Heat of Sorption of Gases in Glassy Polymers: Prediction via Applying

Article pubs.acs.org/jced

Heat of Sorption of Gases in Glassy Polymers: Prediction via Applying Physical Properties of the Penetrants and Polymers Ali Ghadimi,† Somayeh Norouzbahari,‡ and Toraj Mohammadi*,‡ †

Faculty of Petrochemical, Iran Polymer and Petrochemical Institute, P.O. Box 14965-115, Tehran, Iran Research and Technology Center for Membrane Separation Processes, School of Chemical, Oil and Gas Engineering, Iran University of Science and Technology, P.O. Box 16765-163, Tehran, Iran

‡

S Supporting Information *

ABSTRACT: Polymeric gas separation membranes have found widespread application particularly for CO2 capture and natural gas sweetening. Permeability through these membranes depends on solubility (S) and diffusion (D) coefficients of the gas penetrants, for which estimation of the molar heat of sorption (ΔHS), is of high importance to attain the S value. A well-known linear empirical correlation to predict ΔHS in rubbery polymers, has been already proposed by van Amerongen [Rubber Chem. Technol. 1964, 37, 1065]. In this paper, a correlation based on the physical properties of the gas penetrants and polymers has been provided as a basis for the extensively applied ΔHS correlation for the glassy polymers in the literature developed according to van Amerongen’s empirical method. The proposed correlation, takes into account the contribution of the polymer nature, which is more distinct for small molecule, highly supercritical gas solutes, that is, much above their critical temperatures, such as H2 and He. It was found that the physical properties of the penetrants play a significant role in the ΔHS calculation. The developed correlation was satisfactorily validated for various gases including CO2, CH4, C2H4, C2H6, C3H6, C3H8, C4H10, SO2, O2, He, H2, and N2, and glassy polymers consisting of poly(vinyl acetate) (PVAC), poly(ethylene terephthalate) (PETP), poly(vinyl chloride) (PVC), and polycarbonate (PC).

1. INTRODUCTION

Therefore, attaining the solubility coefficient is important to obtain the permeability coefficient of a gas penetrant. The temperature dependency of the solubility coefficient of the gas penetrants, that is, S(T) expressed in m3(STP)/(m3 Pa) through polymeric membranes, is typically given by a van’t Hoff−Arrhenius equation12 that can be also stated as follows:

Polymeric membranes have drawn considerable attention in a variety of fields of gas separation processes in recent years, for instance, CO2 capture from flue gas of the fossil-fuel-fired power plants, natural gas sweetening, production of N2 from air, and H2 recovery in refineries.1−4 Compared to wellestablished separation technologies such as amine-based gas absorption, pressure-swing adsorption, and cryogenic distillation that require constant monitoring and are energyintensive,5−7 separation by means of the membranes offers advantages including ease of operation and scale-up and compactness, and is relatively cost-effective, more specifically for small gas flows.8,9 Permeability of the gas penetrants through dense nonporous polymeric membranes has long been described by the solutiondiffusion model. In this model, molecules of the gas penetrant, initially dissolve into the high pressure side (i.e., upstream) of the membrane followed by their diffusion across the membrane. Finally the diffused molecules desorb from the low pressure side (i.e., downstream) of the membrane.10 The permeability coefficient of a gas component i is generally considered as the product of the respective solubility/sorption coefficient (Si) and a concentration-averaged effective diffusion coefficient (Di) as follows:11 Pi = SiDi

log S(T ) = log S0 − 0.4343(ΔHs/(RT ))

where S0 is a pre-exponential factor in m3 (STP)/(m3 Pa), ΔHS denotes the molar heat/enthalpy of gas sorption/solution in (J/ mol), R stands for the universal gas constant (8.314 J/(mol K)), and T is the absolute temperature in (K). A linear empirical correlation to connect ΔHS of the gas solutes in elastomers (rubbery polymers) to their normal boiling point (Tb) or critical (Tc) temperatures, as the measures of the solute condensability, has been already presented by van Amerongen.13−15 It was demonstrated that this correlation is more accurate when S is correlated versus the Lennard-Jones temperature (energy parameter) of the gas solutes (ε/k expressed in K.16 Indeed ε/k is a more fundamental thermodynamic property compared to Tb and/or Tc of the gas penetrants. Received: December 1, 2016 Accepted: March 21, 2017 Published: March 31, 2017

(1) © 2017 American Chemical Society

(2)

