Multicomponent Adsorption Model for Polar and Associating Mixtures

Article pubs.acs.org/IECR

Multicomponent Adsorption Model for Polar and Associating Mixtures Igor Nesterov, Alexander Shapiro,* and Georgios M. Kontogeorgis Center for Energy Resources Engineering, Department of Chemical and Biochemical Engineering, Technical University of Denmark (DTU), Søltofts Plads, Building 229, 2800 Kongens Lyngby, Denmark ABSTRACT: The multicomponent potential adsorption theory (MPTA) is revisited in this work for polar and associating systems. MPTA is used in combination with the CPA equation of state. Previous studies have shown that both MPTA and other theories present difficulties for complex systems. Some of these problems could be due to the fact that the original MPTA assumes that a given adsorbent has the same adsorption capacity (for example, porous volume) for all the adsorbed substances and is adjusted simultaneously to many data. This is a simplified picture, as experimental data indicate that the adsorption capacities of the various components may also differ. In this paper we develop a scheme for the distribution of the potential, which accounts for the presence of the porous space occupied either by just one component or by both components. These capacities are determined by adjustment of the potentials to experimental data on single-component adsorption. We show that MPTA involving the different adsorption capacities for the different components is capable of predicting binary adsorption data for most of the mixtures considered. In our application of MPTA, we used both the well-known Dubinin−Radushkevich− Astakhov potentials and the potentials directly restored from experimental data by solving the inverse problem. Application of the latter potentials clearly demonstrates the importance of the difference in adsorption capacities. However, the quality of prediction of binary adsorption is similar for both potentials. Thus, we feel that there is no need to use more complex potentials provided that the difference in the individual adsorption capacities is accounted for.

1. INTRODUCTION Prediction of the multicomponent adsorption is important for several areas of chemical engineering, as well as for other fields, e.g., petroleum engineering and environmental applications. A challenge is to make a model capable of predicting the binary and multicomponent adsorption based only on the singlecomponent adsorption data. This is important, since the industrial adsorbents are highly variable, and a correlation of the data for a given adsorbent is of limited use. A new adsorbent would require new measurements of the multicomponent adsorption, which is a difficult and time-consuming task. The modeling is useful for overcoming these difficulties by extraction of all the necessary information from singlecomponent adsorption isotherms and prediction of the multicomponent adsorption equilibria. Several models have been developed for this goal (see overview in Shapiro and Stenby (2002, 2006)1,2). The two traditional adsorption models are the multicomponent Langmuir model3 and the ideal adsorption solution theory (IAST4). Extensive comparison of the two models on available sets of experimental data is performed by Bartholdy et al. (2013)5 (see also references therein). It is demonstrated that while both models are rather successful in predicting equilibria of the relatively simple mixtures like hydrocarbon gases, they both face difficulties when being extended to more complex mixtures. The multicomponent potential adsorption theory (MPTA), proposed by Shapiro and Stenby (1998),7 describes adsorption as segregation of a mixture in the external potential field emitted by the adsorbent. Each component is affected by its own potential, adjusted on the basis of an individual adsorption isotherm. The interaction between the different components in © 2015 American Chemical Society

the mixture is described by an equation of state for the bulk phase. The MPTA is, in its “standard” implementation, as predictive as the IAST and the multicomponent Langmuir models but may be used outside the ranges of their applicability. It is applicable, for example, to adsorption of the supercritical fluids,8 where the traditional approaches cannot be applied in principle. By selection of a right surface potential (like Dubinin−Radushkevich−Astakhov or Steele) and an advanced equation of state (like SBWR or PC-SAFT), the MPTA may be extended onto relatively complex mixtures8 and liquid solutions,9 although with a limited degree of success (Bjørner et al., 20136). Another recently developed predictive model of adsorption is based on application of the density functional theory (DFT) with thermodynamic equations of state. The theory is based on the analysis of the mixture distribution in an external force field, within a confined space representing an average single pore.10−15 The equilibrium conditions are obtained by application of the variational minimization of the basic integrals of statistical mechanics. This is, to some extent, similar to the thermodynamic derivation of the mixture equilibrium conditions in the external field, which may be applied to derive the basic relations of the MPTA.16 Although the similarity of the two approaches might indicate that their ranges of applicability may also be rather similar, the DFT, to the best of our knowledge, has not been applied until now to the fluids with Received: Revised: Accepted: Published: 3039

