Activity Coefficients In Polymer Solutions

2nd Mercosur Congress on Chemical Engineering 4th Mercosur Congress on Process Systems Engineering

ACTIVITY COEFFICIENTS IN POLYMER SOLUTIONS: COMPARISON OF MODELS 1

M.M. Cardoso1, Y.G. Pereira1,2, M. Embiruçu1*, G. Costa1,3

PROTEC/Escola Politécnica - Universidade Federal da Bahia – UFBA Department of Chemical Processes, School of Chemical Engineering, Campinas State University – UNICAMP

2

3

CEPGN - UNIFACS

Abstract. Due to their modular molecular structure, polymers attain specific physical, chemical and thermal properties, which make them suitable for the development of a very large variety of new materials. Polymerization often takes place in solvents, and it is quite important to know how the produced polymer will be distributed between the polymer– rich and solvent-rich phases. In such cases liquid-liquid equilibrium information is necessary, i.e., both polymer and solvent activity coefficients. On the other hand, the removal of solvents or non-polymerized monomers from the produced polymer requires the knowledge of vapor-liquid equilibrium, i.e., the solvent activity coefficients. A great amount of equation of state and activity coefficient models capable of describing phase equilibria in polymer solutions is available today, but only few of these models have been tested for several systems. Thereby, it is useful to investigate the performance of existing thermodynamic models for complex polymer solutions, which still have not been studied considerably. This present work concerns on studying several activity coefficient models: Flory (1953), Oishi and Prausnitz (1978), Elbro et al. (1990), Kontogeorgis et al. (1993), Chen (1993), Vetere (1994), Qian et al. (1991), and the equation of state for determining the activity coefficient of a solvent: Chen et al. (1990) and High and Danner (1990). The evaluation of the models was carried out both at infinite dilution and at finite concentrations, and compared to experimental data. Experimental solvent activity coefficients (infinite dilution) and activities (finite concentrations) were collected from the databases DIPPR and DECHEMA. The database was split up in systems containing non-polar and polar solvents and non-polar and polar polymers in order to validate whether one type of systems create more pronounced problems to the models than others. In this work, using the entire available experimental data, the above models were tested and extensive analyses were performed. We have not found such systematic analysis, mainly with this great number of models. Keywords: activity coefficient, polymer solutions and phase equilibria.

1. Introduction Thermodynamics of polymeric systems play an important role, and is often a key factor, in polymer production, processing and material development, especially for the design of advanced polymeric materials. Many polymeric products are produced “in solution” with a solvent (or a mixture of solvents) and often other low molecular weight compounds (plasticizers, etc.). A problem which often arises is how to remove the low molecular weight constituent(s) from the final product (polymer). The solutions of this problem involves, among other tasks, solving the vapor-liquid equilibrium (VLE) problem, in which the solvent activity needs to be known (at conditions often close to infinite dilution). Polymer systems often exhibit liquid-liquid phase separation, which depends significantly on temperature, pressure, molecular weight, and the molecular weight distribution of the polymer. The activity of the polymer is a predominant factor in LLE calculations for polymer solutions. A great amount of equations of state and activity coefficient models capable of describing phase equilibrium in polymer solutions is available today, but only few of these models have been tested for several

*

To whom all correspondence should be addressed. Address: PROTEC/Escola Politécnica - UFBA - Rua Prof. Aristides Novis, nº. 2, Federação, CEP: 40210-630, Salvador-BA, Brazil E-mail: [email protected], [email protected]

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2nd Mercosur Congress on Chemical Engineering 4th Mercosur Congress on Process Systems Engineering systems. Thereby, it is useful to investigate the performance of the existing thermodynamic models for complex polymer solutions, which still have not been studied considerably. The purpose of this work is to evaluate the performance of nine activity coefficient models: Flory (1953), Oishi and Prausnitz (1978), Elbro et al. (1990), Kontogeorgis et al. (1993), Chen (1993), Vetere (1994), Qian et al. (1991); and the equations of state for determining the activity coefficient of a solvent: Chen et al. (1990) and High and Danner (1990). The evaluation of the models was carried out at infinite dilution (40 systems) and at finite concentrations (63 systems), and compared to experimental data.

