# Mixed-Gas Selectivity Based on Pure Gas Permeation Measurements: An Approximate Model

^{*}

## Abstract

**:**

## 1. Introduction

_{2}/N

_{2}selectivities based on single gas measurement might be higher than mixed-gas selectivities due to the swelling of the polymer in CO

_{2}environment and plasticization effect [10]. The same tendency was observed for hydrocarbons separation through polyalkylmethylsiloxanes composite membranes [9]. The separation of an eight-component mixture of saturated and unsaturated hydrocarbons C1-C4 was studied. It was shown that the values of ideal selectivities for C

_{3}H

_{8}/CH

_{4}and n-C

_{4}H

_{10}/CH

_{4}gas pairs were higher than mixed-gas selectivities. This effect was explained by significant swelling of the membrane material in the hydrocarbon mixture, first of all, due to the presence of n-butane.

_{6}in the feed; the phenomenon was called “pore blocking” or “light gas rejection” effect. Thus, the ideal selectivity ${\alpha}_{12}$, calculated by the ratio of the permeabilities for pure gases 1 and 2 can differ rather significantly from the actual, mixed-gas selectivity ${\alpha}_{12}^{mix}$.

_{2}/C

_{2}H

_{6}[13,14] and CO

_{2}/CH

_{4}[15] mixtures in polyethylene oxide and polydimethylsiloxane, respectively, can be described with acceptable accuracy without regard to diffusional coupling. On the other hand, for pervaporation and gas separation by rigid glassy polymers, consideration of cross-terms in the flux equations seems necessary and, accordingly, the coupling effect cannot be neglected. It has been found that the friction interactions of penetrant-glassy polymer and penetrant-penetrant are of the same order [16,17].

## 2. Basic Equations

## 3. Kedem’s Solution

## 4. Model Development

#### 4.1. Approximations and Flux Equations

#### 4.2. Separation Factor

#### 4.3. Component Permeabilities

#### 4.4. Explicit Form of the Peclét Number

^{3}(STP)/(cm

^{3}·atm), the feed pressure expressed in atm, a correction factor $k\sim 1$ was introduced into the equation to compensate for the approximate nature of equality (48).

^{3}/mol, and $k=1$, then $\mathrm{const}\approx 0.04$. Since component 1 is the preferentially permeating component (${\alpha}_{12}>1$), the coupling effect is maximum in the limit of infinite dilution of component 2 in the feed mixture. In addition, as can be seen from Equation (50), the value of B decreases with increasing diffusivity selectivity. Therefore, one would expect the coupling effect to be weaker for membranes with diffusion selectivity than for membranes with sorption selectivity.

_{2+}, have almost no sorption selectivity (${\alpha}_{S}\sim 1$), and the ideal selectivity values are ${\alpha}_{12}\sim {\alpha}_{D}=2-30$ [8,30]. If one accepts ${\alpha}_{12}={\alpha}_{D}=10$ and ${S}_{2}=10$ cm

^{3}(STP)/(cm

^{3}·atm) (as the average solubility of propane in aromatic polyimides) for a rough estimate, then from Equation (50) one obtains that the Peclét numbers vary from 0.2 to 1.9. For solubility selectivity glassy polymers (polyacetylenes, polynorbornenes, PIM-1) used for C

_{2+}/methane separations, the diffusion selectivity is less than 1, and the sorption selectivity is in a wide range from 5 to ~10

^{3}[8,31]. For typical values ${\alpha}_{12}=10$, ${\alpha}_{D}=0.2$ and ${S}_{2}=4$ cm

^{3}(STP)/(cm

^{3}·atm) (as the solubility of methane in PTMSP at 25 °C) one gets from Equation (50) the range of Peclét numbers from 0.4 to 3.6. Thus, it can be stated that in the hydrocarbon separation by glassy polymers, the Peclét number reaches values of the order of unity (notice that the above estimates were made at a total feed pressure of 10 atm).

