Development and Validation of Pore Structure Models for Adsorption

Langmuir 1999, 15, 6263-6276

6263

Development and Validation of Pore Structure Models for Adsorption in Activated Carbons G. M. Davies† and N. A. Seaton*,‡ Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, United Kingdom, and School of Chemical Engineering, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, United Kingdom Received February 16, 1999 Predicting adsorption over a range of operating conditions and the improvement of the adsorbent itself are two important aspects that arise in the industrial application of adsorption. Both of these aspects can be addressed using molecular simulation techniques in conjunction with an appropriate model of the internal structure of the adsorbent. The internal structure of activated carbons is particularly difficult to model due to the fact that the structure is only locally crystalline and that most of the void volumes within the structure have length scales comparable to small molecules. This paper presents a systematic method to develop suitable models of the internal structure that are based on networks of regularly shaped model pores. Important aspects that are addressed include the realism and consistency of the resulting models. The method is illustrated using the adsorption of pure methane and ethane, and binary mixtures of these components, over a wide range of operating conditions onto four activated carbons.

1. Introduction Molecular simulation is a powerful technique for the prediction of adsorption within porous solids that is based on a model of the structure of the solid. Potential industrial applications of molecular simulation include the prediction of the performance of existing adsorbents, and the identification of optimal pore structures as an input to the development of new adsorbents. This paper addresses the problem of developing a suitable pore structure model for the simulation of adsorption in activated carbons. Most of the existing models of the internal structure can be divided broadly into two main groups: those that attempt to generate a detailed representative section of a material, which we term “representative section” models, and those that are based on networks of regularly shaped model pores, which we term “pore network” models. It is infeasible to experimentally determine a detailed atom-by-atom or equivalent description of an amorphous microporous adsorbent. Representative section models estimate this information on the basis of either simplified models that aim to describe a small section of the adsorbent directly or simulations that specifically mimic the formation of the adsorbent from a well-defined starting material. (The simulations that are used to estimate the structure of an adsorbent are quite distinct from the molecular simulations that are used to calculate the adsorption in these structures.) Examples of these models include those of Kaminsky and Monson1 who have approximated the structure of silica xerogel as an equilibrium packing of uniform spherical adsorbate particles, Gelb and Gubbins2 who have simulated the formation of an amorphous microporous glass, and Segarra and Glandt3 who have * To whom correspondence should be addressed. Telephone: (+44) 131 650 4867. Fax: (+44) 131 650 6551. E-mail: n.seaton@ chemeng.ed.ac.uk. † University of Cambridge. ‡ University of Edinburgh. (1) Kaminsky, R. D.; Monson, P. A. Langmuir 1994, 10, 530. (2) Gelb, L. D.; Gubbins, K. E. Langmuir 1998, 14, 2097. (3) Segarra, E. I.; Glandt, E. D. Chem. Eng. Sci. 1994, 49 (17), 2953.

approximated the internal structure of activated carbons using a simulated packing of a basic crystallite that they proposed. Attempts to simulate the structure of an adsorbent become significantly more difficult as the complexity of the structure of the precursor material and the subsequent processing stages required in the preparation of the adsorbent increases. This is the primary reason Segarra and Glandt3 needed to first propose a basic building block, that of the crystallite structure, before constructing a model of the gross internal structure. It is also the reason they had to resort to simulations that produce essentially random orientations of these crystallites despite the knowledge that the local orientations of lamellae of graphite in the real structure (the crystallites in their model) are most probably a function of the processes used in the adsorbent preparation. Consequently, representative section models are not always an entirely realistic representation of the detailed internal structure. In the case of the models of the silica xerogel and the activated carbon, comparison of pure component adsorption calculated from molecular simulations and the corresponding experimental observations indicate that these models require refinement.1,3 The most significant problem in this approach is determining what these refinements should be and how to implement them in a systematic manner. The pore network models in turn approximate the complex internal void volume of amorphous adsorbents as a collection of regularly shaped model pores. The realism of the pore network model is determined in part by the extent to which the equivalence between the model and the real structure is enforced. Usually, the only enforced equivalence between the pore network model and the real structure is that they both yield the same adsorption isotherm for a single pure adsorbate. The simplest pore network model is the “bundle of straight pores” (or “BSP”) model.4-22 In this model, each pore is considered to be (4) Quirke, N.; Tennison, S. R. Carbon 1996, 34, 1281. (5) Koch, K.; v. Szombathely, M.; Neugebauer, N.; Brauer, P. In Fundamentals of Adsorption; LeVan, M. D., Ed.; Kluwer: Boston, 1996; p 921.

10.1021/la990160s CCC: $18.00 © 1999 American Chemical Society Published on Web 07/08/1999

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accessible to all the adsorptive species in the bulk gas phase. One can imagine (conceptually) a bundle of pores each of which connects to the surface of the adsorbent or (more physically) a connected network of pores in which the connectivity of the network is sufficiently good that the adsorptive species can pass throughout the network. The pore size distribution (PSD) for this model is calculated from the adsorption integral equation:

Ns(T,P,y)i )

∫0∞Fmp,s(w,T,P,y)i f(w) dw

i ) 1...n (1)

where Ns(T,P,y)i is the adsorption of species s at temperature T, pressure P, and bulk gas-phase composition y; Fmp,s(w,T,P,y)i is the “single-pore isotherm”, which is the corresponding adsorption density in a model pore of characteristic dimension w; f(w) is the PSD; and n is the number of adsorption measurements. Based directly on eq 1, the pore size distribution can be formally defined as

f(w) ) dV/dw

(2)

where w is a characteristic dimension of the model pore and V is the pore volume per unit mass of the adsorbent. To calculate the pore size distribution based on eq 1, a suitable method is required to determine the adsorption in model pores. Classical techniques are inappropriate because they cannot account for the important adsorbateadsorbent molecular interactions or steric hindrances that occur due to the proximity of the pore walls in micropores;23 a statistical mechanical method is required. The first statistical mechanical technique employed to calculate the adsorption in a model pore was density functional theory.24 This method simplifies the interactions between adsorbate molecules to provide a computationally fast route for approximating the adsorption in regularly shaped model pores. Quirke and Tennison,4 Koch et al.,5 Niemark et al.,6 Olivier et al.,7 Lastoskie et al.,8 Aukett et al.,9 Jessop et al.,10 Seaton et al.,11 and Sosin and Quinn19 have applied (6) Neimark, A. V.; Ravikovitch, P. I.; Grun, M.; Schuth, F.; Unger, K. K. Langmuir 1995, 11, 4765. (7) Olivier, J. P.; Conklin, W. B.; v. Szombathely, M. In Characterization of Porous Solids (COPS-III). Proceedings of the IUPAC Symposium; Rodriguez-Reinoso, F.; Rouquerol, J.; Sing, K. S. W.; Unger, K. K., Eds.; Elsevier: Amsterdam, 1994; p 81. (8) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4586. (9) Aukett, P. N.; Quirke, N.; Riddiford, S.; Tennison, S. R. Carbon 1992, 30, 913. (10) Jessop, C. A.; Riddiford, S. M.; Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. In Characterization of Porous Solids (COPS-III). Proceedings of the IUPAC Symposium; Rodriguez-Reinoso, F.; Rouquerol, J.; Sing, K. S. W.; Unger, K. K., Eds.; Elsevier: Amsterdam, 1994; p 123. (11) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (12) McEnaney, B.; Mays, T. J.; Chen, X. Fuel 1998, 77 (6), 557. (13) Davies, G. M.; Seaton, N. A. Carbon 1998, 36, 1473. (14) Davies, G. M.; Vassiliadis, V. S.; Seaton, N. A. To be submitted for publication in Langmuir. (15) Lo´pez-Ramoń, M. V.; Jagiello, J.; Bandosz, T. J.; Seaton, N. A. Langmuir 1997, 13, 4435. (16) Gusev, V. Y.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815. (17) Gusev, V. Y.; O’Brien, J. A.; Langmuir, 1997, 13, 2822. (18) Samios, S.; Stubos, A. K.; Kanellopoulos, N. K.; Cracknell, R. F.; Papadopoulos, G. K.; Nicholson, D. Langmuir 1997, 13, 2795. (19) Sosin, K. A.; Quinn, D. F. J. Porous Mater. 1995, 1, 111. (20) Sosin, K. A.; Quinn, D. F.; MacDonald, J. A. F. Carbon 1996, 34 (11), 1335. (21) Wang, K.; Do, D. D. Langmuir 1997, 13, 6226. (22) Do, D. D.; Do, H. D. Stud. Surf. Sci. Catal. 1994, 87, 641. (23) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: New York, 1982. (24) Evans, R.; Tarazona, P. Phys. Rev. Lett. 1984, 52, 557.

