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Adsorption of Binary Hydrocarbon Mixtures in Carbon Slit Pores: A Density Functional Theory Study Suresh K. Bhatia Department of Chemical Engineering, The University of Queensland, Brisbane, Queensland 4072, Australia Received March 31, 1998. In Final Form: August 3, 1998 Adsorption of binary hydrocarbon mixtures involving methane in carbon slit pores is theoretically studied here from the viewpoints of separation and of the effect of impurities on methane storage. It is seen that even small amounts of ethane, propane, or butane can significantly reduce the methane capacity of carbons. Optimal pore sizes and pressures, depending on impurity concentration, are noted in the present work, suggesting that careful adsorbent and process design can lead to enhanced separation. These results are consistent with earlier literature studies for the infinite dilution limit. For methane storage applications a carbon micropore width of 11.4 Å (based on distance between centers of carbon atoms on opposing walls) is found to be the most suitable from the point of view of lower impurity uptake during high-pressure adsorption and greater impurity retention during low-pressure delivery. The results also theoretically confirm unusual recently reported observations of enhanced methane adsorption in the presence of a small amount of heavier hydrocarbon impurity.
Introduction The adsorption of hydrocarbon mixtures on activated carbon is an important subject that has received some attention in recent years. Apart from the traditional interest from the standpoint of separation, the understanding of mixture adsorption is important to some recently proposed applications such as adsorbed natural gas storage.1,2 In this application natural gas is adsorbed onto activated carbon at pressures of about 30 bar, which provides much higher energy storage densities than is possible in the bulk gas at that pressure. This adsorbed natural gas is being investigated as an alternative transportation fuel, and there is therefore much interest in studying the adsorption equilibria and dynamics.3,4 However, natural gas contains a variety of impurities such as sulfur compounds (often intentionally added) and heavier hydrocarbons (predominantly ethane and propane), and the impact of these on the adsorption of methane needs to be investigated. While Talu1 reports a drop in methane storage density in the presence of impurities, under some circumstances an enhancement has also been observed.5 The investigation of this from a theoretical standpoint is therefore a problem of much importance. There exist relatively few theoretical studies in the literature on the adsorption of hydrocarbon mixtures on activated carbon. Most of the studies have been directed at trace component adsorption of hydrocarbons or other halogenated or sulfurous contaminants in nitrogen.6,7 Such trace component studies have also been reported8 for adsorption in buckytubes and MCM-41, and these studies have all shown that trace impurity separation can be (1) Quinn, D. F.; MacDonald, J. A. Carbon 1992, 30, 1097. (2) Talu, O. In Proceedings of the Fourth International Conference on Fundamentals of Adsorption, Kyoto; Elsevier: Amsterdam, 1992. (3) Matranga, K. R.; Myers, A. L.; Glandt, E. D. Chem. Eng. Sci. 1992, 47, 1569. (4) Chen, X. S.; McEnaney, B.; Mays, T. J.; Alcaniz-Monge, J. A. Carbon 1997, 35, 1251. (5) Ahmadpour, A. Ph.D. Thesis, University of Queensland, 1997. (6) Jiang, S.; Gubbins K. E.; Balbuena, P. B. J. Phys. Chem. 1994, 98, 2403. (7) Sowers, S. L.; Gubbins, K. E. Langmiur 1995, 11, 4758. (8) Maddox, M. W.; Sowers, S. L.; Gubbins, K. E. Adsorption 1996, 2, 23.
optimized by tuning the pore size, temperature, and pressure. However, relatively few studies of binary hydrocarbon adsorption at finite concentrations have been conducted,9-12 although other mixtures such as argonkrypton and argon-methane have been examined.13-17 Among these Gusev et al.11 have shown that binary adsorption of methane and ethane in micropores is more sensitive to the pore structure than pure species data. This prompted Jensen et al.12 to propose the variation of binary selectivities with pressure as a basis for estimating pore size distributions. The most convenient vehicle for conducting the above theoretical studies has been the nonlocal density functional theory, several independent versions of which have appeared in the literature.18-20 Among these the Tarazona prescription18 is perhaps the most accurate, matching molecular simulation results even for very small pores. However, it was originally developed for the singlecomponent case, and its adaptation to binary mixtures is algebraically and computationally inconvenient.21 Consequently the approach of Kierlik and Rosinberg19 has been used in most of the subsequent studies. Perhaps the simplest of the density functional theory methods is that due to Denton and Ashcroft20 which, to the author’s knowledge, has hitherto not been utilized for investigating adsorption in carbons. There has however, been some (9) Tan, Z.; Gubbins, K. E. J. Phys. Chem. 1992, 96, 845. (10) Cracknell, R. F.; Nicholson, D.; Quirke, N. In Characterisation of Porous Solids III; Elsevier: Amsterdam, 1993. (11) Gusev, V.; O’Brien, J. A.; Jensen, C. R. C.; Seaton, N. A. In Proceedings of the Fifth International Conference on Fundamentals of Adsorption; Kluwer: Boston, 1995. (12) Jensen, C. R. C.; Papadopoulos, G.; Seaton, N. A.; Gusev, V.; O’Brien, J. A. In Proceedings of the Fifth International Conference on Fundamentals of Adsorption; Kluwer: Boston, 1995. (13) Tan, Z, F.; van Swol, F.; Gubbins, K. E. Mol. Phys. 1987, 62, 1213. (14) Marconi, U. M. B.; van Swol, F. Mol. Phys. 1991, 72, 1081. (15) Sokolowski, S.; Fisher, J. Mol. Phys. 1990, 7, 393. (16) Finn, J. E.; Monson, P. A. Mol. Phys. 1991, 72, 661. (17) Kierlik, E.; Rosinberg, M. L.; Finn, J. E.; Monson, P. A. Mol. Phys. 1992, 75, 1435. (18) Tarazona, P. Phys. Rev. A 1985, 31, 2672. (19) Kierlik, E.; Rosinberg, M. L. Phys. Rev. A 1991, 44, 5025. (20) Denton, A. R.; Ashcroft, N. W. Phys. Rev. A 1991, 44, 8242. (21) Tan, Z.; Marconi, U. M. B.; van Swol, F.; Gubbins, K. E. J. Chem. Phys. 1989, 90, 3704.