2. Description of the models The Flory (1953) model – M1 model – in terms of the solvent activity coefficients is, for a binary system, written as:

ln γ 1 = ln

Φ1 Φ + 1 − 1 + χ ⋅ Φ 22 x1 x1

Where Φ and

χ

are the volume fractions and mole fractions respectively, and

(1)

χ

is the composition–

independent Flory parameter. In the Qian et al. (1991) model – M2 model - the Flory interaction parameter is represented as a product of composition-dependent and temperature-dependent terms. Chen (1993) – M3 model – adopts the Flory expression for the configurationally entropy of mixing and the NRTL theory for the local composition contribution. The Vetere (1994) model – M4 model - relies on the same assumptions as the Chen (1993) model, except that binary interaction parameters are simply related to the difference between the Hildebrand solubility parameters, according to four families on the basis of their polar and non-polar character. The Oishi and Prausnitz (1978) model – M5 model – is a group-contribution activity coefficient model based on the UNIFAC model. The free-volume contribution, derived from the Flory equation of state, was added to the combinatorial and residual terms of the UNIFAC model to account for the free-volume differences between the polymer and solvent molecules. In the Elbro et al. (1990) model – M6 model – the combinatorial and freevolume effects are both included in a combinatorial-free volume expression, similar to the Flory one, where free-volume fractions are used instead of volume fractions. For the polymer solutions with energy interactions, a UNIQUAC residual term is added. The difference between the Kontogeorgis et al. (1993) model – M7 model – and M6 model is that, instead of UNIQUAC residual term, M7 model uses the classical UNIFAC residual term with the linearly temperature-dependent interaction parameters proposed by Hansen et al. (1992). The Chen et al. (1990) equation of state – M8 model – combines the free volume expression from the Flory equation of state, with a local composition expression for the energy of the system derived from arguments similar to the Oishi and Prausnitz (1978) model. In the M8 model, the densities of pure components and the mixture are calculated with the equation of state itself, both for polymer and solvent equation parameters. High-Danner (1990) – M9 model - modified the Panayiotou-Vera equation of state by developing a group contribution approach for the determination of the molecular parameters. With the Chen et al. (1990) and High-Danner (1990) equations of 2

2nd Mercosur Congress on Chemical Engineering 4th Mercosur Congress on Process Systems Engineering state, expressions for the activities coefficients are derived using classical thermodynamics. Through partition function defined by the model, expressions for the activates coefficients can be found in Chen et. al. (1990) (M8) and High-Danner (1990) (M9).

3. Database and results The evaluation of the models was carried out both at infinite dilution and at finite concentrations. Experimental solvent activity coefficients (infinite dilution) and activities (finite concentrations) were collected from the databases Danner and High (1993) (DIPPR) and Wen et. al. (1991) (DECHEMA). The number of collected data points and the types of the systems contained in the database are listed in Table 1 for finite concentrations. The database was split up in systems containing non-polar, weakly polar and strongly polar solvents in order to validate whether one type of systems create more pronounced problems to the models than others. For each pair of polymer-solvent only one temperature was chosen. Absolute Average Deviation (AAD%) of experimental and calculated activities were used as objective function, and the parameters found here were used in infinity dilution calculations either. The database for infinite dilution contains 31 systems for non-polar solvents, 8 systems for weakly polar solvents and 1 for strongly polar solvents. Table 1. Number of systems and data points contained in the database used for finite concentration calculations.

