Adsorption of Carbon Dioxide and Methane and Their Mixtures on an

Oct 2, 1999 - 5.1. PSD from a Fit Including Pure-Component and Binary Experimental Data at T = 293 K. As mentioned before, the quality of a PSD determ...
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Adsorption of Carbon Dioxide and Methane and Their Mixtures on an Activated Carbon: Simulation and Experiment M. Heuchel,*.†,⊥ G. M. Davies,‡ E. Buss,§ and N. A. Seaton† School of Chemical Engineering, King’s Buildings, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JL, United Kingdom, Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, United Kingdom, and Institute of Technical Chemistry, University of Leipzig, D-04103 Leipzig, Germany Received April 12, 1999. In Final Form: August 10, 1999 The aim of this work is to predict the adsorption of pure-component and binary mixtures of methane and carbon dioxide in a specific activated carbon, A35/4, using grand canonical Monte Carlo (GCMC) simulation. Methane is modeled as a one-center Lennard-Jones (LJ) fluid and carbon dioxide as a twocenter LJ plus point quadrupole fluid. Experimental adsorption data for the system have been obtained with a new flow desorption apparatus. The pore size distribution (PSD) for the carbon was determined from both of the experimental CH4 and CO2 isotherms at 293 K. To extract numerically the PSD, GCMCsimulated isotherms for both pure components in slit-shaped pores ranging from 5.7 to 72.2 Å were used. Using only pure experimental CO2 isotherm data, it was not possible to determine a PSD that allowed a reasonable prediction of the pure methane adsorption. However, with both experimental data sets for the pure components, it was possible to derive a PSD that allowed both experimental pure-component isotherms to be fitted. With this PSD and the simulated adsorption densities in single pores, it was possible to predict in good agreement with experiment (i) the adsorption of binary mixtures of CO2 and CH4 and (ii) the adsorption of both pure components at higher temperatures. However, the model was unable to reproduce precisely the experimental pressure dependence of the CO2 selectivity.

1. Introduction A description of the adsorption of carbon dioxide, methane, and their binary mixtures on activated carbons focuses on some interesting problems of current adsorption research. There is first the aspect of technological applications (ref 1 contains a nice introduction). Both carbon dioxide and methane have been implicated as greenhouse gases. One technology for the control of the CO2 emission involves the removal of CO2 from atmospheric emissions by adsorption from gas mixtures. Further, mixtures of CO2 with hydrocarbons occur in natural gas, and they are also important in enhanced oil recovery. Adsorption techniques (membrane and pressure swing methods) together with porous adsorbents, especially carbons, seem promising candidates for these separations. However, it is still very difficult to evaluate activated carbons for a separation because it is difficult to predict their adsorption behavior. What is missing is a suitable model for the internal structure of carbon. If we could model the internal structure of a specific activated carbon, then we could use the model to predict adsorption using molecular simulation. At present, the most commonly used model for the internal structure of a carbon is the pore size distribution (PSD). With a PSD, we assume that the dominant aspect of the internal structure is the proximity of the pore walls. Derived from the highly lamellar structure of graphitic carbon, most often slit-shaped pores are assumed to be a * Author to whom correspondence should be addressed. Telephone: (+49) 341 9736423. Fax: (+49) 341 9736399. E-mail: [email protected]. † University of Edinburgh. ‡ University of Cambridge. § University of Leipzig. ⊥ On leave from Wilhelm-Ostwald-Institute of Physical and Theoretical Chemistry, University of Leipzig. (1) Nicholson, D.; Gubbins, K. E. J. Chem. Phys. 1996, 104, 8126.

reasonable structure model for a single pore. Additionally, recent work2 seems to show that the determination of the pore size distribution, rather than the shape of the model pores, is the crucial factor. For the determination of a PSD, we require (i) some experimental data, (ii) the local isotherms in the model pores that will form the PSD, and (iii) a rigorous solution technique to calculate the PSD. After we have calculated a PSD based on some experimental data, and if the proximity of the pore walls is really the only dominant factor that affects adsorption in the carbon, then we should be able to use the PSD for one species to predict the adsorption at different temperatures, or of another species, or of mixtures. In this paper, we will investigate, by means of a case study, the ability of PSDs to predict pure and binary adsorption of methane and carbon dioxide onto a commercially available activated carbon. The remainder of this paper is divided into the following sections. First, we report the experimental investigation of the system. The next section deals with modeling the adsorption of both components in slit pores using molecular simulation. Then we address the problems of determining a PSD for the carbon. The last section presents fits and predictions on the basis of these PSDs in comparison with experimental results and also discusses the limits of application of this approach. 2. Experimental Section The active carbon investigated in this work was the industrially produced material A35/4, supplied by Carbo Tech, Essen, Germany. The carbon was characterized by a BET surface, (2) Davies, G. M.; Seaton, N. A. Carbon 1998, 36, 1473.

10.1021/la9904298 CCC: $18.00 © 1999 American Chemical Society Published on Web 10/02/1999

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Figure 1. Schematic diagram of the flow desorption apparatus.