density functional theory to the characterization of microporous carbons. More recently, advances in computational power have made Monte Carlo simulation, a more accurate though slower method, a more attractive option. Monte Carlo simulation provides essentially exact solutions for adsorption in model pores, with computational requirements that are easily within the capabilities of modern workstations. The most appropriate molecular simulation technique applied to adsorption in microporous solids is the grand canonical Monte Carlo (GCMC) simulation method.25 Originally applied only to the adsorption of pure components, it has more recently been extended to calculate the adsorption of several species adsorbing simultaneously.26-28 McEnaney et al.,12 Davies and Seaton,13 Davies et al.,14 Lo´pez-Ramoń et al.,15 Gusev et al.,16 Gusev and O’Brien,17 Saimos et al.,18 and Sosin et al.20 have in turn applied GCMC simulations to the characterization of activated carbons. (Sosin et al.20 used the simulated results of Cracknell et al.30 in their calculations.) As already stated, the pore size distribution is proposed as a simplified representation of the complex internal structure of an amorphous microporous adsorbent. The straightforward BSP model is a simplified representation because it essentially assumes that the dominant factor affecting adsorption is the relative proximities of the pore walls to each other. As such, this model does not explicitly account for any other aspects of the internal structure that could affect adsorption. For example, it ignores the effects of pore junctions and the fact that some pores may be inaccessible from the bulk gas phase for some adsorbates due to the pore network connectivity. Also, it does not account for any form of chemical heterogeneity or possible surface defects within the pore walls. The assumption that these factors are relatively unimportant compared to the pore wall proximities might or might not be valid for a particular adsorbent. The realism of the chosen pore structure model can be assessed by considering the consistency of the model, for example by investigating whether a PSD obtained by analyzing the adsorption of one species can be used to predict the adsorption of another species. Gusev et al.16 were the first to consider the consistency of the BSP model by determining a PSD for an activated carbon based on a single methane adsorption isotherm and then using it to predict the adsorption of methane at two other temperatures. The predictions were made using the adsorption integral equation, eq 1, using the previously determined pore size distribution and two sets of simulated model pore isotherms that corresponded to the new temperatures of interest. From the excellent agreement between the predictions and the experimentally determined isotherms they concluded that the pore size distribution represents a consistent characterization with respect to the adsorption of a single pure component. Gusev and O’Brien17 later used a similar pore size distribution based on methane to predict the adsorption of ethane. Although the agreement between the predictions and the experimentally determined isotherms was not as good as (25) Adams, D. J.; Mol. Phys. 1974, 28, 1241. (26) Heffelfinger, G. S.; Tan, Z.; Gubbins, K. E.; Marconi, U. M. B.; Vanswol, F. Int. J. Thermophys. 1988, 9 (6), 1051. (27) Karavias, F.; Myers, A. L. Mol. Simul. 1989, 8 (1-2), 51. (28) Cracknell, R. F.; Nicholson, D.; Quirke, N. Mol. Phys. 1993, 80 (4), 885. (29) Reference deleted at proof stage. (30) Cracknell, R. F.; Gordon, P.; Gubbins, K. E. J. Phys. Chem. 1993, 97, 494. (31) Sandler, S. I. Chemical Engineering Thermodynamics; Wiley: NewYork, 1989.

Pore Models for Adsorption in Activated Carbons

it was for methane, they attributed the discrepancies to either the omission of the quadrupole of the ethane molecule in the GCMC simulations or to the fact that the pore size distribution based on methane becomes unreliable at large pore sizes. The two possible causes of the discrepancies suggested by Gusev and O’Brien17 represent two fundamentally different forms of model failure. The first suggestion, that of the omission of the ethane quadrupole in the molecular simulation, is one possible way in which the model is not sufficiently realistic. The second form of model failure, which is related to the unreliability of the pore size distribution, is an inherent shortcoming of the way the model is posed: the adsorption integral equation, eq 1, is an ill-posed problem. This means that if a pore size distribution can be fitted to a given set of data at all, then in principle an infinite number of pore size distributions will exist that fit the data with comparable accuracy.14 It is also possible that these pore size distributions might not be similar to one another. The major problem with this type of model failure, as Gusev and O’Brien17 implicitly assert, is that another pore size distribution that fits the methane adsorption data with comparable accuracy to the one that they used may have led to a better prediction of the ethane adsorption. For these reasons dealing with the ill-posed character of the adsorption integral equation will form an important part of this paper. Before we can envisage using a model of the internal structure in conjunction with molecular simulation to predict adsorption over a wide range of operating conditions, including the adsorption of mixtures, we need to determine whether it is a sufficiently realistic model of the internal structure to achieve this goal. Equally, any model of the internal structure that is used in adsorbent design will need to be at least sufficiently realistic to account for a wide range of adsorption. This paper presents a procedure to develop such a model of the internal structure. The remainder of the paper is arranged as follows. The following two sections will summarize how molecular simulations can be used to calculate the adsorption in model pores and will present suitable techniques to extract a pore size distribution from the adsorption integral equation. Section 4 then presents our procedure to develop a suitably realistic model of the internal structure. In this section, the technique is demonstrated using the adsorption of methane and ethane and their binary mixtures on four different activated carbons as a case study. 2. GCMC Simulation of Adsorption in Model Pores The GCMC simulations calculate the amount adsorbed by averaging over the molecular configurations within a model pore that are consistent with the specified thermodynamic state. The independent variables of a GCMC simulation are the system volume, which is defined in terms of the simulation cell volume Vsc, the system temperature T, and the chemical potentials of the adsorptives µs. In practice, the chemical potentials of the adsorptives are calculated from the specified temperature, pressure, and composition of the bulk gas phase using an appropriate equation of state; our simulations use the Peng-Robinson equation of state with the parameters summarized in Table 1. The simulation cell is usually rectangular with the top and bottom surfaces of the rectangle representing the pore walls. The “sides” of the rectangle are treated as periodic boundary conditions, so

