# Dry Glass Reference Perturbation Theory Predictions of the Temperature and Pressure Dependent Separations of Complex Liquid Mixtures Using SBAD-1 Glassy Polymer Membranes

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## Abstract

**:**

## 1. Introduction

_{i}is the molecular flux of component i, ρ

_{i}is the number density of species i sorbed in the membrane, Ð

_{i,p}is the Maxwell–Stefan diffusivity of i in polymer p, x

_{p}is the polymer mole fraction in the membrane, k

_{B}is Boltzmann constant, T is temperature in Kelvin, and μ

_{i}is chemical potential of i. As written, Equation (1) is formulated for a flux of units (molecules/area/time). The Maxwell–Stefan diffusivities were extracted from the pure liquid hydraulic permeation data of Mathias et al. [8] via data fitting to Equation (1).

_{j}is defined as the ratio of permeate mole fraction ${x}_{j}^{p}$to the feed or retentate mole fraction ${x}_{j}^{f}$(these experiments were conducted at low stage cuts), see Equation (2) below.

_{j}of 1.4. On the other hand, there is a clear functional dependence of R

_{j}on the pure liquid solubility. We note here that these diffusivities are on a molar basis, not a volumetric basis. We prefer the molar basis as it allows for a clear interpretation of the diffusivities, whereas the volumetric diffusivities commonly employed in polymer-based Maxwell–Stefan formulations require knowledge of solvent-averaged molecular weights of the membrane system [11].

_{avg}is the average diffusivity of guest species in the membrane evaluated through the relation

_{j}without further tuning? We will explore this question in the current work. Of course, if it is assumed that Equation (6) is reasonably correct, then the accurate prediction of the temperature and pressure dependence of the membrane selectivity is necessarily a test of the underlying thermodynamic model. As in our previous application, we employed the dry glass reference perturbation theory [12] (DGRPT) as our base thermodynamic model to calculate the solubility and chemical potential.

## 2. Theory and Model Development: Dry Glass Reference Perturbation Theory

_{g}. In effect, ρ

_{g}has replaced P as a specification to the chemical potential. This is necessary in glassy polymers due to the fact that the thermodynamics pressure P, is not defined in a glassy polymer [9]. Both chemical potentials in the fluid phase and the polymer phase, Equation (9), are evaluated with equilibrium-density-explicit equations of state.

_{g}in the presence of sorbed guest species is not calculated self-consistently in the theory.

_{i}is the component fugacity (equal to the partial pressure in ideal gases), and k

_{s,i}is an empirical coefficient which is adjusted to sorption data. Equations (9) and (10), when combined with a density explicit equilibrium equation of state, complete the NETGP approach.

_{g}) in the presence of the set of guest species densities (ρ

_{k}) and temperature T. The term ${\mu}_{p}^{poly}\left(T,{\rho}_{g}^{o}\right)$ represents the dry reference polymer chemical potential in the absence of sorbed species, i.e., $\left\{{\rho}_{k}\right\}$ = 0. It is assumed that the dry glass density is known. Finally, the derivatives in the expansion are evaluated in the limit of infinite dilution of guest species.

_{o}and atmospheric pressure. The dry polymer modulus γ and thermal expansion coefficient α are dry pure polymer properties and are independent of any sorbed guest species. For bulk phase behavior calculations, γ is best interpreted as a bulk modulus, while for membrane operations, γ is interpreted as a dry Young’s modulus due to the uniaxial stress. Note, the actual modulus of the solvent swollen polymer is different from γ, and is calculated self-consistently in DGRPT.

_{p}= mx

_{pol}μ

^{2}, which provides the magnitude of polar attractions. Here, x

_{pol}is the fraction of segments in a molecule which are polar, and μ is the dipole moment of a polar segment.

_{p}is set by the polymer molecular structure). Hence, these parameters are adjusted to the pure vapor sorption data of at least two vapor species with the polymer. In this work, we employed polymer parameters fit [12] in this way to SBAD-1. Toluene and heptane pure vapor sorption data at 25 °C were employed to extract [12] the polymer m, σ, ε.

