9610
Langmuir 2006, 22, 9610-9618
Modeling High-Pressure Adsorption of Gas Mixtures on Activated Carbon and Coal Using a Simplified Local-Density Model James E. Fitzgerald, Robert L. Robinson, Jr., and Khaled A. M. Gasem* School of Chemical Engineering, Oklahoma State UniVersity, Stillwater, Oklahoma 74078 ReceiVed April 4, 2006. In Final Form: July 25, 2006 The simplified local-density (SLD) theory was investigated regarding its ability to provide accurate representations and predictions of high-pressure supercritical adsorption isotherms encountered in coalbed methane (CBM) recovery and CO2 sequestration. Attention was focused on the ability of the SLD theory to predict mixed-gas adsorption solely on the basis of information from pure gas isotherms using a modified Peng-Robinson (PR) equation of state (EOS). An extensive set of high-pressure adsorption measurements was used in this evaluation. These measurements included pure and binary mixture adsorption measurements for several gas compositions up to 14 MPa for Calgon F-400 activated carbon and three water-moistened coals. Also included were ternary measurements for the activated carbon and one coal. For the adsorption of methane, nitrogen, and CO2 on dry activated carbon, the SLD-PR can predict the component mixture adsorption within about 2.2 times the experimental uncertainty on average solely on the basis of pure-component adsorption isotherms. For the adsorption of methane, nitrogen, and CO2 on two of the three wet coals, the SLD-PR model can predict the component adsorption within the experimental uncertainties on average for all feed fractions (nominally molar compositions of 20/80, 40/60, 60/40, and 80/20) of the three binary gas mixture combinations, although predictions for some specific feed fractions are outside of their experimental uncertainties.
1. Introduction Deep coalbeds contain large quantities of gases such as methane, nitrogen, and carbon dioxide (CO2) through the phenomenon of adsorption. In the U.S. alone, coalbeds are estimated to hold about 135 trillion cubic feet of recoverable gas. This is approximately 14% of U.S. natural gas reserves.1 Coalbed methane recovery accounted for 6% of the U.S. natural gas production in 19972 and 8% in 2003.3 As a result of adsorption in a coalbed reservoir, the methane resides primarily inside the microporous coal structure at a density higher than that in the free gas phase. In adsorption, when a gas (adsorbate) interacts with the surface of the adsorbent, the solidgas interactions increase the fluid density near the solid surface to a value comparable to that of a liquid. This adsorbate density is dependent on temperature, pressure, and molar composition (including gas species other than methane). In the adsorbate phase, a typical reservoir has roughly 90% methane, 8% CO2, and 2% nitrogen, with traces of other hydrocarbons.4 Over the past decade, our objective has been to evaluate/ develop models capable of correlating high-pressure adsorption of the type encountered in CBM reservoirs. To meet the modeling demands of enhanced CBM recovery and CO2 sequestration, which involve multicomponent fluid mixtures, we have sought equilibrium models capable of (a) representing precisely pure and mixture isotherms and (b) facilitating accurate a priori predictions based on gas properties and coal matrix characterization. Our recent results have shown that models originating from three different frameworksstwo-dimensional (2-D) equations * Corresponding author. E-mail:
[email protected]. Phone: (405) 7445280. Fax: (405) 744-6338. (1) Clayton, J. L. Int. J. Coal Geol. 1998, 35, 159-173. (2) Stevens, S.; Spector, D.; Reimer, P. Enhanced Coalbed Methane Recovery Using CO2 Injection: Worldwide Resource and CO2 Sequestration Potential SPE 48881. Proceedings of the Sixth International Oil and Gas Conference and Exhibition in China, 1, 1998. (3) Energy Information Administration. AdVance Summary: U.S. Crude Oil, Natural Gas, and Natural Gas Liquids ReserVes 2003 Annual Report; http:// www.eia.doe.gov/oil_gas/fwd/adsum2003.html; 2005. (4) Mavor, M.; Pratt, T.; DeBruyn, R. Oil Gas J. 1999 April 26, 35-50.
of state (EOS),5 the Ono-Kondo (OK) lattice model,6-8 and the simplified local density/Peng-Robinson model (SLD-PR)9,10s have promising capabilities for representing the adsorption of near-critical and supercritical gases of the type encountered in CBM recovery and CO2 sequestration. In the present work, we report our findings in applying the SLD-PR model to mixture adsorption. Data on the adsorption of gas mixtures are necessary to validate theories for mixture adsorption. For mixed-gas adsorption, few experimental studies have been performed at pressures high enough that observable differences exist between excess and absolute adsorption. However, such high pressures are encountered in enhanced coalbed methane operations, where nitrogen and/or CO2 gas is injected into coalbeds to stimulate production.2,11,12 In this study, we test the model using representative binary gas adsorption measurements on activated carbon (Calgon F-400) and on three coals (Fruitland, Tiffany, and Illinois #6). In addition, the model was tested for a ternary system on the same activated carbon and for binary and ternary mixtures on wet coals, which are the primary interest of the coalbed methane (5) Zhou, C.; Hall, F.; Gasem K. A. M.; Robinson R. L., Jr. I&EC Res. 1994, 33, 1280-1289. (6) Ono, S.; Kondo, S. Molecular Theory of Surface Tension; Springer: Berlin, 1960. (7) Sudibandriyo, M. Generalized Ono-Kondo Lattice Model for High-Pressure Adsorption on Carbon Adsorbents. Ph.D. Dissertation, Oklahoma State University, Stillwater, OK, 2003. (8) Sudibandriyo, M.; Fitzgerald, J. E.; Pan, Z.; Robinson, R. L., Jr.; Gasem, K. A. M. Extension of the Ono-Kondo Lattice Model to High-Pressure Mixture Adsorption. Proceedings of the AIChE Spring National Meeting, March 31April 3, 2003, New Orleans, LA. (9) Fitzgerald, J. E.; Sudibandriyo, M.; Pan, Z.; Robinson, R. L., Jr.; Gasem, K. A. M. Carbon 2003, 41, 2203-2216. (10) Fitzgerald, J. E. Adsorption of Pure and Multicomponent Gases of Importance to Enhanced Coalbed Methane Recovery: Measurements and Simplified Local Density Modeling. Ph.D. Dissertation, Oklahoma State University, Stillwater, OK, 2005. (11) Arri, L. E.; Yee, D. Modeling Coalbed Methane Production with Binary Gas Sorption, SPE Paper 24363, Presented at the SPE Rocky Mountain Regional Meeting, May 18-21, 1992, Casper, WY. (12) Stevenson, M. D.; Pinczewski, W. K. Economic Evaluation of Nitrogen Injection of Coalseam Gas Recovery, SPE Paper 26199, Presented at the SPE Gas Technology Symposium 28-30 June, 1993, Calgary, Alberta, Canada.