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interest in applying the method to studying the structure of uniform as well as nonuniform fluids.22,23 In the present work we report the application of a modified Denton and Ashcroft prescription20 to adsorption in slit-pore carbons, with specific reference to hydrocarbon mixtures. The accuracy of the modification is assessed by comparison with published simulation results for hard sphere mixtures near a hard wall. Subsequently, the method is used to explore the effect of carbon pore size, pressure, temperature, and hydrocarbon mixture composition at finite impurity concentration. The mixtures considered are ethane, propane, or butane in methane, as is pertinent to adsorbed natural gas storage and many problems of hydrocarbon separation.
potential neglecting correlations between molecules. Further, in eq 3, λi is the thermal de Broglie wavelength. The representation of the functional Fhsex[F1,F2] is the major uncertainty of the DFT and is the point of departure for the various versions18-20 of this technique, since exact results for Fhsex[F1,F2] are unknown even in the singlecomponent case (i.e., Fhsex[F]). The prescription used here (DA) is that of Denton and Ashcroft20 and will be subsequently discussed. The equilibrium density distribution Fi(r) is obtained from the unconstrained minimization of the grand potential Ω[F1,F2], leading to the Euler-Lagrange relation
Fi(r) ) zi exp{-βvi(r) + ci(1)(r;[F1,F2]) 2
Mathematical Modeling Central to the approach adopted in the present work is the density functional theory (DFT) for nonuniform fluids, which enables the solution of density profiles under the action of an arbitrary external potential. In its most general form DFT can be used with any arbitrary model of intermolecular potentials; however, in practice algebraic and computational complexity limits the use with somewhat simplified potential models. In what follows, we first present the DFT approach, in light of the DentonAshcroft prescription20 utilized here, before detailing the potential models employed and the resulting bulk equation of state. Density Functional Theory The general DFT is now well established, and a surfeit of applications and prescriptions is now available in the literature.7-9,13-23 Consequently, we shall not burden ourselves here with a detailed discussion of the subject, but instead provide a concise overview of the model used while referring to the lucid review of Evans24 for a more complete description. The starting point of the binary mixture DFT is the grand potential Ω[F1,F2] of a many-particle system having density profile Fi(r), in the presence of an external potential vi(r), given by 2
Ω[F1,F2] ) F[F1,F2] +
∫drFi(r)[vi(r) - µi] ∑ i)1
(1)
Here µi is the chemical potential of component i while F[F1,F2] is the intrinsic Helmholtz free energy comprising three contributions
Fid[Fi] + Fex ∑ hs[F1,F2] + i)1 1
2
ci(1)(r;[F1,F2]) ) -β
∑∑∫drFi(r)∫
2i)1 j)1
δFi(r) δ2Fhsex[F1,F2] δFi(r)δFj(r′)
- r′|) (2)
Fid[Fi] ) kT drFi(r){ln[Fi(r)λi3] - 1}
(3)
In eq 2 Fhsex[F1,F2] is the hard sphere excess, and the third term in the right-hand side represents the contribution from the attractive part of the fluid-fluid intermolecular (22) Patra, C. N.; Ghosh, S. K. J. Chem. Phys. 1997, 106, 2762. (23) Patra, C. N.; Ghosh, S. K. J. Chem. Phys. 1997, 106, 2752. (24) Evans, R. In Fundamentals of Inhomegeneous Fluids; Henderson, D., Ed.; Marcel Dekker: New York, 1992.