#### 4.5. Comparison of the Model with Experimental Data

_{4}H

_{10}/CH

_{4}selectivity is noticeably greater than the ideal value (=3.5) and grows with n-butane concentration in feed (Figure 3a (top)). The component permeabilities (in units of pure gas values), on the contrary, decrease with the increasing concentration of n-butane (Figure 3a (bottom)). Light gas CH

_{4}permeability in mixed-gas conditions decreases as compared to the pure gas value (${\mathsf{\Pi}}_{{\mathrm{CH}}_{4}}^{mix}<{\mathsf{\Pi}}_{{\mathrm{CH}}_{4}}^{0}$), whereas the situation is the opposite for heavier n-C

_{4}H

_{10}(${\mathsf{\Pi}}_{{\mathrm{C}}_{4}{\mathrm{H}}_{10}}^{mix}>{\mathsf{\Pi}}_{{\mathrm{C}}_{4}{\mathrm{H}}_{10}}^{0}$). This behavior is consistent with the second inequality (45).

_{3}H

_{6}/C

_{3}H

_{8}selectivity is almost half that of pure gas selectivity and decreases with increasing concentration of preferentially permeating component (propylene) in feed (Figure 3b (top)). The permeability of both components increases with the propylene concentration in the feed (Figure 3b (bottom)). Permeability behavior in mixed-gas conditions follows the first inequality (45): for more mobile gas C

_{3}H

_{6}permeability decreases compared to pure gas one, while for less mobile gas C

_{3}H

_{8}there is an increase in permeability compared to a pure gas one.

^{3}/mol, and $k=1$. Given the crude approximation (48) for 1–2 diffusivity, the developed model shows more than satisfactory performance.

_{4}H

_{10}/CH

_{4}selectivity of PTMSP membranes significantly exceeds the ideal selectivity, by a factor of 6–18 according to various experimental data [31]. The increase in selectivity is associated with a marked (3–9 times) decrease in CH

_{4}permeability under mixed-gas conditions compared to pure gas conditions [31]. The physical reason for depressing the CH

_{4}permeability is commonly attributed to a blocking mechanism: light gas CH

_{4}diffusion in free volume elements of ”microporous” PTMSP is hindered, blocked by the sorbed molecules of heavier, condensable n-C

_{4}H

_{10}[11,35]. Along with this basic reason, one should also take into account the effect of competitive sorption, which can lead to a change (increase) of solubility selectivity [20,33]. In terms of the developed model, the blocking of methane transport by n-butane is a consequence of frictional interaction between penetrants, in other words, the blocking mechanism is a manifestation of coupling between methane and n-butane fluxes.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## References

