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Sketching a Portrait of the Optimal Adsorbent for CO Separation by PSA Edder J García, Javier Pérez-Pellitero, Christian Jallut, and Gerhard D. Pirngruber Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b04877 • Publication Date (Web): 06 Apr 2017 Downloaded from http://pubs.acs.org on April 10, 2017

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Industrial & Engineering Chemistry Research

Sketching a portrait of the optimal adsorbent for CO2 separation by PSA Edder J. GARCÍA a,c, Javier PÉREZ-PELLITERO a, Gerhard D. PIRNGRUBER a*, Christian JALLUT

b

a

IFP Energies nouvelles, Rond Point de l'échangeur de Solaize, 69360 Solaize, France.

b

Université de Lyon, Université Lyon 1, Laboratoire d'Automatique et de Génie des

Procédés, UMR 5007, CNRS—ESCPE, 43, Bd du 11 Novembre 1918, 69622 Villeurbanne cedex, France. c

Current address: Laboratory of Engineering Thermodynamics, University of Kaiserslautern,

Erwin-Schrodinger-Str. 44, 67663 Kaiserslautern, Germany. * Email: [email protected] KEYWORDS: adsorbent selection criteria, CO2 adsorbents, cage zeolites, carbon sequestration, adsorbent design.

Abstract In silico screening of CO2 adsorbents is a very powerful method for pre-selecting the most promising porous solids for experimental studies. Due to the increasing computational power, it is now possible to investigate a large number of adsorbents in a fairly short time. However, it remains difficult to rationalize structure-performance correlations because the complexity of such porous materials cannot be reduced to a few simple descriptors. In this paper, we present a different approach that was applied to design an optimal adsorbent for CO2 1 ACS Paragon Plus Environment


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separation from gas mixtures. A CO2-CH4 mixture was used as feed. We constructed a simplified model of the porous material by adopting a spherical pore geometry. From the dispersion-repulsion point of view, the spherical pore was modeled by homogenously distributed Lennard-Jones sites. Besides, a charge distribution was introduced to mimic the electrostatic behavior. The selected charge distribution is constituted by negative charges homogeneously distributed over the surface of the pore. The total negative charge is counterbalanced by eight positive discrete charges placed on the corners of the cube inscribed in the sphere. This model can be characterized by two main descriptors: the pore size and the cation charge. For this model, the adsorbate-adsorbent interaction potential due to electrostatic and dispersion-repulsion interactions was determined. Then, the Henry constants of CO2 and CH4 are calculated from statistical thermodynamics and substituted in the Ruthven Statistical Model, which allows calculating the binary adsorption isotherms. Finally, the adsorbed quantities are used to estimate performance indexes of the pressure swing adsorption (PSA) process, such as the working capacity and the separation factor. We show how these two parameters depend on pore size and cation charge. For a medium pressure PSA process, the optimal adsorbent has large pore volume and is highly charged, which provides high working capacity and selectivity. The high cationic charge allows filling the large pore volume efficiently and assures high CO2 selectivity. 1.

Introduction The use of adsorption processes to selectively capture CO2 from gas mixtures – such as

synthesis gas, natural gas, biogas, or flue gas– is part of the panel of solutions for reducing anthropogenic CO2 emissions by Carbon Capture and Storage (CCS). Recently, the urgency of mitigating CO2 emissions was emphasized once again during the COP21 meeting on climate change.1 Pressure Swing Adsorption (PSA) is an efficient and low-energy alternative

2 ACS Paragon Plus Environment

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for CO2 capture and gas upgrading. The CO2-PSA process is based on the selective physical adsorption of CO2 on a porous material. Investment and operating costs of adsorption-based processes depend on the adsorbent properties. The importance of the topic has stimulated an intensive research effort in materials science, which has led to the discovery of numerous new potential adsorbents for CO2.2–14 The quest for better-performing CO2 adsorbents could be rendered more efficient if the properties of the ideal solid for CO2 adsorption could be described. Recently, impressive progress has been made using computational methods to carry out selection from a large number of porous materials and for pre-selecting the most promising candidates as a guideline for experimentalists.15–22 All of these works rely on atomistic simulations of the adsorption isotherms, which are then linked to performance indexes describing the efficiency of the separation process, such as the working capacity, parasitic energy, energy penalty, cost, etc. This method allows investigating new adsorbents independently of their chemical composition and pore structure, which is convenient in studying zeolites, MOFs, ZIFs, and others. Covering such a huge range of structural diversity is a proven strength of the atomistic approach, but it is often challenging to reduce the data to identify general trends, although attempts in that sense have been made. Therefore, we chose a different approach to the problem. Instead of calculating adsorption isotherms from atomistic simulations, we aim at a simpler description of the solid by using idealized pore geometries, for instance spheres and cylinders. These idealized pore models do not fully capture the behavior of complex pore systems, but they allow studying the influence of all properties of the solid in a coherent manner. This strategy leads to a better understanding of the relationship between the properties of the solid and its performance in