(4)
(5)
(6)
According to the DA prescription the one-particle DCF of the inhomogeneous system is approximated by
ci(1)(r;[F1,F2]) ≡ cio(1)(Fj(i)(r),x(i)(r))
(7)
in which cio(1)(Fj(i)(r),x(i)(r)) is the corresponding DCF of component i in a homogeneous fluid, having a weighted density (total) Fj(i)(r) and concentration x(i)(r) associated with component i at any location r. The mixture concentration is always defined as the mole fraction of the larger molecule, by convention chosen to be component 2. The weighted density Fj(i)(r) is defined in terms of weight functions wij satisfying wji ) wij, according to 2
∫dr′Fj(r′)wij(|r - r′|;Fj(i)(r)) ∑ j)1 i ) 1, 2
where Fid[Fi] is the ideal gas free energy at the local density F i:
∫
δFhsex[F1,F2]
cij(1)(r,r′;[F1,F2]) ) -β
2
dr′Fj(r′)Φattr ij (|r
∫dr′Fj(r′)Φijattr(|r - r′|)} ∑ j)1
where β ) 1/kT, and zi ) exp[βµi]/λi3. The quantity ci(1)(r;[F1,F2]) is related to the excess chemical potential of component i in the hard sphere system and is a function of the density profiles. It is termed the one-particle direct correlation function (DCF) of component i and is the first of a hierarchy of DCFs, obtained by successive differentiation of Fhsex[F1,F2], of which the first two are given by
Fj(i)(r) )
2
F[F1,F2] )
β
(8)
which represents nonlocal effects. Physically, Fj(i)(r) is akin to a total density of an effective uniform mixture, associated with component i at position r. While Denton and Ashcroft choose the composition variable x(i)(r) to be constant and equal to the bulk value xb ()xb2), we estimate it from the relative contribution of component 2 to the weighted density, i.e.
x(i)(r) )
1 Fj(i)(r)
∫dr′F2(r′)wi2(|r - r′|;Fj(i)(r)) i ) 1, 2
(9)
Thus, x(i)(r) represents the fraction of the local weighted density Fj(i)(r) contributed by component 2 and should be
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a more suitable value than the bulk quantity xb. As will be seen this provides better correspondence with simulation results, based on comparisons for the case of hard spheres near a hard wall. The above approach represents a departure from other prescriptions18,19 using higher level approximations for ci(1)(r;[F1,F2]), i.e., starting at the level of Fhsex itself, rather than the first differential, and is computationally very convenient. In particular, for the single-component case very simple expressions exist for co(1)(Fo) and c0(2)(r,F0) for the homogeneous fluid, based either on the Carnahan-Starling25 relation or the PercusYevick approximation for hard sphere systems.26 Since extensions to mixtures are less common for the former, which is more accurate, the analytic Percus-Yevick approximation results27 for hard sphere mixtures were used in this study following Denton and Aschcroft.20 To complete the formulation of the theory, it is necessary to specify the weight functions wij. These must satisfy the normalization
∫drwij(r,F0) ) 1,
i,j ) 1, 2
[
lim F(r)fF0
]
δci (r;[F1,F2]) δFj(r′)
σii (Å) ii/k (K)
) cij(2)(|r - r′|;F0,x) (11)
cij(2)(r;F0,x) ∂ci0(1)/∂F0
(12)
Fi(r) ) Fbi exp[-βvi(r) + ci(1)(r;[F1,F2]) - ci0(1)(Fbt,xb) 2
2
∫dr′Fj(r′)Φijattr(|r - r′|) + β∑Fbj∫dr′Φijattr(r′)] ∑ j)1 j)1 (13)
This equation is obtained after rewriting eq 4 for the bulk fluid, where vi(r) ) 0, and eliminating zi. Potential Models The above formulation requires the specification of the potential models for the molecular interactions in the system. The Lennard Jones (LJ) 12-6 pair potential
[( ) ( ) ]
Φij(r) ) 4ij
C3H8
n-C4H10
3.81 148.2
4.443 215.7
5.118 237.1
4.687 531.4
r > rm
) -ij,
r < rm
(15)
of the LJ potential where rm ) 21/6σij is the location of the minimum of the LJ potential. The values of ij and σij for i * j were obtained using the Lorentz-Berthelot mixing rules
ij ) xiijj
(16)
σij ) (σii + σjj)/2
(17)
The hard sphere diameters necessary in evaluating the DCFs are estimated by means of the Barker-Henderson expression
di )
∫0∞[1 - exp(-ui(r)/kT)] dr
(18)
where ui(r) is the repulsive part of the LJ potential
ui(r) ) Φii(r) - Φiiatt(r)
thereby uniquely specifying the weight functions. The final results for the weight functions for planar geometry, based on the PY approximations for the two-particle DCFs, are provided in the Appendix. In actual application it is convenient to rewrite eq 4 in terms of the bulk density as the independent variable, leading to
β
C2H6
Φijattr(r)) Φij(r),
Equations 7, 8, and 11 yield
wij(r;F0,x) )
CH4
bate molecules. The attractive part of the potential is obtained according to the Weeks-Chandler-Anderson28 division
(10)
ensuring that the approximation is exact in the uniform mixture limit F(r) f F0. A further condition is obtained by requiring that the first functional derivatives of ci(1) with respect to the densities yield the exact two-particle DCFs in the uniform limit, i.e. (1)
Table 1. LJ Potential Parameters for Various Fluids fluid parameter
σij r
12
-
σ r
6
(14)
with well depth ij and size parameter σij is used in the present work to represent the interactions among adsor(25) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (26) Wertheim, M. S. Phys. Rev. Lett. 1963, 10, 321. (27) Lebowitz, J. L. Phys. Rev. 1964, 133, A895.