_{ij}between the fluid phase species and the polymer can be adjusted. The binary interaction parameter is used in the combining rule for the cross species ε

_{ij}in terms of the pure component ε

_{a}), fraction of saturated carbon (f

_{sat}= 1 − f

_{a}), and the fraction of carbon which is alkane branches (f

_{br}) in these nine molecules, which will be applied for building correlations between them and the interaction parameters, to be discussed later.

## 3. Results and Discussion:

#### 3.1. Nine Component Mixture

_{j}> 1, and decreases the purity of species with R

_{j}< 1 in the permeate. This is consistent with typical osmotic separations in which higher pressures incrementally compensate against osmotic resistances to permeation.

_{j}decreases for the purified species 1-MN, toluene, and TBB. The small magnitude of this decrease appears to be within the uncertainty of experimental data.

_{j}decreases for the rejected species isocetane and TIPB, while the data appear to show the opposite trend of increasing R

_{j}with increasing temperature. This disagreement may be due to the data uncertainty, considering that both species are dilute in the feed, and small errors in composition measurements could give rise to this apparent discrepancy. The discrepancy may be also due to the model error.

^{−4}K

^{−1}), the model predictions are not significantly affected. To bring the model predicted temperature dependence in line with experiment, we must use α = 1.5 × 10

^{−3}K

^{−1}. However, this value of thermal expansion coefficient is too large to be considered for a dry glassy polymer, which was not applied in this work.

_{j}, it is clear that the model predicted that the temperature dependence of R

_{j}is dominated by the temperature dependence of ${x}_{j}^{m}$.

#### 3.2. Compositional Modelling of a Light Shale Crude Oil

_{p}for each petroleum species. For this, we used the approach of Marshall et al. [26,27] which accurately generates PC-SAFT parameters on the basis of pure component boiling point temperature (T

_{b}), specific gravity (SG), molecular weight (MW) and the fraction of carbon which is aromatic (f

_{a}). The molecular weight and f

_{a}are defined in each SOL-lumped species, and specific correlations for T

_{b}and SG are from ExxonMobil proprietary models. Therefore, the SOL model of composition provided all the required information to derive the PC-SAFT pure component parameters for the crude oil.

_{a}), fraction of saturated carbon (f

_{sat}= 1 − f

_{a}), and fraction of carbon that contains alkane branches (f

_{br}). These fractions are included in Table 3 for the nine hydrocarbons for which we have regressed binary interaction parameters with SBAD-1. We did not include the fraction of naphthenic carbon as there is not currently substantial evidence to suggest that paraffin versus naphthenic interactions with the polymer necessitate descriptors in the k

_{ij}model. This can be seen in Table 3, i.e., both methylcyclohexane and octane have k

_{ij}~0.06. It should be noted that molecular weight dependence is built in theoretically in DGRPT through PC-SAFT.

_{a}, c

_{sat}, and c

_{br}are fitted to the binary interaction parameters in Table 3. The coefficients are listed in Table 5. As shown in the parity plot Figure 6, Equation (15) gives a reasonable correlation of the binary interaction parameters.

_{j}is close to 1 for small molecular weights, which initially increases with an increasing molecular weight going through a maximum and then decreases to ~0 at high molecular weights. The maximum is a result of an initial increase in solubility with increasing molecular weight, and then a subsequent decline in solubility with increasing molecular weight. Double branched alkanes are the only class that do not exhibit the increase in R

_{j}at low molecular weights, or the corresponding maximum.