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the separation process, which leads to general trends that can be used for the design of new adsorbents. As a first step, the overall solid-CO2 interaction within the framework of the simplified model is obtained. Then, this interaction is used to calculate the Henry constant, which controls the initial slope of the adsorption isotherm. Subsequently, the Henry constant is employed in an adsorption model, together with fluid-fluid interaction parameters, to model the isotherm. As a result, the adsorption isotherm is expressed in term of parameters describing the solid and the adsorbates. Therefore, the first task in this approach is to find a suitable model of the solid that allows us to change its properties systematically, and to calculate the overall CO2-solid interactions. The most important intermolecular interactions in the physisorption of CO2 are dispersionrepulsion and electrostatics. The dispersion-repulsion interactions are commonly modeled using a 12-6 Lennard-Jones potential while the electrostatic interactions are modeled by point charges. To simplify the description of the dispersion-repulsion interaction, we distinguish two idealized pore geometries: (1) cage-like pores, which can be modeled by spherical pores; and (2) channel-like pores, which can be modeled by cylindrical pores. For spherical and cylindrical pores, it is possible to analytically integrate the 12-6 Lennard-Jones potential over the pore surface to express the total CO2-solid dispersion interaction as a function of simple descriptors.23,24 These descriptors are the pore size, which in turn determines porosity; the density of atoms on the surface; and the Lennard-Jones parameters. The two latter depends on the chemical composition of the solid, while the pore size is a geometrical descriptor. The quadrupole moment of CO2 strongly interacts with the gradient of the electric field generated by a charged solid. This electrostatic interaction mainly depends on the charge distribution of the solid. In porous materials, such as zeolites or MOFs, the electric field


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within the solid is mainly generated by extra-framework cations while the negative charges are generally more evenly distributed over the framework atoms. Based on this premise, we propose a simple model of charge distribution. For the spherical pore system, the positive charges are placed on the corners of a cube (circumscribed by the sphere), while the negative charges are evenly distributed over the surface of the sphere. For this simple charge distribution, we have shown that the CO2-solid electrostatic interaction is quite similar to that obtained from an all-atom model of a zeolite pore with the same pore radius.25 By numerically integrating the Coulombic interactions in the pore volume, it is possible to calculate the Henry constant due to electrostatic interactions as a function of the pore diameter and the charge of the cations. By combining the dispersion-repulsion and the electrostatic interactions, the total CO2-solid interaction and the corresponding Henry constant can be obtained. Once the Henry constants have been calculated for the simplified model, we can establish the link between the performance of the adsorbent in a CO2-PSA process and the pore diameter and cation charge of the solid. This paper is organized as follows: first, a brief description of the basic principles of the PSA process is presented in Section 2.1. In Section 2.2, the calculation of the parameters used for the adsorbent selection, i.e. the working capacity and the separation factor, is explained. Then, the multicomponent adsorption model used to calculate the working capacity and separation factor is described. In Section 3, we explain the calculation of the Henry constant for the simplified porous solid. The calculation of the dispersion-repulsion and electrostatic interaction are specified in Sections 3.1 and 3.2, respectively. The results are presented in Section 4. In this part, we study the evolution of the Henry constant, the working capacity and the separation factor as a function of pore size and cation charge. Then, a link between the solid properties and the performance in the separation process is presented. For the sake



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of brevity, only the case of the spherical pore system is considered whereas results for cylindrical pores are shown in the Supporting Information. Based on these results, the optimal properties of the adsorbent for CO2 separation from CO2-CH4 mixtures are discussed. Finally, the conclusions are given in Section 5. 2.

Equilibrium representation of a PSA process

2.1 Basic principles of the PSA process This process involves two basic steps.26,27 First, an adsorption step is carried out by introducing the feed mixture into the adsorption column at high pressure (pads). CO2 is fixed on the adsorbent material by dispersion-repulsion and by electrostatic interactions. CO2 must be adsorbed selectively with respect to the other components of the mixture, including CH4, CO, N2, H2, etc. After a certain time, the adsorbent gets saturated by CO2; at that point CO2, starts to breakthrough at the column exit, and the adsorbent requires regeneration. Then, the adsorption step is interrupted, and the adsorbent is regenerated by injecting pure CO2 at a lower pressure (pdes). CO2 and co-adsorbed components are desorbed partially. In the cyclic steady state, the amount of CO2 desorbed during the regeneration step is equal to the amount of CO2 captured in the adsorption step. This amount is called CO2 working capacity (∆nCO2). On the other hand, the ratio of ∆nCO2 and the working capacity of other compounds in the mixture is called separation factor (SF). It depends on the adsorbent's selectivity for CO2. 2.2 Performance indexes of a PSA process: the working capacity and the separation factor We use the working capacity and the separation factor as performance indicators of a PSA process for CO2 separation from CO2-CH4 mixtures. These performance indexes are calculated by considering that the gaseous and adsorbent phases are at equilibrium in the PSA process. The working capacity of component i is defined as 6 ACS Paragon Plus Environment

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∆ = , , − , , (1) where is the adsorbed amount of i (in our case i = CO2 or CH4), p is the pressure and Xi is the molar fraction of i in the gaseous phase. The subscripts ads and des denote the adsorption and the desorption step, respectively. ∆ is the amount of CO2 captured during a PSA cycle and therefore is related to the volume of the adsorption column required to carry out the separation. The ratio of the working capacity of CO2 and that of the competing components (CH4 in our case) is called separation factor SF =

∆nCO 2 ∆nCH 4

(2)

The separation factor is related to the purity of the extracted CO2. At the end of the desorption step, the feed used to regenerate the column is close to pure CO2 (XCO2,des~ 1), so the separation factor can be expressed as SF =

∆nCO 2 nCO 2,ads ⋅ nCO 2,ads nCH 4,ads

(3)

where nCO2,ads and nCH4,ads are the adsorbed amount of CO2 and CH4 during the adsorption step. The ∆nCO2/nCO2,ads ratio gives the adsorbent regenerability while the nCO2,ads/nCH4,ads ratio is proportional to the CO2/CH4 selectivity (α), which for a binary mixture is defined as

α =

nCO 2 X CH 4 nCO 2 (1 − X CO 2 ) ⋅ = ⋅ nCH 4 X CO 2 nCH 4 X CO 2

(4)

