Article Cite This: Langmuir 2017, 33, 11291-11298
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Statistical Mechanical Model for Adsorption Coupled with SAFT-VR Mie Equation of State Luís F. M. Franco,*,†,‡ Ioannis G. Economou,‡ and Marcelo Castier‡ †
School of Chemical Engineering, University of Campinas, Av. Albert Einstein, 500, CEP: 13083-852, Campinas, Brazil Chemical Engineering Program, Texas A&M University at Qatar, P.O. Box 23874, Doha, Qatar
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‡
ABSTRACT: We extend the SAFT-VR Mie equation of state to calculate adsorption isotherms by considering explicitly the residual energy due to the confinement effect. Assuming a square-well potential for the fluid−solid interactions, the structure imposed by the fluid−solid interface is calculated using two different approaches: an empirical expression proposed by Travalloni et al. (Chem. Eng. Sci. 65, 3088−3099, 2010), and a new theoretical expression derived by applying the mean value theorem. Adopting the SAFT-VR Mie (Lafitte et al. J. Chem. Phys., 139, 154504, 2013) equation of state to describe the fluid−fluid interactions, and solving the phase equilibrium criteria, we calculate adsorption isotherms for light hydrocarbons adsorbed in a carbon molecular sieve and for carbon dioxide, nitrogen, and water adsorbed in a zeolite. Good results are obtained from the model using either approach. Nonetheless, the theoretical expression seems to correlate better the experimental data than the empirical one, possibly implying that a more reliable way to describe the structure ensures a better description of the thermodynamic behavior.
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INTRODUCTION The ability of some fluids to adhere onto a certain solid surface constitutes a central phenomenon in nature, commonly referred to as adsorption. Such a phenomenon is governed by the intermolecular interactions between the fluid molecules and the atoms of the solid surface. Taking the solid−fluid interface as a reference frame, the spatial dependence of these fluid−solid intermolecular interactions imposes an inherent inhomogeneity in the system that challenges the very scope of classical equilibrium thermodynamics by establishing a density variation perpendicularly to the interface. This density variation manifests the structural rearrangement exerted on the fluid by the presence of the solid wall. The structural change in the fluid has also implications on the system dynamics, e.g., spatialdependence of transport properties such as diffusion coefficient1,2 and viscosity.3 The modeling of adsorption is rooted on the insightful theoretical formulation established by Langmuir4,5 more than one hundred years ago. Despite very reasonable molecular arguments framed on that seminal work, Langmuir equation solely predicts a single type of adsorption isotherm. Along with extensions to Langmuir’s theory,6 various different theoretical strategies have been proposed. These strategies may be divided into two categories: one that accounts explicitly for the spatial inhomogeneity, and a second that does it implicitly. The first category entails techniques such as molecular simulations,7−14 density functional theory,15−21 and multicomponent potential theory of adsorption (MPTA).22−24 In all these techniques, the molecular spatial distribution within the pore is considered in an explicit manner. The second category entails methods that account for the molecular distribution in an averaged way. The © 2017 American Chemical Society
hypothesis behind such a class of models is that the average density of the confined fluid is sufficient to describe the energetic contribution due to the confinement. Examples of models pertaining to this category are the one proposed by Travalloni et al.,25 who extended van der Waals equation of state to calculate adsorption, and the quasi-two-dimensional fluid approach.26,27 The models in this second category seem the most suitable for industrial applications, although the parameters for the fluid−solid interactions must be fitted to experimental or simulated data. For these implicit inhomogeneous models, the adsorption phenomenon may be seen as a phase equilibrium problem. In the simplest case (only two phases), the bulk phase is described as a homogeneous system containing only the fluid molecules and the adsorbed phase is a