January 15, 2015 March 4, 2015 March 5, 2015 March 5, 2015 DOI: 10.1021/acs.iecr.5b00208 Ind. Eng. Chem. Res. 2015, 54, 3039−3050

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Industrial & Engineering Chemistry Research the same degree of complexity as MPTA.6,10 Detailed comparison of the two models is still to be carried out. The present work describes the further development of the MPTA, in its version combined with the cubic plus association (CPA) equation of state (Kontogeorgis et al., 199617). The CPA is an engineering model for bulk phase equilibria that is reduced to the classical SRK equation for simple fluids like nitrogen and hydrocarbons but also involves the Wertheim terms, making it possible to describe mixtures containing associating compounds. The goal of this work is to extend the MPTA to such polar/associating mixtures. Description of adsorption of the mixtures containing polar and associating compounds has been problematic so far for all adsorption models, including the MPTA. While certain combinations of equations of state and surface potentials may be successful for some experimental data sets,6,18 this success is not guaranteed in all cases, and for other data sets the predictions may be totally off, both quantitatively and qualitatively. In many cases, the MPTA combined with simple equations of state, like SRK, can produce results that are superior to the predictions where complex equations accounting for polarity and association are used. Such a performance of a model usually indicates that there are important physical phenomena affecting the behavior of the system, which are not accounted in the model. One of such phenomena is the different adsorption capacities for the components. The original MPTA presumes that a given adsorbent has the same adsorption capacity (for example, porous volume) for all the adsorbed substances. The adsorption capacity is adjusted for all the components simultaneously, while other potential parameters may be different. Meanwhile, experimental adsorption isotherms of the individual components indicate that the adsorption capacities may also differ. Apparently, the adsorbent contains volumes open for one substance but “blind” for another. There is a part of the porous space where only one component is distributed and another part where both components coexist. In this paper, we develop (to the best of our knowledge, for the first time) a scheme of the distribution of the potential that accounts for the presence of the porous space occupied either by just one component or by both components. The potential distribution is in agreement with the single-component data. By comparison with the experiments, we demonstrate that the MPTA involving the different adsorption capacities for the different components is capable of predicting binary adsorption data for almost all the mixtures considered. Any particular version of the MPTA is based on a certain selection of the surface potentials. In most studies, the traditional Dubinin−Radushkevich−Astakhov (DRA) potentials have been applied,6,18 although the less empirical Steele potential has also been used.8,9 It may also be questioned whether these potentials are “right” and whether selection of a better potential model may help overcome the problems that MPTA and other adsorption theories experience when applied to complex mixtures. In the present paper, we compare the DRA potential with the “best possible” potentials, which are obtained by direct reversion of the single-component adsorption isotherms. A reversion procedure involving regularization has been developed. It is shown that introduction of the fitted potentials slightly improves the predictions. However, it alone is insufficient in rendering MPTA a truly predictive theory. It is demonstrated that while account for the different adsorption capacities significantly improves the

predictivity of the MPTA, introduction of the best-fit potentials, at most, leads to only slight qualitative improvements, resulting in more realistic shapes of the binary adsorption isotherms.

2. MODELS AND METHODS 2.1. Multicomponent Potential Adsorption Theory. The MPTA is based upon the concept by Polanyi, 1914,19 which describes adsorption in terms of behavior of a substance in a potential field emitted by the surface of adsorbent. The original model has been extended onto microporous solids by Dubinin et al.20,21 These models were purely analytical and described adsorption within a simple thermodynamics framework: the bulk phase was considered as an ideal gas, while the adsorbate was represented by an incompressible liquid.22 Extension of the theory onto multicomponent mixtures required thermodynamic numerical computations similar to those carried out for the mixture segregated in a gravity field.1 MPTA describes the adsorption equilibrium between the ncomponent bulk phase and adsorbate at constant temperature T by the following system of n equations involving the surface potentials εi(z):7 μi (xj(z),P(z)) − εi(z) = μ bi (yj b ,Pb) (i = 1, ..., n; j = 1, ..., n − 1)

(1)

where μi is chemical potential of the ith component in the mixture and P(z) its pressure at a distance z from the surface or at a point inside the porous volume indexed by the continuous variable z. The molar fractions of the adsorbate at point z are denoted by xj(z), and the molar fractions in the bulk phase are denoted by yjb. Since the sum of the molar fractions is equal to unity, only n − 1 of them are independent. Hence, eq 1 represents the system of n equations, which, under known pressure and composition of the bulk phase, have to be solved with regard to n unknowns xj(z), P(z) for each point z. The adsorbed (excess) amounts Γi of each component are expressed afterward in terms of the density ρ: Γi =

