Correlating N2 and CH4 Adsorption on Microporous Carbon Using a

Oct 9, 1998 - A new pore size distribution (PSD) model is developed to readily describe PSDs of microporous materials with an analytical expression. R...
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Energy & Fuels 1998, 12, 1071-1078

1071

Correlating N2 and CH4 Adsorption on Microporous Carbon Using a New Analytical Model Jian Sun,† Scott Chen,‡ Mark J. Rood,*,† and Massoud Rostam-Abadi*,†,‡ Environmental Engineering and Science Program, Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, 205 North Mathews Avenue, Urbana, Illinois 61801, and Illinois State Geological Survey, 615 East Peabody Drive, Champaign Illinois 62810 Received August 14, 1998

A new pore size distribution (PSD) model is developed to readily describe PSDs of microporous materials with an analytical expression. Results from this model can be used to calculate the corresponding adsorption isotherm to compare the calculated isotherm to the experimental isotherm. This aspect of the model provides another check on the validity of the model’s results. The model is developed on the basis of a 3-D adsorption isotherm equation that is derived from statistical mechanical principles. Least-squares error minimization is used to solve the PSD without any preassumed distribution function. In comparison with several well-accepted analytical methods from the literature, this 3-D model offers a relatively realistic PSD description for select reference materials, including activated-carbon fibers. N2 and CH4 adsorption is correlated using the 3-D model for commercial carbons BPL and AX-21. Predicted CH4 adsorption isotherms at 296 K based on N2 adsorption at 77 K are in reasonable agreement with experimental CH4 isotherms. Use of the model is also described for characterizing PSDs of tire-derived activated carbons and coal-derived activated carbons for air-quality control applications.

Introduction Activated carbon usually has a heterogeneous pore structure because of the physical and chemical structural complexity and randomness of its original carbonaceous materials.1 The distribution of pore sizes is a critical parameter to characterize the adsorbent, when the adsorption potential (physical interaction between adsorbent and adsorbate) is a dominant factor. Adsorption potential and adsorbate density are strongly dependent on the adsorbent’s pore size (w). In this study, w is defined as the distance between the edges of the carbon atoms in opposite pore walls. The pores are modeled as slits consisting of two infinite graphite planes. Adsorption is enhanced in micropores (w < 20 Å) due to the overlap of the force field created by the opposing pore walls. For applications such as natural gas storage, adsorbents should be designed and prepared in such a way that it has minimum mesopore (20 Å < w < 500 Å) and macropore (w > 500 Å) volumes and a maximum micropore volume. In particular, w for micropores should be near 8 Å, the optimal pore size for CH4 adsorption based on computer simulations.2 As a result, the volumetric CH4 storage capacity of the adsorbent is optimized because of the increase in adsorbate density. † Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign. ‡ Illinois State Geological Survey. (1) Gregg, S. J.; Sing, K. S. W. Adsorption, surface area and porosity; Academic: London, 1984. (2) Matranga, K. R.; Stella, A.; Myers, A. L.; Glandt, E. D. Chem. Eng. Sci. 1992, 47, 1569.

Very useful pore-size distribution (PSD) models have been developed on the basis of N2 adsorption at 77 K. Although the models provide great insights, there is an apparent lack of consistency in the modeled PSD results3 because of their different theoretical foundations and assumptions. Well-accepted PSD characterization methods include MP (by Mikhail et al.),4 DRS (Dubinin-Radushkevich-Stoeckli),5 JC (Jaroniec and Choma),6 HK (Horvath and Kawazoe),7 and, more recently, SNAP (by Seaton et al.)8,9 and DFT (density functional theory by Olivier).10 The MP method is relatively easy to use and applicable for micropores. It is consistent with the multiplelayer adsorption mechanism but does not consider enhanced adsorption potentials in micropores.4 The DR (Dubinin-Radushkevich) equation11 provides more detailed structural information on adsorption in micropores in comparison with BET (Brunauer-EmmettTeller) theory. The DRS method, based on the DR equation, is the first PSD model using a preassumed normal distribution function to describe pore size.5 Using a preassumed distribution function makes PSD modeling easier to use and more expressive with a (3) Russell, B. P.; LeVan, M. D. Carbon 1994, 32, 845. (4) Mikhail, R. Sh.; Brunauer, S.; Bodor, E. E. J. Colloid Interface Sci. 1968, 26, 45. (5) Stoeckli, H. F. J. Colloid Interface Sci. 1977, 59, 184. (6) Jaroniec, M.; Choma, J. Mater. Chem. Phys. 1986, 15, 521. (7) Horvath, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983, 16, 470. (8) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (9) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (10) Olivier, J. P. J. Porous Mater. 1995, 2, 9. (11) Dubinin, M. M. J. Colloid Interface Sci. 1974, 46, 351.