hypothetical averaged phase that encompasses all the structural inhomogeneities of the fluid confined by the adsorbent solid walls. The phase equilibrium criteria are, then, satisfied by the equality of the temperature and the chemical potential of each component in both phases. The mechanical potential equality, however, is excluded from the phase equilibrium criteria, since a pressure difference arises between the two phases. Notwithstanding their efficiency and reasonable accuracy, cubic equations of state, such as the ones used by Travalloni et al.,25,28 provide a poor description of highly dense fluids or selfSpecial Issue: Tribute to Keith Gubbins, Pioneer in the Theory of Liquids Received: August 1, 2017 Revised: September 12, 2017 Published: September 14, 2017 11291
DOI: 10.1021/acs.langmuir.7b02686 Langmuir 2017, 33, 11291−11298
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Langmuir
where u(q1, q2) is the two-body interaction potential between the fluid molecules and the solid wall, q is the generalized position coordinate, and g(q1, q2) is the pair correlation function. The simplest model that one can think of an attractive interaction potential is the square-well potential. Mathematically, such a potential can be expressed as
associating fluids, such as water and alcohols. The Statistical Associating Fluid Theory (SAFT) entails a family of equations of state based on perturbation theory29,30 that provide an accurate description of the thermophysical properties of fluids, both in liquid and vapor phases, and with a special attention to associating compounds. In such a theory, as a consequence of the free energy expansion, the residual Helmholtz free energy is given as a sum of two contributions: the reference fluid Helmholtz free energy and the perturbed Helmholtz free energy. The reference fluid is usually taken as the hard-sphere fluid, for which accurate equations of state are known,31,32 and the perturbed potential includes the dispersion interactions among the segments,33,34 the chain term due to the formation of chains of segments, and the association term due to the highly directional attractive interactions in associating compounds.35−40 Within this framework, two successful equations of state, among several others, are PC-SAFT41 and SAFT-VR Mie.42 Our contribution here is to develop an implicit inhomogeneous model that, ultimately, will be added to a SAFT equation of state to calculate adsorption. We start our theoretical development calculating the residual energy due to the confinement effect. By using the generalized van der Waals framework,43 we calculate the residual Helmholtz free energy contribution due to the confinement effect from the integration over temperature of the calculated residual energy.
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⎧+∞ , if x ≤ 1 uSW (x) ⎪ = ⎨− 1, if 1 ≤ x < λ ε ⎪ if x ≥ λ ⎩ 0,
where x = r/σ is the reduced distance, r is the distance between two nuclei and σ is taken as the parameter of the Mie potential such that, when x = 1, the Mie potential is zero, ε is the depth of the square-well potential, and λ is the dimensionless range of the attractiveness for this interaction potential. Substituting the expression for the square-well potential described in eq 6 into eq 5, one may write
⟨E⟩CONF = −
∫0
β
ACONF = − mNε
⟨Φ⟩β = (2)
⟨E⟩R dβ ]
∫0
β
⟨E⟩R dβ
Ns 2 2V 2
∫ ∫ u(q1, q2) g(q1, q2) dq1 dq2
β
Φ(σ , λ ; Ns , V , β) dβ
(10)
1 β
∫0
β
Φ(σ , λ ; Ns , V , β) dβ
(11)
(12)
A = AIG + AMONO + ACHAIN + AASSOC + ACONF
(13) MONO
is the where A is the ideal gas Helmholtz free energy, A Helmholtz free energy due to the dispersion forces among the monomeric spherical segments (including the hard-sphere potential42,45), ACHAIN is the Helmholtz free energy due to the formation of chains from the spherical segments, and AASSOC is the Helmholtz free energy due to the existence of association sites in the spherical segment. The mathematical expressions for the various components of the Helmholtz free energy for SAFT-VR Mie can be found in the original reference42 and are not presented here. ACONF is the residual Helmholtz free energy due to the confinement effect of the segments. The fact that the chain and the association contribution expressions remain unaffected by the confinement is an implicit assumption here. A more thorough model should consider the impact of the confinement on such terms.