10.1021/la060898r CCC: $33.50 © 2006 American Chemical Society Published on Web 10/17/2006
Modeling High-Pressure Adsorption of Gas Mixtures
industry. The activated carbon was included because the experimental uncertainties in the data are lower than the uncertainty for the coals, thus providing a more rigorous test for the model. Moreover, because coal has a more complex structure than activated carbon, the adsorption on activated carbon serves as a reference for the more complicated adsorption/desorption on water-saturated coal. The isotherms presented here are type I in character, according to the Gibbs classification of isotherms.13 They do not contain evidence of any pore condensation, and many (but not all) isotherms presented here have an adsorption maximum within the experimental pressures of this study. For modeling purposes, water is not taken as an explicit component of the adsorbed fluid. This is the common practice in modeling wet coals. Rather, the assumption is made that the inhibition of gas adsorption due to water is constant at water contents above the equilibrium moisture value and does not affect the ability of the SLD-PR model to predict the mixed-gas adsorption. The computational algorithm employed in this study accounts rigorously for the equilibrium relations of component gas adsorption as well as for the overall and component mass balances. As such, this algorithm allows for the quantitative prediction of how a feed gas partitions itself into the equilibrium compositions, when introduced into the system with a known void volume and an adsorbent of known mass. Specifically, for mixed-gas adsorption at high pressures, we are able to determine quantitatively the component adsorptions of the gas mixture, the equilibrium molar compositions, and the densities of both the adsorbates and the adsorbed phase. In essence, this is similar to the traditional vapor-liquid equilibrium flash calculations where one obtains both the equilibrium and volumetric properties of the system and the amounts of each phase. Although the solution of the governing equations can result in multiple roots, none were encountered in this work, which involved supercritical gas adsorption. Recently, Yan and Yang used our experimental data14 for the binary mixtures of methane, nitrogen, and CO2 on Calgon F-400 activated carbon in a modeling study that employed a nonlocal density functional theory (NDFT) with a pore-size distribution (PSD) model.15 The predictive results for binary adsorption, based on their model characterizations of the pure gas adsorption, were qualitatiVe for the binary systems containing CO2 but reasonable in general. In the present article, we demonstrate that the SLD theory in combination with a modified PR EOS can quantitatiVely predict these binary mixtures using a slit-pore model. The use of a single pore model, rather than the more physically realistic (and more complicated) PSD model, greatly simplifies the calculations necessary for mixed-gas adsorption. Furthermore, for each adsorbent, the SLD-PR model can be applied using a common value for the surface area and an effective slit width for all gas species over all composition and pressure ranges.
2. Modeling a. Definitions of Excess Adsorption. The experimentally accessible data being modeled in this work is the excess adsorption, nEx, which is defined physically as the total amount of gas contained within the system of interest (e.g., coalbed or test apparatus) relative to the amount that would be contained if no adsorption occurred (that is, if an unadsorbed gas phase (13) Donohue, M. D.; Aranovich, G. L. Fluid Phase Equil. 1999, 158-160, 557-563. (14) Sudibandriyo, M.; Pan, Z.; Fitzgerald, J. E.; Robinson, R. L., Jr.; Gasem, K. A. M. Langmuir 2003, 19, 5323-5331. (15) Yan, B.; Yang, X. Chem. Eng. Sci. 2005, 60, 3267-3277.
Langmuir, Vol. 22, No. 23, 2006 9611
occupied the available space throughout the available fluid volume). Mathematically, this may be expressed in experimental quantities for the case of a pure gas as
nEx ≡
ntot - FgasVvoid madsorbent
(1)
where ntot is the total amount of gas (adsorbed and unadsorbed) within the system, Fgas is the density of the unadsorbed (bulk) gas under the experimental conditions, and Vvoid is the void volume within the system (accessible to the adsorbed and unadsorbed material). Furthermore, madsorbent is the amount of adsorbent (e.g., coal) present in the system, and it is used to express the excess adsorption on a per unit mass of adsorbent basis. For illustration, consider a system of fixed volume that contains an adsorbent and into which a gas feed mixture of a given composition is introduced. At equilibrium, the feed gas may be envisioned to be distributed between two phases, the bulk-gas phase and the adsorbed phase. The bulk-gas density and the component mole fractions are defined as Fgas and yi, respectively. Each gas component (i) adsorbs onto the adsorbent. For adsorption from gas mixtures, the excess adsorption of a component i in the mixture, nEx i . is defined as the amount of gas contained within the system relative to the amount that would be contained if all of the void volume were occupied by gas having the equilibrium composition yi,
nEx i )
zintot - VvoidFgasyi madsorbent
(2)
where zi is the overall composition of the gas, ntot is the total amount of gas in the system, and yi is the composition of the equilibrium gas phase. The values of zi and yi will, in general, be different because the adsorbed material will differ in composition from the gas under equilibrium conditions. The equilibrium molar gas composition for each gas, yi, can be calculated by a model if the overall feed composition for each gas, zi, the mass of adsorbent, and the void volume are known. The general method is to assume a set of equilibrium gas compositions and from this to calculate the excess adsorption for each component, nEx i . The assumed equilibrium gas compositions are correct if the following constraint is obeyed for each component:
zi )
nEx i + FgasyiVvoid/madsorbent nEx + FgasVvoid/madsorbent
(3)
For an adsorption experiment based on volumetric principles, the expected experimental error in the excess adsorption, σniEx, is reduced if the largest adsorbent mass possible is placed inside the system container of volume Vcontainer. This minimizes Vvoid/ madsorbent and requires that values of zi be generally different from those of yi. Further details of our experimental techniques are given elsewhere.14 b. Simplified Local Density Model for Mixed-Gas Adsorption. Rangarajan and Lira17 originally articulated the physical premises and assumptions of SLD theory as presented in this work. The model assumes that (1) the chemical potential at any point near the adsorbent surface is equal to the bulk-phase (16) Rangarajan, B.; Lira, C. T. Subramanian R. AIChE J. 1995, 41, 838-845. (17) Chen, J. H.; Wong, D. S. H.; Tan, C. S.; Subramanian R.; Lira C. T.; Orth M. Ind. Eng. Chem. Res. 1997, 36, 2808-2815.