1433

DOI: 10.1021/acs.jced.6b01002 J. Chem. Eng. Data 2017, 62, 1433−1439

Journal of Chemical & Engineering Data

Article

group contribution method. The main reason was that SASA values were originally computed for gas dissolution in water and could not be considered as the exact representative of the van der Waals gas−polymer interactions. De Angelis et al.21 developed a fundamental rigorous interpretation for infinite dilution solubility in glassy polymers through considering the penetrant sorption in three consecutive steps: the gas penetrant condensation, dissolution of the condensed gas in the polymeric matrix at equilibrium, and eventually departure of the polymeric phase from equilibrium to a nonequilibrium glassy state. They employed the lattice fluid (LF) equation of state to calculate the contributions of the condensation and equilibrium mixing enthalpies. Additionally, the nonequilibrium lattice fluid (NELF) model,22,23 was applied to determine the contribution of the glassy polymers nonequilibrium state. Finally, they presented a fundamental explanation for strong dependency of the solubility coefficient on the pure penetrant properties, more specifically its critical temperature (condensability). This dependency was also justified in their previous work,24 in which the role of the polymer fractional free volume was investigated using the NELF model as well. Therefore, on the basis of the aforementioned previous studies, to improve the prediction of ΔHS values in the glassy polymers, the nature of the polymer mainly in terms of its Tg and solubility parameter (δ), should be also taken into account. It is of significant value for low molecular weight, highly supercritical gas penetrants, such as He and H2. In this paper these physical properties were considered as Tb, ε/k, and δ of the gas penetrants along with Tg and δ of the polymer; where attaining this representation for ΔHS is the main objective of this work. Hence, this paper deals with providing a correlation for ΔHS based on the physical properties of the penetrant and polymer as an alternative of the empirical linear correlation for the glassy polymers, that is, eq 3 with about 14% higher precision in terms of the correlation coefficient (R2). Indeed eq 3 has been developed solely based on the fitting of the experimental observations. To achieve this purpose, thermodynamic concepts of condensation and mixing along with numerical optimization techniques were employed. Additionally, the exact contribution of the condensation and mixing enthalpies was determined. It is worthy to note that in the proposed correlation, only polymer and penetrant physical properties appear without any additional adjustment/fitting parameters, which enhances the simplicity of the correlation and makes its application more convenient.

The same procedure has been then applied to derive the similar correlations for ΔHS and S in glassy polymers, that is, the polymers at temperatures below their glass transition temperatures (Tg), using experimental data in the literature.12 Compared to the rubbery polymers, the glassy ones, have found more application in commercial gas separation processes owing to their higher overall permselectivity in recent years.17,18 In brief, the following linear correlations have been proposed for the glassy polymers based on the van Amerongen’s approach:12 10−3ΔHs/R = 0.5 − 0.010ε/k ± 1.2

(3)

log S0 = − 6.65 − 0.005ε/k ± 1.8

(4)

The solubility coefficient can be obtained from eq 2, after substitution of the required ΔHs/R and log S0 values, calculated from eqs 3 and 4, respectively. It is worthy of note that for the glassy polymers, accuracy of the solubility correlation is relatively lower compared to the one for the rubbery polymers.12 This is mainly attributed to neglecting the effect of the polymer nature in the van Amerongen correlation,15 which was originally proposed for the rubbery and not the glassy polymers. A number of other researchers have also proposed different correlations to estimate ΔHs and consequently S of the gas solutes in various glassy polymers. For instance, Bondar et al.19 observed a linear relationship between the logarithm of S and (Tc/T)2, as a measure of the penetrant condensability, in amorphous glassy perfluorodioxole copolymer (AF2400) as follows: log S(T ) = M + N (Tc/T )2

(5)

where M and N represent parameters depending on the polymer. Indeed, they found that over the extremely wide range of Tc from 5 to 677 K that was explored in their study, the conventional linear relationship between the solubility and Tc was not valid anymore. Thereby, they substituted Tc with the reciprocal of the squared reduced temperature, that is, (Tc/T)2 to attain a better description than a linear fit that was reasonable due to the linear dependency of the enthalpy of condensation on the square of the critical temperature.19 However, the resulting correlation led to a lower prediction ability for some small molecules, such as highly supercritical gases such as He, because the effect of the mixing enthalpy (ΔHmix) was neglected, which was the dominant contribution in the enthalpy of sorption (ΔHs) for these solutes compared to the enthalpy of condensation (ΔHCond). This fact was in contradiction with their underpinning simplifying assumption of (|ΔHcond| ≫ |ΔHmix|) for sorption of penetrants in AF2400, and as a consequence their results were affected adversely. Yampolskii et al.20 developed correlations for S(T) as well as for parameters of the dual-mode sorption isotherms, that is, kd and b in amorphous glassy polymers. They presented their correlations on the basis of the molecular surface areas (SA) of the gas penetrants obtained from two different approaches as follows: ln S = a0 + a1[SA]

2. METHODOLOGY 2.1. Development of the Proposed Correlation. In the proposed correlation in this research, relatively familiar concepts were applied to express ΔHS of the gas solutes in the glassy polymers. Generally, ΔHS can be considered as a sum of two contributions12,19,25,26 as follows: ΔHs = ΔHcond + ΔHmix