Figure 2. Pure gas adsorption isotherms on activated carbon A35/4 at T ) 293 K from gravimetric (closed symbols) and flow desorption (open symbols) measurements. The lines represent the fit with the modified Toth isotherm equation of Jensen and Seaton.6 determined from N2 adsorption at 77 K, of 1393 m2/g. Pure gas adsorption isotherms have already been measured gravimetrically with a Sartorius high-pressure microbalance, Model S3DP.3 In this former investigation, a method proposed by Van Ness4 was used to derive both single-component loadings from a data set that contained the total adsorbed masses of the mixture as a function of pressure and gas-phase composition. The method is described in detail elsewhere.5 In the present work, the pure gas adsorption isotherms and the binary mixture equilibria have been redetermined using a new “flow desorption” apparatus. A schematic diagram of the flow desorption apparatus is shown in Figure 1. In a flow desorption experiment, a gas mixture of known composition flows through the regenerated adsorbent bed, i.e., the sample vessel (1), at the desired constant pressure and temperature. After adsorption equilibrium is attained, the adsorber (1) is isolated and connected with the previously evacuated desorption tank (2). The adsorbed molecules are desorbed by heating the adsorber and simultaneously cooling the desorption tank with liquid nitrogen, and the gas phase in the system is almost completely condensed within the desorption tank. Then the desorption tank is isolated and allowed to warm to room temperature overnight. The composition of the gas mixture within the desorption tank is then analyzed by gas chromatography. Finally, the amount of adsorbate and the composition of the adsorbate mixture can be calculated by a mass balance using pressure and temperature measurements and taking into account the gas volumes in different parts of the apparatus. The resulting experimental adsorption values are excess values. Pure gas adsorption isotherms for methane and carbon dioxide on active carbon A35/4 at 293 K are presented in Figure 2. The data measured with the newer flow desorption method agree (3) Buss, E. Gas Sep. Purif. 1995, 9, 189. (4) Van Ness, H. C. Ind. Eng Chem. Fundam. 1969, 8, 464. (5) Buss, E.; Heuchel, M. J. Chem. Soc., Faraday Trans. 1997, 93, 1621.

Heuchel et al.

Figure 3. Adsorption of CO2 and CH4 and their binary mixtures on activated carbon A35/4 at different gas-phase compositions at 293 K and constant pressure P ) 5.3 bar. Closed symbols represent the results from flow desorption experiments; open symbols are based on gravimetrical measurements. Full lines are the predictions from experimental data for the pure components with IAS theory. satisfactorily with the previously measured ones using the traditional gravimetric method.3 From Figure 2, we see that the adsorption of CO2 by the carbon A35/4 is always higher than that of CH4. The experimental data were fitted with the modified Toth isotherm of Jensen and Seaton.6 (The equation and parameters are given in the Appendix.) The binary adsorption equilibrium of CO2 and CH4 on A35/4 has been measured at three equilibrium CO2 gas-phase mole fractions (0.2073, 0.5137, 0.9168) at six total pressures (1.0, 2.5, 5.3, 8.0, 10.0, 15.0 bar) with the flow desorption method. As an example, the loading diagram obtained at P ) 5.3 bar is shown in Figure 3. Again, the flow data are in good agreement with the values determined earlier using the gravimetric method.5 The figure shows further that at this pressure the mixture equilibrium is predicted reasonable well by the ideal adsorbed solution (IAS) theory.7 Similar behavior was found over wide ranges of experimental conditions except for CO2-rich gas mixtures at higher pressures.

3. Molecular Simulation of the Adsorption of Pure Components and Mixtures in Model Pores The most widely used molecular simulation method applied to adsorption problems is the grand canonical Monte Carlo (GCMC) simulation because it allows a direct calculation of the phase equilibrium between a gas phase and an adsorbate phase. The implementation of this simulation method is both well established and well documented (see, for example, refs 8 and 9). The properties of an adsorbate in a model pore of a given volume are calculated by sampling molecular configurations that are consistent with the temperature and the chemical potentials of the components present. Assuming phase equilibrium, the chemical potentials of the adsorbed species are calculated from the specified temperature, pressure, and composition (y) of the bulk gas phase using an appropriate equation of state. In this work, methane has been modeled as a one-center Lennard-Jones (LJ) interaction site. The potential parameters have been used before for the simulation of adsorption.2,10 The linear carbon dioxide molecule was described by a recently developed effective pair potential as a two-center LJ plus point quadrupole (2CLJQ).11 This model represents an alternative to an explicit representa(6) Jensen, C. R. C.; Seaton, N. A. Langmuir 1996, 12, 2866. (7) Myers, A. L.; Prausnitz, J. M. AIChE J. 1965, 11, 121. (8) Allan, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 1987. (9) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (10) Cracknell, R. F.; Nicholson, D.; Tennison, S. R.; Bromhead, J. Adsorption 1996, 2, 193. (11) Mo¨ller, D.; Fischer, J. Fluid Phase Equilib. 1994, 100, 35.

Adsorption of CO2 and CH4 on Activated C

Langmuir, Vol. 15, No. 25, 1999 8697 Table 1. Potential Parameters used in the Simulation molecule

/k, K

σ, Å

l, Å

Q, 10-40 C m2

CO2, 2CLJQ11 CH4, LJ12 carbon LJ,12 a∆ ) 3.35 Å; FC ) 0.114 Å-3.