Xi is the mole fraction of compound i during the feed during the adsorption step. The ideal

adsorbent for a CO2-PSA process should exhibit a high working capacity of CO2, reducing the size of the adsorption column needed to adsorb a given amount of CO2 per cycle. At the same time, the ideal adsorbent should exhibit a low working capacity for the other components of the mixture, increasing the purity of the CO2 product stream. High-purity CO2 is needed to facilitate its compression, transport, storage, and chemical transformation. Therefore, the 7 ACS Paragon Plus Environment


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perfect adsorbent combines high working capacity of CO2 and infinite selectivity, i.e. a zero working capacity for other compounds. Unfortunately, this ideal adsorbent does not exist because high selectivity only can be achieved if the CO2-solid interaction is strong, which renders desorption of CO2 difficult, thereby reducing the working capacity of CO2. On the contrary, a weak CO2-solid interaction favors regeneration, but the working capacity of CO2 and selectivity are penalized. Therefore, the search for the best adsorbent for the PSA process implies identifying the properties of the solid that lead to the best trade-off between selectivity, regenerability, and CO2 working capacity. In a recent work, we developed a mathematic formalism that allows extracting quantitative information from the qualitative trends mentioned above.28 For a series of zeolites with similar pore size but different polarity, we defined the strength of the CO2 adsorption (expressed in terms of the Henry constant) that yields the best compromise between the maximization of the working capacity of CO2 and the minimization of the working capacity of the other compounds. The remaining challenge is to make the link between the optimal Henry constant, defined in terms of the working capacity and the separation factor, and the main descriptors of the solid, such as pore size and cation charge. To simplify the calculation of the working capacity and the separation factor, we make the approximation that the entire adsorption column is in equilibrium with a CO2-CH4 mixture during the adsorption step and with pure CO2 during the desorption step. Therefore, the existence of a mass transfer zone is neglected. Thermal effects (temperature rise and fall during adsorption and desorption) are also ignored. By making these simplifications, the working capacities of CO2 and CH4 and the separation factor can be calculated from the multicomponent adsorption isotherm. Adsorption isotherms are commonly expressed in terms of the fractional loading of the pore space (θ). For a given set of operating conditions, the working capacity is calculated


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from the difference between the fractional loadings during the adsorption and desorption step, according to: ∆ = , , , , − , ,

(5)

Using equations 3 and 4, we obtain the separation factor as a function of the ratio of the fractional loadings. SF =

∆ ∆ = ∙∙ ∆ 1 −

(6)

2.3 The RSM adsorption isotherm model The objective of our work is to link the working capacity and the separation factor to structural descriptors of the adsorbent. For this purpose, the Ruthven Statistical Model (RSM)29,30 is the most appropriate choice among the available multicomponent adsorption models for cages-like pores because the RSM is derived from statistical thermodynamics and describes the adsorption isotherm of a set of freely mobile adsorbate molecules in a cage. The single-component RSM equilibrium equation is s s max   s 2ε ff   ( Kp ) s (bs max ) s  s    1 −  exp  Kp (bs max ) + ∑  s max kT   ( s − 1)! s=2 1   s max    θ=   s 2 s s max s max    s ε   ( Kp ) (bs max ) s ff  1 −  exp 1 + Kp (bs max ) + ∑  s max kT   s! s =2  s max    

(7)

where θ is the coverage, i.e. the fractional loading of the pore volume, and K is the Henry constant,

which

characterizes

the

adsorbate-fluid

interaction.

Adsorbate-adsorbate

interactions are taken into account for via a mean-field Sutherland potential, characterized by the depth of the potential well (εff). The saturation of the pore volume (smax) is given by the ratio of the pore volume (v) and the excluded volume of the adsorbate molecules (b). smax = v / b

(8)



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In Ruthven's original work, the pore volume corresponds to the volume of a single zeolite cage, and smax is the number of molecules that fits into a single cage. In this paper, we use an arbitrary multi-cage system, i.e. smax was arbitrarily fixed to 100. Thus, we take a multi-cage section of the solid where 100 molecules fit. We have shown that the multi-cage model is equivalent to the single-cage description if the number of adsorbed molecules per cage is large.31 The main advantage of this method is that it allows applying the RSM in solids with different pore shape, for instance cylindrical pores. The RSM can be readily extended to multicomponent adsorption.30 In the case of a binary mixture, the coverage of component 1 is given by: j max s max   ( K1 pX 1 ) s ( K 2 pX 2 ) j (b1smax ) s (b2 jmax ) j K pX b s + Z ff , sj ( ) ∑ ∑  1 1 1 max  ( s − 1)! j! 1  j=2 s=2  θ1 = j max s max s j s j  ( K1 pX 1 ) ( K 2 pX 2 ) (b1smax ) (b2 jmax ) smax  Z ff , sj  1 + K1 pX 1 (b1smax ) + K 2 pX 2 (b2 jmax ) + ∑ ∑ s j ! ! j =2 s=2  

(9)

where j + s ≥ 2, b1 and b2 are the excluded volumes of the adsorptive molecules 1 and 2, X1 and X2 are the molar fractions of molecules 1 (CO2) and 2 (CH4) respectively. The term Zff,sj is the configurational integral of the fluid-fluid interactions. In a system with s molecules of type 1 and j molecules of type 2 interacting through a Sutherland potential, the configurational integral is given by: sb jb   Z ff , sj = 1 − 1 − 2  v v  

i+ j

β  exp  s 2 ⋅ b1 ⋅ ε ff ,1 + s ⋅ j ⋅ b12 ⋅ ε ff ,12 + j 2 ⋅ b2 ⋅ ε ff , 2  v 

(

)

(10)

where β = 1/kT. The crossed terms εff,12 and b12 are obtained from Lorentz-Berthelot combining rules. Here, we use the same fluid-fluid interaction parameters applied in a previous work (Table 1).28,31 For CO2, the single-center LJ parameters were optimized to reproduce the liquid-vapour equilibrium while the point charges were taken from the EPM2 model by Harris and Yung.32


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Table 1. Pair interaction parameters.