Langmuir, Vol. 15, No. 19, 1999 6265 Table 1. Summary of the Constant Used in the Peng-Robinson Equation of Statea param

value

critical temp, K methane ethane critical pressure, MPa methane ethane accentricity factor methane ethane interaction param a

190.6 305.4 4.600 4.884 0.008 0.098 -0.003

Constants are from Sandler.31 Table 2. Summary of the Adsorbate and Adsorbent Parameters Used in This Worka parameter

methane param σMM, Å MM/kB, K ethane param σEE, Å EE/kB, K bond length, Å

value 3.81 148.2 3.512 139.8 2.353

param

value

adsorbent param σCC, Å CC/kB, K ∆, Å σC, Å-3

3.4 28.0 3.35 0.114

a Methane parameters from Hirschfelder et al.;37 ethane parameters from Cracknell et al.;36 adsorbent parameters from Steele.38 kB is the Boltzmann constant.

that the system that is being simulated is considered to be surrounded by replicas of itself in the directions parallel to the pore walls. The algorithms implemented in the simulations to sample the molecular configurations are both wellestablished and well-documentedssee, for example, refs 32-34 or refs 28, 35, and 36; the reader is referred to these sources for details. A central part of these algorithms is the calculation of the potential energy of any given molecular configuration. This is determined using suitable interaction potentials which represent the attraction and repulsion between fluid molecules and between an adsorptive molecule and the pore wall. In this work methane has been modeled as a single-site “united atom” and ethane as a two-site molecule. Since each of the sites can be treated as being nearly spherical and nonpolar, the interaction between the sites of each adsorptive molecule can be described using the LennardJones interaction potential:

vff(r) ) 4ff[(σff/r)1/2 - (σff/r)6]

(3)

where σff is the site diameter of the interaction site, ff is the potential well depth for the each site, and r is the distance between two sites. This potential can also be used to calculate the interaction between the sites of different adsorbates by adopting the standard Lorentz-Berthelot rules. The Lennard-Jones parameters used in this work are summarized in Table 2. In order to determine a suitable adsorptive-adsorbent interaction potential, the adsorbent and the shape of the (32) Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: San Diego, 1996. (33) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford Science Publications: Oxford, U.K., 1987. (34) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (35) Cracknell, R. F.; Nicholson, D.; Quirke, N. Mol. Simul. 1994, 13, 161. (36) Cracknell, R. F.; Nicholson, D. J. Chem. Soc., Faraday Trans. 1994, 90 (11), 1487. (37) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; Wiley: New York, 1954.

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model pores to be used in the pore size distribution analysis need to be specified. Because we have specifically considered activated carbons, we have adopted slit-shaped model pores since they not only represent a physically plausible pore shape but also have been shown to simplify the interpretation of the pore size distribution.13 The interaction between a site on an adsorptive molecule and a single semiinfinite pore wall of graphite is given by Steele’s 10-4-3 potential:38

vsf(z) ) 2πsfFsσsf2∆

[(

) ( )

2 σsf 5 z

10

-

σsf z

4

σsf4

]

3∆(z + 0.61∆)3

(4)

where Fs is the number of carbon atoms per unit volume in the graphite layer, ∆ is the separation distance between the layers of graphitic carbon, and, z is the distance between the site in an adsorbate molecule and the adsorbent surface. Once again sf and σsf are determined using the standard Lorentz-Berthelot rules. The parameters used in the Steele potential are also summarized in Table 2. Equation 4 describes the interaction between a site in an adsorbate molecule and one pore wall. Since slit-shaped pores have two pore walls, the combined potential is calculated using

Vsf(z) ) vsf(z) + vsf(Z - z)

(5)

where z is the distance from one of the walls and, Z is the distance between the pore walls (i.e. the pore width). The x and y dimensions of the simulation cell, parallel to the pore walls, were each set to 57.15 Å which corresponds to 15 times the molecular diameter of methane. The adsorption in slit-shaped pores of different widths was calculated by varying the z dimension of the simulation cell from 1.6 to 20 times the molecular diameter of methane in successive simulations. The initial configuration of molecules for the first point on the isotherm in each simulation was generated by placing five molecules within the simulation cell such that they did not overlap. For subsequent points on the isotherm, which were either at higher pressures or different bulk-phase compositions, the final configuration of the previous isotherm point was used as the initial configuration. The Monte Carlo choices involved in a grand canonical Monte Carlo simulation are to move, create, or destroy a molecule or, in a mixture, to swap the identity of a molecule. In each step one of these was chosen with equal probability. For each point on the isotherm, the system was allowed to equilibrate for between 2 × 105 and 5 × 105 steps. After equilibration the simulation continued for between 1 × 106 and 2 × 106 steps in order to calculate the adsorption. The total time required to simulate the adsorption for all the conditions analyzed in the remainder of this paper, over a range of 26 slit-shaped model pore widths, amounted to several weeks using a Sun UltraSparc-Enterprise-2 workstation. (About 600 isotherms in total were simulated.) Two points are worth noting with respect to the time taken to simulate the adsorption in the model pores. Firstly, the analyses reported in this paper cover four activated carbons over a wide range of operating conditions. This is the reason so many adsorption isotherms (38) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, U.K., 1974.

needed to be simulated. For a much more limited analysis involving a single activated carbon the amount of simulation time would only amount to a few days. Secondly, simulating the adsorption in model pores at a given set of conditions is usually a one-off expense. Therefore, should we need to analyze a new activated carbon for which adsorption measurements have been made corresponding to the conditions that we have already simulated, we would be able to analyze it immediately. Naturally, should we wish to incorporate an improved interaction potential or investigate any other physical property of the micropores themselves then the simulations would need to be rerun. The single-pore isotherm, Fmp,s(w,T,P,y)i is obtained from the average number of molecules of each species within a small section of the model pore. These data need to be corrected to account for a subtle difference between simulated and experimentally determined extents of adsorption. Simulated adsorption is termed “absolute adsorption” because it represents the total adsorption for each species within a model pore. In contrast, adsorption experiments measure “excess adsorption”, the difference between absolute adsorption and the amount of adsorptive that would be present if there were no interactions between the adsorptive and the adsorbent. The simulated extents of adsorption for model pores are converted to excess isotherms using bf abs bf nex s (w,T,P,y) ) ns (w,T,P,y) - ysFs (T,P)V

(6)

where nex s (w,T,P,y) is the excess number of molecules of species s in a small section of a model pore of width w at temperature T, pressure P, and bulk gas-phase composition y, nabs s (w,T,P,y) is the absolute (i.e. simulated) number of molecules under these conditions, Fbf s (T,P) is the bulk fluid density of species s at the temperature and pressure concerned and Vbf is the volume in the model pore accessible to the bulk adsorptive. The volume accessible to the bulk fluid in the model pore has been assumed to tend to zero at the smallest pore in which adsorption takes place. Using this assumption the accessible volume for the bulk fluid in a slitshaped pore can be defined as

Vbf(w) )

{

w < wspw 0 (w -wspw)a w e wspw

}

(7)

where w is the pore width of interest, wspw is the smallest pore in which adsorption takes place and a is the area of one of the pore walls. (In terms of the simulation cell dimensions, a ) xy.) Once the simulated adsorption has been converted into an excess quantity it can be expressed as an adsorbate density by identifying the volume in which adsorption takes place. We define this volume to be the volume of the simulation cell itself. The simulation cell represents the small section of the model pore in the molecular simulation and extends to the centers of the carbon atoms in the first layer of graphite in the pore walls. As we have previously remarked,13 this definition of the adsorption volume differs from the “effective volume” used by some other workers.4,9,16,39,40 The main advantage of our definition over an effective volume is that it is independent of the adsorbing species; it simplifies comparisons between pore size distributions based on different adsorbates (calculated independently) and is a necessity when calculating a pore (39) Nicholson, D. J. Chem. Soc., Faraday Trans. 1996, 92 (1), 1. (40) Kaneko, K.; Cracknell, R. F.; Nicholson, D. Langmuir 1994, 10, 4606.