125.317 148.2 28.0

3.0354 3.812 3.40

2.1216

-12.2

Table 2. Summary of the Constant Used in the Peng-Robinson Equation of Statea

Figure 4. Schematic representation of (i) the relative size of the molecular models, (ii) the single interactions between two 2CLJQ molecules for CO2, (iii) a slit pore of width W with a configuration of carbon dioxide molecules, and (iv) the geometric quantities occurring in the quadrupole-quadrupole potential for CO2.

molecule

TC, K

PC, bar

ω

CO2 CH4 CO2-CH4

304.2 190.6

73.83 45.99

0.224 0.012

a

kij21

0.09

Constants are from Smith et

al.20

tion of the charge distributions in CO2.10 The general form of the potential is

Equation 3 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 by using

u2CLJQ ) u2CLJ + uQ

uslit(z) ) usf(z) + usf(W - z)

(1)

where u2CLJ is the two-center LJ potential characterized by the parameters , σ, and l, where l is the distance between the two LJ centers on one molecule (see Figure 4). On the centers of these symmetric linear molecules, i.e., on the midpoints between the two LJ centers, point quadrupoles are embedded which interact with the potential

uQ )

3 Q2 [1 - 5(ci2 + cj2) - 15ci2cj2(c - 5cicj)2] 4 r5

(2)

where Q is the quadrupole moment and r is the distance between the centers of the molecules. Θi and Θj are the polar angles of the molecular axis with respect to the line joining the molecular centers of molecules i and j, and φij is the difference in the azimuthal angles (see Figure 4), ci ) cos Θi, cj ) cos Θj, and c ) cos Θi cos Θj + sin Θi sin Θj cos φij. Hence, the 2CLJQ potential model is characterized by the four parameters , σ, l, and Q. The microporous carbon A35/4 is assumed to be characterized by a (at the moment unknown) PSD of slitshaped model pores. We have adopted the slit-shaped model pore since it not only represents a physically plausible pore shape but it is also the simplest model pore that can fit the experimental data for adsorption on carbons.2 The interaction between an adsorbate site and a single semiinfinite pore wall of graphite is given by Steele’s 10-4-3 potential:12

usf(z) ) 2πsfFsfσsf2∆

[( ) ( ) 2 σsf 5 z

10

-

σsf z

4

σsf4

]

3∆(z + 0.61∆)3

(3)

where Fs is the number of carbon atoms per unit area 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 a carbon atom in the adsorbent surface. The parameters sf and σsf, and the other cross LJ parameters, are determined using the standard Lorentz-Berthelot combining rules. The parameters used in the simulation are summarized in Table 1. (12) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974.

(4)

where z is the distance from one of the walls and W is the pore width, in particular, the distance between the nuclei of the first layer of carbon atoms on opposing pore walls (see Figure 4). The chemical potential was calculated from the temperature, pressure, and composition of the gas phase using the Peng-Robinson equation of state, with the parameters given in Table 2. Technically, the simulations were performed in the following way. A standard rectangular simulation cell has been adopted to represent the slit-shaped model pores with the usual periodic boundary conditions in the x and y directions. The x and y dimensions of the simulation cell 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 1.5 to 20 times the molecular diameter of methane in successive simulations. The initial configuration for the simulation of each isotherm was generated by randomly placing five molecules in the simulation cell. 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 GCMC simulation involved attempts to move, create, or destroy a molecule or 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 before collecting data. After equilibration, the simulation continued for between 2 × 106 and 3 × 106 steps in order to calculate the average values of the extent of adsorption. Complete isotherms for the model pores were determined using successive simulations in which either the pressure or the bulk-phase composition was varied. The simulations were performed on a Sun Ultra 10 workstation. The total time to calculate an isotherm depended on the actual values for pressure and pore width. As an example, it took about 2.5 h to determine an isotherm with 15 data points between 0.1 and 20 bar for an equimolar mixture in a pore with a width of 30 Å. The actual quantity calculated in our GCMC simulations is the absolute adsorption density, i.e., the average number of molecules of each species within a small section of the model pore. For comparison with experimental data, which represent excess values, the simulated values had to be

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corrected. The simulated extents of adsorption for model pores can be converted to excess isotherms by using

Niex(W,T,P,y) ) Niabs(W,T,P,y) yiFibulk(T,P)Vbulk(W) (5) where Niex(W,T,P,y) is the excess number of molecules of species i in a model pore of width W at temperature T, pressure P, and bulk-gas-phase composition y, Niabs(W,T,P,y) is the absolute (i.e., simulated) number of (T,P) is the bulk fluid molecules at these conditions, Fbulk i density of species i at the temperature and pressure concerned, and Vbulk is the volume in the model pore of width W accessible to the bulk fluid. The subtracted value in eq 5 represents the number of molecules of component i in a reference bulk volume. This volume is related to the total volume of the simulation cell, VSC(W) ) AW, where A is the area of a surface pore wall in the simulation cell. In calculating Vbulk, one has to consider that there exists for every component a smallest pore in which adsorption can take place. Using this assumption, the accessible volume for the bulk fluid in a slit-shaped pore can be defined as2.