_{j}is supported by the two-dimensional gas chromatography (2D-GC) results of Thompson et al. [3]. Figure 9 plots 2D-GC measurements of R

_{j}versus retention time for the n-alkane series in the light shale crude oil separation. Increasing the retention time corresponds to an increasing molecular weight. Thompson et al. [3] reported the retention time of n-octane (MW = 114.23 g/mol) to be ~20 min. In Figure 9, the retention time of 20 min is just to the right of the maximum, which occurs at ~15 min, corresponding to n-heptane. The model predicts (Figure 8) that n-nonane exhibits the maximum separation factor for the n-alkane series. This prediction is remarkably accurate (C

_{9}vs. C

_{7}), given the fact that the alkane carbon numbers in the SOL input range from C

_{2}–C

_{70}. In addition, using 2D-GC, Thompson et al. measured the enrichment of n-alkanes in the permeate stream to be ~21%. This is in good agreement with the DGRPT model prediction of ~23%, as shown in Table 6.

_{c}, is calculated as

_{c,p}is the weight fraction of a full molecular class in the permeate, and w

_{c,f}is the weight fraction of a full molecular class in the feed. As can be seen, each class is enriched, except for the double branched alkanes, which are strongly rejected.

## 4. Conclusions

_{p}and the binary interaction parameters between the nine hydrocarbon species and the polymer. These parameters were determined solely from pure component vapor phase sorption measurements at 25 °C. However, the model was able to make robust predictions for the temperature and pressure dependence of complex liquid mixture separations, using only room temperature vapor sorption measurements to parameterize the model.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Diagram of SBAD-1 repeat unit [8].

**Figure 3.**Separation coefficients of the nine-component mixture in the SBAD-1 membrane [3] versus Maxwell–Stefan diffusivities [9] (

**top**) and sorption data from pure component liquids [8] (

**bottom**). The membrane separation was done at a temperature of 22 °C and trans-membrane pressure 40 bar. Dashed lines are included as visual guides. MCYC6 = methylcyclohexane, TBB = tert-butylbenzene, TIPB = 1,3,5-triisopropylbenzene, 1-MN = 1-methylnaphthalene.

**Figure 4.**Separation coefficients (Equation (2)) for the SBAD-1 membrane separation of a complex liquid feed composition listed in Table 4. Symbols are experimental data [3] and curves are predictions using Equation (6). In the top panel, the 40 bar experimental data are for T = 22 °C and the remaining data are at 25 °C.

**Figure 5.**Top: Separation coefficients (Equation (2)) for the SBAD-1 membrane separation of a complex liquid feed composition listed in Table 4, at a fixed feed pressure of 50 bar. Symbols are experimental data [3] and curves are predictions using Equation (6). (

**Bottom**): Same as (

**top**) except plotting model predictions of the ratio of mole fraction in membrane at the feed interface to the feed.

**Figure 6.**Parity plot of correlation predicted binary interaction parameters (Equation (15)) versus data values from Table 3.

**Figure 8.**Model predicted separation coefficients versus molecular weight for several molecular classes at 130 °C and 55 bar.

**Figure 9.**2D-GC measurements [3] of the n-alkane separation coefficients versus retention time in the light shale crude oil separation.

octane | cis-decalin |

1-methylnaphthalene (1-MN) | isocetane |

toluene | tert-butylbenzene (TBB) |

Methylcyclohexane (MCYC6) | 1,3,5-triisopropylbenzene (TIPB) |

isooctane |

**Table 2.**Pure component polar PC-SAFT parameters for nine hydrocarbons and SBAD-1. The molecular weight of SBAD-1 is assumed to be MW = 100,000 g/mol. MCYC6 = methylcyclohexane, TBB = tert-butylbenzene, TIPB = 1,3,5-triisopropylbenzene, 1-MN = 1-methylnaphthalene.