Langmuir, Vol. 15, No. 19, 1999 6267

Figure 1. Calculated excess adsorption isotherms of methane in slit-shaped model pores (T ) 308 K).

Figure 2. Calculated excess adsorption isotherms of ethane in slit-shaped model pores (T ) 308 K).

size distribution based on different adsorbates simultaneously. The excess adsorbate density for model pores is thus calculated using

Fmp,s(w,T,P,y) )

nex s (w,T,P,y) Vsc

Figure 3. Calculated binary excess adsorption isotherms of methane in slit-shaped model pores. Adsorption is from a 50:50 methane:ethane bulk gas mixture (T ) 308 K).

Figure 4. Calculated binary excess adsorption isotherms of ethane in slit-shaped model pores. Adsorption is from a 50:50 methane:ethane bulk gas mixture (T ) 308 K).

(8)

where Vsc is the volume of the simulation cell. We calculated excess adsorbate densities on the basis of our simulated absolute isotherms using eqs 6-8 and the Peng-Robinson equation of state with the parameters summarized in Table 1 to determine the bulk fluid density. The minimum pore width in which adsorption takes place, which is required to determine the accessible volume for the bulk fluid using eq 7, was determined to be approximately 6.1 Å. Figures 1-6 present typical excess isotherms for the adsorption in slit-shaped model pores. Figures 1 and 2 show the excess adsorption of pure component methane and ethane at 308 K for a range of pore sizes. The complex variation between the isotherms for different pore sizes, including the fact that some crossover others, is typical of adsorption in micropores. This is caused by a trade-off between the strength of the adsorbate-adsorbent interaction, which decreases as the pore size increases, and the ability of the micropores to accommodate adsorbate molecules. Therefore, the first significant amount of adsorption takes place in micropores that are large enough

Figure 5. Calculated binary excess adsorption isotherms of methane in slit-shaped pores. Adsorption is from bulk gas-phase mixtures of methane and ethane at 1 bar (T ) 293 K).

to accommodate a single layer of adsorbate molecules. The amount of adsorption is limited by the capacity of the micropores, which only increases appreciably when they are wide enough to accommodate a second layer of adsorbate molecules, one layer along each micropore wall.

6268 Langmuir, Vol. 15, No. 19, 1999

Figure 6. Calculated binary excess adsorption isotherms of ethane in slit-shaped pores. Adsorption is from bulk gas phase mixtures of methane and ethane at 1 bar (T ) 293 K).

This “capacity limitation” is particularly evident in the adsorption of ethane. Some of the isotherms in Figures 1-6 show that adsorption is lower at higher pressures compared to lower pressures. This is a characteristic of excess adsorption and is usually observed when large pores fill to capacity at moderate pressures. In these instances the simulated absolute adsorption does not increase appreciably as the pressure increases. In contrast, the amount of adsorptive that would be present if there were no adsorbateadsorbent interactions increases as the pressure increases. As a direct consequence, the extent of excess adsorption is smaller at higher pressures. This effect is particularly pronounced for the less strongly adsorbing component in binary adsorption, as can be seen in Figure 3. Figures 5 and 6 show the binary adsorption of methane and ethane as a function of the composition of the bulk gas phase. As the bulk mole fraction of methane tends to unity, the adsorption tends to the corresponding pure component adsorption of methane at 293 K and 1 bar. Similarly as the bulk mole fraction of methane tends to zero, the adsorption tends to the corresponding pure component adsorption of ethane at 293 K and 1 bar. 3. Calculation of Pore Size Distributions Several aspects need to be considered when attempting to calculate a pore size distribution from the adsorption integral equation. This section summarizes the most important of these and outlines a suitable solution strategy. (For a more complete analysis see Davies et al.14) The first aspect requiring consideration is the form of the pore size distribution to be used in the analysis. Frequently, relatively straightforward functional forms (an example of which is an n-modal log normal function) are proposed as suitable representations of the pore size distribution. The advantage of this approach is that these functions usually contain only a few parameters, which can be determined in a statistically significant manner (since there are invariably more data points than free parameters). The main disadvantage however is that this constrains the pore size distribution to conform to the particular functional form adopted. This is particularly significant considering that we have no a priori knowledge about the shape of the pore size distribution. This makes it difficult to propose suitable straightforward functional forms. In contrast to functional representations, there are significant advantages to adopting a discrete representa-

Davies and Seaton

tion of the pore size distribution. Discrete representations are inherently more flexible and therefore do not constrain the distribution to a possibly inappropriate form. They are also inherently suited to the numerical inversion of the adsorption integral equation in that numerical inversion involves discretizing the adsorption integral equation. (The adsorption integral equation is inverted numerically because it cannot, in general, be inverted analytically.) In this work we have therefore adopted a discrete representation of the pore size distribution. An initial problem that is encountered when discretizing the adsorption integral equation (which would also arise if a functional form for the pore size distribution were adopted) is that the integration extends to infinity. Davies et al.14 found that the integration to infinity could be successfully incorporated into the analysis by splitting the domain of the pore size distribution into two regions. The first region is the main region of interest and contains several quadrature intervals while the second region extends from the upper limit of the region of interest to an arbitrarily large pore size and only contains a single quadrature interval. This single quadrature interval in the second region is merely used to account for the adsorption that occurs in all pores larger than those in the region of interest. Adopting this method in conjunction with a uniform quadrature interval in the region of interest, we obtain

δwj )

wu - wl m-1

j ) 1...m - 1

(9)

where δwj is the quadrature interval, wu and wl are the upper and lower limits of the pore range of interest, and m is the total number of quadrature intervals in the pore size distribution. The single quadrature interval of the second region is

δwm ) wmax - wu

(10)

where wmax is the maximum limit used in the analysis. The discrete representation of the pore size distribution can be summarized as follows. The quadrature points are

wj

j ) 1...m + 1

(11)

w1 ) wl

(12)

where

wj ) wj-1 + δwj-1

j ) 2...m-1

(13)

wm ) wu

(14)

wm+1 ) wmax

(15)

The pore sizes in the pore size distribution, wj*, have been defined to be the average pore size in each quadrature interval

wj* ) wj +

δwj 2

(16)

Having defined a suitable quadrature, the adsorption integral equation, eq 1, can be written as m