Vbulk(W) ) 0

W < Wspi

Vbulk(W) ) (W - Wspi)A

W > Wspi

(6)

where Wspi is the smallest pore in which adsorption of component i takes place. For methane this value is 6.1 Å and for CO2 5.7 Å. The main argument for eq 6 is that no methane can get into a pore that is smaller than 6.1 Å. Therefore, there would be no “gas-phase” contribution to the excess values defined in eq 5 in pores smaller than 6.1 Å. In a slightly larger pore, the volume available would not immediately jump to the total volume of the simulation cell, VSC(W) ) AW; rather, the only volume that a gas would see would be the one given in eq 6. For binary mixtures, the smaller value of Wspi for CO2 is used. Once the simulated adsorption has been converted into an excess quantity with eq 5, it can be expressed as an adsorbate density by identifying the volume in which adsorption takes place. For consistency with our definition of pore width (i.e., the distance between carbon nuclei on opposing pore walls), we must define this volume to be the volume VSC(W) ) AW of the simulation cell itself. It was previously remarked by Davies and Seaton2 that this definition of the adsorption volume differs from the “effective volume” used by some other workers. The excess adsorbate density for model pores is thus calculated by using

Fi(W,T,P,y) )

Figure 5. Calculated excess adsorption isotherms of carbon dioxide in slit-shaped model pores at T ) 293 K.

Niex(W,T,P,y) VSC

(7)

where VSC is the above-introduced volume of the simulation cell. The bulk fluid density, Fbulk in eq 5 and the chemical i potentials of the components in the bulk phase were calculated with the help of the Peng-Robinson equation of state with the parameters summarized in Table 2. Figures 5 and 6 show the excess adsorption of pure carbon dioxide and methane at 293 K for pore sizes ranging from 5.72 up to 47.6 Å. The possibility of a maximum in the isotherm is characteristic of excess adsorption; this occurs where, at high pressure, an increase in the pressure causes a bigger increase in bulk density than in the density of the adsorbate in the pores (for more information about

Figure 6. Calculated excess adsorption isotherms of methane in slit-shaped model pores at T ) 293 K.

the specific problems of high-pressure adsorption, see e.g., ref 13). A comparison of both figures further shows that the smallest pores (widths up to 6.1 Å) are accessible only to carbon dioxide. Although the length of the CO2 molecule is greater than the diameter of the methane molecule, its diameter is only about 3.0 Å, where the methane molecule has a diameter of 3.8 Å. Therefore, pores with a width between 5.7 and 6.1 Å would be responsible for a molecular sieving effect in mixture adsorption. If we look at Figures 5 and 6, we see a complex variation between the isotherms for different pore sizes, including the fact that some cross over others. This typical result for adsorption in micropores is caused by a superposition of two opposite effects: weaker adsorption with increasing pore size because the strength of the adsorbate-adsorbent interaction decreases as the pore size increases, and higher extent of adsorption because of the ability of larger micropores to accommodate more adsorbate molecules. As a result of “confinement”, the first significant amount of adsorption can be found in micropores that are large enough to accommodate a single layer of adsorbate molecules. Even in this case, 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. The interplay of pore size and pressure on adsorbate density can be still better seen in 3D plots. Figure 7 shows at the top the plots for pure methane (left) and pure carbon dioxide (right). We see clearly the quicker filling of the smallest pores with pressure because of the stronger adsorbate-wall potential. Further, if we look at fixed pressure on the dependence of adsorbate density on pore size, we see two characteristic peaks at pressures over 5 bar. The oscillation of the adsorbate density with increasing pore width is related to the fact that molecules in slit (13) Salem, M. M. K; Braeuer, P.; v. Szombathely, M.; Heuchel, M.; Harting, P; Quitzsch, K. Langmuir 1998, 14, 3376.

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Figure 7. Surface plots of the calculated excess adsorption of methane (left) and carbon dioxide (right) in slit-shaped model pores at T ) 293 K as function of pore width and pressure. Pure-component adsorption (a, b). Single components in binary adsorption at three different gas-phase compositions (c, d, yCO2 ) 0.21; e, f, yCO2 ) 0.51; g, h, yCO2 ) 0.92).

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pores achieve the highest packing density in a whole number of layers. For the linear CO2, the layer structure is more disturbed and the second density maximum is broader. This tendency, to build adsorbate layers in the confined space of slit pores, is disturbed mutually for both components in an adsorbate mixture. The adsorbate density for CO2 in Figure 7d, corresponding to a gas-phase mixture composition of yCO2 ) 0.21 shows, for example in pores up to 7.0 Å, nearly the same density as for pure CO2 (cf. Figure 7b) because this is the pore size range of molecular sieving, where only CO2 can be adsorbed. For larger pore sizes, competition with methane takes place during mixture adsorption and the CO2 adsorbate density is drastically reduced in comparison to adsorption of the pure component. Also, the methane profile for a mixture adsorption at yCO2 ) 0.21, shown in Figure 7c, is reduced, while the layering effect is still pronounced and both density maxima at widths of 7-8 and 10-12 Å at high pressure still have similar values. For adsorption from gas mixtures with higher carbon dioxide concentration (yCO2 ) 0.51, panels e and f; yCO2 ) 0.92, panels g and h), we see the influence of the preferential adsorption of CO2 on the methane adsorbate density (shown in panels e and g). There is a significantly reduced density in the small pores of the first maximum (7-8 Å) in comparison to the value in pores with a width of 10-13 Å, the region of the second density maximum. 4. Determination of Pore Size Distributions After the preceding section has shown how the adsorption in single slit pores can be modeled, we now want to use these results to find a distribution of slit pores as a model for the internal structure of the activated carbon A35/4. The equivalence between the real carbon and our model structure is based on their adsorption behavior. The PSD is therefore calculated from the adsorption integral equation