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chemical potential and (2) the chemical potential at any point above the surface is the sum of the fluid-fluid and fluid-solid interactions. Accordingly, the equilibrium chemical potential is calculated by contributions from these fluid-fluid and fluidsolid interactions. Typically, an integrated potential function such as the 10-4 Lennard-Jones model is used to describe the fluidsolid interactions. Many equations of state (EOS), such as the van der Waals (vdW),16 Peng-Robinson (PR),9,17 ElliotSuresh-Donohue ESD,18,1918-19 and Bender2020 have been used to provide the fluid-fluid potential information. The pore geometry most widely used in a local-density model for carbon adsorbents is a slit with a specified distance L. In this work, the slit width L is defined as the distance between the two orthoganal planes that are tangential to the surfaces of the first graphite planes on opposing sides of the slit. For a slit of width L, the chemical potential is written as
µ(z) ) µff(z) + µfs1(z) + µfs2(L - z) ) µbulk
(4)
where the subscript “bulk” refers to the bulk fluid, “ff” refers to fluid-fluid interactions, and “fs” refers to the fluid-solid interactions. The position within a slit is z, where z is orthogonal to the plane of the solid phase defined as a flat surface formed by the peripheral carbon atoms. A molecule within a slit has fluid-solid interactions with both slit surfaces at distances z and L - z, as seen in eq 4. Detailed information about how the SLD model for slit pores, used in conjunction with the PR EOS, is used for pure gases can be found elsewhere.10 Soule18 used the SLD framework in conjunction with the ESD (Elliot-Suresh-Donohue) EOS21 to model binary gas adsorption. The present modeling differs from Soule’s study in that the model evaluations are conducted using higher-pressure data (14 MPa compared to 2 MPa) for many mixture combinations, including ternary mixture isotherms. For this study, a mass balance for each fluid component is solved simultaneously with the equilibrium criterion for each fluid component, as given by eqs 3 and 4. For pure gas or mixture adsorption, the local density, Fads(z), is a function of position. In addition, in mixture adsorption, the composition, xi, changes with position also. Thus, the excess adsorption on the surface, A, may be described by
nEx i )
∫leftrightsidesideof ofslitslit (Fads(z) xi(z) - Fbulkyi) dz
A 2
(5)
where the local density and composition can be determined from eq 6. For pure gas adsorption, x ) y ) 1. In this study, the PR-EOS was employed to calculate the bulk density and local density. For the Soave-type modifier for the EOS attraction law constant, R(T), in the PR EOS, the MathiasCopeman expression was used with regressed parameters,22 as discussed previously by the present authors.9 The local adsorbed density and the mole fractions of the components at each position (18) Soule, A. D. Studies of Gas Adsorption on Activated Carbon and Cherry Flavor Recovery from Cherry Pits. M.S. Thesis, Michigan State University, East Lansing, MI, 1998. (19) Soule, A. D.; Smith, C. A.; Yang, X.; Lira, C. T. Langmuir 2001, 17, 2950-2957. (20) Ustinov, E. A.; Do, D. D.; Herbst, A.; Staudt, R.; Harting P. J. Colloid Interface Sci. 2002, 250, 49-62. (21) Elliott, J. R.; Suresh, S. J.; Donohue, M. D. Ind. Eng. Chem. Res. 1990, 29, 1476-1485. (22) Herna´ndez-Garduza, O.; Garcia-Sanchez, F.; Apam-Martinez, D.; VazquezRoman, R. Fluid Phase Equilib. 2002, 198, 195-228.
z can be calculated by a local equilibrium relationship
(
ln
)
ˆf ads b(z), Fads(z)) i (x ˆf bulk i
+
Ψfsi (z) + Ψfsi (L - z) )0 kT
(6)
In the adsorbed phase, the fugacity of component i in a mixture, ˆfi, is a function of the local composition, local density, pressure, and temperature. In the bulk phase, the fugacity of component i in a mixture is solved at the bulk density, pressure, and temperature. The fluid-solid potential is a function of the slit geometry and position. Following Chen et al.,17 a partially integrated 10-4 Lennard-Jones potential was used to describe the fluid-solid interactions for each component, which is a truncated version of Steele’s 10-4-3 potential function. Although the fluid may reside anywhere, the density is negligibly small in the distance from the wall to about 3/8(σff). Consequently, for this work, the potential has been set to infinity for positions less than 3/8(σff) from the slit wall:
Ψfsi (z) )
(
4πFatoms(fs)i(σfs)i2
(σfs)i10
-
1
(σfs)i4
4
∑
)
5(z′)10 2 i)1 (z′ + (i - 1)σiss)4 for 3/8(σff)i e z e L - 3/8(σff)i
Ψfsi (z) ) ∞ elsewhere
(7)
Here, fs is the fluid-solid interaction energy parameter and Fatoms ) 0.382 atoms/Å2. The molecular diameter of the adsorbate, the carbon interplanar distance, and the carbon collision diameter are σff, σiss, and σss, respectively. The carbon interplanar distance was adopted to be 0.335 nm,23 and fluid diameters are 0.3758, 0.3798, and 0.3941 nm for methane, nitrogen, and CO2, respectively.24 For convenience, the fluid-solid diameter σfs and the dummy coordinate z′ are defined as σfs ) (σff + σss)/2 and z′ ) z + (σss/2). The pore volume is taken to include the entire volume covered by the slit, (AL)/2. Because the profile of the local-adsorbed density and compositions are symmetric about the pore midpoint, eq 5 is evaluated for one side of the slit only, and the result is multiplied by 2, as shown in eq 8. Also, because the fluid-solid potential was assumed to be infinite at distances of less than 3/8(σff) from the wall surface, the third integral in eq 8 is zero. The second integral is not zero to accommodate the convenient definition of the pore volume as being A(L/2), rather than the fluid-specific definition of A(L - 3/4(σff)i)/2.