Fluid

Solid surface33

σ (Å)

ε (K)

b / (Å3/molecule)

CO2 (fluid-fluid)

3.64

232

60

CO2 (solid-fluid)

3.72

203

CH4

3.73

150

O

3.00

93

62

At low pressures, the selectivity (α) calculated by the RSM model tends towards the ratio of the Henry constants K1/K2.28 If intermolecular interactions are weak, then the selectivity remains constant as a function of pressure. On the contrary, if component 1 has a strong intermolecular interaction compared to component 2, the adsorption selectivity for component 1 increases with the pressure due to the increasing favorable fluid-fluid interactions in the adsorbed phase. For zeolites working in the range of pressure considered here (1 to 5 bar), the CO2/CH4 selectivity can be roughly estimated by the ratio of the Henry constants.28 Therefore, this approach is applied in this work.28 The RSM calculates a fractional loading of the pore volume θ, i.e. the ratio of the pore volume occupied by the adsorbate to the total pore volume. Normally, adsorbed amounts are normalized by the total mass or volume of the adsorbent. To convert the fractional loading into an adsorption capacity per volume of adsorbent (n), we multiplied θ by the porosity of the adsorbent (φ). A correlation between the pore radius and the porosity is presented in the Supporting Information. The key parameters to link the adsorption behaviour, via the working capacity of CO2 and separation factor, and the properties of the solid using the RSM model are the Henry constants. They describe the strength of the interaction between CO2 or CH4 with the solid. In



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the next section, we discuss how to establish the link between the Henry constants and structural descriptors of the solid. 3.

Calculation of the Henry constant from structural descriptors of the solid According to statistical mechanics, the Henry constant (K) is obtained by integrating the

total solid-adsorbent potential Esf over a representative volume of the solid.26

1 % exp)−*+ ,-./!"01= !"#$$ 3455

(11)

where r is the position vector inside the pore. K was normalized by the accessible pore volume (vacc). For CH4, only dispersion-repulsion interactions are taken into account. On the other hand, for CO2, we consider two contributions to the adsorption potential: dispersion-repulsion (Edis) and electrostatic interactions (Eelec). Esf(r) = Edis(r) + Eelec(r)

(12)

The next task is to obtain both solid-fluid interactions as a function of r. The calculations of the dispersion-repulsion and electrostatic interactions are detailed in the following section. 3.1 Contribution due to dispersion-repulsion interaction As stated above, to simplify the description of the solid, a spherical pore is used to model the dispersion-repulsion interactions between the solid and the adsorbate molecules. For spherical or cylindrical pores interacting with a single-center particle, it is possible to integrate analytically the solid-fluid LJ potential over the pore surface to obtain explicit equations of the total dispersion-repulsion potential at a given radial distance (r).23,24 The


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resulting equations explicitly take into account the confinement effect, which depend on the density of the solid atoms and the curvature of the pore.34 The equation to calculate the total pore-fluid dispersion interaction is given in the Supporting Information. Edis is a function of the following parameters: •

The depth of Lennard-Jones potential well (εsf).

•

The density of atoms on the surface of the solid (ρsur).

•

The density of atoms per volume of the solid, which is linked to the porosity.

•

The pore size.

For a given family of adsorbent materials, for example zeolites, εsf and ρsur are fairly constant. Edis is proportional to the product of εsf and ρsur. We have, therefore, defined a dimensionless parameter called surface energy density (SED*) as SED∗ =

< 9:; =:>? @:;

AB

(13)

where σsf is the Lennard-Jones solid-fluid diameter. The surface energy densities of CO2 and CH4 are slightly different because the Lennard-Jones parameters are different for both molecules. For the simultaneous adsorption of CO2 and CH4, we refer to the SED* values for CO2. For zeolites, the typical values of SED* are in the range of 0.79 – 0.94. Only oxygen atoms are considered to model the spherical pore. The LJ parameter for this atom is shown in Table 1. The crossed solid-fluid LJ parameters were calculated using Lorentz-Berthelot combining rules. The total dispersion-repulsion potential was obtained by adding the contributions of a finite number of layers of atoms in the solid around the pore. The layers are separated by a distance that corresponds to the density of the solid. The density of the solid depends on the porosity, which in turn is a function of the pore size (see Equation S3 in the Supporting Information). Thus, aside from SED*, the only remaining independent parameter is the pore size.



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Integrating the adsorption potential over the accessible pore volume yields the Henry constant as a function of the pore radius and the surface energy density.31 3.2 Contribution due to electrostatic forces In zeolites, positive charges are localized on extra-framework cations while negative charges are distributed over the oxygen atoms in the solid framework. The supercage of zeolite A, for example, has a roughly spherical shape and contains eight cations in the corners of a cube. We have demonstrated that the charge distribution of zeolite A can be represented by a simplified charge distribution. This simplified distribution consists of eight positive point charges in the corners of a cube, and the negative charges evenly distributed over the surface of a sphere.25 Surprisingly, the CO2-solid electrostatic potential produced by the simplified charge distribution is quite similar to that obtained by a full-atom model. This simplified representation allows studying the evolution of the electrostatic interactions and the Henry constant as a function of the cation charge (z) and the pore radius. Figure 1 shows a schematic representation of the spherical pore and the simplified charge distribution. The negative charges are homogenously distributed over the sphere while positive charges are placed in the corners of the cube inscribed inside the sphere. The density of negative charges was kept at 0.2 anions/Å2. The crystal structure of the solid is a primitive cube.