(19)
In actual application of the above model to adsorption in carbons, where slit-pore geometry is prevalent, planar symmetry will be assumed. Accordingly, for the solid fluid interaction we use the 10-4-3 potential29
Φsi(z) ) A
[(
) ( )
2 σsi 5 z
10
-
σsi z
4
-
(
σsi4
)]
3∆(0.61∆ + z)3
(20)
where A ) 2πFssiσsi2∆, z is the distance between an adsorbate molecule of component i and the solid surface, ∆ is the interplanar distance in the solid, Fs is the solid density, and the parameters si and σsi follow the LorentzBerthelot rules as in the fluid. The carbon parameter values used were those listed by Steele,29 viz. σss ) 0.34 nm, ss/k ) 28.0K, ∆ )0.335 nm, and Fs ) 11.4 nm-2. Table 1 lists the LJ potential parameters used for the fluids pertinent to the present study based on published values.29,30 Alternately, estimates of the potential parameters for the present exercise may be obtained by fitting the critical or other bulk properties of the fluids by the present DFT. However, such parameters may not necessarily provide more accurate results for the actual temperature and pressure range considered here. Further, since such refinement will not affect the qualitative system behavior, the published literature values were used. Equation of State In the calculations to be reported the bulk phase was characterized by the temperature and mole fraction of (28) Weeks, J. D.; Chandler, D.; Anderson, H. L. J. Chem. Phys. 1971, 54, 5237. (29) Steele, W. A. Surf. Sci. 1973, 36, 317. (30) Reid, R. C.; Prausnitz, J. M.; Poling; B. E. The Properties of Liquids and Gases; McGraw-Hill: New York, 1988.
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component 2. The density of each component in the bulk, necessary for use with eq 13, was then estimated through the bulk equation of state as obtained from the DFT. To this end we used the generic free energy functional, which provides
P ) kTrbt +
Fbt2 2
∑i xi∑j xj∫bulkΦij(r) dr -
∑i xici0(1) + kT∑i xi∫0
kTFbt
Fbt
ci0(1)(F,x2) dF (21)
For the conditions pertinent to the present study it was noted, however, that the bulk fluid was very nearly ideal, and the correction terms in eq 21 were minor. Results and Discussion The above model has been applied in the present work to LJ methane/propane, methane/ethane, and methane/ butane mixtures in activated carbon, using the potential parameters in Table 1. The carbon micropore is idealized as a slit of width H (center to center distance between carbon atoms on opposing pore walls), which is infinite in both lateral directions. Consequently, following the 10-4-3 potential contribution from each side of the slit (cf. eq 20), we have
vi(z) ) Φsi(z) + Φsi(H - z)
Figure 1. Density profiles of a hard sphere mixture near a hard wall. Solid lines represent present calculation results, and symbols the simulation results of Tan et al.21 The dotted curve represents the calculations of Denton and Ashcroft.20 Parameters are d1/d2 ) 1/3, Fb ) 0.03351, and xb ) 0.7144.
(22)
Planar symmetry was assumed in all directions parallel to the slit, so all variables were functions only of the normal coordinate z, measured from one surface of the slit. The integral equations arising from eq 13 were discretized on a uniform grid comprised of 100-300 points along the z direction and were solved iteratively using the conventional technique,20 mixing old and new solutions to promote convergence. What follows is first a demonstration of the improvement offered by the present choice of the mixture concentration (cf. eq 9), by comparison with simulation results for hard sphere mixtures,21 before discussing our results for the LJ systems. Hard Sphere Mixtures near a Hard Wall. Initially the above model was applied to the case of hard sphere mixtures near a hard wall, for which simulation results have been presented by Tan et al.21 The main interest here was in testing the improvement offered by the choice of mixture concentration in eq 9, used in evaluating weighted densities. This differs from the DA choice20 of the bulk value for the mixture concentration. Figures 1 and 2 depict the comparison of the density and component 1 mole fraction profiles predicted by the current model (solid lines) with the simulation results (symbols) of Tan et al.,21 for the case of Fb ) 0.03351, xb ) 0.7144, and d1/d2 ) 1/3. Also superimposed on the figures are the profiles (dashed curves) predicted using a uniform mixture concentration, chosen as the bulk value, for estimating weighted densities as in Denton and Ashcroft.20 The improvement obtained using the present prescription is evident yielding a better match with simulation results at little computational expense. It may be noted that the value of x(i)(r) is iteratively obtained in eq 9, in conjunction with the iterations already being performed for evaluating the weighted densities in eq 8, and the added computational burden is therefore very small. LJ Methane/Propane Mixtures. As indicated earlier, study of the adsorption of hydrocarbon mixtures is of interest because of the relevance to separations, as well
Figure 2. Profiles of mole fraction of component 1, for mixture of hard spheres near a hard wall. Parameter values and other details are the same as in Figure 1.