nEx i )A
L/2 (Fads(z) xi(z) - Fbulkyi) dz + ∫3/8(σ ) 0 0 A∫3/8(σ ) (Fbulkyi) dz - A∫3/8(σ ) (Fads(z) xi(z)) dz ff i
ff i
(8)
ff i
In the bulk phase, we use a linear mixing rule for b and a quadratic mixing rule for a. The fugacity for the bulk phase using the PR-EOS is
( )
(
ˆf bulk i
ln
)
bi pb ) (Z - 1) - ln Z + yiP b RT a 2x2RTb
(
bi b
2 -
)(
∑j yjaij a
ln
)
(1 + Fb(1 + x2))
(1 + Fb(1 - x2))
(9)
Modeling High-Pressure Adsorption of Gas Mixtures
Langmuir, Vol. 22, No. 23, 2006 9613
In the adsorbed phase, we use quadratic mixing rules for both the co-volume b and the attraction constant a. The quadratic mixing rules were included for b in the adsorbed phase because they provided a marginally better fit of the adsorption data. Using these rules, the fugacity in the adsorbed phase is
( ) ˆf ads i (z)
ln
∑j xj(z)bij - b
P
b
Fads(z)RT
)
xi(z)P
ln
(
(
2
P
Fads(z)RT 2
-
Pb RT
)
)(
∑j xj(z) aij(z)
2x2RTb
-
b
(1 + Fads(z)b(1 + x2))
ln
a(z)
(
∑j xj(z)bij - b
2
a(z)
+
)
-1 -
(1 + Fads(z)b(1 - x2))
)
(10)
The formulas for ai(z) depend on the ratio of the slit width, L, to the molecular diameter, (σff)i. Rangarajan et al.16 obtained these formulas by integrating the sum of all two-body interactions between an arbitrarily selected central molecule and all of the other molecules around it. The results of Chen et al.17 were used for the equations describing a as a function of position within a slit. The value of a leads directly to the dependence of the fluid density on the distance from the slit wall (at constant T and p). Combining rules for aij follow the geometric mean. Included in the rule is the flexibility of using an empirical binary interaction parameter (BIP), Cij, to account for inaccuracy in the geometric mean rule.
(aads)ij ) x(aads)i(aads)j(1 - Cij)
(11)
In this article, for simplicity the BIPs are set equal to zero for all component interactions in the bulk phase, but BIPs are regressed for the adsorbed phase. In this work, as well as the previous study9 on pure-gas adsorption, we have adjusted the co-volume, bi, by an empirical fraction Λbi. At high adsorptive loadings, the local fluid density at the wall decreases with increasing values of the parameter Λbi, which is nominally zero. We use a linear combining rule for the co-volume interaction in the adsorbed phase:
(bads)ij )
(
)
bi(1 + Λbi) + bj(1 + Λbj) 2
(12)
In our previous study,9 Λbi were positive values ranging from 0.52 to 0.56 for the pure-gas adsorption on Calgon F-400 activated carbon. Regressed values for Λbi are in general lower in this study because the pore volume included the volume at the solidfluid interface; specifically, the pore volume in the previous and current studies are A(L - σff)/2 and (AL)/2. c. Testing the SLD-PR Model for Mixed-Gas Adsorption. Several general objective criteria can be stated for a successful adsorption model. For mixed-gas adsorption at high pressures, our view is that the model should be able to predict quantitatively the component adsorptions of the gas mixture and the equilibrium molar compositions. In this work, we evaluated both the correlative and predictive capabilities of the SLD-PR model. (23) Subramanian, R.; Pyada, H.; Lira, C. T. Ind. Eng. Chem. Res. 1995, 34, 3830-3837. (24) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids; McGraw-Hill: New York, 1987.