(4)
Thus, to calculate the residual energy due to the effect of confinement, one may calculate the residual Helmholtz free energy through eq 4 with an appropriate model for the confinement effect on the residual energy. Assuming pairwise additivity of the interaction potential imposed by the solid wall on the fluid molecules, the residual energy due to the confinement effect might be calculated as44 ⟨E⟩CONF =
∫0
The final expression for the total Helmholtz free energy for a SAFTlike equation of state, coupled with the extension for confined fluids proposed here, is
(3)
IG
1 β
1 β
ACONF = − mNε⟨Φ⟩β
where Λ is the de Broglie thermal wavelength. The residual Helmholtz free energy is AR =
(9)
And hence:
Assuming that Z(Ns, V, β → 0) = ZIG(Ns, V) = VNs, the canonical partition function, Q(Ns, V, β), may be written as
∫0
∫Ω g(x) dΩ
where N is the number of chains that characterize a certain molecule made out of m spherical segments. The integral that appears in eq 10 can be seen as a temperature average of Φ:
R
⟨E⟩ dβ
Ns 2V
As a result, the final expression for the residual Helmholtz free energy due to the confinement effect is
where Ns is the number of spherical segments, V is the volume, β = 1/ (kBT), kB is the Boltzmann constant, T is the absolute temperature, and Z(Ns, V, β) is the configurational integral. Integrating eq 1 from β = 0 to β, as done in the generalized van der Waals theory:43
V exp[− Ns!Λ3Ns
(8)
Φ(σ , λ ; Ns , V , β) =
(1)
s
Q (Ns , V , β) =
(7)
where:
⎛ ∂ ln Z(Ns , V , β) ⎞ ⟨E⟩R = ⟨E⟩ − ⟨E⟩IG = − ⎜ ⎟ ∂β ⎠N ,V ⎝
β
∫Ω g(x) dΩ
⟨E⟩CONF = − NsΦε
THEORETICAL DEVELOPMENT
Ns
Ns 2ε 2V
where Ω represents the confined space where the fluid is attracted to the solid wall. The energy in eq 5 is no longer the residual energy, because the introduction of the square-well potential imposes a hard-sphere behavior at the limit of β → 0 adopted in eq 2. Equation 7 can be further rearranged as
SAFT equations of state were formulated within the canonical ensemble. Hence to keep consistency, our proposed extension for confined fluids is also developed within the canonical ensemble. The canonical average of the residual energy, ⟨E⟩, is given by
ln Z(Ns , V , β) − ln Z(Ns , V , β = 0) = −
(6)
(5) 11292
DOI: 10.1021/acs.langmuir.7b02686 Langmuir 2017, 33, 11291−11298
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Langmuir An exact representation of Φ would require a complete knowledge of the structure of the confined fluid. The approach we develop here simplifies such a task considering a decoupling of two types of interactions: fluid−fluid and solid−fluid. Such a decoupling allows us to consider that the attractive potential between the solid and the fluid as an extra attractive potential between the fluid molecules when they are at a certain distance from the solid surface. Thus, one would have two different, overlapping, and concurrent structures: one coming from the fluid−fluid interaction, and another imposed by the wall on the fluid. This is an approximation, because the fluid has a single structure but, in terms of the average energy, one may conjecture that the effect of such an approximation vanishes. Therefore, the attraction imposed by the solid on the fluid can be calculated from the structure of a bulk fluid interacting through such a potential subject to the geometrical restraints of the confinement. One must keep in mind that such an approximation underlies the developments shown in the next sections. Models for Φ. Travalloni et al.25 have suggested the following expression for Φ:
Φ = Φ0 + (1 − Φ0)(1 − e
−βε
(2rp/ σ /(2λ − 1)) ⎛ η ⎞ ⎟ ⎜ )⎜1 − ⎟ ηmax ⎠ ⎝
Inserting eq 20 in eq 19: +∞
g (ξ) ≈ g (1) ∑ (− 1)k (k + 1)(ξ − 1)k
Neglecting higher order terms and considering only the linear contribution, eq 21 may be rewritten as g (ξ) ≈ g (1)(3 − 2ξ)
(22)
Thus, one possible theoretical expression for Φ may be written as
Φ = Φ0g(1)(3 − 2ξ)
(23)