as from the viewpoint of the effect of impurities on methane storage. In the present work we have investigated the effect of mixture composition, pressure, pore size, and temperature on the adsorbed density of each component and the selectivity for the heavier component (always taken to be component 2). Parts a and b of Figure 3 depict the predicted variation of the adsorbed densities of methane (solid lines) and propane (dashed lines) with the bulk methane mole fraction, for various pore sizes, at a temperature of 303 K and pressures of 1 and 20 bar, respectively. At 1 bar pressure, the adsorbed phase is predominantly propane, even at very high bulk mole fractions of methane, at least for pore sizes H* ()H/σ1) in the range of 3-5, covering the region of practical interest, as seen in Figure 3a. This is clearly advantageous from a separation viewpoint and is due to the stronger fluidsolid potential experienced by the propane molecules. It also has implications for the delivery cycle in methane
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Figure 3. Variation of adsorbed densities with composition for a binary mixture of methane (s) and propane (- - -), for various carbon slit widths at 303 K, and pressure of (a) 1 bar (b) 20 bar.
Figure 4. Density profiles of (a) methane and (b) propane in a binary mixture, in a carbon pore of slit width H* ) 4 at 303 K and 1 bar bulk pressure. xb represents the bulk mole fraction propane.
storage, suggesting that adsorbed propane impurities are less likely to be released during the desorption. However, this aspect needs to be examined further with studies of desorption dynamics as well. An interesting feature of the mixture adsorption at 1 bar pressure is that for a pore size of H* ) 4, the amount of methane goes through a maximum at very low value of the propane mole fraction (xb2 = 0.05), with about 20% excess adsorption of methane as compared to adsorption of pure CH4. This rather curious behavior has recently been experimentally observed by Ahmadpour5 though no consistent explanation was offered. In these experiments Ahmadpour studied the binary adsorption isotherm of hydrocarbons (methane-ethane, methane-propane, and methane-butane) on activated carbons at 303 K and 0.667 bar pressure and observed such a maximum for methanepropane mixtures. The current results suggest that this maximum is due to the stronger methane-propane interaction, in comparison to the methane-methane interaction potential, leading to enhanced methane adsorption at low bulk mole fractions of propane. This is clearly an undesirable feature from the separation viewpoint. At a pressure of 20 bar, however, this feature is not present as seen in Figure 2b, with the methane adsorbed steadily increasing with bulk methane fraction. Further, it can be seen that at the higher pressure the methane adsorbed increases severalfold while the propane density is only marginally increased. Nevertheless, even small amounts of propane in the bulk gas (>1%) lead to
considerable reduction in methane capacity, indicating the importance of very high methane purity in storage applications. Parts a and b of Figure 4 depict density profiles of methane and propane, respectively, in a pore of width 4σ1, for various bulk mole fractions of propane at 303 K and 1 bar pressure. The methane profile shows a peak at a distance of about 3.54 Å (=σsi) from each wall, and the dramatic decrease due to the presence of a significant amount of propane (xb2 ) 0.4) is evident. Interestingly at xb2 ) 0.4 the methane density at the center exceeds that at xb2 ) 0.04, again due to the stronger effect of the methane-propane interaction at the higher bulk propane fraction of 0.4. Even at this level the propane density at the center is much less than that at its peak, as seen in Figure 4b, but it nevertheless enhances the local methane density. As for the case of methane the propane profiles have their peaks at a distance of approximately σs2 from the wall. However, the peak density reduces much more weakly with drop in bulk propane mole fraction than the corresponding drop for methane with increase in xb2. This is not unexpected considering the much higher critical temperature (369.8 K) of propane (real) and the propensity for capillary condensation at 303 K. Indeed the propane densities achieved at this temperature, as seen in Figure 3a, are quite significant and almost comparable to its saturated liquid density of about 11 mmol/cm3. Parts a and b of Figure 5 depict the variation of adsorbate propane-to-methane ratios with bulk composition at 1 and
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Figure 5. Variation of adsorbate propane to methane molar ratio with bulk composition for carbon pores of various slit widths at 303 K and pressure of (a) 1 bar (b) 20 bar. Dashed curve represents the bulk variation.
the size that barely accommodates one layer of propane molecules, i.e.
H ) σss + σ22
Figure 6. Variation of adsorbate propane to methane molar ratio with slit width at 303 K, bulk mole fraction propane of 0.04, and pressures of 1 and 20 bar.
