NIST’s refprop® to calculate the z-factors used in generating the isotherms. It also logs the pressure and temperature data at specified intervals thereby allowing the user to perform own calculations. In this work, the data logged by the equipment was used to calculate an isotherm with z-factors obtained from each of the equations of states identified in section 2. Some schematics and descriptions of the HPVA-II 200® unit, its data interpretation approach and typical experimental procedures have been discussed by Zhang, et al. 16; Zou and Rezaee 29, Zou, et al. 32. As much as possible, we have also used similar notations as these authors to make comparison of our analysis and theirs easy for readers. Figure 1 shows the schematic of a typical pressure-time profile for a complete adsorption-desorption experiment using the HPVA-II 200. The pre-adsorption steps are usually performed to obtain the volume of the free-space needed for subsequent calculations and to stabilize the sample cell at the experimental temperature. For any two consecutive adsorption steps ݅ െ ͳƬ݅ (as illustrated in figure B1 in appendix B), the differential and cumulative amounts adsorbed (at any step‫)ݎ‬, per unit mass of the adsorbent, can be calculated using equations 3.1 and 3.2 below (details of derivation are available in appendix B). οܸ෨ ௔ௗ௦ǡ௜ ൌ

௜ ௜ ௜ ௜ ܸௌ்௉ ܲ௠஺ ܲ௦௖ ܲ௦௖ ܸ௠௔௡ ܲ௠஻ ܸ௠௔௡ ܸ௦௖ ܸ௨௦ ܲ௦௖௜ ܸ௟௦ ቈቆ ௜ ௜ െ ௜ ௜ ቇ െ ቆ ௜ ௜ ൅ ௜ ௜ ൅ ௜ ௜ ቇ ܴ‫ܯ‬௦ ܶ௠஺ ܼ௠஺ ܶ௠஻ ܼ௠஻ ܶ௦௖ ܼ௦௖ ܶ௨௦ ܼ௨௦ ܶ௟௦ ܼ௟௦

൅ቆ

ܲ௦௖௜ିଵ ܸ௦௖

௜ିଵ ܲ௦௖ ܸ௨௦

௜ିଵ ܶ௦௖ ܼ௦௖

௜ିଵ ܶ௨௦ ܼ௨௦

൅ ௜ିଵ

൅ ௜ିଵ

ܲ௦௖௜ିଵ ܸ௟௦ ௜ିଵ ܶ௟௦௜ିଵ ܼ௟௦

ቇ቉ ͵Ǥͳ

௥

ܸ෨௥௔ௗ௦

ൌ ෍൫οܸ෨ ௜ ൯ ͵Ǥʹ ௜ୀଵ

ܲ௠஺ ǡ ܶ௠஺ Ƭܼ௠஺ in equation 3.1 are respectively the initial pressure, temperature and z-factor of the reference cell (of volume, ܸ௠௔௡ ) before dosing gas into the sample cell, ܲ௠஻ ǡ ܶ௠஻ Ƭܼ௠஻ represent the conditions of the reference cell after dosing. ܲ௦௖ ǡ ܶ௦௖ Ƭܼ௦௖ are the equilibrium pressure, temperature and 4|Page ACS Paragon Plus Environment

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corresponding z-factor in the sample cell (of volume ܸ௦௖ ), ܶ௨௦ Ƭܼ௨௦ are the temperature and z-factor in the upper stem (of volume, ܸ௨௦ ) of the sample holder while ܶ௟௦ Ƭܼ௟௦ are the temperature and z-factor in the lower stem (of volume ܸ௟௦ ) of the sample holder. ܸௌ்௉ , ܴ and ‫ܯ‬௦ are respectively the molar volume of gas at standard conditions, gas constant (ൎ ͺǤ͵ͳͶͷ‫ି ݈݋݉ܬ‬ଵ ‫ି ܭ‬ଵ ) and mass of the adsorbent. One important assumption is made in using equations 3.1 & 3.2 is that ܲ௦௖௔ௗ௦ǡ଴ ൌ Ͳ since the system is expected to be fully evacuated prior to the first adsorption step. These equations are applicable to both adsorption and desorption isotherms since both are typically plots of amounts adsorbed per unit mass against the equilibrium pressures. The equipment uses a helium expansion method to measure the volume of free space which includes the volumes of the lower and upper parts of the stem (that is, ܸ௟௦ &ܸ௨௦ respectively) connecting the sample cell to the reference cell. The details of this measurement technique have been discussed by Zhang, et al. 16, Zou and Rezaee 29, Zou, et al. 32 and are also discussed in appendix B (kindly see the accompanying supporting information for publication). The volume free space at ambient condition (ܸ஺௠௙௦ ) is given as follows: ܸ஺௠௙௦ ൌ ܸ௦௖ ൅ ܸ௟௦ ൅ ܸ௨௦ ͵Ǥ͵ Where: ܸ஺௠௙௦ ൌ