(6)

(7)

where ΔHcond and ΔHmix designate the molar enthalpy of condensation and the molar enthalpy of mixing of the condensed gas solutes within the polymer matrix, respectively. In fact, it is presumed that gas dissolution in a polymer, involves condensation of the gas penetrants to a liquid-like density followed by mixing of the condensed gas within the polymer through creating molecular-sized cavities. As described in the

where a0 and a1 are regression coefficients. It was concluded that the correlations using the van der Waals molecular surface areas (WSA) result in slightly better accuracy compared to the ones based on the solvent accessible surface areas (SASA) for SA in eq 6. Indeed WSA present the external surface area of the molecules resulting from substitution of each atom or sets of atoms by spheres of known radii, estimated via the UNIFAC 1434



Article

⎛ ε/k ⎞ ΔHcond = ΔHcond(Tb)⎜ ⎟ ⎝ Tb ⎠

introduction, van Amerongen15 observed that ΔHS is nearly unaffected by the properties of the polymer or in the other words the polymer nature, in the elastomers and this finding was later extended to the glassy polymers, as well.12 Although this presumption is true for more condensable gas solutes, yet for the low molecular weight, highly supercritical ones, such as He and H2, it is confirmed that ΔHS is governed by ΔHmix, and consequently the nature of the polymer cannot be ignored anymore.12,27 Therefore, the effect of both ΔHcond and ΔHmix on ΔHS should be considered for better accuracy, and eq 3 is now rewritten as follows according to eq 7: ΔHcond + ΔHmix = 1000R(0.5 − 0.010ε/k ± 1.2)

The obtained statement for the enthalpy of condensation, is in well accordance with the experimental data. According to eq 12, the following relationship has been derived so far, which is applied as the new O.F. to determine the ΔHmix statement in this section: ⎛ ε/k ⎞ ΔHmix + ΔHcond(Tb)⎜ ⎟ ⎝ Tb ⎠

(8)

− 1000R(0.5 − 0.010ε/k ± 1.2) = 0

where ΔHcond and ΔHmix are representative of the gas solute physical properties and the solute−polymer interactions, respectively. In the following section, the obtained statements of ΔHcond and ΔHmix, have been determined in two consecutive steps: initially the ΔHcond relation was obtained followed by determination of that of ΔHmix. 2.2. Determination of ΔHcond and ΔHmix Relations. As already stated, ΔHcond represents enthalpy of condensation of the pure gas solutes into the liquid phase or for the solutes that are above their critical temperatures, to a condensed density. It is worth mentioning that ΔHcond merely depends on the physical properties of the gas solutes and there is no relation between ΔHcond and the nature of the polymers. For more condensable solutes, such as higher hydrocarbons as well as many organic vapors, ΔHcond is dominant in calculation of ΔHS, that is (|ΔHcond| ≫ |ΔHmix|) in the both glassy and rubbery polymers.18,28 So with neglecting the effect of ΔHmix, eq 8 would be expressed as follows: ΔHcond ≅ ΔHs = 1000R(0.5 − 0.010ε/k ± 1.2)

(12)

(13)

Initially, since for the low molecular weight gas solutes, such as H2 and He, ΔHcond is very small (|ΔHcond| ≪ |ΔHmix|), ΔHS is subsequently governed by ΔHmix. Hence, the relation of ΔHmix should be in direct accordance with the experimental values of ΔHS for these highly supercritical gases. Moreover, the final statement of ΔHmix for the all gas solutes, from the highly supercritical to the more condensable ones, should properly satisfy eq 13. Once more, a numerical optimization technique was applied to obtain the ΔHmix relation. After a complete exploration, it was found that the following statement has the highest consistency with the specified O.F. in eq 13: ⎡ Tg − T ⎤ ⎥(δ2 − δ1)2 ΔHmix = V2⎢ ⎢⎣ Tg ⎥⎦

(14)

where V designates the molar volume in (cm3/mol), and δ is the solubility parameter in (J/cm3)0.5. The subscripts 1 and 2 refer to the polymer and the gas solute, respectively. In fact, eq 14 is analogous to Hildebrand equation32 as given in eq 15:

(9)

ΔHmix = V2v12(δ2 − δ1)2

This equation can be viewed as an axillary equation which is applied to attain the ΔHcond relation. Now, we define a dimensionless parameter as the dimensionless enthalpy of condensation (ΔHcond ′ ) as expressed in eq 10. It should be noted that similar dimensionless parameters have been also introduced for ΔHcond in the literature:29,30

