Synergistic and Antisynergistic Intracrystalline Diffusional Influences on Mixture Separations in Fixed-Bed Adsorbers

Abstract

Separations of mixtures in fixed-bed adsorbers are influenced by factors such as (1) selectivity of adsorption, S _ads, (2) diffusional time constants, Đ _i /r _c , and (3) diffusion selectivity, Đ ₁/Đ ₂. In synergistic separations, intracrystalline diffusion of guest molecules serves to enhance the selectivities dictated by thermodynamics of mixture adsorption. In antisynergistic separations, intracrystalline diffusion serves to reverse the hierarchy of selectivities dictated by adsorption equilibrium. For both scenarios, the productivities of the desired product in fixed-bed operations are crucially dependent on diffusional time constants, Đ _i /r _c ; these need to be sufficiently low in order for diffusional influences to be effective. Also, the ratio Đ ₁/Đ ₂ should be large enough for manifestation of synergistic or antisynergistic influence. Both synergistic and antisynergistic separations have two common, distinguishing characteristics. Firstly, for transient uptake within crystals, the more mobile component attains supraequilibrium loadings during the initial stages of the transience. Such overshoots, signifying uphill diffusion, are engendered by the cross-coefficients Γ _ij (i ≠ j) of thermodynamic correction factors. Secondly, the component molar loadings, plotted in composition space, follow serpentine equilibration paths. If cross-coefficients are neglected, no overshoots in the loadings of the more mobile component are experienced, and the component loadings follow monotonous equilibration paths. The important takeaway message is that the modeling of mixture separations in fixed-bed adsorbers requires the use of the Maxwell–Stefan equations describing mixture diffusion employing chemical potential gradients as driving forces.

1. Introduction

Microporous crystalline porous materials such as metal–organic frameworks (MOFs) and zeolites are finding applications as adsorbents in several different mixture separations of industrial interest. Such separations are commonly conducted in fixed-bed devices in pressure swing adsorption (PSA) processes that are operated in adsorption and desorption phases, operated in a cyclic manner. The separation performance in a fixed-bed adsorber is usually determined by the thermodynamics of mixture adsorption equilibrium, determined in practice using the ideal adsorbed solution theory (IAST). , The adsorption selectivity for a binary mixture of components 1 and 2 is defined by

$$S_{\mathrm{ads}}=\frac{q_1/q_2}{p_1/p_2}$$ 1

where p ₁ and p ₂ are the partial pressures in the feed mixture; q ₁ and q ₂ are the component molar loadings in the adsorbed phase. Industrially important examples of such equilibrium-based separations include H₂ purification, production of purified oxygen, and separation of xylene isomers. −

In several recent developments, the separation performance is additionally influenced by intracrystalline diffusion of guest constituents. − For production of purified N₂ from air, kinetically driven separations are achieved with carbon molecular sieve (CMS) and LTA-4A zeolite. ,− For removal of N₂ from natural gas, the use of Ba-ETS-4 allows the selective uptake of N₂ by effective size-exclusion of CH₄. , For separation of C₂H₄/C₂H₆ mixtures, the crystal structure of UTSA-280 essentially excludes the saturated alkane from the pores. For separation of C₃H₆/C₃H₈ mixtures, the diffusional influence in CHA zeolite and ZIF-8 serves to override the adsorption selectivity that favors propane. ,,

For separation of C₂H₂/CO₂ mixtures, it is desirable to select MOFs that have adsorption selectivity in favor of CO₂, allowing the desired product C₂H₂ to be recovered, with purities in excess of 99%, in the adsorption cycle in a PSA process scheme. A particularly promising adsorbent is the ultramicroporous Y-bptc in which intracrystalline diffusion also favors CO₂; this adsorption/diffusion synergy results in high productivity of purified C₂H₂ (see Figure S26).

The selective capture of propyne from propene is important in the context of preparing polymer-grade propene feedstocks. Jiang et al. report the efficacy of SIFSIX-Cu-TPA (also named as ZNU-2-Si) for this separation task. A particularly interesting feature is that the separation displays synergism between adsorption and diffusion (see Figure S63).

The experimental data on separation of ethanol/1-propanol and ethanol/1-hexanol liquid feed mixtures in a fixed bed packed with SAPO-34, which has the same structural topology as CHA zeolite, shows that the component that is preferentially adsorbed in ethanol. , The rationalization of these experimental data can be traced to the synergy between adsorption and diffusion, both favoring ethanol (see Figures S23, S24, and S25).