10.1021/ef980108o CCC: $15.00 © 1998 American Chemical Society Published on Web 10/09/1998

1072 Energy & Fuels, Vol. 12, No. 6, 1998

Sun et al.

mathematical formula. However, it inevitably restrains the final PSD results. Furthermore, the DR equation works only at moderate- and high-adsorptive pressures but does not reduce to Henry’s law at low pressures (near zero coverage).12 The JC method is an improved version of the DRS method to describe a polydispersed PSD. It chooses a gamma distribution as the distribution function.6 The gamma distribution defines the pore size in the domain from 0 to +∞. Nevertheless, the JC method inherits the characteristics of the classical DR equation. The HK method is another analytical PSD method but appears to underestimate pore sizes13,14 because of the assumed progressive pore-filling mechanism. SNAP is one of the latest PSD methods based on numerical results from mean-field density functional theory (MFT).8,9 MFT is a complicated theory and needs extensive computations to implement. SNAP also uses a preassumed log-normal distribution function to model N2 adsorption at 77 K and a bivariable polynomial to correlate the numerical results for local isotherms. DFT (Micromeritics Co.) is one of the latest PSD methods based on nonlocal density functional theory.10 Instead of using a preassumed distribution function, DFT employs a regularization technique to solve the generalized adsorption isotherm (GAI),8,15 which results in a discrete PSD. Another version of DFT (Quantachrome Co.) is based on local density functional theory, which neglects adsorbate-adsorbate molecular interactions. Nonlocal theory considers both the interactions between adsorbate-adsorbate and between adsorbent-adsorbate molecules. Theoretically, nonlocal theory offers a more realistic simulation of adsorption in micropores.9 The objective of this study is to model and correlate N2 and CH4 adsorption on microporous carbon through a conventional analytical approach. A unique 3-D isotherm equation, which relates the adsorbate filling fraction to adsorption potential energy, is derived using statistical mechanical principles. A PSD model is then developed on the basis of this 3-D equation to determine the PSDs of adsorbents without requiring a preassumed distribution function. Prediction of the CH4 adsorption isotherm is carried out by correlating N2 and CH4 adsorption for select adsorbents. Description of the 3-D Model The DR Equation and 2-D Isotherm Equation. The DR equation11 takes the form

[ ( )]

θ ) exp -

A βE0

2

(1)

where θ is the pore-filling fraction of the adsorbent’s micropores, A is the differential molar work, β is the affinity coefficient, and E0 is the adsorption characteristic energy. Figure 1 describes θ using the DR equation for N2 adsorption at 77 K in a 15 Å slit pore with β ) 0.33 for N2 and E0 ) 2k/w (k ) 12 kJ-Å/mol).11 The DR equation does not reduce to Henry’s law at zero-filling fraction, and it assumes a continuous pore filling for N2 (or any other adsorbate). Also presented (12) Dubinin, M. M. In Progress in Surface and Membrane Science; Academic: New York, 1975; pp 1-70. (13) Kaminsky, R. D.; Maglara, E.; Conner, W. C. Langmuir 1994, 10, 1556. (14) Kruk, M.; Jaroniec, M. in Extended Abstracts 23rd Biennial Conference on Carbon, State College, PA, 1997; p 106. (15) McEnaney, B. Carbon 1988, 26, 267.

Figure 1. DR and assumed isotherm for N2 adsorption at 77 K in a 15 Å slit pore. The modeled isotherm is described by the dashed line. in Figure 1 is the modeled N2 adsorption isotherm for the 15 Å pore at 77 K. The modeled adsorption process proceeds as follows: as the adsorptive relative pressure (P/P0) increases from zero, adsorption occurs initially on the pore walls. The thickness of the adsorbed layer increases until the filling pressure is reached, at which point adsorption occurs as volume filling, resulting in a discontinuous jump in the isotherm. Adsorbate phase transition occurs from gas to liquid at the filling pressure. The adsorption density increases continuously upon complete pore filling due to the compression of the adsorbate liquid-like phase.16 In an attempt to seek the theoretical basis and improve the characteristics of the DR equation, Chen17 derived a 2-D general isotherm equation for adsorption of gases on microporous solids from statistical mechanical principles

