Ind. Eng. Chem. Res. 2000, 39, 527-532
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Application of IAST in the Prediction of Multicomponent Adsorption Equilibrium of Gases in Heterogeneous Solids: Micropore Size Distribution versus Energy Distribution Kean Wang, Shizhang Qiao, and Xijun Hu* Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
This article investigates the effect of surface heterogeneity and energy-matching scheme on the prediction of adsorption equilibria on activated carbon. The ideal adsorbed solution theory (IAST) is employed to evaluate the local adsorption equilibria on each energy site while the energetic heterogeneity of the system is represented by two forms: (1) uniform energy distribution, which employs the cumulative energy scheme to match different adsorbates in the adsorbed phase; and (2) micropore size distribution, which invokes the adsorbate-pore interaction matching scheme. The adsorption equilibria of hydrocarbon gas mixtures measured on two commercial activated carbons are used to compare the two models. It is found that uniform energy distribution can be insufficient in the prediction of multicomponent adsorption equilibria. On the other hand, the model assuming micropore size distribution as adsorption energetic heterogeneity and the adsorbate-pore interaction energy-matching scheme presents relatively stable prediction results. Introduction The adsorption energetic heterogeneity plays an important role in adsorption equilibria and kinetics on adsorbents such as activated carbon.1,2 The two predictive models often seen in the literature for the description of adsorption equilibria on a heterogeneous adsorbent are (1) the heterogeneous ideal adsorbed solution theory or HIAST model3 and (2) the heterogeneous extended Langmuir (HEL) model.4 These two models are predictive in the sense that only pure component equilibrium information is required and no binary data are necessary (for correction purpose) in the simulation of multicomponent equilibria. The HIAST model employs IAST to evaluate the multicomponent equilibria on each energy site. The overall adsorption equilibria is the integral of the local equilibria over the complete range of energy distribution. Valenzuela et al.3 proposed a bimodal energy distribution with a cumulative energy-matching (CEM) scheme for the heterogeneous adsorbed phase. They studied the equilibrium data of CO2-H2S-C3H8 systems on H-modenite reported by Talu et al.5 and concluded that “HIAST generates predictions that are sometimes better but never worse than IAST”. Hu and Do6 studied the same system with different equilibrium models and pointed out that the conclusion on the goodness of HIAST in representing experimental data should be treated with care because its prediction on multicomponent equilibria depends on the choice of energy distribution and the appropriate determination of the single-component isotherm parameters. Because HIAST invokes IAST on each energy site, it is iterative in nature and very computationally demanding. The HEL model is explicit in form and offers the advantage of simplicity in the simulation of multicomponent equilibria. Kapoor et al.4 studied the binary adsorption * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: +852 2358 7134. Fax: +852 2358 0054.
equilibria of a number of systems with a uniform energy distribution and a CEM scheme and they found that the HEL model presents stable performance, comparable to IAST in many cases. However, the uniform energy distribution and the CEM scheme bears some empiricism and uncertainty in determining the matching energies between different adsorbates. The functional form of energy distribution and the energy-matching scheme for different adsorbates in the adsorbed phase are fundamental issues which may significantly influence the prediction result of adsorption equilibria and consequently affect the related adsorption kinetics.7 Hu and Do6 demonstrated that HIAST can, to some degree, simulate nonideal adsorption behavior. Moon and Tien8 investigated the significance of an energy-matching scheme in HIAST and demonstrated that, by proper selection of the method of energymatching, HIAST is able to simulate some nonideal adsorption behavior. The other energy-matching scheme was proposed by Hu and Do,9 which is based on the assumption that the size distribution of the slit-shaped micropore or MPSD is the source of energetic heterogeneity of activated carbon. They proposed that the matching energies between different adsorbates should be related to the interaction strength of each adsorbate in a local micropore and this constitutes the physical criteria for this issue. This scheme, termed as the adsorbate-pore interaction scheme, is capable of representing some nonideal adsorption behavior arisen from the sizeexclusion phenomena.10 When this energy-matching scheme is embedded into the HEL model, the resulting model is termed the MPSD model, which is also noniterative and has been shown to be comparable to IAST in the prediction of the multicomponent adsorption equilibria of hydrocarbon gases on activated carbons.11 However, the local extended Langmuir isotherm violates the thermodynamics if the maximum adsorption capacities differ from species to species. In this article, IAST will be used to evaluate the local
10.1021/ie990548i CCC: $19.00 © 2000 American Chemical Society Published on Web 12/31/1999