Ns(T,P,y) )

∫0∞FMC,s(W,T,P,y) f(W) dW

(8)

where Ns(T,P,y) is the adsorption of species s (in particles per gram of adsorbent) at temperature T, pressure P, and bulk-gas-phase composition y, FMC,s(W,T,P,y) is the MCsimulated adsorption density (in molecules per nm3) in a model pore of width W, and f(W) is the PSD (in, e.g., cm3/ g/Å). The PSD defined with eq 8 represents the change of volume with W

dV f(W) ) dW

(9)

and in integrated form, we can express the specific volume of the activated carbon as

V)

∫0∞ f(W) dW

(10)

We note that eq 8 has been written in such a way that either the PSD can be determined from a single adsorption isotherm (as is the usual practice) or it can be determined from two or more pure-component adsorption isotherms or it can be determined also from mixture data. Previous research has shown that very often predictions of adsorption equilibrium can be made over a range of conditions (i.e., at different temperatures and pressures, and for different adsorbates) using only a single adsorption

isotherm14-16 for the PSD determination. This is, of course, very attractive because it allows predictions to be made on the basis of a small number of experimental inputs. However, in some cases, predictions based on a single pure-component isotherm can be significantly in error.16 If a PSD can be fitted to some pure and binary data simultaneously, then it is more likely to be an appropriate model also for other components or mixtures. Therefore, one of the first things we will consider in the next section is whether a PSD is a suitable model. To achieve this, we will determine if it can be fitted to pure and binary data at 293 K. A further argument for using more than one isotherm concerns the numerical solution of eq 8. We solve the adsorption integral equation by discretizing it. Implicitly, this means that the more data we have, the smaller the quadrature we can use. This improves the “resolution” of the PSD. Mathematically, eq 8 is a Fredholm integral equation of the first kind. Its solution, i.e., the determination of the PSD f(W), requires specific numerical techniques.17,18 Our numerical approach has been discussed in detail in two recent works.16,19 Here, we will only briefly repeat the main steps of the procedure. The solution of eq 8 consists of finding a function f(W) from a data set of adsorption values Ns(Pi), i ) 1, ..., n, of isotherm points. For the numerical solution, the PSD is represented discretely at m pore sizes Wj* which have been defined to be the average pore size in each quadrature interval δWj with

Wj* ) Wj +

δWj 2

(11)

where Wj, j ) 1, ..., m + 1, are the quadrature points. The integral eq 8 can then be written as m

N(Pi) ≈

FMC,s(Wj*, Pi)δWj f(Wj*) ∑ j)1

(12)

which can be written more compactly in matrix form

N ) AWf

(13)

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

(14)

A ) [FMC,s(Wj*,T,P,y)i]i)1,...,n; j)1,...,m

(15)

W ) diag[δWj]j)1,...,m

(16)

f ) [f(Wj*)]j)1,...,m

(17)

where

Having suitably discretized the adsorption integral eq 8, the next important consideration is to determine a suitable solution procedure. Davies et al.19 have shown that the (14) Gusev, V. Y.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815. (15) Gusev, V. Y.; O’Brien, J. A. Langmuir 1997, 13, 2822. (16) Davies, G. M.; Seaton, N. A. Development and validation of pore structure models for adsorption in activated carbons. Langmuir, in press. (17) v. Szombathely, M.; Bra¨uer, P.; Jaroniec, M. J. Comput. Chem. 1992, 13, 17. (18) Wilson, J. D. J. Mater. Sci. 1992, 27, 3911. (19) Davies, G. M.; Seaton, N. A.; Vassiliadis, V. S. The calculation of pore size distributions of activated carbons from adsorption isotherms. Langmuir, submitted. (20) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics; McGraw-Hill: New York, 1996. (21) Sandler, S. I. Chemical and Engineering Thermodynamics; Wiley: New York, 1989.

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detailed solution procedure depends on whether we are interested in establishing the existence of a solution, i.e., a physically meaningful pore size distribution with strictly nonnegative values, or calculating a representative pore size distribution. 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 zero (in practice, we use a tolerance of 1 × 10-6): m

R)

m

[Ns(Pi) - ∑FMC,s(Wj*,Pi)δWj f(Wj*)]2 ∑ i)1 j)1

(18)

which can be expressed with eqs 14-17 into the compact matrix form16

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

(19)

Using the experimental data points at 293 K for both pure isotherms alone and in combination, it was possible to reduce the residual R, defined in eq 18 or 19, to zero when the number of quadrature intervals, m, is set equal to the number of data points, n. After this confirmation of the existence of a solution, we can now calculate a representative pore size distribution. 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 needs to be given to the solution procedure. This 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. We use here the so-called regularization (see refs 17 and 18), a method in which the result is stabilized by incorporation of additional constraints. Here they are based on the smoothness of the pore size distribution; i.e., we add an additional term to the residual R defined in eq 19: T

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

(20)

where R is a strictly nonnegative smoothing parameter, called the regularization parameter, and S is a suitable discrete representation of a function that measures the smoothness of the pore size distribution. The criterion is based on the assumption that a real carbon 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. As a numerical measure of the smoothness, we have adopted the integral of the square of the second derivative of the pore size distribution: m

S)

[f′′(Wj*)]2δWj ∑ j)1

(21)

The solution of eqs 20 and 21 produces a biased estimate of the solution. Stated in another way, regularization forces a slightly worse fit to the data in order to determine smoother pore size distributions. Also notice that eq 20 reduces to the standard residual defined in eqs 18 and 19 if the smoothing parameter R is set to zero.