The methods used to evaluate the model parameters of such correlations or predictions of CBM mixed-gas adsorption for various case studies are described later. Model parameters generated from pure-component adsorption isotherms are necessary for the prediction of mixture adsorption. As discussed elsewhere10 for pure-gas adsorption, the SLD-PR model can correlate the isotherms of many systems involving wet and dry coals, activated carbons, and zeolites over broad pressure and temperature ranges. To correlate precisely pure-gas adsorption over broad pressure ranges, however, the SLD-PR co-volume must be modified empirically for each adsorbate. Furthermore, the fluid-solid interaction energy for each adsorbate must be regressed to correlate pure-gas isotherms because values predicted from theory do not work in the SLD-PR model. For a specified adsorbent, a common value for surface area and for effective slit width may be used for all gas species over all temperature and pressure ranges. This finding is in contrast to an earlier study where the adsorbent surface area was allowed to vary for each gas.9 Thus, the SLD-PR model requires (2N + 2) model parameters (N values of fs and of Λb, plus L and A) to describe pure-gas adsorption on each adsorbent, where N is the number of gas species. To perform an adsorption flash calculation, the temperature, pressure, feed mole fractions, mass of adsorbent, and void volume are required input information. Once performed, the flash calculation provides the component excess adsorption and the equilibrium mole fraction in the bulk gas phase for each component. In this work, half of the adjusted slit width, (L - 3/4(σff)i)/2, was subdivided into 50 intervals. At each interval, the local density and the adsorbed mole fractions were calculated by simultaneously solving eq 6 for all components and using the mole fraction summation requirement, ∑i xi ) 1. The adsorbed mole fractions were initialized to the feed mole fractions (i.e., xi ) zi for all components i). The solution, however, is contingent on equilibrium (not overall) mole fractions, yi, because they are needed to solve for the bulk-phase fugacity in eq 10. The bulk mole fractions were initialized with the available experimental values to speed the calculation (although any reasonable initial values could be used). Once the local density and the adsorbed mole fractions were calculated for each interval, a trial adsorbed amount was calculated by numerical integration using Simpson’s rule in accordance with eq 8. The next step is to evaluate eq 3 for each gas component. If eq 3 is not satisfied for each component, then a new set of equilibrium mole fractions is used to calculate the next trial adsorbed amount. The procedure is repeated until eq 3 and the adsorbed mole fraction summation are satisfied. Newton’s method with numerical derivatives was used to solve eqs 6 and 10. For the systems considered, convergence was always reliable for eq 10, and if suitable values of fs and L were chosen, it was reliable for eq 6. For the optimization of parameters, the objective function WAAD (weighted-average absolute deviation) was used to correlate data with the SLD-PR model. The function represents the average of the weighted deviations in predicted excess adsorption
WAAD )
1
NPTS
∑ abs NPTS i)1
(
)
ncalcd - nexptl σexptl
(13)
where NPTS is the number of data points, ncalcd and nexptl are the calculated and the experimental excess adsorption, and σexptl is the expected experimental uncertainty for datum i. For the data
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Table 1. Compositional Analysis of Adsorbents Used in This Study Calgon F-400
Fruitland
Illinois #6
Tiffany well #1
Tiffany well #10
carbon % hydrogen % oxygen % nitrogen % sulfur % ash %
88.65 0.74 3.01 0.40 0.73 6.46
Ultimate 68.63 4.27 0.89 1.57 4.19 20.45
71.47 5.13 9.85 1.46 1.27 10.81
47.78 2.62 6.19 0.92 0.57 49.71
56.75 2.77 5.16 1.02 0.52 47.74
vol. matter % fixed carbon %
3.68 89.86
Proximate 20.2 30.61 59.35 55.90
15.48 34.82
15.35 36.91
analysisa
a
Huffman Laboratories, Inc., Golden, Colorado.
used in this work, values of σexptl were estimated from propagation of error calculations in each individual experimental study. To test the efficacy of the SLD-PR model for mixtures, we considered two case studies. The data sets employed are from measurements done at Oklahoma State University; they include mixture adsorption on dry Calgon F-400 activated carbon, wet Fruitland coal, wet Tiffany coal, and wet Illinois #6 coal. Ultimate and proximate analyses for the adsorbents are provided in Table 1. Of particular interest is the adsorption data set for dry Calgon F-400, which consists of the most extensive and reliable (as ascertained by expected experimental uncertainties) measurements for high-pressure binary mixture gas adsorption. Two cases were studied for each adsorbent mentioned above, as follows: Case I. Model parameters for the pure gases were obtained by simultaneous regression of the data for all gases to obtain common values of L and A for all gases as well as the individual parameters characterizing each gas (fs and of Λb for each pure substance) for a total of eight parameters. Mixture adsorption was then predicted on the basis of these eight pure-component parameters. (All BIP values, Cij,were set equal to zero.) This case tests the ability of the model to predict mixed-gas adsorption behavior solely from knowledge of the adsorption behavior of the pure substances in the mixture. Case II. In this case, the pure substance and mixture data were regressed simultaneously to determine the parameters of case I plus the interaction parameters, Cij, for a total of 11 parameters. This case tests the ultimate ability of the model to represent (correlate) both pure and mixed-gas adsorption behavior (for the mixing rules used in this work). The results of the case studies for each of four adsorbents are given below.
3. Adsorbents a. Calgon F-400 Activated Carbon. A total of 335 independent adsorption data points were available for the adsorption of pure
Figure 1. SLD model of methane adsorption in methane/nitrogen mixtures on Calgon F-400 activated carbon at 318.2 K (solid line, regressed Cij; dashed line, Cij ) 0.0).
Figure 2. SLD model of nitrogen adsorption in methane/nitrogen mixtures on Calgon F-400 activated carbon at 318.2 K.
gases and mixtures on this activated carbon. Table 2 presents the parameters regressed for the SLD model for case studies I and II. Figures 1-6 show the model isotherms for both cases. In these Figures, designations such as “Methane/Nitrogen 80/20” indicate that the feed gas to the experiment was 80 mol % methane/ 20 mol % nitrogen. In general, the SLD-PR model is capable of correlating the adsorption data considered in this study within their experimental uncertainties (represented by error bars in all Figures in this work). Overall, the SLD-PR model correlates all of the adsorption data within one experimental standard deviation. Model predictions, based solely on pure-substance adsorption data, are within twice the experimental uncertainties for the methane/nitrogen mixture isotherms, and they are within five times the experimental uncertainties for the CO2-containing mixtures; however, for the gas component adsorbed in its richer mixtures (i.e., 80/20, 60/40), the error is generally less. The SLD-PR predicts the component mixture adsorption at 2.2 times
Table 2. Regressed Parameters for the SLD-PR Model for Activated Carbon and Three Coals fs/k (K) N2
adsorbent
surface area (m2/g)
slit length (nm)
CH4
Calgon F-400 Fruitland coal Illinois #6 coal Tiffany coal
621 63.4 56.3 36.0
1.20 1.02 1.35 1.06
Scenario 1: 71.9 56.1 44.9 51.7
Calgon F-400 Fruitland coal Illinois #6 coal Tiffany coal
604 73.3 54.7 49.4
1.09 1.16 1.33 1.02
CO2
CH4
Λb N2
CO2
Regressed Values Used for Prediction 48.7 86.7 0.33 0.51 0.12 31.7 58.4 -0.23 0.00 -0.32 19.5 63.6 0.08 0.17 -0.06 30.3 64.0 -0.19 0.09 -0.30
Scenario 2: Regressed Values for Correlation 70.5 49.7 76.9 0.23 0.43 56.7 29.3 64.5 -0.07 0.18 43.2 20.2 65.5 0.04 0.27 47.4 23.8 60.9 0.02 0.08
0.00 -0.16 -0.05 -0.11
CH4 -N2
Cij CH4 -CO2
N2 -CO2
0 0 0 0
0 0 0 0
0 0 0 0
0.08 -0.54 0.59 -0.08
0.00 -0.07 -0.07 -0.42
-0.25 -0.50 -0.51 -0.82
Modeling High-Pressure Adsorption of Gas Mixtures
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Figure 3. SLD model of methane adsorption in methane/CO2 mixtures on Calgon F-400 activated carbon at 318.2 K.