ln

F2s g

F2 Λ

-

7 2η η 9 ln(1 - η) + + 8 (1 -η) 8 (1 - η)2 R(T)η +

Φ ) 0 (2) kBT

where F2g and F2s are the adsorbate surface number densities in the gas phase and 2-D adsorbed phase, respectively; Λ is the de Broglie thermal wavelength; η is the surface coverage; R(T) is a function of temperature T;17 and kB is the Boltzmann constant. The 2-D adsorbed phase is subject to a force field represented by a mean potential (Φ). The general isotherm reduces to the DR equation for adsorption in micropores and to Henry’s law at zero coverage when η ) 0. The 3-D Isotherm Equation. Although the 2-D equation provides a convincing theoretical basis for the DR equation, it poses difficulties when using it with the GAI for describing PSDs. The adsorbate density obtained by eq 2 describes the surface coverage of adsorbate in the pores. It does not provide practical information about 3-D adsorption density, and thus cannot be used to obtain PSD information. In this study, we introduce the Carnahan-Starling (CS) equation of state for hard spheres.18,19 (16) Aukett, P. N.; Quirke, N.; Riddiford, S.; Tennision, S. R. Carbon 1992, 30, 913. (17) Chen, S. G.; Yang, R. T. Langmuir 1994, 10, 4244. (18) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (19) Carnahan, N. F.; Starling, K. E. AIChE J. 1972, 18, 1184.

N2 and CH4 Adsorption on Microporous Carbon

Energy & Fuels, Vol. 12, No. 6, 1998 1073

1 + η + η 2 - η3 PV ) NkBT (1 - η)3

(3)

where N is the number of adsorbed molecules in the pore, P and V are the pressure and volume of the adsorbed phase, η ) (π/6)Fσ3 is defined as the “fraction of packing”, σ is the diameter of a hard sphere, and F ) N/V is the number density of the spheres. For a real fluid, a perturbation term attributed to the attractive force between the adsorbate molecules needs to be introduced into eq 3 to become the Carnahan-StarlingDeSantis (CSD) equation of state.20

Pν 1 + η + η2 - η3 a ) 3 kBT k T(v + b) (1 - η) B

(4)

where v is the molar volume of the hard spheres and a and b are the temperature-dependent parameters given by the CSD equation. The 3-D equation is then derived similarly to Chen.17

Fs 8η - 9η2 + 3η3 1 a 4η ln g + + ln(1 + 4η) + kBT b 1 + 4η F (1 - η)3 Φ ) 0 (5) kBT

(

)

Equation 5 is the 3-D adsorption isotherm equation for a given pore size and geometry (e.g., slits), with a mean force field, Φ. The classical DS inverse relationship,12 2k/w, is used for its simplicity to evaluate the mean force field as a function of pore size. Fg and Fs are the volume number densities of the gas and adsorbed phases, respectively. The second term in eq 5 describes the short-range repulsive force between adsorbate molecules, while the third term represents the long-range attractive force between adsorbate molecules. The attractive force between adsorbate molecules, however, is omitted during this PSD calculation because it made little impact on the adsorption density. The fourth term refers to the interaction between adsorbate and adsorbent. For low adsorption density (close to bulk gas density), the packing fraction is close to zero (so are the second and third terms in eq 5), thus the equation reduces to Henry’s law

(

Fs ) exp -

)

Φ Fg ) HnP kBT

(6)

with

(

Hn ) exp -

)

Φ kBT

(7)

In comparison with the 2-D equation, it is necessary to introduce a 3-D equation of state so that the 3-D adsorption density can be obtained by the definition η ) (π/6)Fσ3. Therefore, the N2 and CH4 adsorption density and pore size distribution based on N2 adsorption can be determined. Modified DR Equation and Adsorption Volume Exclusion. For N2 adsorption at 77 K, the nature of adsorbatephase transition from gas to liquid (Figure 1) should be included in the PSD calculation. Pore filling in micropores and capillary condensation in mesopores for N2 adsorption at 77 K is not considered in eq 5. This leads to very small adsorption densities in large pores, even at high adsorptive pressures because the adsorption potentials in these pores are very small. An unrealistically high pore volume will be obtained as a result of solving the GAI. In reality, pore filling or condensation occurs and the adsorption densities should be close to the value of liquid N2 after the critical pressures are (20) De Santis, R.; Gironi, F.; Marrelli, L. Ind. Eng. Fundam. 1976, 15, 183.