Before we begin to estimate PSDs for our carbon from experimental data, we want to mention the three criteria we have applied to determine a reliable distribution function. To determine the optimal degree of smoothing, i.e., the optimal regularization parameter R, we have adopted two methods. The first is the so-called L curve, i.e., a plot of the error of the fit against the regularization parameter R. It has been found that the error remains constant as the smoothing parameter is increased below some threshold value, and above it, the error increases rapidly. The L curve is used to identify the R value belonging to this threshold, which is taken to be the optimal extent of smoothing. A suitable measure for the error of the fit is E ) R/n, where R is the residual given in eq 18 and n is the number of data points. As a second method, we have used the generalized crossvalidation (GCV) function, proposed by Wilson.18 This method also allows us to derive an optimal value for R. The GCV method is based on the observation that a good choice of the smoothing parameter should enable us to predict any one of the n experimental data points from a PSD that is determined using the remaining n - 1 data points. The optimal parameter is the one that determines (with the minimum of least-squares deviation) every single data point based on a PSD which was determined with the remaining n - 1 data points. Further information can be found in refs 16 and 19. In fitting the PSDs, we have applied the concept of the window of reliability, introduced by Gusev et al.,14 which helps to identify the pore width above which the simulated model isotherms become too similar to be able to place a high confidence in the exact location of the peaks in the PSD. This upper pore size value is a function of the adsorbate used, the temperature, and the maximum pressure at which adsorption was measured. In our experiments, the maximum pressure was 17 bar and the temperature 293 K. The window of reliability for CH4 or CO2 adsorption under these conditions extends up to approximately 25 Å so that the PSD is unreliable above this pore size. It can be seen from Figure 7a that, e.g., the methane isotherms over 25 Å become almost indistinguishable. 5. Determination of PSDs from Experimental Data Points and Prediction of Adsorption Behavior 5.1. PSD from a Fit Including Pure-Component and Binary Experimental Data at T ) 293 K. As mentioned before, the quality of a PSD determined for a particular sample depends sensitively on the amount of experimental input. To test the model, we have firstly determined a PSD which fits best all pure and binary data points of the system at T ) 293 K. From all 88 data points, the PSD shown in Figure 8 has been derived. It contains 88 pore sizes in a region of interest between 5.5 and 25.0 Å. The preceding considerations have already shown that no information about larger pores can be detected from the data with statistical significance. The optimal regularization parameter R ) 0.001 was found with the GCV function and the L curve, both shown in Figure 9. Both error estimates show the characteristic cusp at the optimal regularization parameter. The fitted PSD in Figure 8 shows a very pronounced peak between 5 and 6 Å. This is a strong hint for pore space in the active carbon A35/4 which shows molecular sieving properties for the CO2-CH4 mixtures, i.e., that the CO2 molecule (with the smaller cross section) can enter pores that are

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Figure 8. Pore size distribution for A35/4 activated carbon based on a fit to pure and binary CO2 and methane data at 293 K (88 data points).

Figure 9. Error estimates versus regularization parameter R for the PSD fit of Figure 8. The general cross validation (GCV) and the L curve show a characteristic cusp at the optimal value R ) 0.001.

Heuchel et al.

Figure 12. Best fit curves for selectivity of CO2 for several gas-phase mole fractions of CO2 at different pressures.

Figure 13. Pore size distribution for A35/4 activated carbon based on a fit to only pure CO2 data at 293 K (24 data points).

at least for active carbonssuntypical pressure dependence of the selectivity for carbon dioxide

SCO2 )

Figure 10. Best fit curves for pure-component adsorption of CO2 and CH4 on A35/4 activated carbon based on the pore size distribution shown in Figure 8.

Figure 11. Best fit curves for binary adsorption of CO2 and CH4 on A35/4 activated carbon based on the pore size distribution shown in Figure 8.