Ns(T,P,y)i ≈

Fmp,s(wj*,T,P,y)i f(wj*) δwj ∑ j)1

which can be written more compactly as

(17)


Langmuir, Vol. 15, No. 19, 1999 6269

N ) AWf

(18)

N ) (Ns(T,P,y)i)i)1...n

(19)

A ) (Fmp,s(wj*,T,P,y)i)i)1...n,j)1...m

(20)

W ) diag(δwj)j)1...m

(21)

f ) (f(wj*))j)1...m

(22)

where

Having suitably discretized the adsorption integral equation, the next step is to determine a suitable solution strategy. Davies et al.14 have shown that the detailed solution strategy depends on whether we are interested in establishing the existence of a solution or calculating a representative pore size distribution. Establishing the existence of a solution is important for two reasons. Firstly, in some cases it is impossible to calculate a representative pore size distribution for a given set of data even though many pore size distributions can be fitted to the data. This situation arises when more quadrature intervals need to be introduced into the analysis than there are data points before a physically meaningful pore size distribution can be calculated. (A physically meaningful pore size distribution is one that is strictly nonnegative.) When this occurs, it is an indication that more data are required before a representative (i.e. statistically significant) pore distribution can be calculated. Secondly, if no discrete pore size distribution based even on a fine quadrature can be fitted to the data, we are able to conclude that there are no pore size distributions compatible with the data. This is a clear indication that the model omits at least one significant factor that affects adsorption and is therefore a good test of the realism of the model. The existence of a physically meaningful solution can be determined using minimization routines or least squares algorithms to establish whether the following residual can be reduced to below an acceptable tolerance in the error:

As explained above, representative pore size distributions can be calculated only when the number of data points is equal to or exceeds the number of quadrature intervals. Even in these cases, special attention is required because the adsorption integral equation is inherently an ill-posed problem. This means that small perturbations in the data may lead to substantially different pore size distributions being calculated. A suitable method is therefore required to stabilize the numerical calculations such that the resulting pore size distribution is relatively insensitive to small perturbations in the data. The most commonly adopted method for stabilizing the result is to incorporate additional constraints that are based on the smoothness of the pore size distribution. The smoothness is introduced into the analysis on the basis of the assumption that a real material is most likely to exhibit a relatively smooth distribution of pore sizes and that most of the pores will be centered around a few dominant pore sizes. This method of stabilization, termed regularization, has been described in detail by Wilson,41 Szombathely et al.,42 Merz,43 and Wahba.44 Regularization is incorporated into the analysis by including a term, which is a measure of the smoothness of the pore size distribution, into the residual defined in eq 25:

RReg ) (N - AWf)T(N - AWf) + RS

(26)

where R is a strictly nonnegative smoothing parameter and S is a suitable discrete representation of a function that measures the smoothness of the pore size distribution. Notice that regularization produces a biased estimate of the solution. Stated in another way, regularization forces a slightly worse fit to the data in order to generate a smoother PSD. Note that eq 26 reduces to the standard residual defined in eq 25 if R ) 0. The most commonly adopted measure of the smoothness is the integral of the square of the second derivative of the pore size distribution. Rewriting eq 25 in the form of a summation and using this measure of smoothing, we get n

n

R)

∑ i)1

m

[Ns(Pi) -

∑ j)1

Fmp,s(wj*,Pi) δwj f(wj*)]2 (23)

such that

f(wj*) g 0

j ) 1...m

(24)

The residual defined by eq 23 can be written more compactly as T

R ) (N - AWf) (N - AWf)

(25)

In practice, it is most convenient to first determine whether the residual defined in eq 23 can be reduced to below a specified tolerance when the number of quadrature intervals, m, is set equal to the number of data points, n. If this can be achieved, we can then progress to calculating a representative pore size distribution. However, if the residual cannot be reduced to below the specified tolerance initially, the realism of the model can still be assessed by determining whether this can be achieved by increasing the number of quadrature intervals. If the residual cannot be reduced to below the specified tolerance for a fine quadrature, we can conclude that the pore network model omits at least one significant factor affecting adsorption.

RReg )

∑[Ns,(T,P,y)i i)1

m

Fmp,s(wj*,T,P,y)i δwj f(wj*)] ∑ j)1

m

2

f′′(wj*)2 δwj ∑ j)1

+R

(27)

The second derivative of the pore size distribution can be approximated using a finite difference formula:

f′′(wj*) )

[

]

f(wj+1*) - f(wj*) f(wj*) - f(wj-1*) / 1/2δwj+1 + 1/2δwj 1/2δwj + 1/2δwj-1 [1/2δwj+1 + 1/2δwj] (28)

This approximation to the second derivative requires values for f(w0*), δw0, f(wm+1*), and δwm+1. We define these quantities as follows: (41) Wilson, J. D. J. Mater. Sci. 1992, 27, 3911. (42) v. Szombathely, M.; Brauer, P.; Jaroniec, M. J. Comput. Chem. 1992, 13 (1), 17. (43) Merz, P. H. J. Comput. Phys. 1980, 38, 64. (44) Wahba, G. Stat. Decision Theory Relat. Top. 1982, 111 (2), 383. (45) Gusev, V. Y. Personal communication, 1997. (46) Szepesy, L.; Illes, V. Acta Chim. Hung. Tomus 1963, 35, 37. (47) Szepesy, L.; Illes, V. Acta Chim. Hung. Tomus 1963, 35, 53. (48) Szepesy, L.; Illes, V. Acta Chim. Hung. Tomus 1963, 35, 245. (49) Costa, E.; Sotelo, J. L.; Calleja, G.; Marron, C. AIChE 1981, 27, 5.

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Davies and Seaton

f(w0*) ) 0

(29)

δw0 ) δw1

(30)

f(wm+1*) ) 0

(31)

δwm+1 ) δwm

(32)

Reintroducing vector notation, eq 28 can be written as

f′′(wj*) ) Df

(33)

where the components of D are obtained directly from eq 28. Equation 27 can now be written more compactly as

RReg ) (N - AWf)T(N - AWf) + RfTDTWDf

(34)

As Wilson41 has demonstrated, minimizing the residual of eq 34 corresponds to minimizing (assuming that the data points are all of equal importance)

RReg ) -2NTAWf + fT(WTATAW + RDTWD)f (35) Equation 35 can be solved for a given value of R using standard nonlinear minimization routines. In this work we have adopted two methodssthe so-called “L” curve and the approximation to the generalized cross-validation score function proposed by Wilson41sto estimate the optimal value of the smoothing parameter. L curves are essentially a plot of some measure of the error of the fit to the data against the smoothing parameter. It has been found that the error usually remains fairly constant or increases only marginally as the smoothing parameter is increased until some threshold value is reached, beyond which the error increases rapidly. These curves are used to identify this threshold value, which is taken to be the optimal extent of smoothing. A suitable error of the fit, E, is based on the least squares residual

E(R) )

1

n

∑(N(T,P,y)i -

ni)1

m

Fmp,s(wj*,T,P,y)i δwj f(wj*))2 ∑ j)1

(36)

The generalized cross-validation method in turn is based on the observation that a good choice of the smoothing parameter is one that would enable us to predict any one of the n experimental data points from a pore size distribution that is determined using the remaining n 1 data points. This method therefore determines the value of the smoothing factor that results in pore size distributions based on any of the n - 1 remaining data points that on average predict the omitted data point the most successfully. We have adopted the approximation to the generalized cross-validation score function proposed by Wilson41 that is appropriate when calculating strictly nonnegative pore size distributions:

GCV(R) ) (1/n)(N -AWf)T(N - AWf) [1 - (1/n)Tr(AW(WTATAW + RDTWD)-1WTAT)]2

(37)

where f is determined from eq 35. The optimal smoothing parameter is determined as that one which minimizes the cross-validation function.