(15)

where ΔHcond(Tb) designates the enthalpy of condensation of a given gas solute at its normal boiling point temperature, where its values for the selected gas solutes in this work, were taken from the literature.31 Combining eqs 9 and 10 gives

where υ1 denotes the polymer volume fraction and, bearing in mind the diluteness of the solutions of the gases in the polymers, it can be considered equal to 1. On the basis of the experimental values of ΔHS for He and H2 in some glassy polymers,33 it was found that for these highly supercritical solutes, the values of ΔHmix calculated via applying eq 14, results in better accuracy when the normal boiling point temperatures (Tb) of the gas solutes are employed. Additionally, more investigations revealed that for all the selected gas solutes in this work, estimation of ΔHmix at their Tb values can properly satisfy eq 13. Therefore, eq 14 was eventually considered as follows for ΔHmix:

′ ) − 1000R(0.5 − 0.010ε/k ± 1.2) = 0 (ΔHcond(Tb) × ΔHcond (11)

(16)

′ = ΔHcond

ΔHcond(T ) ΔHcond(Tb)

(10)

⎡ Tg − Tb ⎤ ⎥(δ2(Tb) − δ1)2 ΔHmix = V2(Tb)⎢ ⎢⎣ Tg ⎥⎦

Determination of an appropriate formulation for the dimensionless ΔHcond ′ was carried out by means of the numerical optimization technique. For this purpose, an inhouse computer code was developed to examine different dimensionless forms for ΔHcond ′ that satisfy eq 11, which was defined as an objective function (O.F.) to be minimized in the derivative free search method of a genetic algorithm. It should be noted that in this stage, the data of the gas solutes under consideration, except the highly supercritical ones including He and H2, were applied. After a thorough exploration, the best formulation forΔH′cond was found to

( ); thereby, ΔH

be

ε/k Tb

cond

where V2(Tb) and δ2(Tb) represent the molar volume and the solubility parameter of the gas solute at its Tb. The dependency of the solubility coefficient of the gas penetrants on Tg of the polymers has been observed in the literature as well,34,35 which is implied in eq 16 of this work. Another point which accounts for the effect of the polymer nature on the solubility coefficient in eq 16 is the solubility parameter of the polymer, δ1. Moreover,[(Tg − T)/Tg] appearing in eq 14 can be ascribed to the concept of the nonequilibrium excess free volume, that is, (Vg − Vl) observed in the glassy polymers, which is the main contributor of the differences between the sorption phenomenon in the glassy and rubbery polymers.17 Indeed, for a given

can be considered as follows: 1435



Article

Table 1. Some Physical Properties of the Selected Gas Solutes31,37

a

parameter

CO2

CH4

C2H4

C2H6

C3H6

C3H8

C4H10

SO2

O2

He

H2

N2

Tc (K) pc(bar) Vc (cm3/mol) Tb (K) ε/k (K) ΔHcond(Tb) (kJ/mol)

304.2 73.83 94.0 217a 195.2 −16.4a

190.6 45.99 98.6 111.4 148.6 −8.19

282.3 50.40 131.0 169.4 224.7 −13.53

305.3 48.72 145.5 184.6 215.7 −14.69

365.6 46.65 188.4 225.5 298.9 −18.42

369.8 42.48 200.0 231.1 237.1 −19.04

425.1 37.96 255.0 272.7 531.4 −22.44

430.8 78.84 122.0 263.1 335.4 −24.94

154.6 50.43 73.4 90.2 106.7 −6.82

5.2 2.28 57.3 4.2 10.22 −0.08

33.19 13.13 64.1 20.4 62.2 −0.9

126.2 34.00 89.2 77.3 71.4 −5.57

Triple-point temperature.

polymer−solute pair, the concentration (Ci = Sipi) and consequently the solubility of the sorbed gas in the glassy polymer, can be considered as a function of the excess free volume implied in the Langmuir sorption capacity constant (C′H), according to the so-called dual mode sorption model. Alternatively, the fractional excess free volume in a glassy polymer can be expressed as follows:36 ⎡ Vg − Vl ⎤ ⎥ = (αl − αg)(Tg − T ) ⎢ ⎣ Vl ⎦

both low and high molecular weight gases. Some physical properties of the selected gas solutes, the molar enthalpies of condensation at their Tb, along with the properties of the selected glassy polymers are given in Tables 1 and 2, respectively. Since CO2 has no normal boiling point and consequently enthalpy of condensation at this temperature, CO2 triple point was applied instead in Table 1. Table 2. Properties of the Selected Glassy Polymers

(17)

parameter

where Vg and Vl represent the specific volumes of the polymer in the glassy and a hypothetical rubbery states, respectively. (αl − αg) is the change of the thermal expansion coefficient associated with Tg. As evidenced by eq 17, at the temperature equal to zero, fractional excess free volume reaches its maximum value. Division of eq 17 to its maximum gives [(Tg − T)/Tg] that appeared in eq 14 as follows: ⎡ (Vg − Vl )/Vl ⎤ ⎡ Tg − T ⎤ ⎥ ⎢ ⎥=⎢ ⎢⎣ ((Vg − Vl )/Vl )max ⎥⎦ ⎣⎢ Tg ⎥⎦