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multicomponent adsorption equilibria. The adsorption energetic heterogeneity will be represented by a micropore size distribution. This model is denoted as IASTMPSD and will be tested with the binary adsorption equilibrium data of gases on two commercial activated carbons (Norit and Ajax). The results will also be compared to those obtained from the IAST-ED model, which uses the local IAST with a uniform energy distribution.6 Theory Energetic Heterogeneity and the Energy-Matching Scheme. Two approaches dealing with the surface heterogeneity of activated carbon are considered here. The first one directly assumes an energy distribution function for the heterogeneous surface; that is, if E(k) is the local adsorbate-adsorbent interaction energy and F[k,E(k)] is the energy distribution function for species k, the overall adsorbed concentration of the species can be expressed as the integral of the local isotherm, Cµ[k,E(k)], over the complete energy range:
Cµ(k) )
∫0∞Cµ[k,E(k)] F[k,E(k)] dE(k)
(1)
A uniform distribution is assumed in this paper to account for the adsorption energetic heterogeneity, that is,
F[k,E(k)] )
{
1 Emax(k) - Emin(k) 0
for Emin < E < Emax
(2)
elsewhere
With the uniform energy distribution and the assumption that the ordering of the sites from low to high energy is the same for all species, the matching energies between different species in the adsorbed phase for a multicomponent system can be described by the CEM scheme,4 that is,
E(i) - Emin(i) Emax(i) - Emin(i)
)
E(j) - Emin(j) Emax(j) - Emin(j) i,j ) 1, 2, 3, ..., NC (3)
The second approach argues that the energy distribution arises because of the structural heterogeneity, or micropore size distribution (MPSD) in the case of activated carbon. With this approach, eq 1 can be equivalently written in terms of the MPSD of the adsorbent, f(r), as
Cµ(k) )
∫r∞ Cµ[k,E(k,r)] f(r) dr min
(4)
where r is the pore half-width, rmin is the minimal pore size accessible to the adsorbate, E(k,r) is the adsorbatepore interaction energy (for a physical adsorption process this energy can be taken as the negative of the adsorption potential minimum in the local micropore).12 For a physical adsorption process on activated carbon, the adsorption potential can be represented by the 10-4 potential:
up(k,z) )
{ [( ) (
2 σsk 3 5 z
5 /sk
10
) ] [( ) ( ) ]}
σsk + 2r - z
10
σsk 4 + z σsk 4 (5) 2r - z
where z is the distance between the adsorbate molecule and one of the pore walls, /sk is the potential well depth for a single lattice plane, and σsk is the LennardJones collision diameter determined from the LorentzBerthelot rule. With this approach, the energy matching between different adsorbates follows an adsorbate-pore interaction scheme which has the following features: (1) sizeexclusion phenomena, different adsorbates compete with each other only in pores they have access, and (2) the matching energies between different species are related to their own interaction strength with the local micropore. Although the assumption of the slit-shaped micropore and spherical L-J fluids for all adsorbate molecules are oversimplified for a real system, this approach offers a number of advantages over the traditional energy distribution approach in studying adsorption equilibria. The implications of the MPSD model have been discussed in the reference.11 Models for the Prediction of Multicomponent Adsorption Equilibria. The adsorption equilibria of a gas mixture on every energy site, that is, Cµ[k,E(k)] in eqs 1 and 4, is evaluated by the local IAST3 in which the local pure component isotherm of each species is described by the heterogeneous Langmuir equation:
b0(k) exp[E(k)/RT]C(k) C0µ[k,E(k)] ) Cµs(k) 1 + b0(k) exp[E(k)/RT]C(k)
(6)
where Cµs is the saturated adsorbed concentration, b0 is the affinity constant at zero energy level, C is the adsorbate concentration, R is the gas constant, and T is the temperature. If the uniform energy distribution and CEM scheme are employed in eq 1, the overall HIAST model is termed as the IAST-ED model which is the combination of eqs 1-3 and 6. For a pure component system, the assumption of the heterogeneous Langmuir isotherm and the uniform energy distribution will lead to the UniLan equation as the overall isotherm, that is,
Cµ )
(
Cµs 1+b h esp ln 2s 1+b h e-sp
)
(7)
h and s are the mean and where b h 0 ) b0 exp(E/RT) and E the square root of variance of energy distribution, respectively. If MPSD is employed as the source of surface heterogeneity and the pore-adsorbate interaction scheme is used to describe the matching energies between different species,11 this HIAST model is denoted as the IASTMPSD model. In this study, the MPSD is assumed to follow the nonnegative Γ distribution:
f(r) )
qν+1rνe-qr Γ(ν + 1)
(8)
The IAST-MPSD model is the combination of eqs 4-6 and 8. Experimental Section Pure component adsorption isotherms of methane, ethane, and propane were measured on Norit activated carbon and Ajax activated carbon by a volumetric method, respectively. For each adsorbate, adsorption
Ind. Eng. Chem. Res., Vol. 39, No. 2, 2000 529 Table 1. Adsorption Isotherm Parameters for Methane, Ethane, and Propane on Two Activated Carbons with Local Langmuir and Uniform Energy Distribution (A ) Ajax; N ) Norit) Cµs (mmol/g) species
283 K
303 K
333 K
363 K
methane (N) 8.35 7.82 7.01 (A) 10.8 10.15 ethane (N) 9.47 8.47 7.51 (A) 15.4 14.8 14.4 propane (N) 4.51 4.07 3.63 (A) 13.9 14.1 14.8
Figure 1. Adsorption isotherm of pure gases on Norit activated carbon. (s, uniform energy distribution model; - - -, MPSD).