- Galizia, M.; Chi, W.S.; Smith, Z.P.; Merkel, T.C.; Baker, R.W.; Freeman, B.D. 50th Anniversary Perspective: Polymers and Mixed Matrix Membranes for Gas and Vapor Separation: A Review and Prospective Opportunities. Macromolecules
**2017**, 50, 7809–7843. [Google Scholar] [CrossRef] - Grushevenko, E.A.; Borisov, I.L.; Volkov, A.V. High-Selectivity Polysiloxane Membranes for Gases and Liquids Separation (A Review). Petrol. Chem.
**2021**, 61, 959–976. [Google Scholar] [CrossRef] - Balҫık, M.; Tantekin-Ersolmaz, S.B.; Pinnau, I.; Ahunbay, M.G. CO
_{2}/CH_{4}mixed-gas separation in PIM-1 at high pressures: Bridging atomistic simulations with process modeling. J. Membr. Sci.**2021**, 640, 119838. [Google Scholar] [CrossRef] - Yi, S.; Ghanem, B.; Liu, Y.; Pinnau, I.; Koros, W.J. Ultraselective glassy polymer membranes with unprecedented performance for energy-efficient sour gas separation. Sci. Adv.
**2019**, 5, eaaw5459. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhang, C.; Fu, L.; Tian, Z.; Cao, B.; Li, P. Post-crosslinking of triptycene-based Tröger’s base polymers with enhanced natural gas separation performance. J. Membr. Sci.
**2018**, 556, 277–284. [Google Scholar] [CrossRef] - Sutrisna, P.D.; Hou, J.; Zulkifli, M.Y.; Li, H.; Zhang, Y.; Liang, W.; D’Alessandro, D.M.; Chen, V. Surface functionalized UiO-66/Pebax-based ultrathin composite hollow fiber gas separation membranes. J. Mater. Chem. A
**2018**, 6, 918–931. [Google Scholar] [CrossRef] - Vaughn, J.T.; Harrigan, D.J.; Sundell, B.J.; Lawrence, J.A., III; Yang, J. Reverse selective glassy polymers for C3+ hydrocarbon recovery from natural gas. J. Membr. Sci.
**2017**, 522, 68–76. [Google Scholar] [CrossRef] - Iyer, G.M.; Liu, L.; Zhang, C. Hydrocarbon separations by glassy polymer membranes. J. Polym. Sci.
**2020**, 58, 2482–2517. [Google Scholar] [CrossRef] - Grushevenko, E.A.; Borisov, I.L.; Knyazeva, A.A.; Volkov, V.V.; Volkov, A.V. Polyalkylmethylsiloxanes composite membranes for hydrocarbon/methane separation: Eight component mixed-gas permeation properties. Sep. Purif. Technol.
**2020**, 241, 116696. [Google Scholar] [CrossRef] - Liu, L.; Chakma, A.; Feng, X. CO
_{2}/N_{2}separation by poly(ether block amide) thin film hollow fiber composite membranes. Ind. Eng. Res.**2005**, 44, 6874–6882. [Google Scholar] [CrossRef] - Srinivasan, R.; Auvil, S.R.; Burban, P.M. Elucidating the mechanism(s) of gas transport in poly [1-(trimethylsilyl)-1-propyne] (PTMSP) membranes. J. Membr. Sci.
**1994**, 86, 67–86. [Google Scholar] [CrossRef] - Koros, W.J.; Chern, R.T.; Stannett, V.; Hopfenberg, H.B. A Model for Permeation of Mixed Gases and Vapors in Glassy Polymers. J. Polym. Sci. Polym. Phys. Ed.
**1981**, 19, 1513–1530. [Google Scholar] [CrossRef] - Ribeiro, C.P., Jr.; Freeman, B.D.; Paul, D.R. Modeling of multicomponent mass transfer across polymer films using a thermodynamically consistent formulation of the Maxwell-Stefan equations in terms of volume fractions. Polymer
**2011**, 52, 3970–3983. [Google Scholar] [CrossRef] - Krishna, R. Describing mixture permeation across polymeric membranes by a combination of Maxwell-Stefan and Flory-Huggins models. Polymer
**2016**, 103, 124–131. [Google Scholar] [CrossRef] - Genduso, G.; Litwiller, E.; Ma, X.; Zampini, S.; Pinnau, I. Mixed-gas sorption in polymers via a new barometric test system: Sorption and diffusion of CO
_{2}-CH_{4}mixtures in polydimethylsiloxane (PDMS). J. Membr. Sci.**2019**, 577, 195–204. [Google Scholar] [CrossRef] - Krishna, R. Using the Maxwell-Stefan formulation for highlighting the influence of interspecies (1−2) friction on binary mixture permeation across microporous and polymeric membranes. J. Membr. Sci.