20 bar, respectively (solid lines), in pores of various sizes at 303 K, with the dashed curves corresponding to the bulk fluid. As already discussed the selectivity for propane is much better at the lower pressure. These figures, however, indicate somewhat complex behavior at low bulk methane mole fractions with the possibility of optimum pore size for maximum adsorbed propane/methane ratios at both pressures. This is indeed confirmed in Figure 6 depicting the variation of this ratio with slit width, at pressures of 1 and 20 bar, for a bulk propane fraction of 0.04 at 303 K. In both cases oscillatory behavior of the propane selectivity is noted, with the first peak occurring at a dimensionless slit width of about 2.4 and the second at about 3.4. Such peaks have previously been reported for the infinite dilution limit by Jiang et al.,6 using the Kierlik and Rosinberg theory, though in this limit they found the second and subsequent peaks to be absent at low pressures. Clearly, these peaks suggest an optimum pore size of about 2.4σ1, or about 9.1 Å, for the separation, in good agreement with the value of Jiang et al. However, at the higher pressure of 20 bar, it would appear that carbons having a larger pore size of about 13 Å, corresponding to the second peak, are equally effective. As observed by Jiang et al.6 the first peak corresponds to
(23)
so that pores smaller than this would seem not to be effective in the separation from an equilibrium viewpoint. However, pores of this size could still act as molecular sieves providing effective kinetic separation. It may be inferred from Figure 5 that similar behavior to the above occurs even for the lower methane bulk fractions at 1 bar, indicating the importance of choosing the optimal pore size at any bulk composition for the separation. This behavior is, however, not expected at high pressures for the lower bulk methane fractions as seen in Figure 5b. Parts a and b of Figure 7 depict the variation of the adsorbed methane density and propane/methane ratio with pressure, respectively, for various bulk mole fractions of propane, in a pore of width H ) 4σ1, at 303 K. The rather large drop in methane density due to the presence of a small amount of propane in the bulk is evident at pressures greater than about 20 bar with a 3-fold decrease even for xb2 ) 0.05, in Figure 7b. Even more interesting is the effect of pressure on the propane/methane ratio, depicted in Figure 7b, yielding an optimal pressure of about 1 bar for xb2 ) 0.05 and 0.5 bar for xb2 ) 0.1 for the separation at this pore size (4σ1) and temperature. With increase in pressure the methane density increases strongly (cf. Figures 3a and 3b and 7a) while the propane density increases only weakly (cf. Figure 3a and Figure 3b) being already comparable to the saturated liquid density. This leads to the optimality with respect to pressure. A similar feature has also been reported by Jiang et al.6 for the infinite dilution limit, though the effect is much less prominent under this limit as may also be anticipated from Figure 7b. Similar behavior has been reported by Maddox et al.8 for carbon nanotubes, but with a much lower optimal pressure of about 0.02 bar. Parts a and b of Figure 8 depict the effect of temperature on the variation of adsorbed propane-to-methane ratio with bulk methane fraction at 1 and 20 bar pressure, respectively, in a pore of width 4σ1 at 303 K. At 1 bar there appears to be an optimum temperature for the separation at low bulk methane fractions, although the effect is small with little variation of the adsorbed propaneto-methane ratio between 273 and 318 K. At high bulk methane fractions (i.e., small amounts of propane), however, the ratio varies monotonically with temperature, with a severalfold decrease on increasing temperature from 273 to 318 K. Lower temperatures are thus beneficial
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Figure 7. Effect of pressure on (a) adsorbed methane density and (b) adsorbed propane-to-methane ratio, for various bulk mole fractions of propane at 303 K in a carbon pore of slit width H* ) 4.
Figure 8. Effect of temperature on the variation of adsorbate propane-to-methane ratio with bulk composition, in a carbon pore of slit width H* ) 4 and bulk pressure of (a) 1 bar (b) 20 bar.
for low-level propane impurity separation and also for the desorption stage of the methane storage and delivery cycle with more of the impurity propane being retained in the carbon. On the other hand at high pressures the adsorbed propane-to-methane ratio is relatively unaffected by temperature at high bulk methane fractions, as seen in Figure 8b. Consequently, temperature is not a crucial factor in separation of low-level propane impurity from the viewpoint of selectivity, although the amount adsorbed may be expected to increase at lower temperatures. At lower bulk methane fractions, however, Figure 8b indicates increase of selectivity with temperature, suggesting better separation at higher temperatures. This is because of larger reduction in the amount of methane adsorbed compared to propane with temperature increase. Nevertheless, this is a less interesting region of operation for separation because of the large energy costs at high pressures. Methane/Ethane Mixtures. Ethane is perhaps the most predominant among the hydrocarbon impurities in natural gas, and its effect on storage as well as separation is therefore of much concern. Nevertheless, being smaller and more volatile that propane, its effect may be expected to be less. Parts a and b of Figure 9 depict the densities of adsorbed methane and ethane at 1 and 20 bar, respectively, at 303 K in pores of various sizes. At 1 bar pressure it is seen in Figure 9a that ethane is the dominant species below about 80% bulk methane fraction and is clearly less strongly adsorbed than propane (cf. Figure
3a). From a separation viewpoint it would appear that small pore sizes are beneficial, and this is confirmed in Figure 10a showing the highest adsorbed ethane/methane ratio at H ) 3σ1 ) 11.4 Å pore width in the range of sizes studied. Thus an optimal pore size is not evidenced in the range of sizes investigated as opposed to the case of propane where such a size was found. Nevertheless, from eq 23 one would expect an optimal pore size (center to center distance between carbon atoms on opposing walls) of H* ) 2.05, which is below the range of practical interest for most carbons have larger pore widths, usually in the range investigated. The pore size of 11.4 Å (having the same interpretation as H) is therefore recommended from the practical viewpoint. This size is also the most attractive from the storage viewpoint in that the least amount of ethane is released during delivery, as evidenced from Figures 9a and 10a. Further, at high pressures the adsorbed ethane density and selectivity is a weak function of pore size, as seen in Figures 9b and 10b, while the methane adsorbed increased more strongly with reduction in pore size, again pointing to the size of 11.4 Å being the most suitable from the viewpoint of methane storage. Interestingly this is also the size recommended by Matranga et al.3 for maximizing methane delivery during a storage and release cycle in the absence of impurity, making it appropriate from both viewpoints. Further, at this size propane uptake is minimized at the high pressure during storage, as seen in Figure 6. It should be noted that the size of 11.4 Å recommended here does not
6238 Langmuir, Vol. 14, No. 21, 1998
Bhatia
Figure 9. Variation of adsorbed densities with composition for a mixture of methane (s) and ethane (- - -) for various carbon slit widths at 303 K and pressures of (a) 1 and (b) 20 bar.