PVAC PETP PVC PC

(18)

306 345 360 423

25.6 20.4 21.1 20.2

Table 3. Values of (10−3ΔHs/R) in K Obtained from eq 3 and the Developed Correlation in This Work, eq 19 along with the Experimental Data (expt)33,40 polymer

3. RESULTS AND DISCUSSION 3.1. Final Formulation of the Developed Correlation. Regarding eqs 8, 12, and 16, the following relation was attained as the final expression to obtain values of the heat of sorption of gas solutes in the glassy polymers in this work:

PVAC

PETP

⎡ ⎛ ε/k ⎞ 10 ΔHS/R = 10 /R ⎢ΔHcond(Tb)⎜ ⎟ ⎢⎣ ⎝ Tb ⎠ −3

⎤ ⎡ Tg − Tb ⎤ ⎥(δ2(Tb) − δ1)2 ⎥ + V2(Tb)⎢ ⎥⎦ ⎢⎣ Tg ⎥⎦

δ1 (J/cm3)0.538

The molar volumes and the solubility parameter of the gas solutes at their normal boiling points were calculated using the well-known Peng−Robinson equation of state (EoS)39 as described in the Supporting Information. In Table 3, calculated

Therefore, [(Tg − Tb)/Tg] in eq 16 may be regarded as the ratio of the fractional excess free volume obtained at Tb to its maximum value.

−3

Tg (K)

polymer

33

PVC

PC

(19)

As evident from eq 19, ΔHS is correlated as a function of the solute physical properties, that is, ΔHS(Tb), ε/K, Tb, δ2(Tb) and the polymer nature as implied in its Tg and δ1. 3.2. Validation of the Developed Correlation. To validate the developed correlation proposed in this work, various gas solutes including carbon dioxide (CO2), methane (CH4), ethylene (C2H4), ethane (C2H6), propylene (C3H6), propane (C3H8), n-butane (C4H10), sulfur dioxide (SO2), oxygen (O2), helium (He), hydrogen (H2), and nitrogen (N2), and the glassy polymers consisting of polyvinyl acetate (PVAC), polyethylene terephthalate (PETP), polyvinyl chloride (PVC), and polycarbonate (PC), were selected. The selection of the gas solutes was carried out in a way to include

eq 3 this work expt eq 3 this work expt eq 3 this work expt eq 3 this work expt

He

H2

N2

O2

CO2

0.40 1.90 1.06 0.40 1.17 0.13 0.40 1.26 1.10 0.40 1.15 1.26

−0.12 0.62 1.23 −0.12 0.18

−0.21 −0.05 −0.81 −0.21 −0.38 −2.88 −0.21 −0.34 0.84 −0.21 −0.38 −1.35

−0.57 −0.68 −0.55 −0.57 −0.87 −1.40 −0.57 −0.87 0.16 −0.57 −0.85 −1.55

−1.45 −1.46

−0.12 0.23 0 −0.12 0.17 0.21

−1.45 −1.63 −2.9 −1.45 −1.59 −0.95 −1.45 −1.59 −2.61

values of (10−3ΔHS/R) using eq 3, have been compared with the obtained values from eq 19 along with the experimental data for some of the selected gas solutes (not all of them have been listed for the sake of conciseness). As can be seen from Table 3, the developed equation in this study is of fair capability to predict the values of ΔHS for the selected gas solutes and the glassy polymers. Moreover, the relation based on the van Amerongen’s approach (i.e., eq 3), due to neglecting the effect of the polymer nature, has given a same result for each gas solute regardless of the different glassy polymers. However, the developed correlation in this work (eq 19), accounts for the differences in the nature of the various 1436



Article

Figure 1. Comparison of the predicted values for (10−3ΔHS/R) from eq 3 and the developed correlation in this work (eq 19) for various gases and glassy polymers.

Table 4. Values of (10−3ΔHcond/R) for the Selected Gas Solutes from eq 12 parameter

CO2

CH4

C2H4

C2H6

C3H6

C3H8

C4H10

SO2

O2

He

H2

N2

10−3ΔHCond/R (K)

−1.77

−1.31

−2.16

−2.06

−2.94

−2.35

−5.26

−3.82

−0.97

−0.023

−0.33

−0.62

Table 5. Values of (10−3ΔHmix/R) for the Selected Gas Solutes from eq 16 10−3ΔHmix/R (K) polymer