Figure 7. Schematic outline of the main stages that are involved in model development.

The final aspect that needs to be considered when calculating pore size distributions is the confidence that can be placed in the exact locations of the peaks at the larger pore sizes. The locations of these peaks are difficult to determine due to the fact that the isotherms essentially become indistinguishable at large pore sizes. This is a consequence of the fact that for sufficiently large pores adsorption occurs essentially independently on the two pore walls, so that pores of different width have very similar adsorption signatures. Gusev et al.16 introduced the concept of a “window of reliability” which identifies the pore size above which the model pore isotherms become too similar to be able to place a high confidence in the exact location of the peaks. This pore size is a function of the adsorbate used, the temperature, and the maximum pressure at which the adsorption was measured. The window of reliability also impacts on the “region of interest” over which the uniform quadrature is carried out. The region of interest should extend to at least the upper limit of the window of reliability. Otherwise there would be the possibility of missing out on a significant part of the pore size distribution that could be determined reliably. Although the region of interest in the analysis can extend beyond this limit, it must be recognized that the exact locations of any peaks beyond the window of reliability are less well-defined than those within it. 4. Model Development Figure 7 shows the main stages in the development of a pore structure model that could be used for predicting adsorption in porous solids using GCMC simulation. The intended use of the model depends on the application of interest. For our case study, the use might be specified as “prediction of the adsorption of methane, ethane, and mixtures of these species on microporous carbons from room temperature to 373 K and at pressures up to ∼32 bar”. We propose use of the BSP model (the simplest pore network model). If this model turns out to be unsatisfactory, we will refine the model and test the performance of this refinement.


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Table 3. Summary of the Pure Component and Binary Adsorption Data for Four Activated Carbons Used To Assess the Realism of the BSP Pore Size Distribution

ref

type of activated carbon

Gusev and O’Brien45

BPL

Szepesy and Illes45-48

Nuxit

Costa et al.49

AC40

Richter et al.50

AK

data used in our analysis pure component methane adsorption at 308, 333, and 373 K pure component ethane adsorption at 308, 333, and 373 K binary adsorption from a 50:50 bulk gas-phase mixture of methane and ethane at 308, 333, and 373 K binary adsorption from a 75:25 bulk gas-phase mixture of methane and ethane at 308, 333, and 373 K pure component methane adsorption at 293, 313, 333, and 363 K pure component ethane adsorption at 293, 313, 333, and 363 K binary adsorption from bulk gas-phase mixtures of methane and ethane at 1 bar and 293 K pure component methane adsorption at 293 K pure component ethane adsorption at 293 K binary adsorption from bulk gas-phase mixtures of methane and ethane at 0.1 bar and 293 K pure component methane adsorption at 303 K pure component ethane adsorption at 303 K binary adsorption from bulk gas-phase mixtures of methane and ethane at 5 bar and 20 bar and 303 K

In practice, it is desirable to make predictions of adsorption based on as small a set of experimental data as possible. In terms of the BSP model, the PSD would be obtained by analyzing a small set of experimental adsorption data and would then be used to predict the adsorption for a range of conditions. However, when assessing the realism of the model, the fullest possible data set should be used. For example, in our case study we are interested in developing a pore network model of the internal structure that can be used to predict pure component and multicomponent adsorption. Any model that can fulfill this requirement must at least be capable of being fitted to a range of pure component and multicomponent adsorption data. The method outlined in the previous section can be used to determine whether the BSP model can be fitted to a given set of data. Note that although a BSP pore size distribution that can be fitted to such a set of data would be strong evidence in favor of the realism of the model, it does not prove realism. However, in contrast, if a BSP pore size distribution cannot be fitted to such a set of data, it is proof of the lack of realism of the model. A consequence of fitting the PSD simultaneously to a wide range of data is that we are able to differentiate between model failures that are due to a lack of realism and those due to the ill-posed character of the model. This is achieved by first establishing whether any pore size distribution is compatible with all the data. The ill-posed character is not an issue when determining the existence of a pore size distribution: it is only an issue when more than one pore size distribution exists that are compatible with the data. This can be emphasized by considering eq 26. The residual defined by this equation is a measure of the fit to a given set of data. v. Szombathely et al.42 have shown that the best achievable fit to the data is obtained when the smoothing parameter is set to zero, which is essentially a least squares fit to the data. If the residual itself can be reduced to zero under these conditions, then we can achieve a perfect fit to the data. Equally, if the residual can be reduced to below a specified tolerance, then at least one acceptable fit, at least from a practical or engineering perspective, can be achieved. If no acceptable fit to the data can be obtained, we can immediately conclude that the model is not sufficiently realistic. This conclusion is unambiguous and not affected by the illposed character. In contrast, if the residual can be reduced to below a specified tolerance, then any subsequent model failures must be due to the ill-posed nature of the problem. Once a model has been demonstrated to be sufficiently realistic we can then progress to determining how much and what type of data are required in order to determine

any unknown parameters in the model. This relates directly to the overall utility of the model. A model that requires extensive data or even a limited amount of data that are typically not available is not that useful from an engineering perspective. The amount of data required is intimately linked to the ill-posed character of the model. Even if a particular model (the BSP model, for example) can be fitted to a wide range of data, it might be difficult to obtain an adequate description of the PSD based on only part of the data. As we will see below, this is where regularization and methods for identifying optimal smoothing parameters become particularly important. The final aspect of model development is to demonstrate that the model is reliable for at least a few representative case studies that are similar to those to which the model will be applied routinely. This highlights the need for typical pure component and multicomponent adsorption data for at least a few adsorbents that are used industrially. In order to establish whether the BSP model is sufficiently realistic for the purpose of our case study (adsorption of methane and ethane on activated carbons), we now determine whether it can be fitted to the pure and binary adsorption of methane and ethane over a range of temperatures for four different activated carbons. The data used in the analyses and their corresponding references are summarized in Table 3. In all cases the main region of interest was taken to lie between 6.1 and 40.0 Å. The choice of 40 Å as the upper limit of the region of interest was based on the fact that the region included the microporous pore range and extended a fair way into the mesoporous pore range. The upper limit used in the analyses was 100.0 Å. In each case a pore size distribution with 100 quadrature intervals was initially fitted to the data. If the fit was poor, then the number of quadrature intervals was increased to ensure that no pore size distributions exist that can be fitted to the data. In cases where the fit was acceptable, representative pore size distributions were calculated on the basis of either 100 or as many quadrature intervals as there were data points, whichever was the smaller. Figures 8-19 show the best fits to the data and representative pore size distributions for those cases where the fits were acceptable from an engineering perspective. The best fits to the experimental data show that no pore size distribution can be fitted accurately to the adsorption data for the BPL activated carbon. This is particularly evident from the fits to the adsorption of methane from methane/ethane mixtures. In contrast to the BPL activated carbon, pore size distributions are clearly capable of accounting for the available adsorption data for both the Nuxit and AC40 activated carbons. The fit to the last

6272 Langmuir, Vol. 15, No. 19, 1999

Figure 8. Best achievable fits to the pure component adsorption of methane and ethane onto BPL activated carbon. (Note that all the fits shown in Figures 8-18 are based on a pore size distribution calculated using all the methane and ethane adsorption data for the respective carbons.)