too small to allow CH4 to enter. Figures 10 and 11 show the achieved fit of pure and binary experimental data at 293 K. The representation of the pure data is excellent. For the mixture adsorption also, the fit is in general satisfying. The largest deviation appears for the CO2 adsorption from a gas mixture with low CO2 content. The good correlation of simulated and experimental results allows one to draw the conclusion that our model describes the basic physics of the real system. Nevertheless, there are limits to the model. Let us look at the most sensitive parameter with respect to mixture separation, the selectivity. Our accurate flow experiments unveiled a peculiarity of the system CO2-CH4 on A35/4. We found in the experiment (see points in Figure 12) ans

xCO2yCH4 xCH4yCO2

(22)

where x and y represent the mole fractions of the components in the adsorbed phase and in the bulk phase, respectively. The experimental data points in Figure 12 show for the lowest concentration of CO2 in the gas phase (yCO2 ) 0.2073) that the selectivity decreases slightly from 3.1 to 2.8 for a pressure increase from 1 to 15 bar. For a nearly equimolar gas mixture (yCO2 ) 0.5137), it increased slightly from 3.4 to 3.7, and for a gas mixture rich in CO2 (yCO2 ) 0.9168), it increased substantially from 3.6 to 8.9. The calculated separation factors, SCO2, shown as curves in Figure 12, agree well with the experimental values at low pressure. But the simulation predicts, at constant T and yCO2, a slight decrease of selectivity with pressure. In contrast, the experiments show especially for a high mole fraction of CO2 in the gas phase an increase with pressure. The discrepancy arises because our model is not capable of describing for the adsorbate mixture the slight decrease in partial CH4 loading with pressure found experimentally for yCO2 ) 0.9168. 5.2. Predictions with a PSD Fitted Only from Pure CO2 Data at T ) 293 K. In the following, we want to see to which extent we can predict adsorption with the model, using only a limited set of data as input. First we used only the pure CO2 data (24 data points) as input for the determination of the PSD. Similar GCV scores and L function plots, as shown in Figure 9, lead us to the PSD shown in Figure 13. It has peaks at similar pore sizes as the PSD in Figure 8, but the frequency of the pores between 5 and 6 Å is now much smaller. We predicted with this PSD pure methane adsorption on A35/4 (Figure 14) and also the binary adsorption at 293 K, shown in Figure 15. The result of this attempt is poor. We see we are not able to make predictions for the active carbon based on a single isotherm. The pure methane adsorption in Figure 14 is overpredicted. One reason for this unsatisfactory result could be the limited number of quadrature points for the

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Figure 14. Predicted pure methane and fitted pure CO2 adsorption on A35/4 activated carbon based on the pure size distribution shown in Figure 13.

Figure 17. Best fit curves for pure-component adsorption of CO2 and CH4 on A35/4 activated carbon based on the pore size distribution shown in Figure 16.

Figure 15. Predicted binary adsorption of CO2 and CH4 on A35/4 activated carbon based on the pore size distribution shown in Figure 13.

Figure 18. Predicted binary adsorption of CO2 and CH4 on A35/4 activated carbon based on the pore size distribution shown in Figure 16.

Figure 16. Pore size distribution for A35/4 activated carbon based on a fit to pure CO2 and CH4 data at 293 K (52 data points).

pore sizes. Therefore, we have repeated the PSD determination with a representative distribution for 100 pore sizes. Although this new PSD (not shown here) looked more similar to the PSD in Figure 8, which was determined from 88 pure and mixture data points, it does, however, have many more large pores (which play only a small role in the adsorption of CO2 at those conditions). Therefore, the prediction of the pure methane adsorption was not improved (again severely overpredicted). The binary predictions were slightly improved. In summary, from our experimental CO2 data alone, it was not possible to derive a satisfactory model for the active carbon A35/4; we return to this observation later. 5.3. Predictions with a PSD Fitted from Pure CO2 and CH4 Data at T ) 293 K. As a next step, we tried to determine whether a more powerful PSD could be recovered using both pure-component data sets for methane and CO2 together. We used as many quadrature points as there are pure-component data points (52). The estimated PSD is shown in Figure 16. The result is similar to the distribution we got from the fit of all the pure and binary experimental data at T ) 293 K (see Figure 8). The determined PSD in Figure 16 shows again the strong peak in the molecular sieving range at 5-6 Å. The distribution function allows us to fit very well the experimental data points for both pure isotherms (see Figure 17). Assuming the PSD in Figure 16 and by using our simulation results for mixture adsorption in single-slit pores, we then

predicted the binary adsorption at 293 K. The result is shown in Figure 18. The agreement between model and experiment is much improved in comparison with the former prediction in section 5.1 using only the CO2-based PSD. The results are not at all bad. The improvement by using both pure isotherms as input seems to be a particular attribute of the ill-posed character of the adsorption integral eq 8. Adsorption in very small pores, where it is difficult for molecules to enter the pores, is essentially indistinguishable from adsorption in very large pores, where there are only weak adsorbate-wall interactions. This is a particularly nasty form of illposedness because it means that pores of very different sizes can be mixed upsi.e., very small pores can be identified as very large pores and vice versa. If we try to then predict the adsorption of another species, this can cause things to go wrong (as is shown, e.g., in Figure 14). In fact, this just described behavior is also the reason why, when we include methane into the calculation, we get information about the very small pores of CO2seven the ones too small for methane to enter. The methane adsorption data as additional information prevent the large pores from getting mixed up with the smaller pores for its own adsorption of CO2. This “tells” the fitting routine that there are not that many large pores. Since large pores give similar adsorption to very small pores, the fitting routine swaps some of the large pores for small poress hence, the huge increase in the number of small pores in the PSD of Figure 16 compared to the PSD of Figure 13. This “linear dependence” is discussed in more detail in ref 19. Finally, we should mention a further effect of the additional experimental methane data. The larger number of data points stabilizes generally the numerical solution of the adsorption integral equation. 5.4. Predictions of Pure Component Adsorption at Higher Temperatures. The reliability (and also usefulness) of our model rests on its power to predict adsorption at different conditions. Figures 19-21 show predictions (and fits) of pure-component adsorption at 293, 313, and 333 K. The experimental data points shown represent the gravimetric measurements.3 If some of the