CO2

CH4

C2H4

C2H6

C3H6

C3H8

C4H10

SO2

O2

He

H2

N2

PVAC PETP PVC PC

0.31 0.14 0.18 0.18

0.41 0.15 0.18 0.15

0.30 0.10 0.13 0.11

0.31 0.11 0.14 0.12

0.26 0.10 0.13 0.13

0.30 0.13 0.17 0.16

0.19 0.12 0.17 0.19

0.020 0.001 0.001 0.000

0.29 0.10 0.12 0.10

1.93 1.19 1.28 1.17

0.95 0.51 0.56 0.50

0.57 0.24 0.28 0.24

As shown in Table 5, values of the enthalpy of mixing are near zero for SO2 penetrant. This takes place because the SO2 solubility parameter (i.e., δ2(Tb) = 20.11 (J/cm3)0.5) as given in Table S2 in the Supporting Information, is very close to the solubility parameters of the considered glassy polymers in this work (see Table 2). Therefore, their squared difference appeared in eq 16, and as a consequence the values of the mixing enthalpy are very small. One can note that the values of linear regression of (10−3ΔHcond/R) values presented in Table 4, which can be considered as (10−3ΔHS/R − 10−3ΔHmix/R), according to eq 7 versus ε/k of the correspondent gas solutes as given in Table 1, results in the following linear correlation with the correlation coefficient (R2) equal to 0.97:

glassy polymers and consequently outperforms eq 3 in this respect. Nonetheless, eq 3 might be applied as the first approximation of penetrant solubility in the glassy polymers. For better clarity, the results presented in Table 3 have been illustrated in Figure 1, where all the gas solutes are included. As one can see from this figure, the correlation proposed in this work, is in satisfactory consistency with the predictions of eq 3, which has been empirically developed based on the van Amorengen’s method. This agreement confirms the accuracy of the hypothesis employed to derive eq 19 to estimate the heat of sorption of various gas solutes in the glassy polymers in this work. 3.3. Verification of the Assumptions Made To Derive ΔHcond and ΔHmix Relations. Although the success of the assumptions made to derive the statements of ΔHcond and ΔHmix is the best indication of their validity, it has been also ascertained in this section as follows. In Tables 4 and 5, values of the (10−3ΔHcond/R) and (10−3ΔHmix/R), both expressed in K, for the selected gas solutes and the glassy polymers are listed and have been calculated via application of eqs 12 and 16 that were proposed in this work, respectively.

10−3ΔHcond /R = 0.1783 − 0.0103ε /k

(20)

As evidenced by eq 20, the slope of the straight fitted line has the same value of 0.010 as in the literature correlation given in eq 3. As stated earlier, van Amerongen did not include the effect of the polymer nature in his developed correlations for the rubbery polymers; thereby, the intercept of eq 3 is the same for all the polymers. Therefore, to assess the validity of the second assumption made in deriving ΔHmix relation, the 1437



Article

Figure 2. Contribution of the absolute values of 10−3ΔHcond/R and 10−3ΔHmix/R in 10−3ΔHs/R for various gas solutes in PVAC.

average of the (10−3ΔHmix/R) values given in Table 5 was obtained as follows: avg(10−3ΔHmix /R ) = 0.3150

on the van Amerongen’s approach, to rationally estimate ΔHS of the gas penetrants in the glassy polymers. It was demonstrated that ΔHS in these polymers, depends on the Lennard-Jones temperature ε/k, the normal boiling point temperature Tb, the enthalpy of condensation ΔHcond(Tb), the molar volume V2(Tb), and the solubility parameter δ2(Tb) of the gas solutes, as well as the glass transition temperature Tg and the solubility parameter δ1 of the glassy polymers. One of the most important features of the proposed insight, is its capability to individually determine the exact contribution of ΔHcond and ΔHmix to the heat of sorption. Moreover, in development of the correlation in this work, no additional fitting/adjustment parameters were applied, which enhances its application. As a final remark, it can be concluded that although for the most gas solutes, ΔHS is mainly governed by the solutes physical properties, for the small and highly supercritical ones, the nature of the polymer plays a significant role, and this suggests the necessity to incorporate this effect in the ultimate correlation, as was done in this work.

(21) −3

Summation of eq 20 with eq 21; that is, (10 ΔHcond/R + 10−3ΔHmix/R) yields 10−3ΔHS/R = 0.4933 − 0.0103ε /k ≅ 0.5 − 0.010ε /k (22)

As expected, eq 22 is similar to eq 3; hence, the consistency of the proposed statements for the both ΔHcond and ΔHmix and finally ΔHS is confirmed. As an additional remark, the normal boiling point of the gas solutes has been confirmed to have a meaningful relation with the solubility coefficient in the polymers, regardless of the rubbery or the glassy ones.12,15 The obtained results in this research, have also ascertained this dependency. Although eq 3 is expressed as a function of ε/k, yet the two constant coefficients of this correlation have a close relation with Tb of the solutes, and as observed above, all of the introduced terms for ΔHcond (i.e., ΔHcond(Tb) and ε / K ) and for