The main objective of this Review is to discuss and underscore the key factors that underlie both synergistic and antisynergistic separations. By detailed analyses of various mixture separations in fixed-bed adsorption devices, we highlight some common characteristics of both synergistic and antisynergistic separations. Such characteristics include transient overshoots in the uptake of the more mobile guest and serpentine equilibration trajectories. The secondary objective is to stress the need to employ the Maxwell–Stefan (M–S) formulation of multicomponent diffusion in which the driving forces for intracrystalline transport of guest species are the corresponding gradients in the chemical potentials. ,,

The Supporting Information provides (a) the derivation of the M–S equations, (b) computational modeling details of transient mixture uptakes within crystals and adsorber breakthroughs, (c) unary isotherm data fits used for IAST calculations, and (d) guest diffusivities.

2. Transient Overshoots in Binary Mixture Uptake

We begin by highlighting an important signature of kinetic separations, either synergistic or antisynergistic, by examining published experimental data sets on transient mixture uptake in adsorbent crystals.

Using infrared microscopy (IRM), Titze et al. have determined the uptake of n-hexane (nC6)/2-methylpentane (2MP) mixtures in MFI crystals exposed to an equimolar gas phase mixture at constant total pressure; see Figure a. The transient equilibration of nC6 displays a distinct overshoot in its component loading during early stages of the transience; the overshoot is in excess of the final equilibrated loading. The mixture adsorption equilibrium favors the linear isomer (S _ads = 1.4). There is synergy between adsorption and diffusion because nC6 has a higher mobility than 2MP, by a factor of 100, within the 5.5 Å sized channels of MFI zeolite (structural details are provided in Figures S6 and S7).

1.

(a) Experimental data on transient uptake of nC6(1)/2MP(2) mixtures in MFI zeolite at 298 K. , (b) Experimental data on transient uptake of the N₂(1)/CH₄(2) mixture within LTA-4A crystals, exposed to binary gas mixtures at partial pressures p ₁ = 50.9 kPa and p ₂ = 49.1 kPa at 194 K. , (c) Experimental data for transient uptake of ethanol/1-propanol mixtures within SAPO-34. (d–f) Experimental data , for spatial-averaged transient uptake of (d) 1:1 (e) 2:1, and (f) 3:1 CO₂(1)/C₂H₆(2) gas mixtures within crystals of DDR zeolite at 298 K. , The continuous solid lines are simulations based on eq . The dashed lines are the simulations based on eq .

The uptake of N₂(1)/CH₄(2) mixtures in crystallites of LTA-4A zeolite have been reported by Habgood; see Figure b. , The “pencil-like” 4.4 Å × 3.3 Å sized nitrogen molecule hops lengthwise across the 8-ring windows of LTA-4A (structural details are provided in Figure S38). The spherical 3.7 Å methane molecule is constrained in its intercage hopping and has a diffusivity that is significantly lower than that of N₂. , Due to its higher polarizability, the adsorption strength of CH₄ is higher than that of N₂. At short contact times, the pores of LTA-4A are significantly richer in the more mobile N₂, manifesting in an overshoot in component loading in excess of the final equilibrium value. With the progression of time, the N₂ in the pores is displaced by CH₄ that has a higher adsorption strength. ,

Measurements of the uptake of ethanol/1-propanol mixtures in SAPO-34 zeolite (structural details are provided in Figures S20 and S21) are shown in Figure c. The more mobile ethanol exhibits an overshoot during the early stages of the uptake transience. ,

Figure d–f shows the experimental uptakes , of CO₂/C₂H₆ gas mixtures of different compositions within crystals of DDR, a cage-type zeolite (structural details are provided in Figures S29 and S30). ,− The cross-sectional dimension of CO₂ is smaller than that of C₂H₆, and therefore the intercage hopping rate of CO₂ is significantly higher than that of C₂H₆, by a factor of 100–1000. The mixture adsorption equilibrium favors the saturated alkane due to higher polarizability. During early transience, the more mobile CO₂ attains supraequilibrium loadings.

For elucidation and quantification of the six data sets in Figure , we need to solve the set of partial differential equations describing the time derivative of the component molar loadings in a spherical crystallite of radius r _c:

$$\rho \frac{\partial q_i(r,t)}{\partial t}=-\frac{1}{r^2}\frac{\partial }{\partial r}(r^2{N}_i)$$ 2

The M–S equation describing the dependence of the intracrystalline fluxes N _i on the gradients of component chemical potentials is ,,

$$N_i=-\rho {Đ}_i\frac{q_i}{RT}\frac{\partial {\mu }_i}{\partial r}$$ 3

The M–S diffusivity Đ _i equals the corresponding diffusivity for a unary system, determined at the same pore occupancy; the diffusivity for any species i in a mixture remains invariant to the choice of the partner species. Other variables in eqs and are described in the Nomenclature section.