Figure 2. Local isotherms for slit pores of various sizes for N2 adsorption at 77 K calculated using the 3-D equation and modified DR equation. achieved for the corresponding pore sizes. Therefore, the modified DR equation21 is used to calculate the N2 adsorption density after pore filling. The modified DR equation can be formulated as

[ ( )]

F ) Fl exp -

A βE0

2

(8)

where F is the density of the adsorbed phase and Fl is the density of saturated liquid N2. This modification assumes that the increase of adsorbate density after pore filling is continuous because of the compression of N2 in the liquid-filled pore. The fraction of pore filling in eq 8 is θ ) F/Fl where θ approaches 1 at P/P0 ) 1 (complete pore filling). A correlation between the pore-filling pressure and the critical pore size in the 3-D model was obtained from MFT8,9 to describe the discontinuous jump in the N2 adsorption isotherm.22 In other words, the adsorption densities prior to and after pore filling are calculated by the 3-D equation and by the modified DR equation, respectively. The modified DR equation is used only for pores >10 Å, because pore filling does not likely occur in pores smaller than the size equivalent to three layers of N2 (∼10 Å), resulting from volume exclusion. Volume exclusion becomes important when adsorption occurs in small pores. For example, a 6 Å pore can only accommodate one layer of N2 because the molecular diameter of N2 is 3.5 Å,23 while a 7 Å pore can accommodate two layers of N2. As a result, the adsorption density in a 7 Å pore is higher than that in a 6 Å pore. Volume exclusion is included in the 3-D equation when calculating local isotherms and PSDs. Calculating Local Isotherms. Local isotherms used in the GAI for N2 adsorption at 77 K (Figure 2) are calculated for select pore sizes between 6 and 70 Å using the 3-D equation (prior to pore filling) and the modified DR equation (after pore filling). As the pressure is increased from zero, the thickness of the adsorbed layer increases until the filling pressure is reached. Pore filling occurs at the filling pressure, resulting in a discontinuous jump in the isotherm. The N2 adsorption density in some ultramicropores at P/P0 ) 1 is 24% larger than the value of liquid N2. This is not unusual because the enhancement of the adsorption potential is so strong in ultramicropores that N2 can become that dense. (21) Sun, J.; Rood, M. J.; Rostam-Abadi, M. in Extended Abstracts 23rd Biennial Conference on Carbon, State College, PA, 1997; p 348. (22) Sun, J.; Brady, T. A.; Rood, M. J.; Lehmann, C. M.; RostamAbadi, M.; Lizzio, A. A. Energy Fuels 1997, 11, 316. (23) Hirschfelder, J. O.; Curtis, C. F.; Bird, B. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954.

1074 Energy & Fuels, Vol. 12, No. 6, 1998

Sun et al. Table 1. BET and t-Plot Areas and DR Pore Volumes for ACFCs-15 and -25 ACFC sample

ACFC-15

ACFC-25

BET area [m2/g] t-plot pore area [m2/g] DR micropore volume [cm3/g] total pore volumea [cm3/g]

730 1293 0.372 0.379

1860 2284 1.137 1.023

a Total pore volume read directly from experimental N adsorp2 tion isotherm at P/P0 ) 0.98.

Figure 3. Experimental N2 77 K isotherms for ACFCs-15 (O) and -25 (0) plotted on log scale to highlight the low-pressure region. Two isotherms are fitted with JC equation (lines). Parameters for gamma distribution are calculated as R ) 73.2 and 0.898; ξ ) 1.1771 × 108 and 274 931; W0 ) 0.343 and 0.910 for ACFCs-15 and -25, respectively. In fact, grand canonical Monte Carlo simulation16 indicated that the N2 packing density at 77 K in ultramicropores can be 20% greater than the liquid value at P/P0 ) 0.56. PSD Determination Based on N2 Adsorption at 77 K. To obtain the PSD, the GAI is formulated as,8,15

n(Pr) )