Polymer BR HDPE LDPE PAA PD PDD PDMS PEO(PEG) PH PIB PMAA PMMA POD PP PPOX PS PVAC PVC IR PVME Total

Non-polar solvents Systems Data Points 4 26 4 30 3 26 1 17 1 5 6 76 2 23 1 21 6 67 1 8 1 6 2 17 1 11 4 51 2 11 1 8 2 12 2 26 44 441

Weakly polar solvents Systems Data Points 1 3 1 8 2 20 3 30 2 14 2 18 2 14 13 107

Strongly polar solvents Systems Data Points 1 7 1 6 1 7 2 9 1 6 6 35

The results, expressed as absolute average deviation, for nine models applied to 63 systems, for finite concentrations are shown in Table 2. The following models are based on the group contribution approach and are, thus, purely predictive: M4, M5, M7, M8 and M9. M1 model needs one parameter to be estimated, models M3 and M6 need two parameters estimation and M2 model needs five parameters estimation. The presence of 3

2nd Mercosur Congress on Chemical Engineering 4th Mercosur Congress on Process Systems Engineering empty spaces in Table 2 indicates the inexistence of necessary properties of group(s) to the calculation. The GCVOL model (Elbro et. al. 1991) is used to estimate the polymer liquid volumes. The GCVOL model is applied to solvents (up to the normal boiling point), and waxes, as well. Spencer and Danner (1972) model was also used to calculate the solvent molar volume. The best model to calculate the solvent molar volume was selected to each system, in order to minimize activity errors. Nevertheless, no pattern of behavior, according to system types, was recognized. The van der Waals volume is calculated using the group increments given by Bondi (1968). Polymer and solvent solubility parameters, needed in M4 model, were calculated by group contribution using the method of Fedors (1974), when not available in literature. Table 2. Mean percentage deviation between experimental and predicted solvent activities at finite concentration

Systema BR + Benzene (4)

M1

M2

M3

AAD / %b M4 M5 M6

1.06

1.16

1.16

21.02

4.61

2.42

M7

M8

M9

4.57

57.18

11.22

BR + Cyclohexane (4)

0.93

0.93

0.93

14.33

9.21

0.58

6.55

42.22

10.79

BR + Ethyl benzene (12)

4.85

4.85

4.85

4.85

26.55

4.47

11.98

37.76

29.44

BR + n-Hexane (6) HDPE + Cyclohexane (6) HDPE + n-Decane (10)

1.82 3.63 5.16

1.83 3.63 5.16

1.82 3.64 5.20

29.38 17.02 38.94

7.92 31.10 47.12

2.04 3.64 4.99

6.74 19.26 34.57

22.96 3.66 30.85

6.80 32.16 41.57 10.70

HDPE + n-Hexane (8)c

5.84

5.84

5.86

8.62

21.39

5.73

6.47

7.14

HDPE + Isooctane (6)c

3.93

3.93

3.94

18.21

5.90

4.22

16.29

-

LDPE + n-Butane (7)

1.44

1.46

1.27

45.06

63.58

1.77

58.98

-

-

LDPE + n-Hexane (11)c

7.85

7.82

8.00

61.59

71.94

7.61

68.68

-

-

LDPE + Isobutane (8)c

5.93

5.93

5.99

17.61

50.68

5.80

39.79

-

PAA + Water (7)

30.68

30.66

30.64

67.69

*

35.80

57.21

-

-

PD + Toluene (17)

3.48

3.48

3.49

4.58

4.25

4.08

5.37

-

-

PDD + Toluene (5)

8.04

8.04

8.03

23.80

14.80

7.99

13.94

-

-

PDMS + Benzene (16) PDMS + Cyclohexane (23) PDMS + n-Heptane (10) PDMS + n-Hexane (16) c PDMS + Methyl Ethyl Ketone (3) PDMS + n-Octane (12) PDMS + Toluene (15) PEO(PEG) + Benzene (12) PEO(PEG) + Chloroform (11) PEO(PEG) + Water (5)