We relate the ∂μ _i /∂r to the loading gradients ∂q _i /∂r by defining thermodynamic correction factors Γ _ij :

$$\frac{q_i}{RT}\frac{\partial {\mu }_i}{\partial r}=\sum _{j=1}^n{\mathrm{\Gamma }}_{ij}\frac{\partial q_j}{\partial r};{\mathrm{\Gamma }}_{ij}=\frac{q_i}{p_i}\frac{\partial p_i}{\partial q_j};i,j=1,2,\mathrm{...},n$$ 4

The Γ _ij are determinable from the IAST description of the mixture adsorption equilibrium. For binary mixtures, we may combine eqs and to obtain the following explicit expression, using two-dimensional matrix notation, for the intracrystalline fluxes in terms of the gradients in the molar loadings:

$$\left(\begin{array}{l}{N}_1\\ N_2\end{array}\right)=-\rho \left[\begin{array}{ll}{D}_{11}& D_{12}\\ D_{21}& D_{22}\end{array}\right]\left(\begin{array}{l}\frac{\partial q_1}{\partial r}\\ \frac{\partial q_2}{\partial r}\end{array}\right);\left[\begin{array}{ll}{D}_{11}& D_{12}\\ D_{21}& D_{22}\end{array}\right]=\left[\begin{array}{ll}{Đ}_1& 0\\ 0& {Đ}_2\end{array}\right]\left[\begin{array}{ll}{\mathrm{\Gamma }}_{11}& {\mathrm{\Gamma }}_{12}\\ {\mathrm{\Gamma }}_{21}& {\mathrm{\Gamma }}_{22}\end{array}\right]$$ 5

The cross-coefficients Γ₁₂ and Γ₂₁ cause the flux of species i, N _i , to be influenced by the gradient of the molar loading of the partner species j: ∂q _j /∂r. In order to appreciate the significance of such “coupling” effects, Figure presents calculations of the elements Γ _ij for 20/80 N₂(1)/CH₄(2) mixture adsorption in LTA-4A zeolite as a function of the total pressure, p _t. We note that the Γ₁₂ and Γ₂₁ are increasingly significant in relation to the main elements Γ₁₁ and Γ₂₂ with increasing values of p _t.

2.

Calculations of the elements Γ₁₁, Γ₁₂, Γ₂₁, and Γ₂₂ for 20/80 N₂(1)/CH₄(2) mixture adsorption in LTA-4A zeolite at 194 K.

The continuous solid black lines in Figure are radial-averaged component loadings

$${\mathrm{q̅}}_i(t)=\frac{3}{r_{\mathrm{c}}^3}{\int }_0^{r_{\mathrm{c}}}{q}_i(r,t)r^2\mathrm{d}r$$ 6

obtained by numerical solution of eqs , , and . For the set of six experiments in Figure , the attainment of supraequilibrium loadings by the more mobile guest components are quantitatively portrayed by the M–S diffusion equations. Such overshoots are signatures of the occurrence of uphill diffusion. ,

Additional evidence of uphill diffusion of more mobile partners (component 1) is available for N₂(1)/CH₄(2) uptake in Ba-ETS-4 (see Figures S49 and S50), O₂(1)/N₂(2) uptake in LTA-4A zeolite (see Figure S55), C₃H₆/C₃H₈ uptake in CHA zeolite (see Figure S61), C₃H₆/C₃H₈ uptake in ZIF-8 (see Figure S62), benzene­(1)/ethylbenzene­(2) uptake in H-ZSM-5 (see Figure S64a), and benzene(1)/p-xylene­(2) uptake in H-ZSM-5 (see Figure S64b). ,,

At low pore occupancies, Γ _ij → δ _ij , where δ _ij (i = j) = 1 and δ _ij (i ≠ j) = 0 is the Kronecker delta; in this case eq simplifies to yield a set of flux relations that are uncoupled:

$$N_i=-\rho {Đ}_i\frac{\partial q_i}{\partial r};i=1,2$$ 7

The dashed black lines in Figure are simulations based on the simplified eq , asserting Γ _ij = δ _ij ; we note that this simplification is unable to anticipate the overshoots in the component loadings of the more mobile partners in all six data sets. The first important message to emerge is that thermodynamic coupling influence is of essential importance in kinetic separations, both synergistic and antisynergistic.