∫0

∞

[xi(z) ρ(xj(z),P(z)) − yi b ρb ] dz

(2)

The chemical potentials and other thermodynamic functions in eqs 1 and 2 are found on the basis of an equation of state, which is used in exactly the same form as for thermodynamic equilibria of the bulk fluids. In such a way, the model separates the bulk phase and the surface interactions. Throughout this work, the cubic-plus-association (CPA) EoS has been applied.17 The pure-component parameters of CPA are the literature parameters adjusted on the basis of the vapor pressure and liquid density data.25 They were not modified to fit the adsorption data. No interaction parameters (kij parameters) were used. The surface potentials εi(z) are presumed to be the same for a component i in the mixture and for the pure component. Then they can be extracted from the experimental data on adsorption of a pure substance. With the known potentials, multicomponent adsorption may be purely predicted on the basis of eqs 1 and 2. Thus, the procedure of MPTA consists of two parts. First, the surface potentials are selected and their parameters are adjusted on the basis of the data for singlecomponent adsorption. Next, the binary or multicomponent adsorption data (normally, adsorption selectivities) are predicted. 3040

DOI: 10.1021/acs.iecr.5b00208 Ind. Eng. Chem. Res. 2015, 54, 3039−3050

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Industrial & Engineering Chemistry Research

Figure 1. Adsorption isotherms (a) and surface potentials (b) for benzene adsorbed on alumosilicate molecular sieve MS13X: (dashed line) with the DRA potential; (solid line) with the restored potential. Experimental data are from Konno et al.23

Figure 2. Adsorption isotherms (a) and surface potentials (b) for methanol adsorbed on carbon molecular sieve MS5A: (dashed line) with the DRA potential; (solid line) with the restored potential. Experimental data are from Konno et al.23

Figure 3. Adsorpiton isotherm (a) and surface potentials (b) for methanol adsorbed on activated carbon: (dashed line) with the DRA potential; (solid line) with the restored potential. Experimental data are from Konno et al.23

interaction, and β is a parameter related to heterogeneity of the porous space, equal to 2 in most cases (and in all the cases considered in the present paper). As it can be seen from Figures 1−3 (dashed lines), the DRA potential has an inflection point. Such s-shaped potential can be used to fit the experimental data well enough for many systems,5,6 especially for the components where surface excesses reach saturation within the experimental pressure range, like benzene in Figure 1. DRA potentials can also handle the data where saturation is not achieved within the experimental range (Figure 2). But this ability has its limits, as shown in Figure 3, where MPTA/DRA cannot reproduce experimental data for methanol on activated carbon, producing the surface excess curve that tends to saturation, while the data

2.2. Choice of the Surface Potentials. 2.2.1. Dubinin− Radushkevich−Astakhov (DRA) Potentials. The conventional approach used with MPTA is based upon the use of some predefined expressions for surface potentials. The parameters of these functions are adjusted to experimental data. In the previous studies the Dubinin−Radushkevich− Astakhov (DRA) potentials have been extensively used.2,6 The DRA potential describes the distribution z(ε) of the porous volume z as a function of the surface potential ε: ⎛ ε ⎞β z(ε) = z 0 exp⎜ − ⎟ ⎝ ε0 ⎠

where z0 is the adsorption capacity (accessible porous volume), ε0 is the characteristic adsorption energy of the fluid−solid 3041


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Figure 4. Restoration of the potential for benzene on alumosilicate molecular sieve applying the Tikhonov method with the different regularization parameters λ: (a) function ϕ(ε); (b) potential curve z(ε); (c) fitting of the experimental data; (dashed line) λ = 4000 (high degree of smoothing); (solid line) λ = 255.52 (best fit). Experimental data are from Konno et al.23

Figure 5. Potential and adsorption curves for benzene and acetone adsorbed on alumosilicate molecular sieve MS13X (restored potentials) before (a) and after (b) they are forced to the same value of z0. Experimental data are from Konno et al.23

integral equation for the unknown function ϕ(ε) = dz/dε. The dependence of density on chemical potential is calculated from an equation of state, while the dependence Γ(Pb) is known from an experiment on adsorption of pure components on porous medium. The equation of state is used again to convert Γ(Pb) into the dependence Γ(μb). Equation 3 is the Fredholm integral equation of the first kind with the kernel K = ρ(μb+ε) − ρ(μb). Once ϕ(ε) is determined, the potential function is found from the following differential equation:

do not exhibit such a tendency. The experimental data are in all cases from the work of Konno et al.,23 as discussed below. 2.2.2. Restored Potentials. The reliability of computations with the predefined model potentials (like the DRA or Steele) depends on the choice of the potential function, which is always questionable. As an alternative, we suggest direct restoration of the surface potentials from the single-component adsorption isotherms. Instead of regressing the parameters in the equation for the DRA potential, the potential function may be found by solution of the integral equation Γ(μ b ) =