Figure 10. Variation of adsorbate ethane-to-methane ratio with bulk composition, for various carbon slit widths at 303 K, and bulk pressure of (a) 1 and (b) 20 bar.
correspond to that measured by adsorption methods, due to the influence of size of the adsorbate molecules. To circumvent this ambiguity Neimark and Ravikovitch31 suggest calibrating the theoretical and experimental pore sizes for helium as a standard. In our work, however, all pore sizes have the same interpretation as H, given earlier. Figure 9a shows the unusual effect of maximum in adsorbed methane density at the pore size of H* ) 3, observed earlier for propane at H* ) 4 (cf. Figure 3a). For the more volatile ethane the density increases strongly with decrease in pore size, and consequently the effect of its interaction with methane is most pronounced at the smaller pore size at which the maximum occurs. This effect was, however, not observed for methane/ethane mixtures by Ahmadpour5 because of larger pore size carbons being used. Figure 11 depicts the variation of adsorbed methane density and ethane/propane ratio with pressure, for various bulk ethane mole fractions, at a pore size H* ) 4 at 303 K. Clearly small amounts of ethane reduce methane storage capacity significantly, though much less than propane (cf. Figure 7). As for the case of propane an optimal pressure is found for maximum selectivity at any bulk mole fraction of ethane. However, the optimal pressures are significantly higher, having values of 8.25 and 6.25 bar at bulk ethane mole fractions of 0.05 and 0.1, respectively. Thus, for ethane separation high pressures (31) Neimark, A. V.; Ravikovitch, P. I. Langmuir 1997, 13, 5148.
are more suitable provided they are properly optimized for the bulk ethane level. Nevertheless, the improvement above 1 bar pressure, at about 40% selectivity enhancement, is smaller than for propane but is still significant. Methane/Butane Mixtures. Butane is generally much less significant as an impurity in natural gas or other methane sources. However, it is much less volatile than ethane or propane, making it potentially more detrimental to most applications. Accordingly the effect of small amounts of butane in methane was investigated here. Figure 12 shows the adsorbed methane and butane densities as a function of bulk methane fraction, in pores of different size at 1 bar pressure at 303 K. Clearly butane, even in very small amounts strongly reduces methane adsorption at all the pore sizes investigated. Further, in pores of size H* ) 3 (i.e., 11.4 Å) butane is retained at high densities even at bulk mole fractions methane up to 0.9999, though this low level of butane does not reduce the methane adsorption significantly. Thus, in carbons having this pore size butane released during methane delivery, after high-pressure storage, may be expected to be the least, again supporting this as the appropriate pore size. As in the case of the other gases, the methane adsorption goes through a maximum at large bulk methane mole fractions for H* ) 5. In this case, however, the maximum occurs at about 0.999 bulk mole fraction methane and may be experimentally hard to observe, except in very accurate targeted measurements. Again, this was not
Adsorption of Binary Hydrocarbon Mixtures
Langmuir, Vol. 14, No. 21, 1998 6239
Figure 11. Variation of adsorbed (a) methane density and (b) ethane to methane molar ratio with pressure, for various bulk mole fractions of ethane at 303 K in a carbon pore of slit width H* ) 4.