3. Serpentine Equilibration Trajectories

To gain further insights into the characteristics of kinetic separations, we plot the experimentally measured loadings of the more mobile partner, q ₁, as a function of the loading of the tardier partner, q ₂; see red spherical symbols in Figure . In all six cases we note that the component loadings follow serpentine paths to equilibrium. If eq is invoked by asserting Γ _ij = δ _ij , the approach to equilibrium is monotonous in nature, as shown in the dashed black lines in Figure .

3.

Equilibration trajectories in composition space for (a) nC6(1)/2MP(2) mixtures in MFI zeolite at 298 K, (b) N₂(1)/CH₄(2) mixture in LTA-4A crystals at 194 K, (c) ethanol/1-propanol mixtures in SAPO-34 at 298 K, and (d–f) CO₂(1)/C₂H₆(2) mixtures within crystals of DDR zeolite at 298 K. The red circles represent experimental data on component loadings from Figure . The dashed lines are the trajectories based on eq . The red lines represent the fast eigenvectors. The blue lines represent the slow eigenvectors. The green lines represent the trajectory followed if Đ ₁ = Đ ₂.

To obtain further insights, we determine the two eigenvalues λ₁ and λ₂ of the two-dimensional matrix $\left[\begin{array}{ll}{D}_{11}& D_{12}\\ D_{21}& D_{22}\end{array}\right]=\left[\begin{array}{ll}{\mathit{Đ}}_1& 0\\ 0& {\mathit{Đ}}_2\end{array}\right]\left[\begin{array}{ll}{\mathrm{\Gamma }}_{11}& {\mathrm{\Gamma }}_{12}\\ {\mathrm{\Gamma }}_{21}& {\mathrm{\Gamma }}_{22}\end{array}\right]$ . Since component 1 is the more mobile partner, λ₁ > λ₂. The green-colored straight lines in Figure are the trajectories that are followed in the happenstance in which the M–S diffusivities of both species are equal to each other, i.e., Đ ₁ = Đ ₂, and λ₁ = λ₂; the mixtures separations are purely based on mixture adsorption thermodynamics.

The experimentally determined loading trajectories are bounded by the two corresponding eigenvectors. During the early stages of the transience, the equilibration is influenced by the fast eigenvector (detailed derivations are provided in the Supporting Information):

$$\begin{array}{l}(q_2-q_{20})=e_{12}(q_1-q_{10})\\ e_{12}=\frac{1}{2}(-\frac{{\mathrm{\Gamma }}_{11}}{{\mathrm{\Gamma }}_{12}}+\frac{{Đ}_2}{{Đ}_1}\frac{{\mathrm{\Gamma }}_{22}}{{\mathrm{\Gamma }}_{12}}+\sqrt{{(\frac{{\mathrm{\Gamma }}_{11}}{{\mathrm{\Gamma }}_{12}}-\frac{{Đ}_2}{{Đ}_1}\frac{{\mathrm{\Gamma }}_{22}}{{\mathrm{\Gamma }}_{12}})}^2+4\frac{{Đ}_2}{{Đ}_1}\frac{{\mathrm{\Gamma }}_{21}}{{\mathrm{\Gamma }}_{12}}})\end{array}$$ 8

In eq , q ₁₀ and q ₂₀ are the initial component loadings. The fast eigenvectors are indicated by the straight red lines in Figure a–f.

As equilibrium is approached, the equilibration trajectory follows the path dictated by the slow eigenvector

$$\begin{array}{l}(q_2-q_2^{*})=e_{22}(q_1-q_1^{*})\\ \frac{1}{e_{22}}=\frac{1}{2}(\frac{{Đ}_1}{{Đ}_2}\frac{{\mathrm{\Gamma }}_{11}}{{\mathrm{\Gamma }}_{21}}-\frac{{\mathrm{\Gamma }}_{22}}{{\mathrm{\Gamma }}_{21}}-\sqrt{{(\frac{{Đ}_1}{{Đ}_2}\frac{{\mathrm{\Gamma }}_{11}}{{\mathrm{\Gamma }}_{21}}-\frac{{\mathrm{\Gamma }}_{22}}{{\mathrm{\Gamma }}_{21}})}^2+4\frac{{Đ}_1}{{Đ}_2}\frac{{\mathrm{\Gamma }}_{12}}{{\mathrm{\Gamma }}_{21}}})\end{array}$$ 9

In eq , q ₁ and q ₂ are the equilibrated loadings; the slow eigenvectors are indicated by the straight blue lines in Figure a–f. We note that the slopes of the fast and slow eigenvectors, e ₁₂ and e ₂₂, are dependent on the ratio Đ ₁/Đ ₂. Indeed, it is possible to estimate Đ ₁/Đ ₂ from the tangents to the experimentally measured equilibration paths, along with estimates of Γ _ij .