∫0

εmax

[ρ(μ b +ε) − ρ(μ b )]ϕ(ε) dε

(3)

dz = ϕ(ε) dε

This equation is transformed (eq 2) for the surface excess of a single-component substance. It should be considered as an 3042

or

dε 1 = dz ϕ(ε)

(4) DOI: 10.1021/acs.iecr.5b00208 Ind. Eng. Chem. Res. 2015, 54, 3039−3050

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Figure 6. Potential curves for benzene and methanol adsorbed on activated carbon ACG2X (restored potentials) for the case where uniform distribution of the capacity difference is applied: (a) potentials calculated for the pure components; (b) potentials used to calculate adsorption at the pore space shared by both components, where the potential of benzene is scaled; (c) the scaled potential curve to calculate adsorption of benzene at the space occupied by this component only.

To solve eq 3, the surface excess Γ(μb) is approximated by a ratio of polynomials. Then Tikhonov’s regularization method from the Regtools 4.0 package24 is applied to the equation. With this method the solution of integral equation is found as the solution of the following least-squares problem: min

⎛ b ⎞ ⎛A⎞ ⎜ ⎟x − ⎜ ⎟ ⎝ λL ⎠ ⎝ λLx*⎠

2

occupied by one component only. The potential curve for this component should somehow be divided between the part of the space where this component is alone and the space that it shares with another component. A rule for such splitting must be introduced as another assumption of the model. (A more general approach would be to distribute the whole adsorption space by the values ε1, ε2 for both components. However, such a general approach would result in the loss of predictivity for the binaries, which is one of the main goals of using MPTA.) In this work we propose the following approach for accounting for the difference between the adsorption capacities of the various components. We assume that for the component with a larger adsorption capacity the distribution of the adsorption potential in the extra space is the same as in the space shared with another component. More precisely, consider the substances 1 and 2 with adsorption capacities z01 and z02 such that z01 > z02. Within the part of the porous space A of the total capacity z02 both components coexist, while within the space B of capacity z01 − z02 only the first component is present. The potential function z(ε) for the first component may be expressed as z01ζ1(ε), where ζ1(0) = 1. Correspondingly, the potential function for the second component has the form of z02ζ2(ε). We assume that in the part A of the porous space the potential functions of the first and of the second components have the form of z02ζ1(ε) and z02ζ2(ε), correspondingly. Thus, in this part they have a common adsorption capacity z02. In part B of the porous space only the first component can exist, and its adsorption function is (z01 − z02)ζ1(ε). The potential function for the first component has the same shape but is differently scaled in parts A and B of the porous space (Figure 6). The adsorbed amounts for component 1 are calculated separately in parts A and B and then added. The binary adsorption computations are applied for part A, while single-component adsorption isotherms are computed in part B.

(5)

where A and b are the matrices of quadrature coefficients to represent the kernel and right-hand side integrals ∫ ε0max∫ μ0 b,maxK(ε,μb) dμb dε and ∫ μ0 b,maxΓ(μb) dμb correspondingly, L is the regularization matrix and λ is the regularization parameter that represents the degree of regularization and must be specified prior to the solution. The L-curve approach is used to find the optimal regularization parameter as the point of maximal curvature on the dependence of solution seminorm || Lxλ|| on residual norm ||Axλ − b||. Shapes of the function ϕ(ε) and potential curve are plotted in Figure 4 for the two different values of λ to show how variation of the regularization parameter affects the solution and the quality of description of experimental data. 2.2.3. Adsorption Capacities. The total adsorption capacity z0 is the maximum porous volume occupied by a component. It is clear from Figures 1−3 that values of z0 may be different for the different components on the same adsorbent. A substantial condition in the previous studies was that all the components in the mixture occupy the same accessible volume, that is, that their adsorption capacities are equal. The potential functions for the substances on the same adsorbent were optimized together, with a common value of z0, for which the average deviation from the experimental data for both components is minimal (Figure 5b). This approach will be referred further as “common adsorption capacity”. Meanwhile it was previously stated6 that if the difference between the components’ adsorption capacities is not being taken into account, errors in the prediction of adsorption may occur. For example, Monsalvo and Shapiro8 stated that such difference in the components’ adsorption capacities is the cause of overestimation of the CO2 surface excess at high pressures predicted for adsorption of N2−CO2 and CH4−CO2 mixtures on activated carbon. Consider a binary mixture of components with different adsorption capacities. There will be a part of porous space