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Heuchel et al. Table 3. Fitted Parameters for Pure-Component Adsorption on Active Carbon A35/4 isotherm CH4, 293 K CO2, 293 K

Figure 19. Predicted pure adsorption of CO2 (at 313 and 333 K) and methane (at 293, 313, and 333 K) on A35/4 activated carbon based on the pore size distribution shown in Figure 13. The fit of pure CO2 adsorption (dotted line) at 293 K was used for the PSD determination. The experimental data are from gravimetrical measurements.3

Figure 20. Predicted pure adsorption of CO2 and methane at 313 and 333 K on A35/4 activated carbon based on the pore size distribution shown in Figure 16. The fit of pure CO2 and methane adsorption at 293 K (dotted lines) was used for the PSD determination. The experimental data are from gravimetrical measurements.3

Figure 21. Predicted pure adsorption of CO2 and methane at 313 and 333 K on A35/4 activated carbon based on the pore size distribution shown in Figure 8. The fit of pure CO2 and methane adsorption at 293 K (dotted lines) was used (with additional binary measurements) for the PSD determination. The experimental data are from gravimetrical measurements (see ref 3).

data were used in the determination of the PSDs, then it is a fit to the data; otherwise, it is a prediction. Figure 19 shows the predictions for pure-component adsorption on A35/4 with the PSD based on the fit to only pure CO2 data at 293 K (see Figure 13). In Figure 19, we see that, compared with a very accurate fit of the CO2 data points at 293 K, all other adsorption isotherms for CO2 and CH4 are strongly overpredicted. A significant improvement results if the predictions are undertaken with the PSD, shown in Figure 16, and which was derived from a fit which included additionally the pure methane data at 293 K. On the basis of this PSD, we can predict quite nicely the adsorption of pure CO2 and CH4 at 313 and 333 K. (Figure 20 contains also the fits for both pure components at 293 K.) We see further that the CO2 adsorption at 313 and 333 K is still slightly overpredicted. However, a further improvement is possible, especially for the CO2 adsorption at 313 and 333 K, when assuming

a, K, mmol/g mmol/(g bar) 14.91 24.12

1.361 4.649

c 0.5306 0.4641

κ, % error 10-3/bar in fit 4.383 26.97

0.36 1.94

the PSD shown in Figure 8 as a model for the active carbon A35/4. This PSD was derived from a fit including all (pure and mixture) experimental points at 293 K and contained, therefore, most of the “experimental information” about the system we had at 293 K. Consequently, it allows the best prediction of pure-component adsorption at 313 and 333 K. 6. Conclusions In a case study, we have outlined the potential of simulations to derive models for the internal structure of specific active carbons which allow a high degree of prediction. As a specific result concerning the adsorption of CO2 and CH4 on the active carbon A35/4, we have found that A35/4 contains a significant amount of pores with a width lower then 6.1 Å. These pores are accessible only to CO2, and they are therefore responsible for a partial molecular sieving effect in the mixture adsorption. Several general conclusions can be drawn. We have found that if more experimental information is used for the PSD fit, the resulting distribution allows more accurate predictions for mixture adsorption, and the extension to other temperatures is more reliable. For the adsorption of the system CO2-CH4 on an active carbon at 293 K or higher temperatures, it is probably necessary to use the pure experimental data of both species at one temperature to derive a reliable PSD. This would be a distribution that then allows predictions of mixture equilibria, similar to the results in section 5.2, and the prediction of purecomponent adsorption at higher temperatures, as shown in section 5.4. In contrast, for methane-ethane on active carbons, where the size difference of the ethane molecule (length about 5.8 Å) to methane is still greater than for the CO2 molecule (length about 5.1 Å) but where both components are of similar chemical nature, it was possible to derive essential PSDs only from the experimental methane adsorption isotherm data at a single temperature.16 Finally, we mention that the model approach could be extended in considering network effects. To describe the binary methane-ethane adsorption in active carbon BPL, Davies and Seaton16 had to assume an excluded volume for ethane as a result of the pore connectivity in BPL. Because for our carbon good agreement was found without considering network effects, this was not needed here. Acknowledgment. M.H. would like to thank the EU for a Marie Curie Fellowship (ERBFMBICT972461). The experimental work has been funded in part by Deutsche Forschungsgemeinschaft, whose support is gratefully acknowledged. Appendix: Fit of Experimental Data We have fitted our experimental single-component adsorption data by the semiempirical isotherm equation proposed by Jensen and Seaton6

[ (

n(P) ) KP 1 +

KP a(1 + κP)

)]

c -1/c

where n(P) is the adsorbed amount, K is the Henry’s law

Adsorption of CO2 and CH4 on Activated C

constant, a is measure of capacity, and κ represents an adsorbed-phase compressibility. The parameter c determines the curvature of the isotherm. Especially in the region of higher pressures, this isotherm equation shows an improvement over the Toth equation, which is com-

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monly used to represent adsorption isotherms analytically. Table 3 shows the values determined for the parameters. LA9904298