4. Antisynergistic Separation of N₂(1)/CH₄(2) Mixtures

To meet pipeline specifications (<4% N₂), the selective capture of nitrogen from natural gas streams, primarily containing CH₄, is required in practice because the nitrogen content in several reserves may be as high as 20%. For smaller natural gas reserves, PSA technology is economically preferred to cryogenic distillation because the feed mixtures are available at high pressures. ,, It is desirable to use adsorbents in PSA units that are selective to N₂ such that CH₄ can be recovered as raffinate in the adsorption cycle. For most known materials, adsorption selectivity for separation of N₂/CH₄ mixtures is in favor of CH₄ due to its higher polarizability.

We analyze separations of 20/80 N₂(1)/CH₄(2) mixtures in a fixed-bed adsorber using LTA-4A zeolite operating at a total pressure of 100 kPa and T = 194 K. Figure a shows the transient breakthrough simulations for 20/80 N₂(1)/CH₄(2) mixtures through a fixed-bed adsorber packed with LTA-4A crystals operating at 194 K and total pressure p _t = 100 kPa. The y-axis is the % CH₄ at the adsorber outlet. The x-axis is Q ₀ t/m _ads, where Q ₀ is the volumetric flow rate of the gas mixture at the inlet to the fixed bed at actual temperature and pressure conditions. The continuous black lines are simulations taking due account of intracrystalline diffusion using eq , with parameters Đ ₁/r _c = 1.56 × 10^–5 s^–1; Đ ₁/Đ ₂ = 21.7. The blue lines are the simulations based on eq . For the target CH₄ purity of say 96%, we can determine the moles of 96%+ pure CH₄ produced from a material balance on the adsorber. Expressed per kilogram of LTA-4A zeolite in the packed bed, the respective productivities are 0.93 and 0.11 mol kg^–1, with 96%+ CH₄ purity. Use of the simplified eq asserting Γ _ij = δ _ij results in a severe underestimation of the CH₄ productivity in the PSA unit.

4.

(a, b) Transient breakthrough of the 20/80 N₂(1)/CH₄(2) mixture in a fixed-bed adsorber packed with LTA-4A crystals operating at 194 K and total pressure p _t = 100 kPa. The y-axis is the % CH₄ at the adsorber outlet. The x-axis is Q ₀ t/m _ads, where Q ₀ is the volumetric flow rate of the gas mixture at the inlet to the fixed bed at actual temperature and pressure conditions. The black lines are simulations based on eq . The blue lines are the simulations based on eq . In (a) we maintain Đ ₁/r _c = 1.5 × 10^–5 s^–1; Đ ₁/Đ ₂ = 21.7. In (b) the two simulations include thermodynamic coupling and maintain the ratio Đ ₁/Đ ₂ = 21.7; the black and red lines represent two different severities of diffusional influences: Đ ₁/r _c = 1.56 × 10^–5 s^–1; Đ ₁/r _c = 6.09 × 10^–6 s^–1.

Having established the importance of including the thermodynamic correction factors in PSA simulations, we proceed to examine the influence of the severity of diffusional limitations. Figure b compares the LTA-4A adsorber breakthrough simulations for 20/80 N₂(1)/CH₄(2) mixtures for two different scenarios: Đ ₁/r _c = 1.56 × 10^–5 s^–1; Đ ₁/r _c = 6.09 × 10^–6 s^–1; for both scenarios we maintain Đ ₁/Đ ₂ = 21.7 and include the thermodynamic coupling influences. Expressed per kilogram of zeolite in the packed bed, the respective productivities are 0.93 and 1.62 mol kg^–1, with 96%+ CH₄ purity. More severe diffusional limitations, signified by lower values of Đ ₁/r _c , result in higher CH₄ productivity. Indeed, if the diffusional severity is diminished, kinetic separations are not achievable.

A further point to note is that for kinetic separations to be effective with LTA-4A, the operating temperatures need to be below 220 K; see simulations in Figures S44 and S45. In current industrial practice, the adsorbent used is Ba-ETS-4 (also termed CTS-1), which is effective at ambient temperatures; see simulation data in Figures S46–S51.