3. RESULTS AND DISCUSSION The predictive ability of the different versions of the MPTA presented in section 2 was tested on a range of binary mixtures. The test systems were selected from the experimental work of Konno et al.,23 as the same set of four binary mixtures contains at least one polar or associating component (methanol− benzene, acetone−benzene, acetone−n-hexane, and methanol− acetone) on the three studied adsorbents: activated carbon 3043


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Table 1. Percentage Average Absolute Deviations (%AAD) of Prediction Performed with All the Studied Approaches to MPTA on the Set of Test Systems DRA potentials, common adsorption capacity

restored potentials, common DRA potentials, uniform distribution adsorption capacity of the capacity difference

restored potentials, uniform distribution of the capacity difference

ACG2X methanol−benzene methanol−acetone acetone−benzene acetone−n-hexane

15.48 23.55 12.40 10.40

6.63 10.91 27.71 64.84

methanol−benzene methanol−acetone acetone−benzene acetone−n-hexane

22.91 10.55 72.25 29.84

74.72 6.41 6.13 52.90


14.09 24.04 11.25 6.59

25.50 8.43 4.20 5.90

17.04 16.56 11.25 7.76

7.19 14.40 7.06 34.13

19.83 6.18 76.44 25.02

11.06 24.60 19.49 40.86

13.88 25.67 6.62 6.21

13.83 13.64 8.71 4.99

MS5A

MS13X

Table 2. Qualitative Results of Prediction Performed with All the Studied Approaches to MPTA on the Set of Test Systemsa DRA potentials, common adsorption capacity

restored potentials, common DRA potentials, uniform distribution adsorption capacity of the capacity difference

restored potentials, uniform distribution of the capacity difference

ACG2X methanol−benzene methanol−acetone acetone−benzene acetone−n-hexane

QR D QW QR

D QR QR QW


D QR QW QW

QW QR QR QW


QR QR QR QR

QR QR QR D

D QR QR QR

QR QR QR QW

D QR QW QW

QR QR QR QW

QR QR QR QR

QR QR QR QR

MS5A

MS13X

a

QR, qualitatively right; D, doubtful; QW, qualitatively wrong.

vs molar fraction in the bulk) comprises the data between zero and unity. The quantitative results are divided into the three groups: with %AAD below 10%, with %AAD between 10% and 20%, and with %AAD above 20%. For the qualitative comparison, the group called “qualitatively right” contains the cases where the predicted selectivity plot has the same convexity as the experimental data no matter how far it is from the experiment. The opposite group, “qualitatively wrong”, contains cases where convexities of predicted and experimental curves are different. The last group, “doubtful”, contains the cases where the results are indefinite. Quantitative and qualitative results of prediction for all the studied approaches are shown in Tables 1 and 2. 3.1. DRA Potentials with the Common Adsorption Capacity. For the case where the DRA potentials of both components of a binary mixture are forced to have the same value of z0, out of 12 studied systems only one has %AAD below 10%, six systems belong to the 10−20% interval, and five systems have %AAD values over 20%. As the worst case, the adsorption of acetone−benzene mixture on MS5A is predicted with %AAD higher than 50% (Table 1). Qualitatively, there are four systems for which prediction does not follow the trend of

ACG2X, carbon molecular sieve MS5A, and alumosilicate molecular sieve MS13X. This makes, in total, 12 combinations of binary mixtures and adsorbents. For all these systems data on single component adsorption and on binary adsorption selectivity are available, forming a comprehensive and consistent data set that can be used for the evaluation of MPTA in its different variants. All the adsorption isotherms were measured by the volumetric apparatus at the same temperature of 303.15 K. In the previous section the two approaches to build the potential curves (DRA and restored potentials) and the two approaches to the adsorption capacities of the components (“common adsorption capacity” and “uniform distribution of the capacity difference”) were described. In total this gives four combinations that were tested for all 12 systems. The results will be discussed in terms of both percentage deviations and quality of prediction of compositions of the adsorbed mixtures. The average absolute deviation (%AAD) is taken as a quantitative measure, while the reproduction of the tendencies in experimental data is taken as a qualitative measure. The selection of the %AAD is a reasonably correct quantitative measure, since the selectivity plot (molar fraction in adsorbate 3044