ு௘ ு௘ ு௘ ு௘ ு௘ ு௘ ܶ௦௖ ܲ௦௖ ܼ௦௖ ܲ௠஺ ܸ௠௔௡ ܲ௠஻ ܸ௠௔௡ ܲ௦௖ ܸ௨௦ ܸ௨௦ ቆ െ െ ൅ ு௘ ு௘ ு௘ ு௘ ு௘ ு௘ ு௘ ு௘ ு௘ ቇ ͵ǤͶ ܲ௦௖ ܶ௠஺ ܼ௠஺ ܶ௠஻ ܼ௠஻ ܶ௨௦ ܼ௨௦ ܶ௦௖ ܼ௦௖

The parameters in equation 3.4 are defined similar to those in equation 3.1. 4. Mathematical Modeling The experimentally determined excess adsorbed amounts are usually converted to absolute values which are then described by some mathematical model for practical applications 11. In this work, the corrected isotherms are parameterized using the 3-parameter Langmuir model

7, 11, 13, 16

mainly due to its simplicity

and wide acceptance for describing coal gas adsorption isotherms 11. If the excess amount adsorbed, ܸ௘௫௖ is plotted as a function of the equilibrium pressure, P, then the Langmuir model can be defined as: ܸ௘௫௖ ൌ

ܸ௅ ܲ ߩ௕௨௟௞ ൬ͳ െ ൰ ͶǤͳ ܲ ൅ ܲ௅ ߩ௔ௗ௦

Where: VL = Langmuir volume 5|Page ACS Paragon Plus Environment

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PL = Langmuir pressure ߩ௔ௗ௦ = Density of the adsorbed gas phase (assumed to be independent of pressure) ߩ௕௨௟௞ ൌ

ܲ‫ܯ‬ ൌ ሺ‫ݐ݊݁݀݊݁݌݁݀݁ݎݑݏݏ݁ݎ݌‬ሻͶǤʹ ܼܴܶ

M = Molecular weight of methane In this work, the model parameters ሺܸ௅ ǡ ܲ௅ ǡ ܽ݊݀ߩ௔ௗ௦ ሻwere determined by non-linear curve fitting using the trust region reflective algorithm implementation in Scientific Python (SciPy) to allow for the parameters to be bounded to non-negative values. Each of the parameters is thus bounded toሺͲǡ ൅λሻ. The quality of the match with experimental isotherms has been assessed using the coefficient of determination, R2 defined as: ௜ୀ௡ ଶ

ܴ ൌͳെ

௜ ቌ෍൫ܸ௘௫௣௧ ௜ୀଵ

െ

௜ୀ௡ ௜ ଶ ௜ ܸ௙௜௧ ൯ ൙ ෍൫ܸ௘௫௣௧ ௜ୀଵ

ଶ െ ܸത௘௫௣௧ ൯ ቍ ͶǤ͵

௜ ௜ is the experimental amount adsorbed at step݅ሺ݅ ൌ ͳǡ ʹǡ ͵ǡ ǥ ǡ ݊ሻ; ܸ௙௜௧ is the corresponding Where ܸ௘௫௣௧

amount adsorbed calculated with equation 4.1 and ܸത௘௫௣௧ is the mean experimental amount adsorbed given as: ௜ୀ௡ ௜ ܸത௘௫௣௧ ൌ ෍൫ܸ௘௫௣௧ ൯൘݊ ͶǤͶ ௜ୀଵ