5. Synergistic Separation of nC6(1)/2MP(2) Mixtures

For hexane isomers, the values of the research octane number (RON) increase with the degree of branching: n-hexane (nC6) = 30, 2-methylpentane (2MP) = 74.5, 3-methylpentane (3MP) = 75.5, 2,2-dimethylbutane (22DMB) = 94, and 2,3-dimethylbutane (23DMB) = 105. In the context of octane enhancement of gasoline, branched isomers of hexane are preferred components for inclusion in the high-octane gasoline pool. ,, There are a number of adsorbents that have potential use in the separation of linear and branched isomers of hexane isomers. , With the intersecting channel structure of MFI zeolite, the mono- and dibranched isomers locate at the channel intersections that offer more “legroom”, whereas the linear nC6 can locate anywhere in straight and zig-zag channels; see computational snapshots in Figure S8 of the Supporting Information. ,, The hierarchy of adsorption strengths is nC6 > 2MP ≈ 3MP > 22DMB ≈ 23DMB and is governed by entropy effects engendered by differences in molecular configurations and conformations. ,,, The intracrystalline diffusivities follow a similar hierarchy: nC6 ≫ 2MP ≈ 3MP > > 22DMB ≈ 23DMB (see Figure S14 of the Supporting Information).

Uphill diffusion of nC6, as evidenced in Figure a, is beneficial to the hexane isomer separations in PSA units because the desired raffinate phase in the adsorption cycle will be richer in the branched isomers that have higher RON values. Breakthrough simulations were performed for 50/50 nC6/2MP mixture separations in order to confirm this expectation. Figure a plots the dimensionless concentrations at the exit of the adsorber, c _i /c _i0, as a function of a modified time parameter defined by Q ₀ t/m _ads. In the three sets of simulations, the ratio Đ ₁/Đ ₂ = 100, as determined from the transient uptake experiments of Titze et al. (cf. Figure ). Three different values for the diffusional time constants were chosen, Đ ₁/r _c = 1.6 × 10^–6, 1.0 × 10^–5, and 1.6 × 10^–2 s^–1, to investigate the varying degrees of diffusional influence. In practice, stronger diffusional influences are achievable for use of larger sized crystals. The comparisons of the separations achieved with different crystal sizes should be on the basis of the parameter Q ₀ t/m _ads. Increasing values of Đ ₁/r _c signify diminishing influence of intracrystalline transport; indeed, for Đ ₁/r _c → ∞, there are no diffusional limitations. We note that with decreasing Đ ₁/r _c , i.e., stronger diffusional influences, the breakthroughs take on a more distended character and 2MP exits the adsorber at earlier times. Figure b plots the RON of the product gas mixture exiting the adsorber. With stronger diffusion influences, i.e.. lower values of Đ ₁/r _c , there is a longer time interval during which product gas with RON= 74.5 can be recovered at the exit; these time intervals are indicated by the colored arrows in Figure b.

5.

(a) Transient breakthrough for 50/50 nC6(1)/2MP(2) mixtures in MFI zeolite at 298 K and total pressure of 10 Pa. The dimensionless concentrations at the exit of the adsorber, c _i /c _i0, are plotted as a function of a modified time parameter defined by Q ₀ t/m _ads. For the three sets of simulations, Đ ₁/r _c = 1.6 × 10^–6, 1.0 × 10^–5, and 1.6 × 10^–2, s^–1, maintaining the ratio Đ ₁/Đ ₂ = 100. (b) RON of a product gas mixture leaving a fixed-bed adsorber packed with MFI zeolite.

The adsorption/diffusion synergy is also effective for the separation of hexane isomers using Fe₂(BDP)₃ as the adsorbent; for details see Figures S17–S19 of the Supporting Information.

6. Synergistic Separation of C₂H₂(1)/CO₂(2) Mixtures

The recovery of ethyne (C₂H₂) from mixtures with CO₂ is of industrial importance because C₂H₂ is an important feedstock for chemical synthesis and has wide applications. Due to the closeness of the boiling points, distillation separations need to operate at cryogenic temperatures and high pressures. Adsorption separations are also challenging because both molecules possess zero dipole moments and approximately the same quadrupole moment. The molecular dimensions of C₂H₂ (3.32 Å × 3.34 Å × 5.7 Å) are remarkably close to that of CO₂ (3.18 Å × 3.33 Å × 5.36 Å). Since the desired product C₂H₂ needs to be recovered at high purities, typically >99%, it is preferable to choose adsorbents in which CO₂ is preferentially adsorbed, such that C₂H₂ is recovered as raffinate in the adsorption cycle of PSA operations. MOFs that have adsorption selectivity in favor of CO₂ include ZU-610a, Ce­(IV)-MIL-140-4F, and Y-bptc. Of particular interest is the ultramicroporous Y-bptc for which the adsorption selectivity S _ads = 3.75, in favor of CO₂. The diffusion selectivity, determined experimentally from unary uptakes, Đ ₂/Đ ₁ = 114, also favors CO₂.