where Φ0 accounts for the geometric features of the pore. The other two terms in eq 23, g(1) and ξ, must be evaluated from specific models. Model for g(1). Gil-Villegas et al.45 proposed the following expression for g(1) of square-well fluids:
g (1) =
⎡ λ ∂g ∂g ⎤ 1 − η/2 eff + εgeff + ε(λ 3 − 1)⎢ − η eff ⎥ 3 ∂η ⎦ (1 − η) ⎣ 3 ∂λ (24)
where for 1.1 ≤ λ ≤ 1.8:
(14)
where Φ0 is the value of Φ for a hypothetical situation in which no correlation between particles is found, i.e., for g(x) = 1: ⎛ 2rp/σ + 1 − 2λ ⎞ ⎟⎟ Φ0 = 1 − ⎜⎜ ⎝ 2rp/σ − 1 ⎠
(21)
k=0
geff =
1 − ηeff /2 (1 − ηeff )3
(25)
ηeff = c1η + c 2η2 + c3η3
(26)
⎛ c1 ⎞ ⎛ 2.25855 − 1.50349 0.249434 ⎞ ⎛ 1 ⎞ ⎟ ⎜ ⎟ ⎜c ⎟ ⎜ ⎜⎜ 2 ⎟⎟ = ⎜⎜− 0.669270 1.40049 − 0.827739 ⎟⎟ × ⎜⎜ λ ⎟⎟ ⎝ c3 ⎠ ⎝ 10.1576 − 15.0427 5.30827 ⎠ ⎝ λ 2 ⎠
(27)
d
(15)
where rp is the characteristic length of the pore, e.g., for a cylindrical pore rp is the pore radius; d is the confinement dimensionality: d = 1 for slit pores, d = 2, for cylindrical pores, d = 3 for spherical pores, and noninteger values of d indicate fractal pores. Moreover, η = Nsπσ3/6V is the packing fraction and ηmax is the maximum packing fraction for that specific pore geometry. Travalloni et al.25 have also correlated an empirical expression for ηmax:
Figure 1 shows a comparison between the values of g(1) calculated using eq 24 and the results of Monte Carlo simulations for
3
ηmax =
∑ δj exp[ωj(1 − 2rp/σ )] (16)
j=1
where δ1 = 0.606316, δ2 = −0.250942, δ3 = 0.311601, ω1 = 0, ω2 = 0.3104305, and ω3 = 2.006885 for cylindrical pores. The question remains if it is possible to derive a theoretical expression for Φ. Deriving an exact expression for Φ is rather impractical for it requires an analytical expression for the pair correlation function. Nevertheless, taking eq 9 and applying the mean value theorem to it, one can write Φ = g (ξ)
Ns 2V
∫Ω dΩ
(17)
which leads to
Φ0 ≡
Ns 2V
∫Ω dΩ
Figure 1. Values of the radial distribution function for a supercritical square-well fluid (λ = 1.5) at the contact (x = 1) as a function of the packing fraction for (βε)−1 = 1.5, 2.0, 2.5, and 3.0. Closed circles, Monte Carlo simulations results with standard deviations smaller than the symbol size. Continuous lines, model proposed by Gil-Villegas et al.45
(18)
The confined space represented as Ω must be restricted to the interval between 1 and λ, which is the region where the solid walls attract the fluid molecules, if one considers a square-well potential. Therefore, one might consider that g(ξ) can be written as a Taylor series expansion around ξ = 1: +∞
g (ξ) =
∑ k=0
k 1 ⎛ ∂ g (x) ⎞ ⎜⎜ ⎟⎟ k! ⎝ ∂x k ⎠
supercritical square-well fluids (λ = 1.5). The Monte Carlo simulations were carried out for the bulk square-well fluid and the details of such simulations can be found in a subsequent section. Despite the deviation at low densities, the overall agreement is acceptable, showing that such an expression is a reasonable model to calculate g(1). The disagreement at low densities occurs because of an internal inconsistency of the model, which does not satisfy the following ideal gas limit:
(ξ − 1)k x=1
(19)
We can safely assume that at contact (x = 1) the pair correlation function of a square-well fluid can be well approximated by46 g (x) ≈
g (1) x2
lim g (1)e−βε = 1
(20)
η→0
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(28) DOI: 10.1021/acs.langmuir.7b02686 Langmuir 2017, 33, 11291−11298
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Langmuir Nonetheless, for confined fluids, one would be more interested in what happens at higher densities and, hence, this model seems to be reasonably accurate. Model for ξ. Solving eq 22 for ξ, one gets:
ξ=
g (ξ) ⎤ 1⎡ ⎢3 − ⎥ 2⎣ g (1) ⎦
each translation trial, obeying periodic boundary conditions.47,48 The acceptance ratio was set to 0.5. All the systems were equilibrated during runs of 1.08 × 108 steps. For the production step, 5 blocks of 5.4 × 106 steps each were run. The radial distribution function was then calculated from the stored position coordinates.