■

Tb

ASSOCIATED CONTENT

* Supporting Information

ΔHmix (i.e., V2(Tb), (Tg − Tb)/Tg and δ2(Tb)), have been determined at Tb of the gas solutes. Finally, in an attempt to elucidate the contribution of each condensation and mixing enthalpies in ΔHS, their values given in Tables 4 and 5 were compared. As evidenced, ΔHcond has the main contribution in determination of ΔHS with the exception of highly supercritical gas solutes consisting of He and H2 and also to some extent N2. This is shown in Figure 2, where the absolute values of the three enthalpies are depicted for each gas solute in PVAC and the exact contribution of each |ΔHcond| and |ΔHmix| on |ΔHS| is conspicuous. Indeed for the highly supercritical gas solutes with very low ΔHcond values, the effect of the polymer nature which is implied in ΔHmix would be more pronounced and cannot be ignored. In this work, this effect was taken into account in terms of Tg and δ of the polymer in ΔHmix of the gas solutes.

S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.6b01002. Molar volumes of the selected gas solutes and their solubility parameters both at their normal boiling point temperatures (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +98(21)77240496. Fax: +98(21)77240495. E-mail: [email protected]. ORCID

Toraj Mohammadi: 0000-0003-0455-3254 Notes

The authors declare no competing financial interest.

■

4. CONCLUSIONS In this paper, a correlation was provided that can also serve as an alternative for the widely applied empirical correlation based

REFERENCES

(1) Baker, R. W.; Lokhandwala, K. Natural Gas Processing with Membranes: An Overview. Ind. Eng. Chem. Res. 2008, 47, 2109−2121.

1438



Article

(25) Stannett, V.; Hopfenberg, H.; Petropoulos, J. Diffusion in Polymers; North Carolina State Univ.: Raleigh, 1972. (26) Prabhakar, R. S.; Merkel, T. C.; Freeman, B. D.; Imizu, T.; Higuchi, A. Sorption and Transport Properties of Propane and Perfluoropropane in Poly(dimethylsiloxane) and Poly(1-trimethylsilyl1-propyne). Macromolecules 2005, 38, 1899−1910. (27) van der Vegt, N. F. A.; Kusuma, V. A.; Freeman, B. D. Basis of Solubility versus TC Correlations in Polymeric Gas Separation Membranes. Macromolecules 2010, 43, 1473−1479. (28) Freeman, B.; Yampolskii, Y.; Pinnau, I. Materials Science of Membranes for Gas and Vapor Separation; Wiley, 2006. (29) Meyra, A. G.; Kuz, V. A.; Zarragoicoechea, G. J. Universal behavior of the enthalpy of vaporization: an empirical equation. Fluid Phase Equilib. 2004, 218, 205−207. (30) Velasco, S.; Roman, F. L.; White, J. A.; Mulero, A. Prediction of the enthalpy of vaporization of metals and metalloids. Fluid Phase Equilib. 2006, 244, 11−16. (31) Majer, V.; Svoboda, V.; Kehiaian, H. V. Enthalpies of vaporization of organic compounds: a critical review and data compilation; Blackwell Scientific Oxford: Boston, 1985; Vol. 32. (32) Barrer, R. M.; Barrie, j. A.; Slater, j. Sorption and Diffusion in Ethyl Cellulose. Part 111. Comparison between Ethyl Cellulose and Rubbe. J. Polym. Sci. 1958, 27, 177−197. (33) krevelen, D. W. V. Properties of Polymers-Their Correlation with Chemical Structure; Their Numerical Estimation and Prediction from Additive Group Contributions, 4th, ed.; Elsevier: Amsterdam, 2009. (34) Paterson, R.; Yampol’skii, Y.; Fogg, P. G. T.; Bokarev, A.; Bondar, V.; Ilinich, O.; Shishatskii, S. IUPAC-NIST Solubility Data Series 70. Solubility of Gases in Glassy Polymers. J. Phys. Chem. Ref. Data 1999, 28, 1255−1450. (35) Mi, Y.; Zhou, S.; Stern, S. A. Representation of gas solubility in glassy polymers by a concentration-temperature superposition principle. Macromolecules 1991, 24, 2361−2367. (36) Petropoulos, J. H. Some fundamental approaches to membrane gas permeability and permselectivity. J. Membr. Sci. 1990, 53, 229−258. (37) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The properties of gases and liquids, 5th ed.; McGraw-Hill: New York, 2001. (38) Hansen, C. M. Hansen Solubility Parameters-a User’s Handbook, 2nded.; Taylor & Francis Group: Boca Raton, 2007. (39) Peng, D.-Y.; Robinson, D. B. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59−64. (40) Costello, L. M.; Koros, W. J. Effect of structure on the temperature dependence of gas transport and sorption in a series of polycarbonates. J. Polym. Sci., Part B: Polym. Phys. 1994, 32, 701−713.