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Figure 7. The s-shaped prediction curves obtained on the basis of the DRA potentials with common adsorption capacity: (a) ACG2X/acetone− benzene; (b) ACG2X/acetone−n-hexane; (c) MS5A/acetone−benzene; (d) MS5A/acetone−n-hexane. Experimental data are from Konno et al.23

Figure 8. Adsorption of the methanol−benzene mixture on carbon, showing the correlation with the DRA potentials and common adsorption capacity: (a) ACG2X/methanol−benzene; (b) MS5A/methanol−benzene. Experimental data are from Konno et al.23

experimental data and two doubtful systems. Out of the four systems, which do not reproduce the experimental trends, three have %AAD above 20% and one has %AAD between 10 and 20% (Table 1). The latter is acetone−benzene on ACG2X, for which prediction yields an s-shaped curve, contrary to the data trend (Figure 7a). In general, the application of the DRA potentials may often result in s-shaped curves for systems that do not clearly exhibit experimentally such a tendency. The mixtures of acetone− nonpolar component, adsorbed on activated carbon or a carbon molecular sieve, exhibit such an s-shape curve (Figure 7), while for the mixture of methanol−benzene the s-shape is much less pronounced (Figure 8). Quantitatively, the deviations are within 20% for three binary systems out of four when adsorbed on both activated carbon and alumosilicate molecular sieve. The methanol−acetone

mixture is always problematic, while acetone−n-hexane shows the best results. Results for the alumosilicate molecular sieve are slightly better in comparison to activated carbon. Deviations increase for MS5A in comparison to the other two adsorbents mentioned above, with the only satisfactory result is for the methanol−acetone mixture, but the deviation is about 70% for acetone−benzene. Generally, the predictions of the MPTA for the binary mixtures containing polar components are unstable when the DRA potentials with the common adsorption capacities are applied. The scheme needs further improvement. 3.2. Restored Potentials with Common Adsorption Capacity. It may be questioned whether the failure of the approach to MPTA described above is attributed to the shapes of the applied model potentials. In order to answer this question, we investigate the results for the MPTA with the 3045


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Figure 9. Systems for which prediction of adsorption with restored potentials and common adsorption capacity results in the worst predictions (% AAD above 50%): (a) ACG2X/acetone−n-hexane; (b) MS5A/acetone−n-hexane; (c) MS5A/methanol−benzene. Experimental data are from Konno et al.23

Figure 10. Systems for which prediction of adsorption with restored potentials and common adsorption capacity results in %AAD between 20% and 30%: (a) ACG2X/acetone−benzene; (b) MS13X/methanol−benzene. Experimental data are from Konno et al.23

65%. The results for the carbon molecular sieve have two systems with %AAD of 53% and 75%, while the other two systems have %AAD below 10%. Thus, the performance pattern over adsorbent types is similar to the one for the DRA and common adsorption capacity with the best quality for alumosilicate molecular sieve, decreasing quality for activated carbon and even worse results for the carbon molecular sieve systems (Table 1). Unlike the results obtained with the common adsorption capacity and the DRA potentials, the results with the restored potentials are either rather precise (%AAD below 10%) or totally off. We believe that this difference is due to reaction of the potentials on forcing them to the common adsorption capacity. The DRA potentials may slightly change their shape by varying ε0 and partly compensating for the consequences of the “unphysical” operation of bringing them to the common adsorption capacity. Meanwhile, the restored potentials are rigidly determined by the experimental data. Thus, the restored potentials, adapted to a common value of z0, may only make a good prediction or fail. Comparison between the results provided by the two types of potentials with the common adsorption capacity approach makes it possible to conclude that there is a flaw in the scheme with adsorption capacities of two different components forced to have the same value. This flaw should be overcome by introduction of individual adsorption capacities in the cases where they are clearly different for the two components. 3.3. DRA Potentials with Uniform Distribution of the Capacity Difference. As can be seen from the overview