0.48 15.35 0.03 0.27 1.90 0.03 1.61 2.05 33.87 1.25

0.61 15.33 0.03 0.27 1.92 0.03 1.63 3.33 33.88 1.25

0.61 15.40 0.03 0.27 1.88 0.03 1.05 3.28 33.88 1.24

27.39 4.53 45.80 15.91

14.40 31.69 0.23 0.33 1.25 0.16 2.50 13.04 34.17 7.66

0.63 16.46 0.03 0.18 1.56 0.04 2.01 4.14 33.72 0.37

11.24 27.49 0.40 19.01 5.05 2.01 2.50 5.74 42.09 6.81

2.20 60.75 -

31.09 34.10 1.00 23.67 7.95 0.76 2.77 15.49 6.18

PH + Toluene (21)

4.62

4.62

4.34

5.62

5.62

5.40

4.59

-

-

PIB + Benzene (26)

2.78

2.78

2.76

6.14

28.40

3.08

7.51

30.77

30.02

PIB + n-Butane (7)

1.28

1.28

1.28

9.02

4.19

1.29

26.29

10.73

41.22

PIB + Cyclohexane (10)

4.25

4.26

4.24

9.49

6.73

4.47

10.42

4.86

13.82

PIB + Isobutane (7) c

0.75

0.76

0.72

12.26

7.94

0.90

28.37

6.86

47.08

PIB + n-Octane (5)

0.16

0.16

0.16

0.45

1.96

0.20

2.33

0.66

2.91

PIB + n-Pentane (12)

0.38

0.41

0.28

3.64

8.92

0.37

8.58

6.29

10.52

PMAA + Water (7) c PMMA + Methyl EthylKetone (8) PMMA + Toluene (8) POD + Toluene (6) PP + Diethyl Ketone (11) c PP + Diisopropyl Ketone (9) PP + n-Hexane (10) c PP + Carbon Tetrachloride (7) PPOX(PPG) + Benzene (11)

16.76 3.45 2.60 7.66 23.52 10.81 3.95 3.55 1.07

16.75 3.45 2.59 6.73 23.61 10.85 3.95 3.57 1.07

16.78 3.45 2.60 3.01 4.58 4.83 3.95 1.19 1.07

54.06 13.47 9.08 25.74 57.52 9.53 22.32 20.73 6.42

* 3.75 6.68 17.73 59.01 18.92 12.33 38.22 1.41

17.59 3.40 4.12 7.82 23.78 9.19 4.01 5.26 0.88

21.96 32.74 5.07 10.90 48.92 15.56 4.15 22.59 10.04

5.55 23.65 34.60 36.25 -

19.61 18.09 82.42 -

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2nd Mercosur Congress on Chemical Engineering 4th Mercosur Congress on Process Systems Engineering

M1

M2

M3

AAD / %b M4 M5 M6

14.29 0.85 4.00 1.05 4.50 1.70 3.59 1.35 0.42 0.79 7.93 6.96 4.00 3.05 2.79 5.44 5.67 8.28 7.34 8.03 1.72 4.48 0.23

14.38 0.84 4.04 1.39 4.91 1.76 3.60 1.34 0.60 0.80 7.93 6.98 4.00 3.84 2.78 5.70 5.66 8.31 7.39 8.07 1.72 3.23 0.23

2.56 0.84 0.20 1.38 4.91 1.71 3.58 1.36 0.25 0.79 7.93 6.86 3.99 11.00 1.38 5.74 5.66 8.09 7.01 7.86 1.71 1.94 0.23

25.04 0.84 1.84 1.30 18.76 1.34 10.25 5.42 2.02 0.61 8.48 10.82 10.83 4.98 5.16 16.09 6.04 4.75 10.69 21.68 8.72 73.01 3.37