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RESULTS AND DISCUSSION The phase equilibrium criteria for the adsorption are given by the equality of temperature and the equality of the chemical potential in both phases:
(29)
where g(ξ) can be calculated from the radial distribution function obtained via Monte Carlo simulations as g (ξ) =
3 λ −1 3
∫1
λ
g (x)x 2 dx
(30)
μads (λ , ε ; T , ρads ) = μ bulk (T , ρbulk )
To correlate the values of ξ obtained through molecular simulations, we propose the following temperature-independent expression:
ξ=
⎛ λ + 1⎞ ⎛ λ − 1⎞ ⎜ ⎟ + ⎜ ⎟tanh[(α λ + β )η + α λ + β ] 1 2 1 2 ⎝ 2 ⎠ ⎝ 2 ⎠
(32)
where μads is the adsorbed phase chemical potential, ρads = Nads/ Vads is the number density of the adsorbed phase, μbulk is the bulk phase chemical potential, and ρbulk = Nbulk/Vbulk is the number density of the bulk phase, calculated at a specified bulk pressure, pbulk. Although in this work we extend the SAFT-VR Mie42 equation of state, the model derived here can be, in principle, coupled to any version of SAFT. Both interaction potential parameters, λ and ε, must be fitted to experimental data. The objective function for such fitting procedure was chosen to be the Average Absolute Relative Deviation (AARD) of the adsorbed amount, Γ = ρadsvp, where vp is the specific pore volume:
(31)
where α1 = −4.3154, α2 = 1.0397, β1 = 11.021, and β2 = −3.2542. The adoption of an expression such as eq 31 in the model introduces an empirical flavor to it. One can no longer call such a model a purely theoretical model. Nevertheless, for the sake of simplicity, we will refer to it, in this Article, as the theoretical model, acknowledging here the degree of empiricism introduced by eq 31. Figure 2 shows a comparison between Monte Carlo simulations for supercritical square-well fluids (λ = 1.5) and the proposed empirical
AARD =
1 Ndata
Ndata
calc Γ exp k − Γk Γ exp k
∑ k=1
(33)
49
The Nelder and Mead simplex method was used to search for the AARD minimum value. Adsorption of Methane, Ethane, and Propane on MSC5A. SAFT-VR Mie parameters42 for the fluid−fluid interactions of methane, ethane, and propane are shown in Table 1. MSC5A is a carbon molecular sieve with specific surface area of 650 m2·g−1 and a specific pore volume of 0.56 cm3·g−1.50 Therefore, assuming a cylindrical geometry, the pore radius of MSC5A was considered to be 1.72 nm.25 Figure 3 shows the comparison between our calculations using SAFT-VR Mie combined with the empirical expression for Φ (eq 14) as well as with the theoretical expression for Φ (eq 23) and the experimental data obtained in the literature50 for the adsorption of methane in a carbon molecular sieve (MSC5A) at 303.15 K. The results show excellent agreement between our calculations and the experimental data. Figures 4 and 5 present the experimental data and model calculations for the adsorption of ethane and propane, respectively, in a carbon molecular sieve (MSC5A) at 303.15 K. For both cases, one can see that the model using the theoretical expression for Φ gives a much better correlation of the experimental data than the one employing the empirical expression for Φ.