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Figure 1. Schematic representation of the spherical pore and the simplified charge distribution. Positive charges: green; negatives charges: red. The lines connecting the cations are placed just as a guide for the eye. The semitransparent sphere represents the LennardJones potential used to model the dispersion-repulsion interactions.

The point charges were taken from the EPM2 model by Harris and Yung32, which gives a correct representation of the quadrupole moment of CO2. The electrostatic interaction between the solid and CO2 was calculated by the Ewald summation technique. The average electrostatic interaction (E(r)elec) at a given position r was calculated by averaging all the possible orientations of the CO2 quadrupole using a Boltzmann factor.

E elec

2π

π

0

0

∫ ∫ (r ) =

E elec (Π ' , ω ' , r ) exp(− E elec (Π ' , ω ' , r ) / kT ) dΠ ' dω ' 2π

π

0

0

∫ ∫

(14)

exp( − E elec (Π ' , ω ' , r ) / kT ) dΠ ' dω '

where Π’ and ω’ are the angles describing the orientation of the CO2 quadrupole. Eelec(r) is only a function of the position in the pore, pore size, and charges.



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3.3 Integrating dispersion-repulsion and electrostatic interactions The total CO2-solid interaction was calculated by adding the electrostatic and the LennardJones interactions, see Eq. (12). The contribution of both interactions was added up at each position (r). For CH4, only dispersion interactions were taken into account. The Henry constant was obtained by numerical integration over the accessible volume, i.e. by integrating Eq. (11) from the center of the pore to the accessible pore radius, Racc. To evaluate the integral, the trapezoidal rule was used with a grid space of 0.1 Å. The accessible pore volume is delimited by the accessible pore radius. For spherical pores, the radius (R) is the distance between the atoms in the wall and the center of the pore. On the other hand, the accessible pore radius (Racc) is the difference between R and the Lennard-Jones solid-fluid collision diameter σsf, Racc = R - σsf 4.

(15)

Results and discussion

For all calculations in this work, the temperature was kept at 303 K. During the adsorption step, an equimolar CO2 - CH4 mixture was considered. This composition is typically found in biogas. The pressure during the adsorption step of the PSA cycle was fixed to 5 bar while it was set to 1 bar during the desorption step. 4.1 Henry constants Figure 2 shows the Henry constant of CH4 in a spherical pore as a function of the accessible pore diameter. As the pore diameter decreases, the Henry constant increases due to the confinement effect, which amplifies the dispersion interaction between the solid and CH4. At very small pore diameters, the Henry constant is also expected to drop because repulsion forces become dominating, but this domain is only reached when the accessible pore diameter


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is close to zero. Furthermore, there is a roughly linear relationship between the logarithm of the Henry constant of CH4 and the surface energy density.

Figure 2. Henry constant of CH4 at 303 K for the spherical pore as a function of the accessible pore diameter for different surface energy densities (SED*). Only CH4-solid dispersion interactions are taken into account. In the case of CO2, two types of interactions contribute to the Henry constant: dispersionrepulsion and electrostatics. The dispersion interactions gradually diminish when the pore diameter increases. Concerning the electrostatic interactions, we showed in a previous work that the average adsorption potential goes through a maximum as a function of the pore diameter.25 A brief summary of our earlier work will guide our interpretation of the Henry constants of CO2: for large pores, the symmetry effects cancel out the gradient of the electric field in a significant fraction of the pore volume. Specifically, the gradient of electric field is zero at all inversion points of the unit cell and at the center of the pore. In addition, for large pores, the combined interaction of the cations with CO2 is reduced, due the large distances between the positive charges. Consequently, an important fraction of the pore is not effectively covered by the gradient of the electric field due the large cation-cation distance. On the other hand, for small pore diameters, CO2 can be only located close to the center of the pore where the gradient of the field is zero. Therefore, the CO2-solid electrostatic 17 ACS Paragon Plus Environment


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interaction is weak. As a consequence of both observed effects (for small and large pores), the electrostatic interaction reaches a maximum when the asymmetric regions close to the cations are the dominant contribution to the pore volume and the combined interaction of the cations is maximized. Figure 3 shows the Henry constant of CO2 as a function of the pore diameter for different cation charges and for several surface energy densities.

(a)

(b)

(c)

(d)

Figure 3. Henry constant of CO2 in the spherical model pore as a function of the accessible pore diameter, for different cation charges z and several surface energy densities (SED): (a) SED* = 0.79; (b) SED* = 0.94; (c) SED* = 1.1; and (d) SED* = 1.26.


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At Dacc = 1.28 Å, all Henry constants have the same value independently of the cation charge and SED*. At this restricted pore size, CO2 is necessarily located close to the center of the pore where the gradient of the electric field cancels out for symmetry reasons, as explained above. Therefore, the effect of the cation charge is negligible. Contrary to this situation, at large pore sizes (Dacc = 11.3 and 23.3 Å), the Henry constant of CO2 strongly depends on the cation charge. When the cation charge is 1 e, the Henry constant of CO2 is approximately 12 times higher than that obtained for the system with low charge (z = 0.1 e). Three cation charges were used to represent three bond types. First, z = 1 e, a high-polar solid where the bond between the lattice and the cations is ionic. Second, z = 0.1 e, a nonpolar system where the bond between the lattice and the cations is covalent. Finally, z = 0.5 e, an intermediate polarity where the cation-lattice bond is partially covalent. When the cation charge is low (z = 0.1 e), the Henry constant of CO2 gradually decreases with the pore diameter, as in the case of CH4. For the fully charged system (z = 1 e), the Henry constant initially increases with the pore diameter, and a maximum is reached at an accessible pore diameter of about 6 Å, which corresponds to the diameter that maximizes the electrostatic interaction in a cubic charge distribution.25 Above that value, the overall Henry constant drops because the gradient of the electric field cannot effectively cover the whole pore volume. However, the Henry constant in the charged system remains more than one order of magnitude higher than in the system with z = 0.1 e, indicating a significant contribution of the electrostatic interaction to the overall potential. At its maximum close to 6 Å, the amplification of the Henry constant by the electrostatic interactions is close to three orders of magnitude. When the cation charge is lower than 0.5 e, the impact of the electrostatic forces on the Henry constant is strongly reduced. The maximum at 6 Å disappears at z = 0.1 e and reduces to a small shoulder in the curve for z = 0.5 e. Likewise, when the dispersion