Figure 12. Variation of adsorbed densities with composition for a mixture of methane (s) and butane (- - -) for various carbon slit widths at 303 K and pressure of 1 bar.
observed by Ahmadpour5 whose data points did not lie in the required narrow region. This maximum is not predicted for the smaller pore sizes studied in the present work and provides an enhancement in methane capacity of about 20% at H* ) 5, which is undesirable for separations. Conclusions This study has examined binary mixture adsorption in carbon pores, using the DFT of Denton and Ashcroft,20 from the view of separation of heavier hydrocarbons from methane, and also their effect on methane storage. The results show that even very small impurity concentrations can reduce methane storage capacity substantially, and the methane used must be of very high purity. As in earlier studies6-8 at the infinite dilution limit, the present work shows the existence of optimal pore sizes for maximum selectivity for impurity adsorption. This optimum pore size is wide enough to just accommodate one layer of the impurity. We also find optimality with respect to pressure for maximum selectivity for the impurity at the finite
concentrations explored here. In general because the heavier hydrocarbon impurities studied (ethane, propane, and butane) can capillary condense at ambient temperature (being below their critical temperature), increase in pressure beyond a certain value does not enhance their adsorption while still increasing methane adsorption, leading to the optimality. As a result the optimal pressure also reduces with increase in impurity mole fraction in the bulk. Proper choice of the pressure can lead to severalfold increases in selectivity for propane, but the effect is less for ethane. These features suggest that careful design of hydrocarbon separation processes in terms of adsorbent design and system pressure is necessary to achieve optimal performance. Choice of process temperature is also important with lower temperatures being appropriate for small levels if impurity. However, it should be noted that the results have been obtained using the assumption of spherical molecules and, while indicative of trends, may not be precise. In particular, as the chain length of the hydrocarbon increases, models incorporating multisite potentials may be more appropriate. In such cases molecular orientation must also be considered, for which simulation techniques are more suited than DFT. Since ethane, propane, and butane are not spherical molecules, more precise results may therefore be obtained by simulation methods utilizing more complex potential models. However, it has been suggested7 that the results may not be expected to be qualitatively different and that the optimum pore sizes obtained using the spherical molecule approximation are sufficiently accurate. The large selectivity for heavier hydrocarbons at lower pressures indicated that impurities may be selectively retained during delivery of methane after adsorption at high pressures. This retention is also favored by low temperature. However, studies of delivery dynamics are needed to verify and support these conclusions. It is found that a pore size of 11.4 Å is appropriate for methane storage applications from the point of view of enhanced ethane retention during delivery and maximum methane adsorption at high pressures. This is therefore also the most suitable pore size for ethane separation applications. The present computations also show that the methane adsorption is enhanced by small levels of heavier impurities, due to the enhanced potential, in pores of certain sizes. The prediction is supported by recent experimental results of Ahmadpour,5 and our computations also show
6240 Langmuir, Vol. 14, No. 21, 1998
Bhatia
that the pore size at which the enhancement occurs increases with molecular mass of the impurity. Acknowledgment. This work has been supported by a grant from the Australian Research Council. The author is grateful to Dr. S. K. Ghosh, Bhabha Atomic Research Centre, Mumbai, India, for fruitful discussions.
planar geometry.
wii(z) )
π 2 2 a (d 2 - Z2) + bi(di3 - z3) + d(di5 - z5) , Ai i i 3 5 z e di (A.4)
[
-Cii(r) ) (ai + bir + dr3),
(A.1)
r > di, i ) 1, 2
) 0,
-C12 ) a1,
) a1 +
r e di
) 0,
[
w12(z) )
]
{
2π a1 1 (d 2 - z2) + b (d123 - z3) - λ(d122 B1 2 12 3
]]
z2) + λ2(d12 - z) + 4dλ
[
[
(d124 - z4) - λ(d123 - z3) + 4
] [ ]}
(d125 - z5) 3 2 λ (d122 - z2) - λ3(d12 - z) + d 2 5
λ4(d12 - z) , w12(z) ) 0,
λ e r e λ + d1 (A.2)
r > λ + d1 C21 ) C12
π 2 2 a (d 2 - z2) + bd13 + 2dλd14 + dd15 , B1 1 12 3 5 zeλ
w12(z) )
λ(d124 - z4) + 2λ2(d123 - z3) - 2λ3(d122 - z2) +
reλ
(br2 + 4dλR3 + R4) , r
z > di, i ) 1, 2
) 0,
Appendix Analytical Results for the Weight Functions in Planar Geometry. Equations 10 and 12 may be readily used to obtain the weight functions wij(r;F0,x) in planar geometry, given the PY results for the two-particle DCFs. The latter are given by Lebowitz27 as
]
(A.3)
in which di is the hard sphere diameter of component i, λ ) (d2 - d1)/2 and R ) (r - λ). In the above, the parameters ai, b, bi, and d are complex functions of the densities and hard sphere diameters of the two components, as specified by Lebowitz. Skipping further details of the procedure and algebraic manipulations, which are outlined by Denton and Ashcroft,20 we present here the final results for the weight functions in
λ < z e d12
z > d12
(A.5)
w21(z) ) w12(z)
(A.6)
The constants Ai and B1 follow from the normalization requirement in eq 10 and are given by
Ai ) 4π
[
[
]
aidi3 di3bi ddi6 + + , 3 4 6
(
)
i ) 1, 2 (A.7)
(
)
a1d123 λ d1 λ d1 + 4dλd14 + + + bd13 + B1 ) 4π 3 3 4 4 5 λ d1 (A.8) dd15 + 3 6 LA980354S
(
)]