Figure a shows transient breakthrough of 50/50 C₂H₂(1)/CO₂(2) mixtures through a fixed-bed adsorber packed with Y-bptc crystals operating at 298 K and total pressure p _t = 100 kPa. The dimensionless concentrations at the exit of the adsorber, c _i /c _i0, are plotted as a function of a modified time parameter defined by Q ₀ t/m _ads. For the four sets of simulations we choose Đ ₁/r _c = 4.25 × 10^–6, 1.18 × 10^–5, 4.72 × 10^–5, and 4.25 × 10^–4 s^–1, maintaining the ratio Đ ₂/Đ ₁ = 114, as determined experimentally. With lower values of the diffusional constants Đ ₁/r _c , i.e., stronger diffusional influences, the breakthrough of the desired product C₂H₂ occurs earlier, resulting in higher productivity of purified C₂H₂ per kilogram of adsorbent in the bed. In Figure b, the productivities of 99%+ pure C₂H₂ are plotted as a function of Đ ₁/r _c . We note that with decreasing Đ ₁/r _c , the C₂H₂ productivities increase from 1.05 to 1.35 mol kg^–1.

6.

(a–d) Transient breakthrough of 50/50 C₂H₂(1)/CO₂(2) mixtures through a fixed-bed adsorber packed with Y-bptc crystals operating at 298 K and total pressure p _t = 100 kPa. In the sets of simulations in (a, b): Đ ₁/r _c = 4.25 × 10^–6, 1.18 × 10^–5, 4.72 × 10^–5, and 4.25 × 10^–4 s^–1, maintaining the ratio Đ ₂/Đ ₁ = 114. In the sets of simulations in (c, d): Đ ₁/r _c = 1.18 × 10^–5 s^–1, and the ratio Đ ₂/Đ ₁ = 200, 114, 50, 10, and 2. In (a, c), the dimensionless concentrations at the exit of the adsorber, c _i /c _i0, are plotted as a function of a modified time parameter defined by Q ₀ t/m _ads. In (b, d), the C₂H₂ productivities are plotted as a function of (b) Đ ₁/r _c and (d) Đ ₂/Đ ₁.

In the sets of breakthrough simulations in Figure c, we maintain Đ ₁/r _c = 1.18 ×10^–5 s^–1, with varying ratios of diffusivities of CO₂ to C₂H₂: Đ ₂/Đ ₁ = 200, 114, 50, 10, and 2. We note that with increasing values of Đ ₂/Đ ₁, the C₂H₂ productivities increase from 0.1 to 1.35 mol kg^–1; see Figure d. We conclude that both Đ ₁/r _c and Đ ₂/Đ ₁ have a significant impact on the productivities of purified C₂H₂. From the breakthrough experiments of He et al., for Y-bptc, with Đ ₂/Đ ₁ = 114, the C₂H₂ productivity is calculated to be 1.52 mol kg^–1, in reasonable agreement with the simulations.

7. Antisynergistic Separation of CO₂(1)/C₂H₆(2) Mixtures

The separation of CO₂/C₂H₆ mixtures to produce purified C₂H₆, while capturing CO₂, is relevant in the context of natural gas processing. Due to azeotrope formation, current technologies for CO₂/C₂H₆ separations require use of extractive distillation that is energy demanding. An energy-efficient alternative to extractive distillation is to use PSA technology, drawing inspiration from the antisynergy evidenced in the uptake experiments in Figure d–f.

Figure shows the transient breakthrough of 50/50 CO₂(1)/C₂H₆(2) mixtures through a fixed-bed adsorber packed with DDR crystals operating at 298 K and total pressure p _t = 40 kPa. In the two sets of simulations, the ratio Đ ₁/Đ ₂ = 1333, as determined experimentally (cf. Figure d). Two different values of the diffusional time constants were chosen: Đ ₁/r _c = 1.25 ×10^–3 and 0.5 s^–1. For the case in which diffusional influences are of diminished significance, Đ ₁/r _c = 0.5 s^–1, the separation selectivity favors C₂H₆ because of mixture adsorption equilibrium; in this case CO₂ exits the bed earlier. This is undesirable from the viewpoint of natural gas processing because we need to recover C₂H₆ as raffinate. With significantly stronger diffusional influences, choosing Đ ₁/r _c = 1.25 ×10^–3 s^–1, uphill diffusion of CO₂ ensures that it enters the pores preferentially, while C₂H₆ is rejected into the gas phase as raffinate, as is desired in practice. The stronger antisynergy between adsorption and diffusion serves to reverse the selectivity in favor of CO₂.

