# 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

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**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 | - | - |

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**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