5. Results and Discussion 5.1 Sample Characterization with Nitrogen Adsorption Figure 2a shows the measured nitrogen adsorption and desorption isotherms of the coal sample used in the study. The sample exhibits nitrogen adsorption isotherm characterized by the absence of monolayer coverage but notable multilayer coverage 34 up to relative pressure of about 0.8 and capillary condensation indicated by the sharp increase in amount adsorbed thereafter

35

. A minimal hysteresis of type H3 (IUPAC

classification) characterized by an adsorption isotherm of type II and a lower closure of the hysteresis loop indicative of non-rigid aggregates of slit-like particles suggesting the presence of a network of macropores 34

. The pore-size distribution (PSD) shown in figure 2b indicates negligible micropore volume accessible

to nitrogen (hence, the lack of monolayer coverage observed in the isotherm). The pores are roughly bimodally distributed with dominant pore sizes of approximately 4nm and 25nm. Mesopores (pore sizes up


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to 50nm) accounts for over 90% of the pore volume while the remaining < 10% of the pore volume can be attributed to available macropores. 5.2 Methane-Coal Adsorption Isotherms by HPVA – Effects of EOS 5.1.1

Volume of Free Space

Figure 3a shows the variation in the ambient volume of free-space calculated by different EOSs relative to the value obtained from NIST-refprop. These deviations are directly related to the differences in the z-factor of helium gas calculated by the different EOSs. As shown in figure 3b, PR-EOS, SBWR-EOS and LK-EOS gave higher helium z-factors than NIST-refprop and thus their corresponding ambient volumes of free space are less proportionately. On the other hand, PR-Peneloux-EOS, SRK-EOS and SRK-Peneloux-EOS gave lower z-factors of helium compared to NIST-refprop and consequently, their resultants ambient volumes of free space are proportionately larger. All EOSs (except Lee-Kesler) deviated to less than 0.1% compared to the reference EOS. In fact, the deviations are within -0.025% to +0.009% for the cubic EOSs and 0.09% for SBWR-EOS. The greater deviation (< 0.25%) observed with Lee-Kesler may be due to its use of linear interpolation to calculate the z-factor of a fluid from those (z-factors) of a simple fluid and a reference fluid. The inverse relationship observed between the z-factor and ambient volume of free space, for each EOS, follows from the gas law (݊ ൌ ܸܲ Τܼܴܶ). For the same, pressure, volume and temperature, the amount of gas (n) in a system is inversely related to the z-factor of the gas which (as shown in figure 3b) is dependent on the choice of EOS. It should be noted that the deviations in volume of free space from NIST-refprop’s are not to be considered errors but rather they serve to warn that any reported volume of free space may not be taken as absolute but is highly sensitive to z-factor (and by implication, EOS). This has implications for the calculated isotherms especially at higher pressures where the amounts of gas adsorbed have been reported to be highly sensitive to the volume of free space 36-37. Similar observation is made in this paper as shown in figure 4 in which a change of ± 0.25% in the ambient volume of free space resulted in‫Ͳͳט‬Ψchange in the amount adsorbed at the last equilibrium pressure tested in this paper. While this may be argued to be within the range of uncertainty for this type of measurements, we envisage that the effect may be more pronounced in a truly microporous material (such as shale rock). It is illustrated in this paper (see figure 5) that this behavior is not necessarily connected with pressure equilibrium or any leakage in the system, as is generally believed 9

, but is predominantly controlled by the EOS used in calculating the z-factor. 5.1.2

Methane Adsorption Isotherms

To minimize the error in calculated isotherms due to variation in volume of free space, z-factor of methane was calculated with the same EOS used in calculating the z-factor of helium for calculating the volume of 7|Page ACS Paragon Plus Environment