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interactions are relatively stronger, due to a high surface energy density, the maximum in the curve Henry constant vs. pore diameter becomes less pronounced. Figure 3 shows that the maximum in Henry constant of CO2 achieved for the spherical pore with a cubic charge distribution is close to 6 Å, independently of the surface energy density. However, this maximum in KCO2 does not guarantee that 6 Å is the optimal pore diameter for CO2 adsorption. As mentioned in the introduction, the Henry constant must be tuned to an intermediate value that provides the best compromise between working capacity, regenerability, and selectivity. In the next section, we first turn our attention to the working capacity of CO2. 4.2 Working capacity By substituting the Henry constants of CO2 and CH4 into the RSM (equation 9), we can calculate the working capacity of CO2 for the operating conditions of the PSA process. Figure 4 presents the CO2 working capacity (expressed as fractional loading of the pore volume) as a function of the cation charge and the pore diameter. Each pore diameter gives an optimal cation charge. The optimal charge increases with the pore diameter, i.e. when the dispersion interactions decrease. The reduction in the dispersion interaction is compensated by the electrostatic interactions, i.e. by increasing the cation charge.


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(a)

(b)

Figure 4. CO2 working capacity (expressed as fractional loading of the pore volume) as a function of cation charge, for different pore diameters. (a) SED* = 0.79; (b) SED* = 0.94. In addition, a comparison of Figure 4(a) and (b) shows that increasing the surface energy density, i.e. increasing the dispersion interactions, shifts the optimal charge to a lower value. In this case, the electrostatic forces necessary to compensate the higher dispersion interaction are lowered. As a conclusion, the relevant magnitude is the overall binding force of CO2, resulting from the sum of dispersion and electrostatic interactions, i.e. the relevant parameter governing the working capacity is the Henry constant of CO2. In Figure 5, we represent the working capacity of CO2 as a function of the Henry constant. Independently of the pore diameter, except for the smallest one, the working capacity peaks at a Henry constant of 9.5 x 10-3 molecules bar-1 Å-3. For the operating conditions considered here, this value offers the best compromise between a strong adsorption and an untroublesome regeneration. Below the optimal Henry constant, the adsorption of CO2 is too weak. Above the optimal Henry constant, the regeneration of CO2 becomes too difficult.



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(a)

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(b)

Figure 5. Working capacity (expressed as fractional loading of the pore volume) as a function of the Henry constant of CO2. (a) SED* = 0.79; (b) SED* = 0.94. We emphasize that other operating conditions, such as feed composition, temperature, pressure, etc., will lead to a different optimal Henry constant. For example, in the case of a Vacuum Swing Adsorption (VSA) process, in which the desorption step occurs under vacuum, a stronger adsorption of CO2 would be more favorable, i.e. a higher value of the optimal Henry constant would be obtained.21 The curves of the working capacity as a function of the Henry constant of CO2 for different pore diameters are not fully superimposed (Figure 5). This effect is more important for high SED* values. When the pore diameter decreases, the maximal working capacity decreases due to the competitive co-adsorption of CH4. As the pore diameter decreases, the Henry constant of CH4 increases, which produces a stronger competition of CH4 and CO2 for adsorption sites. A stronger adsorption of CH4 reduces the adsorbed amount of CO2. At low pore diameters, this effect not only reduces the maximal working capacity, but also shifts the position of the maximum to higher Henry constants of CO2. A stronger adsorption of CO2 is required to compensate the stronger competitive adsorption of CH4. This phenomenon


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remains, however, a second order effect. As a first approximation, we can consider that the optimal working capacity only depends on the Henry constant of CO2. In Figure 6, the CO2 working capacity is plotted as a function of the pore diameter for three different cation charges. For z = 0.8 and 0.5 e, there is an optimal pore diameter. Above this optimal diameter, the pore is not effectively covered by the gradient of the electric field. For z = 1 e, this optimal pore diameter is beyond the range of pore diameters studied here. This confirms the conclusion reached in our earlier study (where the co-adsorption of CO2 was not taken into account):25 for highly charged systems, very large pore diameters (> 2 nm) maximize the working capacity.

Figure 6. CO2 working capacity (fractional loading of pore volume) as a function of pore diameter, for different cation charges. SED*= 0.94. In Figures 3, 4, and 5, the working capacity was plotted as a fractional loading of the pore volume. At the end of section 2, we mentioned that the relevant parameter is the volumetric working capacity, which is obtained by multiplying the fractional working capacity by the porosity of the solid. In the case of zeolites, the porosity increases with the pore diameter, see Eq. S3 un the Supporting Information. Hence, if the fractional working capacity (∆θ) is



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replaced by the volumetric working capacity in Figure 6, the larger pore diameters become even more favored. 4.3 Separation factor The separation factor is the product of the CO2/CH4 selectivity and the regenerability of CO2. Figure 7 shows the CO2/CH4 selectivity as a function of the accessible pore diameter for three different charges (z = 0.5, 0.8, and 1 e). For z < 0.5 e, the CO2/CH4 selectivity is lower than 10 (not shown). Selectivity reaches its maximum at a pore diameter of approximately 6 Å independently of the cation charge. At this maximum, the selectivity abruptly increases with the cation charge due to electrostatic interactions. As the pore diameter increases, the effective volume covered by the gradient of the electric field decreases, reducing the CO2/CH4 selectivity.