Figure 2. Values of ξ for a supercritical square-well fluid (λ = 1.5) as a function of the packing fraction for different values of (βε)−1. Open squares, Monte Carlo simulations results for (βε)−1 = 1.5. Open circles, Monte Carlo simulations results for (βε)−1 = 2.0. Open uptriangles, Monte Carlo simulations results for (βε)−1 = 2.5. Open down-triangles, Monte Carlo simulations results for (βε)−1 = 3.0. Continuous lines, empirical model expressed by eq 31. expression for ξ. The agreement is good and the dependence on temperature is indeed negligible. Monte Carlo Simulations. Monte Carlo simulations for a system with 108 spherical particles interacting via a square-well potential (eq 6) were carried out using an in-house code. The initial configuration was set to the one corresponding to a face centered cubic crystal. The Metropolis algorithm in the canonical ensemble was employed for
Table 1. Values of SAFT-VR Mie Parameters for Fluid−Fluid Interactions component 42
methane ethane42 propane42 carbon dioxide51 nitrogen51 water53
m
σ/Å
λr
λa
(ε/kB)/K
(εHB ab /kB)/K
Kab/Å3
1.0000 1.4373 1.6845 1.6936 1.4214 1.0000
3.7412 3.7257 3.9056 3.0465 3.1760 3.0555
12.650 12.400 13.006 18.067 9.8749 35.823
6 6 6 6 6 6
153.36 206.12 239.89 235.73 72.438 418.00
0.0 0.0 0.0 0.0 0.0 1600.0
0.0 0.0 0.0 0.0 0.0 496.66
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Figure 3. Adsorption isotherm of methane on a carbon molecular sieve (MSC5A) at 303.15 K. Closed circles, experimental data.50 Continuous line, SAFT-VR Mie + CONF with the theoretical expression for Φ. Dashed line, SAFT-VR Mie + CONF with the empirical expression for Φ.
Figure 6. Adsorption isotherms of ethane on a carbon molecular sieve (MSC5A) at different temperatures. Closed circles, experimental data.50 Continuous lines, SAFT-VR Mie + CONF with the theoretical expression for Φ. Dashed lines, SAFT-VR Mie + CONF with the empirical expression for Φ.
parameters fitted at 303.15 K. Both models are less accurate at the low and high temperatures compared to 303.15 K, but the model using the theoretical expression for Φ gives a better qualitative description of the data. Table 2 summarizes the fitted parameters and AARD values. Despite the similarities on the parameter values for both models, λ seems to be almost independent of the carbon number of the light alkane using a theoretical expression for Φ. This suggests that, should one keep such a value fixed, there would be a single parameter to be fitted, which means that the theoretical expression is a better candidate for applications. Fixing λ = 1.45 for the hydrocarbon-MSC5A interactions, the values of ε remain similar and the AARD values, even though larger, remain small. Besides that fitting aspect, a much stronger evidence is that the theoretical expression gives systematically smaller values of AARD. Indeed, a better description of the fluid structure should improve the accuracy of the calculation of a thermodynamic property. Figure 7 illustrates the % AARD surface as a function of both parameters for the adsorption of ethane in MSC5A at 303.15 K, using the theoretical expression for Φ. Despite the mathematical complexity of the model, in particular of the model for the fluid−fluid interaction, the smoothness of the surface suggests an easy search for the minimum. Thus, an algorithm as that proposed by Nelder and Mead49 seems to be sufficient to perform a reliable calculation. Adsorption of Carbon Dioxide, Nitrogen, and Water on Zeolite 13X. SAFT-VR Mie parameters for the fluid−fluid interactions of carbon dioxide,51 nitrogen,51 and water52 are shown in Table 1. Zeolite 13X was considered as having spherical pores with 0.865 nm radius and a specific pore volume of 0.3964 cm3·g−1.53 Figures 8 and 9 show the adsorption isotherms for carbon dioxide and nitrogen, respectively, at different temperatures. The extended SAFT-VR Mie equation of state with the theoretical expression for Φ was fitted to the experimental data obtained in the literature.54 No calculations were performed with the Φ expression of Travalloni et al.25 since it was shown, in the previous example, to be inferior to the theoretical expression developed here. Table 3 summarizes the fitted parameters for the fluid−solid interactions, as well as the AARD values. The AARD values indicate that the correlation ability of the model is remarkable, especially for nitrogen. Although the fluid−solid parameters are not the same for all temperatures,
Figure 4. Adsorption isotherm of ethane on a carbon molecular sieve (MSC5A) at 303.15 K. Closed circles, experimental data.50 Continuous line, SAFT-VR Mie + CONF with the theoretical expression for Φ. Dashed line, SAFT-VR Mie + CONF with the empirical expression for Φ.