Figure 6. SLD model of CO2 adsorption in nitrogen/CO2 mixtures on Calgon F-400 activated carbon at 318.2 K.
Figure 4. SLD model of CO2 adsorption in methane/CO2 mixtures on Calgon F-400 activated carbon at 318.2 K.
Figure 7. SLD model prediction of gas-phase compositions for methane/nitrogen feed mixtures on Calgon F-400 activated carbon at 318.2 K.
Figure 5. SLD model of nitrogen adsorption in nitrogen/CO2 mixtures on Calgon F-400 activated carbon at 318.2 K.
the experimental uncertainty on average for all mixture isotherms. As shown in Figures 1-6, the experimental uncertainties are small for many isotherms; thus, predictive errors of two or three times the uncertainty translates to relatively small absolute deviations from the experimental isotherm. Furthermore, it is worth mentioning that the percentage errors would be substantially less if the data were expressed in terms of absolute (rather than excess) adsorption. The SLD-PR model can correlate adsorption isotherms that exhibit a maximum as a function of pressure. Of particular interest is nitrogen adsorption in the 40/60 and 20/80 nitrogen/CO2 mixtures (Figure 5). These two isotherms each exhibit a maximum and an inflection point and become concave up with pressure, finally crossing over and becoming negative. The correlative model is least accurate for methane adsorption at high methane concentrations in methane/nitrogen mixtures. Also, for pure CO2 at high pressures, the model overpredicts the
Figure 8. SLD model of ternary adsorption of 10/40/50 methane/ nitrogen/CO2 mixture on Calgon F-400 activated carbon at 318.2 K.
data. The models quantitatively predict the gas-phase composition as a function of pressure, as shown for methane/nitrogen mixtures in Figure 7. Other mixtures have similar quantitative agreement. As shown in Figure 7, the gas-phase composition approaches the feed gas composition at higher pressures because the amount of gas adsorbed at higher pressures is less significant relative to the amount in the gas phase. For systems with higher void volumes, the gas-phase composition is closer to the feed composition. Adsorption measurements also were conducted for a ternary 10/40/50 methane/nitrogen/CO2 feed mixture. These measurements facilitate model testing of ternary mixture adsorption solely from pure-component measurements or from correlated binary isotherms. For those cases, Figure 8 depicts the SLD model predictions for the component excess isotherms of methane, nitrogen, and CO2. The model predictions of gas adsorption for
9616 Langmuir, Vol. 22, No. 23, 2006
Figure 9. SLD model of methane adsorption in methane/nitrogen mixtures on wet Fruitland coal at 319.3 K (solid line, regressed Cij; dashed line, Cij ) 0.0).
Figure 10. SLD model of nitrogen adsorption in methane/nitrogen mixtures on wet Fruitland coal at 319.3 K.
CO2 and methane are on average within the experimental uncertainty for each isotherm, and for nitrogen, the predictions are within, on average, three times the experimental uncertainty. b. Wet Fruitland Coal. Although calculations were performed for three coals, only the results for Fruitland will be described here (for brevity and because the results are similar, in most cases, for the three coals). Details for the other coals appear in the Supporting Information. Hall published pure and binary measurements of methane, nitrogen, and CO2 on Fruitland coal at 319.3 K.25,26 However, many of the reported adsorption measurements containing CO2 were calculated using inadequate gas-phase density predictions. As a result, Hall’s raw data were used to recalculate the adsorption on Fruitland coal and estimate the uncertainty for each datum. In addition to Hall’s data, pure-gas adsorption on Fruitland coal measured by others, as depicted in a previous study,9 was included in the data regression analysis. Figures 9-14 depict the model results for all of the experimental data. For clarity, only select data for the pure-gas adsorption are depicted in the Figures. The SLD can predict the component adsorption within the experimental uncertainties on average for the three binary gas mixtures, although some specific isotherms are outside their experimental uncertainties. For example, the amount of nitrogen adsorbed in a methane/nitrogen mixture (Figure 10) or the nitrogen adsorbed in a nitrogen/CO2 mixture (Figure 1) is underpredicted, especially for mixtures that are (25) Hall, F. E., Jr. Adsorption of Pure and Multicomponent Gases on Wet Fruitland Coal. M.S. Thesis, Oklahoma State University, Stillwater, OK, 1993. (26) Hall, F.; Zhou, C.; Gasem, K. A. M.; Robinson, R. L., Jr; Yee D. Adsorption of Pure Methane, Nitrogen, and Carbon Dioxide and their Binary Mixtures on Wet Fruitland Coal, SPE Paper 29194, Presented at the SPE Eastern Regional Meeting, Nov 8-10, 1994, Charleston, WV.
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Figure 11. SLD model of methane adsorption in methane/CO2 mixtures on wet Fruitland coal at 319.3 K.
Figure 12. SLD model of CO2 adsorption in methane/CO2 mixtures on wet fruitland coal at 319.3 K.