∫

∞

0

F(Pr,w) f(w) dw

(9)

where n(Pr) is the amount of adsorbed N2 at a relative pressure Pr ()P/P0) that is obtained directly from the experimental adsorption isotherm, F(Pr,w) is the adsorbate density at Pr, which can be calculated using the 3-D adsorption isotherm (eq 5) and modified DR equation (eq 8) by varying Φ (i.e., the effective pore size w), and f(w) is the distribution of pore volume as a function of w. Equation 9 can be broken down into a set of linear equations and solved using least-squares error minimization (Appendix). Materials Used. Two activated-carbon fiber-cloth (ACFC) samples (ACFC-15 and ACFC-25) obtained from American Kynol, Inc. (New York, NY) were chosen for their relatively pure chemical composition (no ash) and uniform pore structure. Among these two ACFCs, ACFC-15 had the shorter activation time (lower burnoff and higher yield) compared with ACFC-25.24 Norit Row, a commercially available activated carbon manufactured by American Norit Co., was used to compare results from DFT and the 3-D model. N2 adsorption data at 77 K and DFT results for Norit Row were obtained from Kruk.25 Other commercial activated carbons used in this study included BPL manufactured by Calgon Carbon Co. and AX-21 from Mega-Carbon Co. The 77 K N2 adsorption isotherm of BPL and AX-21 was measured with a Micromeritics ASAP2400 (P/P0 ) 10-3-1), and used to calculate the CH4 adsorption isotherm.

Results and Discussion PSD Characterization of Activated Carbon Fibers. The experimentally determined N2 isotherms at 77 K for the ACFC samples are provided in Figure 3. The resulting physical properties, such as BET surface (24) Foster, K. L.; Fuerman, R. G.; Economy, J.; Larson, S. M.; Rood, M. J. Chem. Mater. 1992, 4, 1068. (25) Kruk, M.; Jeroniec, M. Adsorption 1996, 3, 209.

Figure 4. (a) Experimental (symbols) and calculated (lines) N2 adsorption isotherms at 77 K and (b) PSDs for ACFCs-15 (O) and -25 (0) by the 3-D model.

area, t-plot pore area, and DR pore volume, that were derived from those isotherms are summarized in Table 1. PSDs for ACFC-15 and ACFC-25 obtained by the N2 isotherm and 3-D model are presented in Figure 4. The micropore volume and pore volume for w < 100 Å from the 3-D model are 0.363 and 0.988 cm3/g, respectively (Table 2). An increased pore volume and pore widening are expected for ACFC-25 compared to ACFC-15. These features are well-illustrated in Figure 4. The pore volumes determined with the 3-D model (Table 2) have the best agreement with the total pore volumes obtained from the experimental isotherms among the analytical methods reviewed (Table 1). Calculated N2 isotherms using the 3-D model and the corresponding experimental isotherms for the two ACFC samples are compared

N2 and CH4 Adsorption on Microporous Carbon

Energy & Fuels, Vol. 12, No. 6, 1998 1075

Table 2. Summary of PSD Information for ACFCs-15 and -25 by MP, JC, HK, and the 3-D Model ACFC-15

ACFC-25

method

PSD max [Å]

micropore vol [cm3/g]

pore vol [cm3/g]

PSD max [Å]

micropore vol [cm3/g]

pore vol [cm3/g]

MP JC HK 3-D

5.1 9.5 5.3a 7.0

0.318 0.343 0.333 0.363

0.322 0.343 0.336 0.363

7.7 15.5 5.8 9.0

1.066 0.504 0.801 0.988

1.109 0.865 0.843 1.070

a

Actual maximum should be less.

Figure 6. PSD results for Norit Row (near micropore region) by DFT and the 3-D model.

Figure 5. Comparison of PSDs for ACFC-25 by MP, JC, HK, and the 3-D model.

with good agreement in Figure 4. Additionally, ACFC25 has a roughly 8% nonmicroporous pore volume. In contrast to the unimodal PSDs resulting from JC and HK models, a multimodal PSD for ACFC-25 is revealed by the 3-D model. These modes correspond to the inflections in the experimental isotherm (Figure 5). The MP method indicates a PSD maximum at about 8 Å for ACFC-25 (Table 2). This is the result of the MP method not considering enhanced adsorption in micropores. The adsorption film thickness (related directly to the estimated pore size in the MP method) should be greater in micropores than for nonporous materials at a given relative pressure. The MP method tends to underestimate the pore size for micropores. The PSD obtained by the JC method predicts a single mode and extends further into the mesopore region (maximum at 16 Å). Such a result is presumably caused by use of the DR equation and the initial constraint associated with the preassumed normal distribution for the PSD. Compared with the other models, the HK method gives the smallest PSD maxima for both ACFC samples (Table 2). It does not appear to respond well to pore widening brought about by the extent of activation for ACFC-25. The progressive pore-filling mechanism in the HK method does not include adsorption in a pore until the pore is filled. As a result, adsorption prior to pore filling is artificially added to the adsorption in the pores smaller than that particular pore.13 Such an approach results in underestimation of micropore sizes, especially in ultramicropores (w < 7 Å), where the amount of adsorbed N2 prior to pore filling is significant