**2017**, 540, 261–276. [Google Scholar] [CrossRef] - Dutta, R.C.; Bhatia, S.K. Atomistic Investigation of Mixed-Gas Separation in a Fluorinated Polyimide Membrane. ACS Appl. Polym. Mater.
**2019**, 1, 1359–1371. [Google Scholar] [CrossRef] [Green Version] - Dhingra, S.S.; Marand, E. Mixed gas transport study through polymeric membranes. J. Membr. Sci.
**1998**, 141, 45–63. [Google Scholar] [CrossRef] - Yang, W.; Zhou, H.; Zong, C.; Li, Y.; Jin, W. Study on membrane performance in vapor permeation of VOC/N
_{2}mixtures via modified constant volume/variable pressure method. Sep. Purif. Technol.**2018**, 200, 273–283. [Google Scholar] [CrossRef] - Vopička, O.; De Angelis, M.G.; Sarti, G.C. Mixed gas sorption in glassy polymeric membranes: I. CO
_{2}/CH_{4}and n-C_{4}/CH_{4}mixtures sorption in poly(1-trimethylsilyl-1-propyne) (PTMSP). J. Membr. Sci.**2014**, 449, 97–108. [Google Scholar] [CrossRef] - Ricci, E.; Minelli, M.; De Angelis, M.G. A multiscale approach to predict the mixed gas separation performance of glassy polymeric membranes for CO
_{2}capture: The case of CO_{2}/CH_{4}mixture in Matrimid^{®}. J. Membr. Sci.**2017**, 539, 88–100. [Google Scholar] [CrossRef] - Kedem, O. The role of coupling in pervaporation. J. Membr. Sci.
**1989**, 41, 277–284. [Google Scholar] [CrossRef] - Schmitt, A.; Craig, J.B. Frictional coefficient formalism and mechanical equilibrium in membranes. J. Phys. Chem.
**1977**, 81, 1338–1342. [Google Scholar] [CrossRef] - Krishna, R.; Wesselingh, J.A. The Maxwell-Stefan approach to mass transfer. Chem. Eng. Sci.
**1997**, 52, 861–911. [Google Scholar] [CrossRef] - Ghoreyshi, A.A.; Farhadpour, F.A.; Soltanieh, M. A general model for multicomponent transport in nonporous membranes based on Maxwell-Stefan formulation. Chem. Eng. Commun.
**2004**, 191, 460–499. [Google Scholar] [CrossRef] - Esposito, E.; Mazzei, I.; Monteleone, M.; Fuoco, A.; Carta, M.; McKeown, N.B.; Malpass-Evans, R.; Jansen, J.C. Highly Permeable Matrimid
^{®}/PIM-EA(H2)-TB Blend Membrane for Gas Separation. Polymers**2019**, 11, 46. [Google Scholar] [CrossRef] [Green Version] - Baker, R.W. Membrane Technology and Applications, 2nd ed.; Wiley: Hoboken, NJ, USA, 2004; Chapter 8. [Google Scholar]
- Krishna, R.; van Baten, J.M. A simplified procedure for estimation of mixture permeances from unary permeation data. J. Membr. Sci.
**2011**, 367, 204–210. [Google Scholar] [CrossRef] - Liu, X.; Bardow, A.; Vlugt, T.J.H. Multicomponent Maxwell-Stefan Diffusivities at Infinite Dilution. Ind. Eng. Chem. Res.
**2011**, 50, 4776–4782. [Google Scholar] [CrossRef] [Green Version] - Semenova, S.I. Polymer membranes for hydrocarbon separation and removal. J. Membr. Sci.
**2004**, 231, 189–207. [Google Scholar] [CrossRef] - Yampolskii, Y.; Starannikova, L.; Belov, N.; Bermeshev, M.; Gringolts, M.; Finkelshtein, E. Solubility controlled permeation of hydrocarbons: New membrane materials and results. J. Membr. Sci.
**2014**, 453, 532–545. [Google Scholar] [CrossRef] - Sultanov, E.Y.; Ezhov, A.A.; Shishatskiy, S.M.; Buhr, K.; Khotimskiy, V.S. Synthesis, Characterization, and Properties of Poly(1-trimethylsilyl-1-propyne)-block-poly(4-methyl-2-pentyne) Block Copolymers. Macromolecules
**2012**, 45, 1222–1229. [Google Scholar] [CrossRef] - Raharjo, R.D.; Freeman, B.D.; Sanders, E.S. Pure and mixed gas CH
_{4}and n-C_{4}H_{10}sorption and dilation in poly(1-trimethylsilyl-1-propyne). Polymer**2007**, 48, 6097–6114. [Google Scholar] [CrossRef] - Tanaka, K.; Taguchi, A.; Hao, J.; Kita, H.; Okamoto, K. Permeation and separation properties of polyimide membranes to olefins and paraffins. J. Membr. Sci.
**1996**, 121, 197–207. [Google Scholar] [CrossRef] - Pinnau, I.; Toy, L.G. Transport of organic vapors through poly(1-trimethylsilyl-1-propyne). J. Membr. Sci.
**1996**, 116, 199–209. [Google Scholar] [CrossRef]