Figure 7. CO2/CH4 selectivity for an equimolar mixture as a function of the pore diameter, for different cation charges. SED* = 0.94. As explained in section 2.3, the ratio of the Henry constant of CO2 and that of CH4 was used as an approximation to estimate the CO2/CH4 selectivity. The Henry constant of CH4 gradually decreases as a function of pore diameter. On the other hand, the Henry constant of CO2 peaks at pore diameter of about 6 Å because at this pore size the electrostatic 24 ACS Paragon Plus Environment

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interactions with CO2 are maximized. Therefore, it is expected that the CO2/CH4 selectivity peaks at a pore diameter of about 6 Å. The separation factor as a function of the pore diameter is shown in Figure 8. For a cation charge of 0.5 e, the separation factor peaks at a pore diameter of ~ 7 Å, a value that is slightly higher than the maximum in selectivity. At higher charges, the Henry constant of CO2 is higher, causing a lower regenerability. Therefore, the optimal compromise between selectivity and regenerability is reached at a higher pore diameter. For the highest cation charge (z = 1 e), the optimal pore diameter is greater than 2.28 nm, the largest pore diameter used in this work.

Figure 8. CO2/CH4 separation factor as a function of the accessible pore diameter, for different values of the cation charge. SED* = 0.94. The behavior of the separation factor with the pore diameter can be explained by analyzing the selectivity and regenerability. In contrast to selectivity, regenerability of CO2 is determined by the Henry constant and the pressure range in which the separation is carried out. A high Henry constant of CO2 leads to a steep isotherm that reaches saturation quickly. Therefore, the regenerability is low, unless the separation is carried out at very low pressures. On the contrary, full regenerability is achieved for a linear isotherm, in which the linear range of the isotherm extends over a larger pressure range. 25 ACS Paragon Plus Environment


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For highly charged systems (z > 0.5 e), the Henry constant of CO2 increases up to a pore diameter of ~ 6 Å and then gradually decreases (Figure 3). Regenerability follows the inverse trend, i.e. it reaches a minimum at ~ 6 Å and then steadily increases. As in the case of the working capacity, we can assume that the Henry constant of CO2 is the dominating parameter that dictates at which pore diameter the optimal separation factor is reached. As before, we checked whether we can find a unified curve that represents the separation factor as a function of the Henry constant of CO2 for any combination of pore diameter and cation charge (Figure 9). Indeed, most of the SF vs. KCO2 curves have their maximum at a similar value, i.e. KCO2 ~ 20 x 10-3 molecules bar-1 Å-3. This value is higher than the Henry constant maximizing the working capacity, which indicates that the separation factor requires a stronger adsorption of CO2. As in the case of the working capacity, there is no unified curve. The maximal separation factor increases with the pore diameter due to a weaker adsorption of CH4. In addition, the maximum shifts to a higher Henry constants of CO2 when the pore diameter is low to compensate the stronger adsorption of CH4. When compared to the case of the working capacity, a larger shift is observed for the separation factor, indicating that SF is more sensible to the co-adsorption of CH4.

(a)

(b)


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Figure 9. CO2/CH4 separation factor as a function of the Henry constant of CO2 for different pore diameters. (a) SED* = 0.79; (b) SED* = 0.94. KCO2 was varied by changing the charge z between 0.1e and 1.0e. Although the selectivity varies from values close to 1 up to 1300, the separation factor only varies in a fairly small range from 1 to 6. This is because the CO2/CH4 selectivity strongly increases with the Henry constant, but regenerability strongly decreases with the Henry constant of CO2. These two factors largely compensate each other. 4.4 Best compromise between working capacity and separation factor In the previous sections, we determined the best compromise between pore size and cation charge in terms of working capacity and separation factor. For a CO2-PSA process, the overall optimum is found when working capacity and separation factor are maximized at the same conditions. For z = 0.5 e, Figure 10 indicates that the optimal pore diameter maximizing either the working capacity or the separation factor are luckily fairly close to each other. The separation factor is maximized at lower pore diameter than the working capacity.



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Figure 10. Volumetric working capacity (Δn, expressed as the volume occupied by CO2 in the total volume of the solid) and separation factor as a function of pore size for a cation charge of 0.5 e. SED* = 0.94. Increasing the cation charge shifts both optimum to higher pore diameters, but the two values remain close to each other. For a cation charge of 1 e, the optimum pore diameter is beyond the range of values studied here, i.e. higher than 23 Å. This confirms the conclusion that we had already anticipated in an earlier study of the electrostatic interaction of a cubic charge distribution with CO2.25 Adding dispersion-repulsion forces and taking into account the co-adsorption of CO2 does not change the picture radically. The use of solids with very large pores and highly charged cations is definitively the way to achieve efficient CO2 separation in medium- or high-pressure processes. Where is material science standing with respect to this goal? The synthesis of stable solids with ultra-large pores is still challenging. Several MOFs with extremely high porosities have been synthesized35, but these structures are weakly charged, i.e. they produce a very weak electrostatic field. To the best of our knowledge, the only cationic MOF structures with fairly large pore sizes are still the sod- and rho-ZMOF.36 In the field of zeolites, the synthesis of structures with very large pore size and porosity has made enormous progress in the last years. Several structures have been synthesized with lower framework density than FAU. All 28 ACS Paragon Plus Environment

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of them are silicogermanate zeolites (ITQ-33 = ITT

37

, ITQ-37 = ITV

38

, ITQ-40 = IRY

39

,

ITQ-43 40, ITQ-54 41), i.e. neutral structures. Introducing aluminum and compensating cations in these frameworks remains a daunting task. For practical purposes, the only viable cationic, low-density zeolite structures are still FAU and EMT. The diameter of the supercage in FAU and EMT zeolites is roughly 12.8 Å, but the accessible diameter for CO2 and CH4 is only ~ 9 Å. Therefore, we chose this accessible diameter to compare the evolution of the separation factor and working capacity as a function of cation charge (Figure 11).