Figure 5. Adsorption isotherm of propane on a carbon molecular sieve (MSC5A) at 303.15 K. Closed circles, experimental data.50 Continuous line, SAFT-VR Mie + CONF with the theoretical expression for Φ. Dashed line, SAFT-VR Mie + CONF with the empirical expression for Φ.
Figure 6 shows the temperature dependence of the adsorption phenomenon and the extrapolation capability of both models, for the adsorption of ethane in MSC5A. The data set at 303.15 K was used to fit the interaction parameters. Thus, the results at a lower (278.65 K) and at a higher (323.15 K) temperatures are predicted values using the interaction 11295
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Table 2. Values of the Fitted Fluid−Solid Interaction Parameters for the Extended SAFT-VR Mie for Confined Fluids and AARD Values, Considering the Experimental Adsorption Isotherms of Methane, Ethane, and Propane on MSC5A at 303.15 K50 empirical Φ
a
theoretical Φ
component
λ
(ε/kB)/K
AARD (%)
λ
(ε/kB)/K
AARD (%)
(ε/kB)/Ka
AARD (%)
methane ethane propane
1.966 1.685 1.506
1489.0 2150.1 2487.4
5.0 4.3 11.7
1.480 1.434 1.440
1647.2 2016.3 2111.3
4.5 0.9 5.3
1696.1 2008.0 2094.6
5.1 1.8 5.4
With fixed λ = 1.45.
Figure 9. Adsorption isotherms of nitrogen on zeolite 13X at different temperatures. Closed circles, experimental data.54 Continuous lines, SAFT-VR Mie + CONF with the theoretical expression for Φ.
Figure 7. Contour plot of the percentage average absolute relative deviation for the adsorption isotherm of ethane in a carbon molecular sieve (MSC5A) at 303.15 K as a function of the fluid−solid squarewell potential parameters: λ and ε, using the theoretical expression for Φ.
Table 3. Values of the Fitted Fluid−Solid Interaction Parameters for the Extended SAFT-VR Mie for Confined Fluids and AARD Values, Considering the Experimental Adsorption Isotherms of Carbon Dioxide,54 Nitrogen,54 and Water54 on Zeolite 13X component
T/K
λ
(ε/kB)/K
AARD (%)
carbon dioxide
298.15 318.15 338.15 373.15 413.15 298.15 318.15 338.15 373.15 413.15 313.15
1.248 1.263 1.285 1.331 1.381 1.487 1.545 1.606 1.783 2.044 1.151
1479.6 1496.3 1510.6 1495.2 1406.7 648.88 605.26 563.01 488.07 437.61 1956.3
1.0 1.9 3.0 4.9 3.8 0.41 0.72 0.53 0.59 2.3 28.7
nitrogen
Figure 8. Adsorption isotherms of carbon dioxide on zeolite 13X at different temperatures. Closed circles, experimental data.54 Continuous lines, SAFT-VR Mie + CONF with the theoretical expression for Φ.