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free space. The variations in the z-factors of methane at 25 oC calculated by different EOSs at different equilibrium pressures are shown in figure 6. For each pressure value, SBWR-EOS, followed by LK-EOS, gave the closest z-factor of methane relative to NIST-refprop’s. The deviation from the reference value increases with pressure for each EOS hence, the deviation in the resultant isotherm (shown in figure 7) also increases proportionately with pressure. Following the inverse relationship between z-factor of gas and amount of the gas in a system, it is not surprising that PR, PR-Peneloux and SBWR EOSs gave higher methane adsorption isotherms than the reference EOS since their z-factors are lower (figure 6). For these EOSs, the higher methane adsorption isotherms (compared to the reference EOS) can be seen as the result of the combined effects of volume of free space (actually, z-factor of helium) and z-factor of methane. PR-EOS and PR-Peneloux EOS gave almost the same volumes of free space as the reference EOS but their significant negative deviations in zfactor of methane resulted in great amounts adsorbed at each pressure. Similarly, SBWR-EOS which gave lower volume of free space relative to the reference EOS should ordinarily give much higher amounts of methane adsorbed at each pressure compared to PR-EOS and PR-Peneloux EOSs but for its methane zfactors which are significantly lower (figure 6) compared to these EOSs. On the other hand, SRK-EOS and its volume-shifted form resulted in lower methane adsorption isotherms compared to the reference EOS. This is expected considering that both the volume of free space and z-factor of methane from these EOSs are higher compared to the reference EOS. Lastly, the LK-EOS gave higher methane adsorption isotherm compared to the reference EOS despite its higher methane z-factors. This is because the effect of z-factor is counteracted by the lower volume of free space (compared to the reference EOS). The above observations of combined effects of z-factors of helium (by implication volume of free space) and methane are also true at 40 oC (figure 7). The difference in this case is the reducing effect of temperature on adsorption isotherms. At higher temperature, lower amounts of methane are adsorbed at each pressure because of the reducing effect of temperature on gas density. 5.3 Langmuir Parameters The results of the 3-parameter Langmuir model fit (described in section 4) to each of the adsorption isotherms are summarized in table 2 and the deviations in the values of the Langmuir volume and pressure for each EOS from the NIST-refprop’s are shown in Figure 8. The R2 values in table 2 show that the measured excess adsorption isotherms are reasonably represented by the 3-parameter Langmuir model. For each EOS, the value of the Langmuir volume at 40 oC is lower than the value at 25 oC due to the lower amount adsorbed at the higher temperature. The Langmuir pressure is higher at 40 oC than at 25 oC for each EOS. For each temperature, SBWR-EOS gave the lowest deviations in the values of Langmuir volume and


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pressure while the PR-EOS and PR-Peneloux gave the largest deviations compared to the NIST-refprop. Contrary to the common practice of assuming a constant (temperature independent), adsorbed phase density (often taken from published reports or predetermined from molecular simulation) for fitting the measured isotherms 7, 11, 16, 38, it is observed here that adsorbed phase density is significantly lower at 40 oC compared to the value at 25 oC. This observation is consistent with the reducing effect of temperature on gas density as well as the reducing effect of temperature on amount adsorbed 11, 15-17, 32. It is important to mention that the large value of adsorbed phase density obtained with SBWR-EOS at 25 oC only shows that the amounts adsorbed can be taken as absolute values sinceߩ௕௨௟௞ Τߩ௔ௗ௦ ൎ Ͳ in this case. 6. Conclusions This study represents the first comparison of the performances of different equations of state in highpressure volumetric measurements of methane adsorption on a coal sample. Overall, it can be concluded that no two equations of state gave the same methane adsorption and associated Langmuir parameters. Relative to NIST-refprop (by which we mean a combination of McCarty & Arp EOS for helium and Setzmann & Wagner’s EOS for methane), all the equations of state tested in this paper show varied deviations for the z-factors of both helium and methane and consequently, the respective resultant volumes of free space and methane adsorption isotherms also deviated proportionately. More specifically, the results presented in this paper have shown that: 1. While cubic equations of state gave lower deviations for z-factor of helium (and hence, volume of free space) compared to NIST-refprop, they gave higher deviations for z-factor of methane (and hence, methane adsorption isotherms). Virial-type equations of state gave higher deviations for zfactor of helium but lower deviations for z-factor of methane. 2. The final “calculated” methane adsorption isotherm is controlled by the combined effects of the volume of free space (actually, z-factor of helium) and the z-factor of methane. This is contrary to the common belief that the volume of free space is the single parameter controlling the measured amounts of gas adsorbed. 3. The adsorbed phase density is temperature dependent and is significantly lower for 40 oC that it was for 25 oC. This is in agreement with the reducing effect of temperature on amounts adsorbed. 4. Although the calculated Langmuir parameters show varied deviations from the NIST-refprop’s, these deviations did not reflect the deviations of the individual isotherms from the reference isotherm. Lastly, the focus of this work is not to argue for or against any EOS but to signal that reported volumetrically measured high-pressure gas/solid sorption isotherms may not be taken as “unique” values but considered 9|Page ACS Paragon Plus Environment