(2) George, G.; Bhoria, N.; AlHallaq, S.; Abdala, A.; Mittal, V. Polymer membranes for acid gas removal from natural gas. Sep. Purif. Technol. 2016, 158, 333−356. (3) Ghadimi, A.; Norouzbahari, S.; Sadrzadeh, M.; Mohammadi, T. Improvement in gas separation properties of a polymeric membrane through the incorporation of inorganic nano-particles. Polym. Adv. Technol. 2012, 23, 1101−1111. (4) Tatar, A.; Barati-Harooni, A.; Najafi-Marghmaleki, A.; Mohebbi, A.; Ghiasi, M. M.; Mohammadi, A. H.; Hajinezhad, A. Comparison of two soft computing approaches for predicting CO2 solubility in aqueous solution of piperazine. Int. J. Greenhouse Gas Control 2016, 53, 85−97. (5) Kohl, A. L.; Nielsen, R. Gas Purification; Elsevier Science, 1997. (6) Norouzbahari, S.; Shahhosseini, S.; Ghaemi, A. CO2 chemical absorption into aqueous solutions of piperazine: modeling of kinetics and mass transfer rate. J. Nat. Gas Sci. Eng. 2015, 26, 1059−1067. (7) Norouzbahari, S.; Shahhosseini, S.; Ghaemi, A. Chemical absorption of CO2 into an aqueous piperazine (PZ) solution: development and validation of a rigorous dynamic rate-based model. RSC Adv. 2016, 6, 40017−40032. (8) Rabiee, H.; Soltanieh, M.; Mousavi, S. A.; Ghadimi, A. Improvement in CO2/H2 separation by fabrication of poly(ether-bamide6)/glycerol triacetate gel membranes. J. Membr. Sci. 2014, 469, 43−58. (9) Rahim, N. A.; Ghasem, N.; Al-Marzouqi, M. Stripping of CO2 from different aqueous solvents using PVDF hollow fiber membrane contacting process. J. Nat. Gas Sci. Eng. 2014, 21, 886−893. (10) Baker, R. W. Membrane Technology and Applications; Wiley: New York, 2012. (11) Ghadimi, A.; Sadrzadeh, M.; Shahidi, K.; Mohammadi, T. Ternary gas permeation through a synthesized PDMS membrane: Experimental and modeling. J. Membr. Sci. 2009, 344, 225−236. (12) van Krevelen, D. W.; te Nijenhuis, K. Properties of Polymers: Their Correlation with Chemical Structure; Their Numerical Estimation and Prediction from Additive Group Contributions; Elsevier Science: Amsterdam, 2009. (13) Van Amerongen, G. J. The Permeability of Different Rubbers to Gases and Its Relation to Diffusivity and Solubility. J. Appl. Phys. 1946, 17, 972−985. (14) Van Amerongen, G. J. Influence of structure of elastomers on their permeability to gases. J. Polym. Sci. 1950, 5, 307−332. (15) Van Amerongen, G. Diffusion in elastomers. Rubber Chem. Technol. 1964, 37, 1065−1152. (16) Michaels, A. S.; Bixler, H. J. Solubility of gases in polyethylene. J. Polym. Sci. 1961, 50, 393−412. (17) 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. (18) Ghosal, K.; Freeman, B. D. Gas separation using polymer membranes: an overview. Polym. Adv. Technol. 1994, 5, 673−697. (19) Bondar, V. I.; Freeman, B. D.; Yampolskii, Y. P. Sorption of Gases and Vapors in an Amorphous Glassy Perfluorodioxole Copolymer. Macromolecules 1999, 32, 6163−6171. (20) Yampolskii, Y.; Wiley, D.; Maher, C. Novel correlation for solubility of gases in polymers: effect of molecular surface area of gases. J. Appl. Polym. Sci. 2000, 76, 552−560. (21) De Angelis, M. G.; Sarti, G. C.; Doghieri, F. Correlations between Penetrant Properties and Infinite Dilution Gas Solubility in Glassy Polymers: NELF Model Derivation. Ind. Eng. Chem. Res. 2007, 46, 7645−7656. (22) Doghieri, F.; Sarti, G. C. Nonequilibrium lattice fluids: a predictive model for the solubility in glassy polymers. Macromolecules 1996, 29, 7885−7896. (23) Baschetti, M. G.; Doghieri, F.; Sarti, G. C. Solubility in Glassy Polymers: Correlations through the Nonequilibrium Lattice Fluid Model. Ind. Eng. Chem. Res. 2001, 40, 3027−3037. (24) De Angelis, M. G.; Sarti, G. C.; Doghieri, F. NELF model prediction of the infinite dilution gas solubility in glassy polymers. J. Membr. Sci. 2007, 289, 106−122. 1439


Heat of Sorption of Gases in Glassy Polymers: Prediction via Applying

Recommend Documents