Figure 11. CO2/CH4 separation factor and volumetric working capacity as a function of the cation charge. The pore accessible diameter (Dacc) is 9.3 Å and SED* = 0.94. Figure 11 shows that the separation factor peaks at a charge of 0.8e while the working capacity reaches a maximum at a charge of 0.7e. This is in qualitative agreement with our previous experimental study28, where we had predicted an optimal performance for intermediate to low cation content for FAU-types zeolites. 5.

Conclusions We studied the adsorption properties of a spherical pore containing eight cations placed in

cubic geometry. For this model system, we first calculated the Henry constants of CO2 and 29 ACS Paragon Plus Environment


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CH4 arising from the sum of dispersion-repulsion and electrostatic interactions with the solid. The Henry constant of CH4 decreases as the pore size increases because the confinement effect diminishes. In the case of CO2, where electrostatic interactions add to the dispersion forces, the Henry constant reaches a maximum at the pore size where the electrostatic field efficiently covers the whole pore volume. The optimal working capacity of CO2 (in terms of fractional occupation of the pore volume) is achieved when the Henry constant of CO2 provides the best compromise between strong adsorption and easy desorption. For the example of an equimolar CO2/CH4 mixture in the pressure range of 5 to 1 bar, we predict that this optimum is reached at a Henry constant of KCO2 = 10 x 10-3 molecules bar-1 Å-3. The separation factor peaks at higher values of the Henry constant of CO2, especially when the pore size is small, due to a stronger co-adsorption of CH4. The optimal Henry constant of CO2 can be reached by many combinations of charge and pore size: the weaker confinement in larger pores must be compensated by a higher charge and vice versa. Fixing the adsorbent pore diameter to 9.3 Å, the best performance is achieved for a cation charge of 0.7-0.8e. On the other hand, if the charge is 1e, the best performance is predicted for a pore diameter higher than 23 Å. If this pore diameter is reached in a lowdensity structure, the volumetric capacity of the PSA process is maximized. Thus, lowdensity, highly-charged materials with pore diameters larger than 20 Å are desirable. Large pores and low density provide high volumetric capacity, while the high charge ensures selective adsorption of CO2 in a large fraction of the pore volume. The method presented here enables a rapid identification of the adsorbent properties that maximize the working capacity and separation factor. Even though we limited this paper to spherical pores with cubic charge distribution, the method can be extended to other pore geometries and charge distributions. Different assumptions in the adsorbent model or in the adsorption conditions can shift the optimal pore diameter and charge. For instance, carrying


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out the adsorption process at lower pressure range (VSA) would lead to higher optimal Henry constant, i.e. smaller pore size. A lower pressure range favors desorption and disfavors adsorption, which must be compensated by a stronger adsorbate-adsorbent interaction. Supporting information The supporting information provides a correlation between the pore radius and the porosity, the equation to calculate the total pore-fluid dispersion interaction, and analogous data on Henry constant, working capacity, selectivity and separation factor in a cylindrical model pore with a hexagonal charge distribution. This information is available free of charge via the Internet at http://pubs.acs.org. 6.

Notation b

excluded volume of the adsorbate molecule (Å3/molecule)

Dacc accessible pore diameter (Å) Eelec electrostatic potential Edis dispersion-repulsion potential Esf

solid-fluid adsorption potential

k

Boltzmann constant

K

Henry constant (molecules bar-1 Å-3)

ni

Adsorbed amount of compound i in volume per volume of adsorbent

p

pressure

r

radial distance from the center of the pore

r

position in the pore volume (vector)

Racc accessible pore diameter (Å) s

number of molecules in a pore in the Ruthven Statistical Model

s, j number of molecules in the pore SED surface energy density 31 ACS Paragon Plus Environment


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SF

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separation factor

smax, jmax

maximal number of molecules in the pore

T

temperature

v

volume of the pore in the Ruthven Statistical Model (Å3)

Xi

mol fraction of compound i

WC working capacity (volume adsorbate / total volume ) z

cation charge (e)

Z

configurational integral

Greek letters α

selectivity

β

1/kT

∆

distance between successive layers in the model solid.

εff

fluid-fluid interaction potential in the RSM (K)

εsf

depth of the potential well of the solid-fluid interaction (K)

θ

fractional loading of the pore volume

Π’

angle describing the orientation of CO2 quadrupole

ρs

density of the solid

ρsk skeletal density of the solid ρsur density of atoms at the adsorbent surface (atoms/Å2) σsf

Lennard-Jones solid-fluid diameter (Å)

φ

porosity of the adsorbent (pore volume per volume of the adsorbent particle)

ω’

angle describing the orientation of CO2 quadrupole

Subscripts 1

more strongly adsorbed (heavy) component


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2

more weakly adsorbed (light) component

acc accessible ads adsorption des desorption ff

fluid-fluid

s

solid

sf

solid-fluid

sk

skeletal

sur surface sph spherical

Superscripts * dimensionless quantity

Abbreviations LJ

Lennard-Jones

MOF metal organic framework PSA

pressure swing adsorption

RSM Ruthven statistical model 7. (1) (2) (3)

(4)

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Sketching a Portrait of the Optimal Adsorbent for ... - ACS Publications

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