isotherms at three or more temperatures were measured with the bulk-phase pressure up to 130 kPa. The adsorption equilibria of the binary mixtures among these three adsorbates were measured on Norit activated carbon with a differential adsorber bed (DAB) rig. The bulk-phase pressure was kept constant at 50 kPa. The equilibrium data of these binary gas mixtures were measured on Ajax activated carbon using a volumetric rig with a total bulk-phase pressure of 66.7 kPa. The detailed experimental procedures and the rig setup can be found in the references.13,14 Results and Discussion First, the pure component isotherm data of three adsorbates measured on each carbon sample were employed, respectively, to optimize the parameters with the uniform energy distribution model and the MPSD model. For the MPSD model, rmin is selected as 0.8584σsk for 10-4 potential, referring to a pore size where the adsorption potential equals zero. The collision diameter of each adsorbate is taken from the reference,11 which is the average diameter of the adsorbate (3.85 Å for methane, 3.9 Å for ethane, and 4.3 Å for propane) and the carbon atom (3.4 Å). The Γ distribution parameters are found to be q ) 21.57 Å-1, ν ) 97.87 for Ajax carbon11 and q ) 21.01 Å-1, ν ) 98.14 for Norit carbon.13 The model fittings (solid lines for the uniform energy distribution model and dashed lines for the MPSD model) and experimental data (symbols) of each species on Norit activated carbon are shown in Figure 1. It is
b0 (kPa-1 Emin Emax × 105) (kJ/mol) (kJ/mol) 0.8820 0.2269
0.6708 0.0115
18.79 22.15
0.8620 0.3618
0 0
27.28 27.21
7.624 0.105
11.49 0
26.57 44.30
seen that both models can adequately fit the experimental data. The model fittings for the equilibrium data of three species on Ajax activated carbon follow the same trend, thus, are not shown here. The detailed procedures of model fitting and the results can be found in our previous studies.7,11 The derived isotherm parameters are listed in Table 1 for the uniform energy distribution model and in Table 2 for the MPSD model, respectively. With these parameters, we are ready to simulate the multicomponent adsorption equilibria on Norit activated carbon and Ajax activated carbons with the two models (IAST-ED and IAST-MPSD). Next, the uniform energy distribution and MPSD models are employed, respectively, with the local IAST to evaluate the local adsorption equilibria on each energy (pore) site. The overall adsorption equilibria on an adsorbent are represented by the corresponding HIAST models, that is, IAST-ED and IAST-MPSD. The simulation is first performed for the binary equilibria of hydrocarbon gas mixtures on Norit activated carbon. For the purpose of comparison, the global IAST model is also employed in the simulation with the UniLan equation as the pure component isotherm. Figure 2a shows the model simulations (lines) and experimental data (symbols) for the ethane-propane system on Norit activated carbon at 303 K. We first look at the model assuming local Langmuir with uniform energy distribution (IAST-ED) model. It can be seen that, for propane, the IAST-ED model (solid line) predicts equally well as the global IAST model (dotted line). For ethane, however, the IAST-ED model predicts much worse than the IAST model. The unsatisfactory prediction result of the IAST-ED model suggests that the uniform energy distribution and the CEM can be insufficient in predicting multicomponent equilibria on a heterogeneous adsorbent. This result also confirms the allegation made by Hu and Do6 that “the goodness of HIAST in representing experimental data should be treated with care since its prediction on multicomponent equilibria depends on the choice of energy distribution and the appropriate determination of the single component isotherm parameters”. Then, we turn to the IAST-MPSD model. The prediction result is shown in Figure 2a as dashed-dotted lines. It is seen that the IAST-MPSD model predicts much better than the IAST-ED model, in general. The prediction for propane is much better while that for ethane is comparable to the global IAST model. Because IAST is thermodynamically verified and can be safely assumed to be applicable on each energy site, the difference between the prediction results from IASTED and the IAST-MPSD can only be attributed to the