7.

Transient breakthrough of 50/50 CO₂(1)/C₂H₆(2) mixtures through a fixed-bed adsorber packed with DDR crystals operating at 298 K and total pressure p _t = 40 kPa. The dimensionless concentrations at the exit of the adsorber, c _i /c _{i 0}, are plotted as a function of a modified time parameter defined by Q ₀ t/m _ads. For the two sets of simulations Đ ₁/r _c = 1.25 × 10^–3 and 0.5 s^–1, maintaining the ratio Đ ₁/Đ ₂ = 1333.

8. Conclusions

On the basis of a detailed and careful examination of a large number of synergistic and antisynergistic mixture separations using microporous crystalline adsorbents, two important conclusions may be drawn.

• (1)For transient uptake within crystals, the more mobile component attains supraequilibrium loadings during the initial stages of the transience. For quantification of such overshoots, signifying uphill diffusion, we require the use of the M–S diffusion equations using ∂μ _i /∂r as driving forces. The overshoots are engendered by the cross-coefficients Γ _ij (i ≠ j) of the matrix of thermodynamic correction factors. The component molar loadings, plotted in composition space, follow serpentine equilibration paths, which are also attributable to thermodynamic coupling influences. Use of the simplified eq asserting Γ _ij = δ _ij , ignoring thermodynamic coupling, results in monotonous equilibration trajectories without overshoots.
• (2)Mixture separations in fixed-bed adsorption devices are influenced by a combination of three separate factors: adsorption selectivity, S _ads, diffusional time constants, Đ _i /r _c , and diffusion selectivity, Đ ₁/Đ ₂. Two different scenarios may be delineated. In synergistic separations, S _ads > 1; Đ ₁/Đ ₂ > 1, diffusional influences have the potential of enhancing the separation performance. In the second scenario, S _ads > 1; Đ ₁/Đ ₂ < 1, the antisynergy between adsorption and diffusion may be exploited to reverse the selectivity dictated by mixture adsorption equilibrium. For both scenarios, the productivities of the desired product in fixed-bed operations are crucially dependent on diffusional time constants, Đ _i /r _c ; these need to be sufficiently low in order for diffusional influences to be effective. For synergistic or antisynergistic separations to be effective, the ratio Đ ₁/Đ ₂ should be large enough, typically by more than one or two orders of magnitude.

Glossary

Nomenclature

Latin Alphabet

c_i

molar concentration of species i, mol m^–3

c _i0

molar concentration of species i in the fluid mixture at the inlet, mol m^–3

Đ _i

Maxwell–Stefan diffusivity for molecule–wall interaction, m² s^–1

[D]

Fick diffusivity matrix, m² s^–1

m _ads

mass of adsorbent packed in the fixed bed, kg

n

number of species in the mixture, dimensionless

N_i

molar flux of species i with respect to the framework, mol m^–2 s^–1

p _i

partial pressure of species i in the mixture, Pa

p _t

total system pressure, Pa

q _i

component molar loading of species i, mol kg^–1

q̅ _i

radial-averaged component loading of species i, mol kg^–1

Q ₀

volumetric flow rate of gas mixture entering the fixed bed, m³ s^–1

r

radial direction coordinate, m

r _c

radius of crystallite, m

R

gas constant, 8.314 J mol^–1 K^–1

S _ads

adsorption selectivity, dimensionless

t

time, s

T

absolute temperature, K

Greek Alphabet

Γ _ij

thermodynamic factors, dimensionless

δ _ij

Kronecker delta δ _ij (i = j) = 1; δ _i (i ≠ j) = 0, dimensionless

λ _i

eigenvalues of Fick diffusivity matrix [D], m² s^–1

μ _i

molar chemical potential, J mol^–1

ρ

framework density, kg m^–3

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/prechem.2c00003.

• Summary of the Maxwell–Stefan theory of diffusion in microporous materials, (b) methodology adopted for numerical solutions to transient uptake within a single crystalline particle, (c) methodology used for transient breakthroughs in fixed-bed adsorbers, and (d) simulation details including input data on unary isotherms and Maxwell–Stefan diffusivities for each case study (PDF)

The author declares no competing financial interest.