Davies and Seaton

Figure 11. Best achievable fits to the pure component adsorption of methane and ethane onto Nuxit activated carbon.

Figure 12. Best achievable fits to the binary adsorption of methane and ethane onto Nuxit carbon from bulk gas phase mixtures at 1 bar and 293 K. Figure 9. Best achievable fits to the binary adsorption of methane and ethane onto BPL activated carbon from a bulk gas mixture of 50:50 methane:ethane.

Figure 13. Representative pore size distribution for Nuxit activated carbon based on a fit to all the methane and ethane data. Figure 10. Best achievable fits to the binary adsorption of methane and ethane onto BPL activated carbon from a bulk gas mixture of 75:25 methane:ethane.

carbon is slightly worse than those of the Nuxit and AC40 activated carbons. These results indicate that the BSP model is not a sufficiently realistic representation of the internal structure of BPL carbons under the conditions of interest. As we are seeking a model for the prediction of adsorption in microporous carbons in general, this outcome

indicates that a refinement of the pore structure model is required. When the realism of the pore size distribution is assessed, it is helpful to consider the extent and type of adsorption data that are used in the analyses. It is worth noting that significantly more data, which covered a wide range of pressures and bulk gas compositions, were available for the BPL activated carbon. This is in contrast


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Figure 14. Best achievable fits to the pure component adsorption of methane and ethane onto AC40 activated carbon.

Figure 17. Best achievable fits to the pure component adsorption of methane and ethane onto AK activated carbon.

Figure 15. Best achievable fits to the binary adsorption of methane and ethane onto AC40 carbon from bulk gas phase mixtures at 0.1 bar and 293 K.

Figure 18. Best achievable fits to the binary adsorption of methane and ethane onto AK carbon from bulk gas phase mixtures at 1 and 5 bar at 293 K.

Figure 16. Representative pore size distribution for AC40 activated carbon based on a fit to all the methane and ethane data.

Figure 19. Representative pore size distribution for AK activated carbon based on a fit to all the methane and ethane data.

to the limited binary data for the Nuxit and AC40 activated carbons that only spanned a single relatively low pressure for each carbon (1 and 0.1 bar, respectively). The slightly worse fit to the data of Richter et al.50 could, by this argument, be due to the fact that there were more binary data for this carbon. This highlights an important aspect (50) Richter, E.; Schutz, W.; Myers, A. L. Chem. Eng. Sci. 1989, 44 (8), 1609.

of testing pore network models for realism. Testing whether a model is capable of being fitted to pure and binary data is a much more demanding and revealing test than using only pure component data. For example, Davies and Seaton51 found that a bundle of straight pores’ pore size distribution can be fitted to the same pure component methane and ethane adsorption data of Gusev45 at 308 K. On the basis of such an analysis we would have concluded

6274 Langmuir, Vol. 15, No. 19, 1999

that the BSP model is sufficiently realistic to capture the important factors that affect adsorption. It is only when the binary data are included into the analysis that the deficiency is revealed. Although the original conclusion is valid for pure component adsorption, we specifically require a model that can be used to predict multicomponent adsorption. We now consider how the results obtained from assessing the realism of the BSP model can be used in developing a refined model of the internal structure. A notable feature of all the best fits to the adsorption data is that in most cases it is the binary methane adsorption that is significantly in error. This result suggests that the methane adsorption from bulk gas-phase mixtures in the model structure is insignificant. Plots similar to those shown in Figure 3, which show the methane adsorption in slit-shaped model pores from bulk gas mixtures, reveal that extremely little methane adsorbs from mixtures in slit-shaped pores. In contrast, the extent of ethane adsorption from bulk mixtures is almost as high as that from pure bulk ethane. This can be seen by comparing Figure 4 to Figure 2. This is the reason the straightforward BSP pore size distribution is unable to account for the significant extents of experimentally determined methane adsorption. Suitable refinements to the BSP pore size distribution are therefore those that would either increase the methane adsorption or decrease the ethane adsorption from binary mixtures. One physically plausible refinement to the bundle of straight pores’ pore size distribution, which has recently been investigated by Lo´pez-Ramoń et al.,15 would be to include a measure of the pore network connectivity. Network connectivity is a measure of the internal topology within a microporous material. It accounts for the fact that the pores in a real network are interconnected and, as a result, that some are inaccessible from the bulk gas phase. If some of the pores in activated carbon are inaccessible to ethane due to the connectivity, then methane would have more pores to adsorb in compared to ethane. Of particular interest is the fact that, when adsorption from a bulk gas-phase mixture is considered, methane would be able to adsorb in these “extra” pores without any competition from ethane. Since this would lead to higher extents of methane adsorption from mixtures, we have investigated the effect of including a measure of connectivity into the model. The connectivity of a pore network may be quantified in terms of the mean coordination number, Z, which is a measure of the average number of pores that intersect at each pore junction. The lower the value of Z, the fewer paths exist between any pore and the surface of the adsorbent, and the greater the probability that the presence of constrictions in the pore network will render that pore inaccessible to at least one of the adsorptive species. The pore network connectivity is closely related to percolation theory (see for example ref 52). For a given network, percolation theory provides a relationship between two variables for each component: Xs and As. The bond occupation probability, Xs, is the fraction of the total pore volume that species s could adsorb in if there were no bottlenecks (i.e. for an infinitely connected network or

Davies and Seaton

Figure 20. Illustration of the concept of two accessible volumes within an activated carbon.

one in which all the pores were accessible from the bulk gas phase). The accessibility, As, is the actual fraction of the total pore volume that species s can adsorb in. Note that the accessibility corresponds directly to the aspect of the pore structure that we wish to investigate. Although methane and ethane have similar “end-on” molecular radii, ethane will be unable to negotiate a sharp turn at a pore junction which might permit the passage of methane. It is therefore plausible that ethane could be excluded from part of the internal volume accessible to methane. Figure 20 illustrates the concept of two accessible volumes within an activated carbon: the one accessible to both ethane and methane and the other accessible only to methane. Note that the condition for equilibrium within each of these volumes is that the adsorbed species must exhibit the same chemical potential that they possess in the bulk gas phase. An equivalent condition is that they exhibit the same fugacity in each accessible volume as they do in the bulk gas phase. Note that different combinations of adsorbates adsorb within each of the accessible volumes. This has two important but related consequences. Firstly, we need to calculate suitable model pore isotherms for each of the accessible volumes. For example, for the system being considered we require a set of isotherms for the binary adsorption of methane and ethane for the fraction of the internal volume in which both of these species can adsorb. We also require pure component methane isotherms, even when considering the adsorption from mixtures, for the adsorption within the fraction of the volume that is only accessible to methane. Secondly, it is no longer convenient to express the adsorption as a function of the temperature, bulk gas-phase pressure, and composition. This is because the bulk gas-phase pressure and composition is only directly applicable to the adsorption that takes place in the fraction of the internal volume that is accessible to all the adsorbates. It is therefore more convenient to express adsorption as a function of the temperature and fugacities of the species involved in the adsorption. On the basis of these observations the accessibility can be incorporated into the pore size distribution analysis as follows:

∫0∞Fmp,s(w,T,fhs)i f(w) dw

Ns(T,fhs)i ) As (51) Davies, G. M.; Seaton, N. A. In Advances in Adsorption Science and Technology: The Proceedings of the Fourth China-Japan-USA Symposium on Advanced Adsorption Separation Science and Technology; Zhing, L., Zhenhua, Y., Eds.; South China University of Technology Press: Guangzhou, China, 1997. (52) Sahimi, M. Applications of Percolation Theory; Taylor and Francis: London, 1994.