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highly dependent on the EOS used for data interpretation. Large errors may be incurred if the same EOS is not used, with the same dataset, for volume of free space and amounts adsorbed. We hope that this paper will alert the research community of a need to standardize the data interpretation approach for this type of measurements similar to low-pressure nitrogen physisorption. Adsorption steps

Desorption steps Step 1

Step 2

Increasing pressure steps

Pressure

Pre-adsorption steps (free-space measurements, system evacuation, etc.)

Step n

Decreasing pressure steps Step k

Step 2

Manifold/Ref. Cell Sample Cell

Step 1

Time Figure 1: Typical pressure-time plot for a HPVA adsorption-desorption experiment a.) Low-Pressure Nitrogen Sorption Isotherms

b.) Pore-Size and Pore Volume Distributions 0.010

dV/dD (cm3/g/nm) Norm_cum_vol

3.5

dV/dD, cc/g/nm

3.0

2.5 2.0 1.5

1.0 0.5 0.0 0.000

0.200

0.400 0.600 Relative Pressure, P/Po

0.800

1.000

> 90%

1.0

0.008

0.8

0.006

0.6

0.004

0.4

0.002

0.2

Normalized Pore-Volume, frcation

4.0

Amount Adsorbed, cc STP/g

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 17

0.0

0.000 1

10

100

Pore Size, nm

Figure 2: a) Nitrogen adsorption/desorption isotherms (b) sample’s pore size & pore volume distributions

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Page 11 of 17

ȋȌǦ̷ʹͷ Ǧ

ȋȌΨǦȋ̷ʹͷƬͲǤͷ Ȍ Ǧ

Ǧ

Ǧ

Ǧ

Ǧ

Ǧ

Ǧ

ǦͳǤͲͲΨ

ǦͲǤͷͲΨ ͲǤͲͲΨ ͲǤͷͲΨ Ψ

ͳǤͲͲΨ

ǦͳǤͲͲΨ

ǦͲǤͷͲΨ ͲǤͲͲΨ ͲǤͷͲΨ % Difference in z-factor

Figure 3: (a) Variation in volume of free space by different equations of state (actual free space volumes are in figure B1 of Appendix B) (b) Variation in z-factor of helium gas by EOS (Reference Z-factor = 1.0023)

Effect of Volume of Free-Space on Methane Adsorption EOS = NIST-Refprop ͹ͲͲ

ͲǤʹͷΨ ͲǤʹͷΨ

͸ͷͲ

+10%

͸ͲͲ

ǡȀ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-10%

ͷͷͲ ͷͲͲ ͶͷͲ ͶͲͲ ͵ͷͲ ͵ͲͲ ʹͷͲ ʹͲͲ Ͳ

ʹ

Ͷ ǡ

͸

ͺ

Figure 4: Effect of Volume of free space on methane-coal adsorption isotherm at 25 oC