due to the enhanced adsorption potential. This could be a reason that the HK method underestimates the pore size. MP and HK methods do not provide calculated adsorption isotherms to allow verification of PSD modeling results. The JC method and the 3-D model allow recalculation of adsorption isotherms as presented in Figures 3 and 4, respectively. Recalculation of the adsorption isotherm with the JC and 3-D models provide another method to validate the models’ results. Comparison of PSDs from DFT and 3-D Model. Continuous PSD that results using the DFT method10 provided in Figure 6 for Norit Row is reproduced by normalizing the pore volumes to the corresponding pore size intervals and taking the center point of each size interval as the corresponding pore size. The PSD provided by DFT is usually presented as a discrete bar chart.14,27,28 PSD results for Norit Row obtained by the 3-D model based on the same N2 isotherm are also plotted in Figure 6. A reasonable agreement can be observed between the two methods, although the PSD maximum by DFT is 0.5 Å larger than that by the 3-D model. Optimal Pore Size for CH4 Adsorption. Multiplelayer adsorption does not occur at ambient temperature for CH4 because CH4 is a supercritical gas. Therefore, there must be an optimal pore size associated with the maximum CH4 adsorption density. The densities of adsorbed CH4 at 3.4 MPa (500 psia) and 300 K on an ideal adsorbent with various pore sizes are calculated using eq 5. An ideal adsorbent consists of a slit-shaped pore between two single graphite layers. The affinity coefficient, β, for CH4 is calculated using the following equation

β)

[P′] ) 0.353 [P′]0

(10)

where [P′] and [P′]0 are the parachors of CH4 and (26) Quayle, O. R. Chem. Rev. 1953, 53, 439. (27) Olivier, J. P.; Conkin, W. B.; Szombathely, M. V. In Characterization of Porous Solids III; Elsevier: Amsterdam, 1994. (28) Ryu, Z. Y.; Zheng, J. T.; Wang, M. Z. Extended Abstracts 23rd Biennial Conference on Carbon, State College, PA, 1997; p 138.

1076 Energy & Fuels, Vol. 12, No. 6, 1998

Figure 7. Optimal pore size for CH4 adsorption.

Sun et al.

Figure 8. Experimental CH4 adsorption isotherms (symbols) (Sosin, 1995) and predictions (lines) by the 3-D model for AX21 (O) and BPL (0).

benzene, respectively, which can be obtained from standard references.26 Dependence of adsorbed CH4 density on pore size is plotted in Figure 7. The optimal pore size with a maximum adsorption capacity is ∼8.0 Å, closely matching the results obtained by computer simulation.2 This observation is one check of the theoretical basis used in the model and the assumptions made in the 3-D isotherm derivation. Prediction of CH4 Adsorption Isotherm. Prediction of a CH4 adsorption isotherm at 296 K is another approach to verify the PSD model on the basis of N2 adsorption at 77 K. BPL is used as the target material, whose experimental CH4 adsorption isotherm was obtained from Sosin.29 Although the two BPL samples possibly came from different manufactured batches, they have similar BET surface areas (1100 vs 1030 m2/g by Sosin29). Because the materials are made from the same parent material and have a comparable BET surface area, it is assumed that they have similar PSDs. The 3-D equation was initially used to calculate the CH4 adsorption densities (with volume exclusion but no pore filling because CH4 is supercritical), which are then combined with the modeled PSD for BPL to obtain the correlation between CH4 uptake versus pressure at 296 K. Figure 8 plots the experimental and modeled CH4 isotherms for BPL. The CH4 adsorption isotherm at 296 K is also calculated for AX-21 using the 3-D model. The experimental CH4 isotherm for AX-2129 is compared with the 3-D modeled isotherm in Figure 8. CH4 adsorption is overestimated by the 3-D model in the lowpressure region and underestimated in the high-pressure region. This is possibly because of the use of the DS inverse relationship to calculate the adsorption potential energy for CH4 adsorption. In comparison with experimental results based on a molecular probe study,30 the DS inverse relationship overestimates the adsorption potential (Figure 9). The potential, evaluated by the DS relationship, also decreases rapidly as the pore size increases,31 resulting in underestimated CH4 adsorption density (uptake). This effect could not