(38)

for pure component adsorption where hfs is the fugacity of species s in the bulk gas phase and As is the accessibility of species s to the total internal pore volume. Similarly, for the binary adsorption


∫0∞Fmp,s(w,T,f)i f(w) dw + ∞ (As - As*)∫0 Fmp,s(w,T,fhs)i f(w) dw

Langmuir, Vol. 15, No. 19, 1999 6275

Ns(T,f)i As*

(39)

where f is a vector of the fugacities of the two components in the bulk gas-phase mixture, hfs is the fugacity of species s as before, and s* refers to the excluded species, in our case ethane. In our investigation we have assumed that As ) 1 for methane. This corresponds to recognizing that the analysis only registers pores that either of the adsorptives can enter. There is therefore only one additional parameter in the model: the accessibility of ethane. Note that the way that we have included the connectivity into the model corresponds to the physical situation that arises when the pores in the real structure are arranged randomly in a criss cross fashion. If this was not the case, then the accessibility would be a function of the pore size. We have calculated a measure of the fit to the data using an equation similar to eq 36 that is based on eqs 38 and 39 rather than eq 1. In all of these calculations the smoothing parameter was set to zero to ensure that we calculated the best achievable fit to the data. Figure 21 shows the overall error of the fit as a function of the excluded volume. This figure shows that the overall fit improves as the fraction of the volume that is excluded to ethane increases. The fits to the pure component data are very similar to those shown in Figure 8 over the range of excluded volumes considered and have therefore not been presented. As expected, the fits to the binary adsorption data improve significantly. The fits to the binary data are shown for an excluded volume of 10 and 40% in Figures 22 and 23. Although we recognize that it is unlikely for ethane to be excluded from such large fractions of the internal volume, these figures clearly demonstrate that the concept of an excluded volume significantly improves the model and therefore could be an important attribute of the internal structure. This implies that these models, which could be termed “connected pore network” models, could exhibit a particular potential in engineering calculations. Finally, eqs 38 and 39 only describe pure and binary adsorption within a connected pore network model. Since these models have been shown to improve the characterization of the internal structure of activated carbons, it is worth developing a generalized equation that is capable of describing both pure and multicomponent adsorption in a connected network of pores. This can be achieved by ranking the k components of a bulk gas phase that adsorb within the internal volume of an adsorbent in decreasing order with respect to their accessibility to that volume. Therefore, species 1 will correspond to the component that has the largest accessibility, As)1, and species k will correspond to the component that has the smallest accessibility, As)k. All the k components will be able to adsorb in a fraction of the total internal volume that corresponds to As)k. In this part of the internal volume an equilibrium is established between the k adsorbed components and the k components in the bulk gas phase. For convenience we label the vector of fugacities of the k components in the bulk gas phase fk. Next, we note that all the components except the kth component will be able to adsorb in a fraction of the total internal volume that corresponds to As)k-1 - As)k. In this part of the internal volume an equilibrium is established between the adsorbed phase and the first k - 1 components of the bulk gas phase. For this equilibrium, we label the vector of the fugacities

Figure 21. Error of the overall fit to the adsorption data onto the BPL activated carbon as a function of the fraction of the total internal volume excluded to ethane.

Figure 22. Best achievable fits to the binary adsorption of methane and ethane onto BPL activated carbon from a bulk gas mixture of 50:50 methane:ethane. The fits are for ethane being excluded from 10 and 40% of the total internal volume.

Figure 23. The best achievable fits to the binary adsorption of methane and ethane onto BPL activated carbon from a bulk gas mixture of 75:25 methane:ethane. The fits are for ethane being excluded from 10 and 40% of the total internal volume.

of the first k - 1 components fk-1. This labeling can be continued until the fraction of the internal volume in which only one component adsorbs, As)1, is reached. For convenience, we label the pure component fugacity of species 1, f1. Defining one additional accessibility for convenience,

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Davies and Seaton

As)k+1 ) 0, enables us to summarize the connected pore network model as k

Ns(T,f)i )

(Al - Al+1)∫0 Fmp,s(w,T,fl)i f(w) dw ∑ l)s ∞

(40)

i ) 1...n As expected, for pure and binary adsorption isotherms, eq 40 simplifies to eqs 38 and 39. 5. Conclusions A procedure has been outlined that can be used to develop suitable pore network models of the internal structure of activated carbons. This procedure, which assesses the realism of proposed pore network models before determining the extent and type of data that are required to determine their unknown parameters, has three significant attributes that make it particularly attractive: (1) It is relatively straightforward to determine the realism of a model. This means that proposed models can be evaluated quickly and effectively. (2) The procedure has eliminated the problems of the ill-posed character of the adsorption integral equation in assessing the realism of the model. This is significant because the ill-posed character is one of the most problematic mathematical aspects that needs to be considered when developing pore network models. (3) Results from the realism analysis can be used to identify types and regions of data that a pore network model consistently fails to accurately describe. This is valuable information that can be used to develop more refined models of the internal structure.

We have demonstrated that the BSP model leads to excellent fits to pure and binary adsorption data involving methane and ethane onto Nuxit, AC40, and AK activated carbons. These fits are sufficiently good to move onto determining the type and extent of data that are required to achieve a similar characterization based on much less data. Such analyses show that multicomponent adsorption can be predicted on the basis of PSDs calculated from a single adsorption isotherm.53 Less good agreement was achieved for the adsorption of these components onto BPL activated carbon. The test of the model realism was much more stringent in this carbon due to the extensive binary adsorption data included in the analysis. This indicates that multicomponent data, in addition to pure component data, is a much more demanding test of realism than when only pure component data are used. The results of the fit to the data of the BPL activated carbon have been used to refine the pore network model of the internal structure. Including a measure of the connectivity into the pore size distribution analysis leads to significantly better fits to the data. This is an indication that pore network models are likely to be sufficiently realistic to be useful in engineering calculations. Acknowledgment. G.M.D. gratefully acknowledges financial support from the Bradlow Foundation. The donors of the Petroleum Research Fund, administered by the American Chemical Society, are acknowledged for partial support of this work. The authors also wish to thank Matthias Heuchel for helpful comments during the preparation of this paper. LA990160S (53) Davies, G. M.; Seaton, N. A. To be submitted for publication in AIChE.

Development and Validation of Pore Structure Models for Adsorption

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