Figure 13. SLD model of nitrogen adsorption in nitrogen/CO2 mixtures on wet Fruitland coal at 319.3 K.
more dilute in nitrogen, where the predictions are generally outside the experimental uncertainties. The WAAD for the correlative model (case II) for each component in the three binary mixtures shown are much better than those of the predictive model (case I). Comparing case II to case I, the percentage improvement of the WAAD for the lesser-adsorbed component is better than for the more-adsorbed component. On the basis that the WAAD is less than unity for the predicted adsorption of each component in a binary mixture, the use of BIPs to regress binary mixture adsorption is not readily justified; to do so would be an overfit (unless the experimental uncertainties are smaller than the estimated values). Nevertheless, the use of BIPs allows for the correlation of the lesser-adsorbed component in dilute mixtures within the experimental uncertainties, and in this context, the BIPs are useful.
Modeling High-Pressure Adsorption of Gas Mixtures
Figure 14. SLD model of CO2 adsorption in nitrogen/CO2 mixtures on wet Fruitland coal at 319.3 K.
For adsorption on wet Fruitland coal, the SLD can predict the component adsorption within the experimental uncertainties on average for the three binary gas mixtures, although some specific isotherms are outside their experimental uncertainties. The SLDPR model correlates the binary mixture adsorption behavior of the systems considered to be well within one standard deviation of experimental error. However, the one-fluid mixing rules require large corrections (BIPs) for some binary interactions. The use of BIPs provides correlation of the lesser-adsorbed component in dilute mixtures within the experimental uncertainties. For the two modeling cases conducted, the regressed parameters such as the surface area, pore volume, and Λb can be significantly different even though both modeling cases represented the experimental data within the experimental uncertainties, on average. c. Wet Tiffany Coal. British Petroleum (BP) Amoco provided representative coal samples from Tiffany injection wells #1 and #10, which are located in the San Juan Basin in Colorado, near the New Mexico-Colorado border. Measurements are discussed elsewhere,27 and the compositional coal analyses are provided in Table 1. Adsorption isotherms were measured for pure methane, nitrogen, and CO2 on the wet, mixed Tiffany coal sample. The coal sample was an equal-mass mixture of coals from two wells, and the measurements were conducted at 327.6 K at pressures to 13.8 MPa. Adsorption was measured for methane/nitrogen, methane/CO2, and nitrogen/CO2 binary mixtures on wet, mixed Tiffany coal at 327.6 K and pressures to 13.8 MPa. These measurements were conducted for a single molar feed composition for each mixture. Adsorption also was measured for a single methane/nitrogen/ CO2 ternary mixture on wet, mixed Tiffany coal at 327.6 K and pressures to 13.8 MPa. For brevity, model results for both cases I and II are depicted in figures shown in the Supporting Information.28 These figures indicate that the component methane adsorbed in a methane/ CO2 mixture is the worst predicted. Experimental data for this isotherm show increasing excess adsorption, whereas the model predicts an adsorption maximum at 3.4 MPa. Moderately successful predictions are made for methane/nitrogen and nitrogen/CO2 mixtures. Ternary predictions are only qualitatively successful. The correlative (case II) results overall are much better than the predictive results (case I). Upon comparison of the two cases, pure-component descriptions are worse in case II because they (27) Fitzgerald, J. E.; Pan, Z.; Robinson, R. L.; Gasem, K. A. M.; Reeves, S. Fuel 2005, 84, 2351-2263. (28) Supporting Information.
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are part of the overall regression that includes the mixture data. From the correlative pure and binary regressions, the ternary mixture component adsorption was predicted. Results indicate that the adsorption for each component can be represented within three experimental uncertainties, on average. However, large values of the BIPs are required for methane/CO2 and nitrogen/ CO2 mixtures, as shown in Table 2. Model results for Tiffany coal adsorption improved signifcantly when we introduced a BIP in the b co-volume of eq 10. Nevertheless, for further evaluation, additional data for this system would be required. d. Wet Illinois #6 Coal. The adsorption measurements on wet Illinois #6 were performed by Liang.29 These measurements indicate that both methane and nitrogen adsorption on wet Illinois #6 are about half of that adsorbed on wet Fruitland coal under the same conditions. Although the adsorption measurements for Tiffany are performed at a slightly higher temperature (328.2K) than for Illinois #6 (319.3 K), both methane and nitrogen adsorption amounts are comparable for the two coals (within 15%). Replicate runs were conducted for each pure gas to confirm the precision of the measurements and to investigate the effect of variations in moisture content and coal sample preparation on the adsorption behavior. The Illinois #6 data were acquired using two coal samples of different moisture contents. Both measurement sets indicate that water content values above the equilibrium water content do not significantly affect the adsorption behavior. Details on intralaboratory reproducibility for wet Illinois #6 coal can be found elsewhere;29 for dry Illinois #6 coal, interlaboratory reproducibility has also been investigated.28 Again, model results for both cases I and II are depicted in figures shown in the Supporting Information.28 The SLD model can predict the component adsorption well within the experimental uncertainties on average for the three binary gas mixtures. Such predictions suggest that the expected experimental uncertainties may be smaller than those used. With the use of BIPs, the SLD model improved the overall WAAD mainly through improvements for adsorption in methane/nitrogen mixtures, although such improvements are probably unwarranted upon consideration that the predictions were already well within experimental uncertainties. The nitrogen adsorption in the nitrogen/CO2 mixture isotherms shows behavior that is inconsistent with the adsorption behavior typified by Calgon F-400 activated carbon and Fruitland coal in Figures 5 and 13, respectively. Specifically, the nitrogen adsorption for all feed mixtures rises with pressure after having reached a maximum. The SLD model does not predict such behavior; however, because of the large experimental uncertainties for these isotherms, the SLD model will predict within the expected uncertainties, on average, for each isotherm. This and the discontinuous nature of some of these isotherms suggest that the observed aberrant behavior of the nitrogen adsorption in nitrogen/CO2 mixtures may be an experimental artifact. As shown in Table 2, both the surface area, A, and the pore volume, (AL)/2, are about the same for cases II and I. Parameters fs and Λb are slightly different for each case. The BIPs regressed for Illinois #6 are large compared to the respective BIPs from vapor-liquid equilibrium behavior, although they are comparable in magnitude to the BIPs regressed for Fruitland coal.