be offset by using the DS relationship for N2 adsorption because the modified DR equation is used to calculate the adsorption density after complete pore filling. At high-adsorptive pressures, the adsorption density is close to the value of liquid N2 (Figure 2) when the DS relationship has a negligible contribution to the adsorption density. The prediction may be improved by using more sophisticated and realistic approaches to evaluate the adsorption potential, such as the Steele 10-4-3 potential averaged over the pore for both carbon-N2 and carbon-CH4.32 Other Applications of the 3-D Model. The 3-D model was initially designed to correlate N2 and CH4 adsorption.22,33 Other applications include PSD modeling of tire-derived34 and coal-derived35 activated-carbon adsorbents that are being developed for air-quality control applications.

(29) Sosin, K. A.; Quinn, D. F. J. Porous Mater. 1995, 1. 111. (30) McEnaney, B. Carbon 1987, 25, 457. (31) Chen, S. G.; Yang, R. T. J. Colloid Interface Sci. 1996, 177, 298.

(32) Steele, W. A. The Interaction of Gases with Solid Surface; Pergamon Press Ltd.: New York, 1974. (33) Sun, J.; Rood, M. J.; Rostam-Abadi, M.; Lizzio, A. A. Gas Sep. Purif. 1996, 10, 91.

Figure 9. Evaluation of adsorption potential using DS inverse relationship and McEnaney empirical equation on the basis of molecular probe study.

N2 and CH4 Adsorption on Microporous Carbon

Figure 10. PSD results by the 3-D model for two tire-based carbons, made with 0 and 1.5 h steam activation at 850 °C following charring in N2 at 600 °C for 1 h.

PSD results from the 3-D model for a tire-based carbon sample made with zero activation has a micropore volume of 0.042 cm3/g and no detectable ultramicropores (Figure 10). The same tire-based carbon after 1.5 h of steam activation at 850 °C (additional 11% burnoff) has developed a considerable ultramicropore volume, while the overall micropore volume increases by 500% (0.245 m3/g) (Figure 10). The percent of pore volume that is microporous also increases from 37% to 67%. Development of microporosity through the opening of closed pores in the tire material is unlikely because carbon black, the base carbon material in tires, has a negligible microporosity. This suggests that steam activation generates new micropores, especially ultramicropores by partial gasification. PSD modeling of select coal-based activated-carbon adsorbents was performed to study the effect of pore structure and heteroatoms (e.g., sulfur) on the Hg0 adsorption capacity. The ability of coal-based adsorbents to remove Hg0 from gas streams is important with respect to the development of low-cost adsorbents to remove Hg0 from flue gases generated by coal-fired power plants. The correlation between Hg0 capacity and micropore volume for a series of Illinois-coal (IBC-109)based adsorbents suggest that Hg0 capacity is proportional to the micropore volume, indicating that adsorption of Hg0 onto adsorbent is significantly influenced by physical adsorption (Figure 11). PSDs for two samples prepared from low-sulfur coal (LSC, IBC-109) with a 1.2% total sulfur content on a dry ash-free (daf) basis and high-sulfur coal (HSC, Crown-2) with a 4.1% total sulfur daf basis are almost identical (Figure 12). An apparent difference is the 6% larger micropore volume for the LSC sample (0.336 cm3/ g) compared to the HSC sample (0.316 cm3/g). The Hg0 adsorption capacity for the HSC (1.35 mg/g) is 200% more than the LSC (0.67 mg/g).35 Total sulfur contents are 1.18% (daf) and 2.2% (daf) for LSC and HSC, respectively. It is evident that sulfur in the adsorbent (34) Lehmann, C. M. B.; Rostam-Abadi, M.; Rood, M. J.; Sun J. Energy Fuels 1998, 12, 1095. (35) Hsi, H. C.; Chen, S. G.; Rostam-Abadi, M.; Rood, M. J.; Richardson, C. F.; Carey, T. R.; Chang, R. Energy Fuel 1998, 12, 1061.

Energy & Fuels, Vol. 12, No. 6, 1998 1077

Figure 11. Correlation between Hg0 adsorption capacity and micropore volume obtained by the 3-D model for two serial Illinois coal (IBC-109 and Crown-2)-based carbons.