water
they are consistent with each other and exhibit a trend that embraces what is missing in the proposed model in terms of the temperature dependence. Differently from the experimental data set for light hydrocarbons on MSC5A, the bulk pressure range for carbon dioxide and nitrogen adsorption on zeolite 13X is broader and goes up to 10.0 bar, showing that the model also works at higher pressures. One of the advantages in using an equation of state such as SAFT-VR Mie is the possibility to calculate properties of associating fluids, e.g., water, albeit other strategies to describe such molecules can be considered as well. Figure 10 shows the adsorption isotherm of water vapor on zeolite 13X at 313.15 K. The extended SAFT-VR Mie equation of state with the theoretical expression for Φ was fitted to the experimental data set of Kim et al.53 Although the model is able to correlate the experimental data well for intermediate bulk pressure values,
Figure 10. Adsorption isotherm of water on zeolite 13X at 313.15 K. Open circles, experimental data.53 Continuous lines, SAFT-VR Mie + CONF with the theoretical expression for Φ.
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only a qualitative agreement is observed for very low values of the bulk pressure. The model exhibits a capillary condensation at a pressure higher than the one experimentally observed. The second issue is observed at higher pressures, where the experimental loading starts a second growth. Two hypotheses may explain it: this behavior could correspond to the initial stage of a hysteresis, which cannot be described by our model because it only considers equilibrium states, or this behavior could correspond to a solid material with multiple pore sizes, in which case our model is also limited because it considers a single pore size. The experimental value for the loading at 7.022 kPa for the adsorbed water vapor on zeolite 13X is 17.40 mol·kg−1.53 Adopting a specific pore volume of 0.3964 cm3·g−1, this loading corresponds to an adsorbed phase density of 790.1 kg·m−3, whereas, in the bulk phase at 7.022 kPa and 313.15 K, the NIST value for water density is 0.0487 kg·m−3.55 From it, one can infer that, even at very low bulk pressures, the fluid inside the pore can achieve a dense state, for which a good description of the fluid−fluid interactions is needed. Finally, an implicit inhomogeneous model as the one proposed here may be useful for engineering applications, but since its derivation is based on the molecular interactions between the fluid and the solid, the numerical values of the fitted parameters have a strong physical meaning. From Table 3, one can see that the interactions between water and zeolite 13X are much more short-ranged than the ones with carbon dioxide or nitrogen, but also have a deeper attractive well, which means that the water molecules are strongly attracted by zeolite 13X. However, the extent of such attractive forces is somehow shorter than the ones observed for carbon dioxide or nitrogen.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: lfranco@feq unicamp.br. ORCID
Luís F. M. Franco: 0000-0002-9334-9660 Ioannis G. Economou: 0000-0002-2409-6831 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This publication was made possible by NPRP Grant Number 81648-2-688 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.
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CONCLUSIONS In this proof-of-concept work, we demonstrated a method to extend the SAFT-VR Mie equation of state to calculate adsorption isotherms. This extension depends implicitly on the structural arrangement for the fluid on the adsorbed phase. Two models, one empirical, previously suggested in the literature, and one theoretical, developed here, were coupled to the SAFT-VR Mie equation of state to calculate the adsorption isotherms of methane, ethane, and propane in a carbon molecular sieve, considered as a cylindrical pore. The theoretical expression proposed here seems to provide a better correlation of the experimental data, showing that a better description of the fluid structure results in a more accurate estimation of thermodynamic properties. The adsorption isotherms of carbon dioxide, nitrogen, and water on zeolite 13X, considered as having spherical pores, were also studied. Although good correlation was found for both carbon dioxide and nitrogen, the limitation of the model to describe metastates is discussed for the case of water adsorbed on zeolite 13X. The model can be applied to formulate problems with multiple pore sizes, each of them modeled by the extended equation of state with specific parameters for the fluid−solid interactions. The thermodynamics of confined fluids has still a long journey ahead of it, but we believe this model opens new ways of performing adsorption calculations using a theoretical approach. The extension to fluid mixtures remains to be addressed in future work. 11297
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