Component | m | σ(Å) | ε/k_{b}(K) | α_{p}(D) | Ref. |
---|---|---|---|---|---|

1-MN | 3.163 | 3.998 | 354.70 | 3.6 | [12] |

MCYC6 | 2.675 | 3.989 | 281.63 | 0 | [12] |

TIPB | 5.471 | 3.922 | 255.83 | 2.16 | [12] |

TBB | 3.459 | 3.953 | 284.62 | 2.16 | [12] |

isooctane | 3.144 | 4.091 | 249.63 | 0 | [12] |

isocetane | 5.016 | 4.301 | 266.58 | 0 | [12] |

octane | 3.841 | 3.819 | 242.13 | 0 | [12] |

toluene | 2.612 | 3.814 | 293.33 | 2.16 | [24] |

SBAD-1 | 0.0397 MW | 2.963 | 124.13 | 0.028 MW | [12] |

**Table 3.**Binary interaction parameters [12] between hydrocarbons and SBAD-1. Fraction of aromatic carbon (f

_{a}), fraction of saturated carbon (f

_{sat}= 1 − f

_{a}), and fraction of carbon which is alkane branches (f

_{br}). MCYC6 = methylcyclohexane, TBB = tert-butylbenzene, TIPB = 1,3,5-triisopropylbenzene, 1-MN = 1-methylnaphthalene.

Component | k_{ij} | f_{a} | f_{sat} | f_{br} |
---|---|---|---|---|

octane | 0.0663 | 0 | 1 | 0 |

1-MN | 0.0174 | 0.91 | 0.09 | 0 |

toluene | −0.0051 | 0.857 | 0.143 | 0 |

MCYC6 | 0.0688 | 0 | 1 | 0 |

isooctane | 0.1712 | 0 | 1 | 0.375 |

cis-decalin | 0.1256 | 0 | 1 | 0 |

isocetane | 0.1181 | 0 | 1 | 0.437 |

TBB | 0.1296 | 0.6 | 0.4 | 0.2 |

TIPB | 0.1147 | 0.4 | 0.6 | 0.2 |

**Table 4.**Nine-component mixture for SBAD-1 membrane separation from Thompson et al. [3].

Component | ${\mathit{x}}_{\mathit{j}}^{\mathit{f}}$ |
---|---|

octane | 0.22 |

1-MN | 0.02 |

toluene | 0.171 |

MCYC6 | 0.281 |

isooctane | 0.15 |

cis-decalin | 0.11 |

isocetane | 0.013 |

TBB | 0.022 |

TIPB | 0.016 |

**Table 5.**Coefficients for the general binary interaction parameter model between hydrocarbons and SBAD-1.

c_{a} | c_{sat} | c_{br} |
---|---|---|

0.01252 | 0.0663 | 0.2797 |

**Table 6.**Model predictions of molecular class enrichment in the permeate (Equation (16)) at 130 °C and 55 bar.

Class | % Enrichment |
---|---|

Linear alkanes | 23.1 |

Single branch alkanes | 19.0 |

Double branch alkanes | −43.4 |

Alkyl cyclohexanes | 11.9 |

Alkyl benzenes | 35.3 |

Alkyl cyclohexylbenzenes | 0.3 |

Alkyl naphthalenes | 39.1 |

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**MDPI and ACS Style**

Marshall, B.D.; Li, W.; Lively, R.P.
Dry Glass Reference Perturbation Theory Predictions of the Temperature and Pressure Dependent Separations of Complex Liquid Mixtures Using SBAD-1 Glassy Polymer Membranes. *Membranes* **2022**, *12*, 705.
https://doi.org/10.3390/membranes12070705

**AMA Style**

Marshall BD, Li W, Lively RP.
Dry Glass Reference Perturbation Theory Predictions of the Temperature and Pressure Dependent Separations of Complex Liquid Mixtures Using SBAD-1 Glassy Polymer Membranes. *Membranes*. 2022; 12(7):705.
https://doi.org/10.3390/membranes12070705

**Chicago/Turabian Style**

Marshall, Bennett D., Wenjun Li, and Ryan P. Lively.
2022. "Dry Glass Reference Perturbation Theory Predictions of the Temperature and Pressure Dependent Separations of Complex Liquid Mixtures Using SBAD-1 Glassy Polymer Membranes" *Membranes* 12, no. 7: 705.
https://doi.org/10.3390/membranes12070705