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Table 2. Adsorption Isotherm Parameters for Methane, Ethane, and Propane on Two Activated Carbons with the MPSD Model /sk (kJ/mol)
0 Cµs (mmol/g)
Norit Ajax
q (Å-1)
ν
C1
C2
C3
C1
C2
C3
β × 104 (mol/(g K))0.5/kPa
δ × 104 (K-1)
21 21.6
98.1 97.9
6.13 7.31
4.80 8.04
4.40 6.04
12.5 12.2
18.5 16.9
20.7 19.7
1.82 1.03
1.29 0.18
Figure 3. Selectivity of ethane to propane on Norit activated carbon at 303 K. (s, IAST-ED; ‚‚‚, IAST; - ‚ -, IAST-MPSD). Table 3. Relative Average Error between Model Predictions and Experimental Data of the Ethane-Propane System on Norit Activated Carbon 303 K ARE (%)
C2
333 K C3
C2
363 K C3
C2
C3
IAST-ED 31.2 (3) 8.8 (2) 29.2 (3) 7.0 (2) 20.6 (1) 5.1 (2) IAST 8.8 (1) 9.5 (3) 21.9 (1) 7.5 (3) 24.6 (3) 5.7 (3) IAST-MPSD 10.0 (2) 5.1 (1) 28.0 (2) 4.9 (1) 21.0 (2) 5.0 (1)
Figure 2. Binary adsorption equilibria of ethane-propane on Norit activated carbon. (s, IAST-ED; ‚‚‚, IAST; - ‚ -, IASTMPSD).
assumption of energy distribution and the energymatching scheme within the HIAST model. The better predictions of the IAST-MPSD are in favor of both the choice of MPSD as the source of energy distribution and the pore-adsorbate interaction energy-matching scheme. This is physically expected because the structure of activated carbon is very complicated and the assumption of the uniform energy distribution is an oversimplified solution. What is more, the CEM does not reflect the physical reality of the adsorption process in a micropore.11 On the other hand, the assumption of a MPSD function bears more reality for adsorbents such as activated carbon and the pore-adsorbate interaction scheme sets the physical criteria for matching different adsorbates in a local adsorption site. Figure 2b,c shows the model predictions (lines) and experimental data (symbols) for the ethane-propane system on Norit activated carbon at two other temperatures: 333 and 363 K. We see that as the temperature increases, the adsorption energetic heterogeneity of the system decreases and all the model predictions improve. However, the same trend in the predictability of each model continues, as reflected in the value of the average relative error (ARE) listed in Table 3. The order of the goodness of fit of each model is indicated in the second column (in bracket) of Table 3. For ethane, the ARE generally follows this order: IAST-ED >IAST-MPSD >
IAST. The order of ARE for propane is the following: IAST > IAST-ED > IAST-MPSD. To further demonstrate the effect of energetic heterogeneity and the energy-matching scheme in the prediction of multicomponent equilibria within the frame of the HIAST model, the selectivity of the binary ethane-propane system on Norit carbon at 303 K is investigated. This selectivity is defined as
S)
(x/y)ethane (x/y)propane
The selectivity is then calculated for the experimental data and shown in Figure 3 versus the molar fraction of ethane as symbols. It can be seen that these symbols best agree with the simulation results from the IASTMPSD model (dotted-dashed lines). The global IAST model (dotted lines) simulates a very flat trend in the selectivity while the IAST-ED model is the worst among the three models, which gives a much higher selectivity. Some people may argue that the inferior performance of the IAST-ED model may be due to the insufficient experimental data points (or pressure range) for ethane, which results in the incorrect energy distribution from the optimization process. This argument, however, cannot be fully justified as we see that the MPSD model, for which the parameters were also optimized with the same set of data, predicts well with IAST. Figure 4a shows the model prediction (lines) and the experimental data (symbols) of the methane-ethane system on Norit activated carbon at 303 K. We see that the prediction of IAST-ED (solid line) is good, very comparable to that of IAST (dotted lines) for both adsorbates. This indicates that the energy distribution and the CEM scheme are sufficient in describing the methane-ethane system.