restored model-independent surface potentials. As in the previous section, after reconstruction the potentials have been forced to have a common adsorption capacity. Such an approach results in certain improvement. There are six systems with good and one with satisfactory prediction of the data, compared to only one good and six satisfactory predictions with the DRA potential. On the other hand, predictions with restored potentials result in five systems with %AAD above 20%, as many as for the DRA potentials. The prediction quality within the group with %AAD above 20% is different for the two approaches. While for the DRA potentials only one system exhibits %AAD above 50%, with the restored potentials and common adsorption capacity, three systems exhibit such a bad prediction in the case of restored potentials/ common adsorption capacity (Table 1). These problematic systems are acetone−n-hexane on ACG2X, acetone−n-hexane on MS5A, and methanol−benzene on MS5A (Figure 9). For all of them the predicted content of polar component in the adsorbed phase is extremely high. For the other two systems, with %AAD higher than 20%, the experimental trend is roughly reproduced (Figure 10). The prediction performance of the restored potentials with the same value of z0 with regard to the adsorbents is quite similar to the behavior described for the DRA potentials. Mixtures on alumosilicate molecular sieves provide the best results with three systems with %AAD below 10% and one with %AAD of about 26%. Predictions for activated carbon include one system with %AAD below 10%, one with %AAD between 10% and 20%, and the two systems with %AAD of 28% and 3046


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Figure 11. “The best” systems, with the lowest %AAD provided by DRA potentials with uniform distribution of the capacity difference: (a) ACG2X/ acetone−n-hexane; (b) MS5A/methanol−acetone; (c) MS13X/acetone−benzene; (d) MS13X/acetone−n-hexane. Experimental data are from Konno et al.23

Figure 12. The DRA potential tends to predict s-shaped curves. Shown are the cases of uniform distribution of the capacity difference: (a) ACG2X/ methanol−benzene; (b) MS5A/acetone−benzene; (c) MS5A/methanol−benzene; (d) MS5A/acetone−n-hexane. Experimental data are from Konno et al.23

results in Tables 1 and 2, this approach shows a definite improvement compared to the potentials having common adsorption capacity. There are only three systems for which the predictions result in %AAD above 20% and one system

exhibiting %AAD above 50%. The last system is acetone− benzene on MS5A, the same as for the approach with the same value of z0 for both components. The group with %AAD from 10% to 20% includes five systems, and there are four systems 3047


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Figure 13. Predictions of binary adsorption for the systems with the lowest %AAD obtained with the approach with restored potentials and uniform distribution of the capacity difference: (a) ACG2X/methanol−benzene; (b) ACG2X/acetone−benzene; (c) MS13X/acetone−benzene; (d) MS13X/acetone−n-hexane. Experimental data are from Konno et al.23

Figure 14. Prediction of adsorption of the binary mixtures by the approach with restored potentials and uniform distribution of the capacity difference. Shown are the two “worst” systems for which prediction does not follow the experimental trend: (a) ACG2X/acetone−n-hexane; (b) MS5A/acetone−n-hexane. Experimental data are from Konno et al.23

but this system still has %AAD above 20%. The prediction for the methanol−acetone mixture on MS13X shows an underestimation of the content of methanol in the adsorbed mixture for both cases with DRA potentials, so the prediction of mixtures of two polar components on alumosilicates with DRA potentials needs to be studied further including more data for similar systems. Most of the data for the adsorption on carbon molecular sieves are not well described by any of the studied approaches. As for all cases discussed previously, the approach from this section shows the best performance for the adsorption on alumosilicate molecular sieve. The prediction quality decreases slightly for the activated carbon and decreases further for the carbon molecular sieve. The combination of DRA potentials with the uniform distribution of the capacity difference provides very good prediction for the adsorption of binary mixtures containing

with %AAD below 10% (Figure 11), compared to the one such system for the approach with DR potentials and the same value of z0. The tendency to produce s-shaped selectivity curves remains, although the set of systems with pronounced s-shaped selectivity curves differs from the one described in section 3.1 (Figure 12). The mentioned set of four systems with s-shaped prediction curves includes two cases with predictions that do not follow the tendency of experimental data and two doubtful cases. The rest of the systems are described well. There are two systems for which the prediction quality decreases slightly as compared to the approach with the same adsorption capacities. These are acetone−benzene on MS5A and methanol−acetone on MS13X. The prediction for the acetone−n-hexane mixture on MS5A is improved with introduction of uniform distribution of the capacity difference, 3048