ͳǤͲͲΨ

Energy & Fuels

͹ǤͲͻͲ ͹ǤͲͺͺ

ǡ

͹ǤͲͺ͸

͹ǤͲͺͶ ͹ǤͲͺʹ ͹ǤͲͺͲ ͹ǤͲ͹ͺ

͹ǤͲ͹͸ ͹ǤͲ͹Ͷ ͹ǤͲ͹ʹ ͹ǤͲ͹Ͳ

ʹͺͲ

ʹͺͶ

ʹͺͺ

ʹͻʹ

ʹͻ͸

͵ͲͲ

ǡ

Figure 5: Example of pressure equilibration for methane adsorption at 7 MPa & 25 oC % Difference in Z-factor of methane @ 25 oC byEOS relative to NIST-refprop Ǧ

͹Ǥͳ

Ǧ

͸Ǥʹ

Ǧ

ͷǤ͵

Equilibrium Pressure, MPa

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 17

ͶǤͶ ͵Ǥʹ ʹǤͲ ͳǤͲ ͲǤͷ Ǧ͵ǤͲͲΨ

ǦʹǤͲͲΨ

ǦͳǤͲͲΨ ͲǤͲͲΨ ͳǤͲͲΨ Diff erence in z-factor

ʹǤͲͲΨ

͵ǤͲͲΨ

Figure 6: Variation in z-factor of methane at 25 oC by different equations of state


Page 13 of 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Figure 7: Effects of equation of state on methane adsorption capacity of coal ȋȌΨȋ ǦȌ

ȋȌΨȋ ǦȌ Ǧ

Ǧ

ʹͷ

ʹͷ ͶͲ

ͶͲ Ǧ

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Ψ

ǦͶͲ Ǧ͵Ͳ ǦʹͲ ǦͳͲ

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͵Ͳ

ͶͲ

Ψ

Figure 8: Percentage difference in Langmuir parameters by EOS with respect to NIST-Refprop®


Energy & Fuels 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 17

Table 1: compositions of the coal sample Components Composition (%) Telovitrinte 71.1 Detrovitrinite 4.0 Fusinite 3.3 Semi-fusinite 13.0 macrinite 0.3 Inertodetrinite 4.0 Mineral matter 4.5

Table 2: Langmuir Fitting Parameters (Adsorption Isotherms) for Different Equations of State Equation of State

Langmuir Parameters

NIST-refprop VL, Scf/ton PL, MPa ρads, Kg/m3 R2 VL, Scf/ton PR PL, MPa ρads, Kg/m3 R2 PR-Peneloux VL, Scf/ton PL, MPa ρads, Kg/m3 R2 VL, Scf/ton SRK PL, MPa ρads, Kg/m3 R2 VL, Scf/ton SRKPeneloux PL, MPa ρads, Kg/m3 R2 VL, Scf/ton SBWR PL, MPa ρads, Kg/m3 R2 VL, Scf/ton Lee-Kesler PL, MPa ρads, Kg/m3 R2

25.0 oC 772.9 1.02 689.9 0.9985 943.3 1.34 590.9 0.9969 964.6 1.38 578.8 0.9966 806.6 1.08 183.8 0.9978 815.0 1.10 165.4 0.9977 800.0 1.09 1.30E+09 0.9983 836.8 1.13 742.9 0.9973

40.0 oC 705.8 1.26 92.1 0.9953 860.9 1.63 103.7 0.9918 880.1 1.68 104.6 0.9915 740.0 1.34 69.4 0.9946 748.2 1.35 66.9 0.9947 699.7 1.23 116.6 0.9957 768.4 1.39 98.2 0.9911


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Author Information Corresponding Author * E-mail Address: [email protected] Notes The authors declare no conflicting financial interest. Acknowledgements The authors would like to acknowledge the contributions of Australian Government Research Training Program and Curtin Research Scholarships and the Unconventional Gas Research group at the discipline of Petroleum Engineering, Western Australian School of Mines: Minerals, Energy and Chemical Engineering in supporting this research. Supporting Information x

Appendix A - description of the comparison equations of state

x

Appendix B - description of the adsorption steps

x

Appendix C – volume of free space calculated by different equations of state

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Effect of Equation of States on High Pressure Volumetric

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