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Ajax activated carbon at 303 K are employed to check the predictive capabilities of each model. The experimental data (symbols) and the model prediction results (solid lines for IAST-ED; dashed-dotted lines for IASTMPSD) are shown in Figure 5parts a and b, respectively. It is seen that the predictions from both models simulate the experimental data well (within experimental error), which suggests that the uniform energy distribution and CEM scheme are adequate in describing the equilibrium data of these systems on Ajax activated carbon. Conclusions The adsorption energetic heterogeneity and the energymatching scheme play very important roles in the simulation of adsorption equilibria of multicomponent systems on heterogeneous adsorbents. With the uniform energy distribution and the cumulative energy-matching scheme, the HIAST does not necessarily predict better than the IAST. On the other hand, the IASTMPSD model, which assumes the MPSD as the source of energy distribution and a pore-adsorbate interaction scheme, works well and presents reliable performance for the binary equilibrium data measured on two commercial activated carbons. Acknowledgment
Figure 4. Binary adsorption equilibria of gases on Norit activated carbon. (s, IAST-ED; ‚‚‚, IAST; - ‚ -, IAST-MPSD).
Financial support from the Croucher Foundation and the Research Grants Council of Hong Kong are gratefully acknowledged. Notation
Figure 5. Binary adsorption equilibria of gases on Ajax activated carbon. (s, IAST-ED; - ‚ -, IAST-MPSD).
Figure 4b,c shows the model prediction (lines) and the experimental data (symbols) of the methane-ethane system at 333 K and the methane-propane system at 303 K, respectively. We see that the energy distribution of methane, which is derived from the isotherm data measured at the same pressure range as that of ethane, also proves to be reliable. The simulation results from the IAST-MPSD model for these experimental data are shown in Figure 4a-c as dashed-dotted lines. Their performances are seen to be satisfactory at these experimental conditions. Finally, the binary equilibria data of the methaneethane and methane-propane systems measured on
ARE ) average relative error b0 ) affinity constant at zero energy level (kPa-1) C ) adsorbate concentration (kPa) Cµ ) adsorbed phase concentration (mmol/g) C0µ ) adsorbed phase concentration, evaluated by pure component isotherm (mmol/g) Cµs ) saturated adsorbed concentration (mmol/g) 0 Cµs ) saturated adsorbed concentration at temperature T0 (mmol/g) CEM ) cumulative energy-matching E ) adsorbate-adsorbent interaction energy (kJ/mol) Emax ) maximum energy (kJ/mol) Emin ) minimum energy (kJ/mol) F ) energy distribution function f ) micropore size distribution function HEL ) heterogeneous extended Langmuir HIAST ) heterogeneous ideal adsorbed solution theory IAST ) ideal adsorbed theory M ) molecular weight (g/mol) MPSD ) micropore size distribution q ) distribution parameter (Å-1) R ) gas constant (kJ/(mol K)) r ) pore half-width (Å) rmin ) minimal size of pore accessible to adsorbate S ) selectivity s ) square root of variance of energy T ) temperature (K) T0 ) reference temperature for adsorption capacity (273 K in this study) x ) adsorbed-phase molar fraction y ) bulk-phase molar fraction σsk ) collision diameter (Å) *sk ) potential well depth (kJ/mol) β ) adsorption affinity characterizing adsorbent, β ) b0(k)xMkT in MPSD model
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0 δ ) thermal expansion coefficient of adsorbent, Cµs ) Cµs exp[δ(T - T0)] in MPSD model ν ) distribution parameter
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(9) Hu, X.; Do, D. D. Effect of Pore Size Distribution on the Prediction of Multicomponent Adsorption Equilibria. Fundamentals of Adsorption; LeVan, M. D., Ed.; Kluwer Academic Publishers: Boston, MA, 1996; pp 385-392. (10) Hu, X. Multicomponent Adsorption Equilibrium of Gases In Zeolite: Effect Of Pore Size Distribution. Chem. Eng. Commun. 1999, 174, 201-214. (11) Wang, K.; Do, D. D. Characterizing Micropore Size Distribution of Activated Carbon Using Equilibrium Data of Many Adsorbates at Various Temperatures. Langmuir 1997, 13, 62266233. (12) Jagiello, J.; Schwarz, J. A. Energetic and Structural Heterogeneity of Activated Carbons Determined Using Dubinin Isotherms and an Adsorption Potential in Model Micropores. J. Colloid Interface Sci. 1992, 154, 225-237. (13) Qiao, S.; Wang, K.; Hu, X. Binary Adsorption Equilibrium of Hydrocarbons in Activated Carbon: Effect of Pore Size Distribution. Ind. Eng. Chem. Res., submitted for publication. (14) Ahmadpour, A.; Wang, K.; Do, D. D. Comparison of Models on the Prediction of Binary Equilibrium Data of Activated Carbons. AIChE J. 1998, 44, 740-752.
Received for review July 26, 1999 Revised manuscript received October 21, 1999 Accepted November 12, 1999 IE990548I