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Figure 15. Prediction of adsorption of binary mixtures by the approach with restored potentials and uniform distribution of the capacity difference. Shown are the worst cases for the systems for which prediction follows the experimental trend: (a) MS5A/acetone−benzene; (b) MS5A/methanol− acetone. Experimental data are from Konno et al.23

used for such systems if the components of the mixture are taken with individual adsorption capacities. These capacities are determined by adjustment of the potentials to experimental data on single-component adsorption. The binary adsorption is predicted if the space occupied by only one component (the component with a higher individual capacity) is distributed by potential in the same way as the rest of the space. We have applied the MPTA in combination with the two types of surface potentials: the model Dubinin−Radushkevich− Astakhov potentials and the potentials directly restored from experimental data by solving the inverse problem. Application of the latter potentials clearly demonstrates the importance of the difference in adsorption capacities. However, the quality of prediction of binary adsorption is similar for both potentials. There is no need to go to more complex potentials provided that the difference in the individual adsorption capacities is accounted for. Meanwhile there are some precautions to the application of the method: (1) Application of DRA potentials provides the s-shaped selectivity curves for polar−nonpolar mixtures on carbon (both activated carbon and molecular sieve). This is not always in agreement with experimental data. Application of the restored potentials is preferable for such cases. (2) Both DRA and restored potentials perform relatively poorly for the mixtures adsorbed on carbon molecular sieve irrespective of the approach used. The cause of such behavior is unclear and must be investigated. (3) Prediction of adsorption of acetone−hydrocarbon mixtures on carbon adsorbents made with restored potentials is not satisfactory. Meanwhile, application of the DRA potentials may lead to poor results when adsorption of polar−polar mixtures on a polar (alumosilicate) molecular sieve is considered. Besides this, the two implementations of MPTA perform well on the selected set of systems. More experimental data are needed for further verification of the theory.

polar components. There are only three systems with %AAD above 20%. This makes it possible to recommend such combination to the calculations of adsorption with the exception of polar−polar mixtures on polar surface and any kind of mixture on carbon molecular sieves. Of course, calculations on more systems are needed before offering general recommendations. 3.4. Directly Restored Potentials with a Uniform Distribution of the Capacity Difference. The uniform distribution of the capacity difference is also superior to the common capacity for the restored potentials. There are three systems predicted with %AAD above 20%, five systems with % AAD between 10% and 20%, and four with %AAD below 10% (Figure 13). There are not as many systems with %AAD below 10% as for restored potentials with the same z0, but the results are more stable. The highest %AAD is about 40% (for acetone−n-hexane on MS5A), which is the lowest among all the “high %AAD values” for all cases (Table 1). Qualitatively, there are only two systems for which the prediction curves do not follow the tendencies of experimental data: acetone−nhexane on ACG2X and on MS5A (Figure 14). For all other systems, prediction tendencies shown by experimental data are followed by MPTA as well. Figure 15 shows the systems with the lowest %AAD values if adsorption of acetone−n-hexane is not taken into account. It is clear that for most of the systems prediction follows the experimental trends. Overall, the performance of this approach for the set of the test systems is very good and comparable to the previous approach, so it can also be recommended for adsorption modeling. The results with wrong qualitative description are attributed to carbon molecular sieves and acetone−n-hexane mixture adsorbed on carbon. This approach also exhibits good qualitative performance for 10 out of 12 systems that may be considered as an advantage over the DRA potentials, while both potential types show more or less similar results in terms of standard deviations.

4. CONCLUSION In this work the reliability of the multicomponent potential theory of adsorption (MPTA) is studied in application to the adsorption of binary mixtures containing polar or associating components on different polar and nonpolar adsorbents. The MPTA was applied in combination with the CPA equation of state, which has proven to be adequate for the phase equilibria of the mixtures considered. It is found that the MPTA can be

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 3049


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(22) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces; Wiley: New York, 1997. (23) Konno, M.; Shibata, K.; Saito, S. Adsorption equilibria of hydrocarbon gaseous mixtures containing polar components. Chem. Eng. Jpn. 1985, 18, 398−408. (24) Hansen, P. C. Regularization Tools Version 4.0 for Matlab 7.3. Numer. Algorithms 2007, 46, 189−194. (25) Kontogeorgis, G.; Folas, G. Therodynamic Models for Industrial Applications; Wiley: New York, 2010.

ACKNOWLEDGMENTS The work has been carried out in the framework of the MAPS project sponsored by Statoil (Norway). The authors are grateful to the participants of the MAPS and CHiGP projects, especially to Dr. Even Solbraa, for multiple discussions and useful advice.

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REFERENCES

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Multicomponent Adsorption Model for Polar and Associating Mixtures

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