Figure 12. PSDs for the two Illinois coal-based carbons (lowsulfur coal-based carbon (LSC) and high-sulfur coal-based carbon (HSC)) developed for Hg removal from flue-gas streams.

plays an important role in adsorption of Hg0. The Hg0 capacity appears to be linearly proportional to the percentage of the total sulfur content of the final product. The adsorption of Hg0 onto sulfur-containing adsorbents can also be influenced by chemisorption.35 Further research is occurring in this area. Summary and Conclusions A new analytical PSD model has been developed by solving the generalized adsorption isotherm for N2 adsorption at 77 K using least-squares error minimization. Local isotherms for each single pore size are calculated using a 3-D adsorption isotherm equation derived from statistical mechanical principles. In comparison to select analytical methods from the literature, this 3-D model offers a relatively realistic PSD description for select reference materials. N2 and CH4 adsorption is correlated using the 3-D model for BPL and AX21. Predicted CH4 adsorption isotherms are in reasonable agreement with experimental CH4 isotherms. Acknowledgment. We appreciate N2 isotherm data for ACFCs from Mark Cal of New Mexico Institute of

1078 Energy & Fuels, Vol. 12, No. 6, 1998

Sun et al.

Technology and N2 isotherm data for Norit Row from Michal Kruk and Mietek Jaroniec of Kent State University. This research was sponsored by the Illinois Clean Coal Institute through a grant (DE-FC2292PC92521) from the Illinois Department of Natural Resources and its Coal Development Board and by the U.S. Department of Energy. Financial support was also provided from the Office of Solid Waste Research (OSWR) at the University of Illinois at UrbanaChampaign (OSWR12-7GS) and by Ford Motor Company. Appendix A. Solution to the GAI-PSD Calculation

i

F(Pjr,wi) )

i

wi - wi

l

n(Pjr) )

(a-1)

where F(Prj, wi) (i ) 1, 2, ..., l; j ) 1, 2, ..., m) are the

F(Pjr,wi)V(wi), ∑ i)1

j ) 1,2,...,m (a-2)

where V(wi) is the (unknown) volume of ith pore size interval per gram of carbon. A least-squares error minimization criterion is used to determine the optimal solution for this linear equation set

E)

Equation 9 can be broken down into a set of linear equations. The number of equations, m, can be the number of data points from the experimental isotherm (usually up to 50 points for N2 adsorption). Each data point provides each linear equation with one experimental value of the adsorbed N2 amount n(Prj) (n is a function of Pr, j ) 1, 2, ..., m) for each corresponding relative pressure Prj. The adsorbate density F(Prj,w) at each Prj is calculated using eq 5 for various pore sizes w from a minimum of 4 Å to a maximum of 100 Å in grids of 1 Å (within pore sizes of 4-20 Å), 2 Å (20-40 Å), and 10 Å (40-100 Å). wmax is chosen, to a certain extent arbitrarily, as 100 Å, as the samples in this study are highly microporous and have a negligible macropore volume. The number of unknowns in each linear equation depends on the resolution of the pore size interval, and the largest number is (unlimited to) 33 in this study. In other words, PSD can be constructed in the form of histograms over several pore size intervals l e 33). The average adsorbate density for intervals 20 Å < w < 100 Å are calculated by

∫ww F(Pjr,w) dw

average densities in pores of ith pore size interval wi at jth relative pressure Prj. wi and wi are the lower and upper limit of the ith pore size interval. Thus, eq 9 was transformed into a set of linear equations:

1

m

∑ m j)1

(

l

n(Pjr) -

)

F(Pjr,wi)V(wi) 2 ∑ i)1

(a-3)

where E is the mean-square error per fitted point and m is the number of the data points from the experimental N2 adsorption isotherm. A constraint is imposed requiring all V(wi) be g0. Total micropore volume is the summation of the total pore volumes over all the (17) microporous size intervals. 17

V0 )

V(wi) ∑ i)1

(a-4)

All of the calculations were performed with a personal computer containing a Pentium-120 processor using Excel for Windows. In particular, for the calculation of the adsorption density for each pore size for each experimental adsorptive pressure, a macro is recorded so that it can run automatically for the number of data points from the experimental isotherm (10 e m e 50). PSD determination by least-squares error minimization is carried out using Solver in Excel. Multiple PSD solutions are possible in some cases because of the instability of the least-squares technique. However, a unique solution is obtained by correlating the peak positions of the PSD profile with the inflections of the experimental isotherm for N2 at 77 K and by comparison of the overall structure of the experimental isotherm with the resulting modeled adsorption. EF980108O