4. Discussion The SLD-PR model can provide quantitative predictions of binary gas adsorption and the equilibrium gas compositions in (29) Liang, X. Adsorption of Pure and Multi-Component Gas on Wet Coal. M.S. Thesis, Oklahoma State University, Stillwater, OK, 1999.
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high-pressure adsorption for the systems encountered in this study. The model can be useful for the prediction of mixed-gas adsorption. For the Calgon F-400 activated carbon, predictions of mixedgas adsorption, based on model characterizations of pure-gas adsorption, are quantitatively better using a single-pore model system than predictions utilizing a pore-size distribution.15 The pore model for this study was constrained to yield a surface area and slit length that are identical for all gases. This appears to be an excellent assumption. In a case study by Yan et al.,15 the total pore volume with a PSD of slit pores was not constrained to be identical for each gas. They reported pore volumes of 0.734, 0.923, 1.492 cm3/g for nitrogen, methane, and CO2, respectively, when the PSDs were regressed individually for each gas. In the second case study, the pore volumes were regressed simultaneously to be identical for the three gas pairs. (For example, the pore volume for CO2-CH4 mixtures is different than that for CO2-N2 mixtures.) The values of the pore volumes were notably different from the set generated in the first case. In contrast, the pore volumes obtained in this study are 0.373 and 0.329 cm3/g for cases I and II, respectively, and are identical for all three gases. For the SLD-PR model, the ability to use identical pore volumes for each gas may be attributed to the model flexibility gained through regressing both Λbi and fs. In principle, the co-volume empirical adjustment can be eliminated by using a more accurate EOS for describing the hard-sphere density limit, and fs should be inferred from theory. Regression of these empirical parameters may have compensated for inaccuracies in some of the assumptions of the SLD-PR model. Such assumptions include the validity of the localized criterion for equilibrium, the slit pore geometry, and the ablity of PR-EOS to describe the local fugacity in the pore (as described by eq 10). As discussed in our earlier article,9 the pore volume for the Calgon activated carbon can be estimated by the assumption that at sufficiently high pressures no further adsorption occurs. At these pressures, a plot of nEx versus Fgas should become linear with the negative slope of the pore volume. For pure CO2 and ethane, the pore volumes (calculated directly from the experimental adsorption data) were shown to be 0.376 and 0.384 cm3/ g, respectively. Ustinov et al.31 studied the mixed-gas high-pressure adsorption on Norit R1 Extra activated carbon involving binary gas mixtures of methane, nitrogen, and CO2. In his study, a distribution of elements over adsorption volume (EAV) as a function of the fluid-solid potential energy was used to characterize the puregas isotherms. The EAV and thus the pore volume are identical (30) Goodman, A. L.; Busch, A.; Duffy, G. J.; Fitzgerald, J. E.; Gasem, K. A. M.; Gensterblum, Y.; Krooss, B. M.; Levy, J.; Ozdemir, E.; Pan, Z.; Robinson, R. L., Jr.; Schroeder, K.; Sudibandriyo, M.; White, C. M. Energy Fuels 2004, 18, 1175-1182. (31) Ustinov, E. A.; Staudt, R.; Do, D. D.; Herbst, A.; Harting, P. J. Colloid Interface Sci. 2004, 275, 376-385.
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for all three gases. The predictive and correlative results show good-to-excellent fits. The approach of the EAV is similar to that employed by the SLD-PR: characterization of the adsorbent involves optimization of an EAV distribution function or, under the SLD-PR, optimization of the fs parameter within the 10-4 potential. However, the regression of the empirical parameter Λbi does not share a strict similarity with the EAV model by Ustinov et al.30 because the use of Λbi involves a modification of the fluid potential in the slit pore and not the fluid-solid potential. Nevertheless, the use of the Λbi parameter may be unnecessary if a more accurate EOS than the PR-EOS is used.10 As discussed here, successful characterization of the adsorbent can be accomplished with two optimization methodologies: (a) the use of a set of parameters that determine how the fluid-solid and fluid-fluid potential energy changes as a function of position from an idealized surface or (b) the use of a distribution function with respect to the fluid-solid potential energy in generic phase space. In both cases, the pore volumes should be constrained to be identical for each gas, unless there is evidence to the contrary.
5. Conclusions Modification of the PR-EOS co-volume through the use of Λbi allows the correlation and prediction of the component excess adsorption for mixed-gas adsorption. The modified SLD-PR can represent adsorption isotherms encountered in CBM production and CO2 sequestration, including those exhibiting excess adsorption maxima. For a specified adsorbent, the SLD-PR model can be applied using a common value for the surface area and an effective slit width for all gas species overpressure and composition ranges. For the adsorption of methane, nitrogen, and CO2 on dry activated carbon, the SLD-PR can predict the component mixture adsorption to about 2.2 times the experimental uncertainty on average solely on the basis of pure-component adsorption isotherms. For the adsorption of methane, nitrogen, and CO2 on wet coals (excluding Tiffany coal), the SLD-PR model can predict the component adsorption within the experimental uncertainties on average for all feed fractions of the three binary gas mixture combinations, although some predictions (for a specific feed fraction) are outside their experimental uncertainties. The SLD-PR model can correlate the binary mixture component adsorption of methane, nitrogen, and CO2 on coals and activated carbon within one standard deviation of experimental error. The one-fluid mixing rules appear to be adequate for the task of correlation; however, the large values of the binary interaction parameters (Cij) for some binary interactions raise questions regarding the physical reality of the rules. Supporting Information Available: SLD models of gas adsorption on wet Tiffany coal and on wet Illinois #6 coal. This material is available free of charge via the Internet at http://pubs.acs.org. LA060898R