**Figure 1.**Separation factor for the binary gas mixture (molar fraction ${x}_{1}=0.1$) with the ideal selectivity ${\alpha}_{12}=15$ as a function of the membrane Peclét number: (

**a**) sorption-selective membrane (${\alpha}_{S}=30$, ${\alpha}_{D}=0.5$); (

**b**) diffusion-selective membrane (${\alpha}_{S}=1.5$, ${\alpha}_{D}=10$). Solid lines: the pressure ratio r = 0, dotted lines: r = 0.1.

**Figure 2.**Calculated permeability of component 1 (red lines) and component 2 (blue lines) for the binary gas mixture (molar fraction ${x}_{1}=0.1$) with the ideal selectivity ${\alpha}_{12}=15$ as a function of the membrane Peclét number: (

**a**) sorption-selective membrane (${\alpha}_{S}=30$, ${\alpha}_{D}=0.5$); (

**b**) diffusion-selective membrane (${\alpha}_{S}=1.5$, ${\alpha}_{D}=10$). Solid lines: the pressure ratio r = 0, dotted lines: r = 0.1.

**Figure 3.**(

**a**) Mixed-gas n-C

_{4}H

_{10}/CH

_{4}selectivity and relative permeabilities ${\tilde{\mathsf{\Pi}}}_{i}={\mathsf{\Pi}}_{i}^{mix}/{\mathsf{\Pi}}_{i}^{0}$ of the mixture components through PTMSP membrane versus n-C

_{4}H

_{10}feed molar fraction; (

**b**) Mixed-gas C

_{3}H

_{6}/C

_{3}H

_{8}selectivity and relative permeabilities of the mixture components through polyimide 6FDA-TrMPD versus C

_{3}H

_{6}feed molar fraction. The results of selectivity and permeability calculations are given for pressure ratio r = 0 (solid lines) and r = 0.05 (dashed lines). Experimental selectivities and permeabilities for 2 mol % n-C

_{4}H

_{10}/98 mol % CH

_{4}/PTMSP and 50 mol % C

_{3}H

_{6}/50 mol % C

_{3}H

_{8}/6FDA-TrMPD systems are shown by the symbols.

**Table 1.**Permeabilities and solubility coefficients (Henry’s constants), and their ratio for n-C

_{4}H

_{10}/CH

_{4}(2 mol %/98 mol %) and C

_{3}H

_{6}/C

_{3}H

_{8}(equimolar mixture) in PTMSP and 6FDA-TrMPD membranes, respectively. Diffusivity selectivities are shown in the last column.

Polymer | ${\mathsf{\Pi}}_{1}$ (barrer) | ${\mathit{S}}_{1}$ (cm ^{3}(STP) cm^{−3}·atm^{−1})
| ${\mathsf{\Pi}}_{1}/{\mathsf{\Pi}}_{2}$ | ${\mathit{S}}_{1}/{\mathit{S}}_{2}$ | ${\mathit{D}}_{1}/{\mathit{D}}_{2}$ |
---|---|---|---|---|---|

n-C_{4}H_{10} | n-C_{4}H_{10}/CH_{4} | ||||

PTMSP ^{1} | |||||

pure | 49,000 | 1156 | 3.5 | 286.1 | 0.012 |

mixed | 68,000 | - | 31 | - | - |

C_{3}H_{6} | C_{3}H_{6}/C_{3}H_{8} | ||||

6FDA-TrMPD ^{2} | |||||

pure | 30 | 17.5 | 11 | 1.2 | 8.8 |

mixed | 20 | - | 6.0 | - | - |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Malakhov, A.O.; Volkov, V.V.
Mixed-Gas Selectivity Based on Pure Gas Permeation Measurements: An Approximate Model. *Membranes* **2021**, *11*, 833.
https://doi.org/10.3390/membranes11110833

**AMA Style**

Malakhov AO, Volkov VV.
Mixed-Gas Selectivity Based on Pure Gas Permeation Measurements: An Approximate Model. *Membranes*. 2021; 11(11):833.
https://doi.org/10.3390/membranes11110833

**Chicago/Turabian Style**

Malakhov, Alexander O., and Vladimir V. Volkov.
2021. "Mixed-Gas Selectivity Based on Pure Gas Permeation Measurements: An Approximate Model" *Membranes* 11, no. 11: 833.
https://doi.org/10.3390/membranes11110833