Systema PPOX(PPG) + Water (5) PPOX + Methanol (4) PS + Acetone (8) PS + Carbon Tetrachloride (14) PS + Chloroform (11) PS + CycloHexane (11) PS + Methyl Ethyl Ketone (11) PS + n-Propyl Acetate (11) PS + Toluene (15) PVAC + Acetone (5) PVAC + Benzene (8) PVAC + n-Propanol (6) c PVAC + n-Propyl Chloride (3) c PVAC + Vinyl Acetate (9) PVC + 1,4-Dioxane (9) PVC + Di-n-Propyl Ether (9) PVC + Toluene (8) IR + Benzene (5) IR + Acetone (5) IR + Ethyl acetate (9) IR + Carbon Tetrachloride (7) PVME + Trichlorine Methane (13) PVME + Benzene (13)

* 0.44 5.28 1.30 7.36 2.90 3.90 5.30 6.78 3.76 9.75 6.92 * 4.14 57.92 34.71 23.93 * * * * 18.27 2.53

4.48 0.10 3.83 1.18 5.29 1.20 3.55 1.66 1.14 1.62 8.21 6.72 3.69 3.17 2.81 6.09 7.78 7.03 7.07 6.02 1.96 5.09 0.67

M7

M8

M9

32.29 1.02 33.16 4.44 5.54 9.39 57.19 1.29 2.25 18.89 9.57 43.07 7.35 17.28 8.56 17.78 12.53 15.84 11.55 10.53 15.33 4.22 4.60

28.27 12.96 8.15 2.08 30.01 7.06 14.39 4.89 13.66 24.19 13.43 12.08 36.77 21.51 -

35.43 5.49 26.95 13.65 5.54 16.30 15.00 66.36 3.00 32.04 6.27 -

(Number of data points) a

BR, polybutadiene; HDPE, high density polyethylene (linear polyethylene); LDPE, low density polyethylene; PAA, poly(acrylic acid);

PD, polydecene-1; PDD, polydodecene-1; PDMS, poly(dimethylsiloxane); PEO(PEG), poly(ethylene oxide); PH, polyheptene-1; PIB, polyisobuthylene; PMAA, poly(methacrylic acid); POD, polyoctadecene-1; PP, polypropylene; PPOX(PPG), poly(propylene oxide); PS, polystyrene; PVAC, poly(vinyl acetate); PVC, poly(vinyl chloride); IR, polyisoprene; PVME, poly(vinyl methyl ether). b

AAD = (1/N) ∑(│acalc. – aexp.│/ aexp.) x 100

c

dilute concentration data

* unrealistic behavior (huge deviations)

Table 3 summarizes the analyses of the results generated by all the models. As stated earlier, theses results are separated for three groups of solvents, with solvent activities at finite concentrations. We should be careful when compare results obtained predictive models (no-adjusted parameters) and models with adjusted parameters. Table 3. Summary of prediction results of finite activity coefficients by different models for polymer-solvent systems.

Model M1 No. of systems AAD% M2 No. of systems AAD% M3 No. of systems AAD% M4 No. of systems

Non-polar solvents

Weakly polar solvents

Strongly polar solvents

44 4.11

13 5.85

6 11.80

44 4.12

13 5.95

6 11.81

44 3.91

13 4.13

6 9.82

39

13

6

5

2nd Mercosur Congress on Chemical Engineering 4th Mercosur Congress on Process Systems Engineering

Model

Non-polar solvents

Weakly polar solvents

Strongly polar solvents

AAD% 16.72 13.10 M5 No. of systems 41 11 AAD% 16.31 17.99 M6 No. of systems 44 13 AAD% 4.40 5.67 M7 No. of systems 44 13 AAD% 14.44 21.42 M8 No. of systems 23 9 AAD% 19.89 19.18 M9 No. of systems 25 9 AAD% 17.90 26.37 * several systems showed unrealistic behavior (huge deviations).

29.06 * * 6 10.84 6 27.06 1 24.19 2 36.27

The difference between the M1 and M2 models is the amount of parameters to be estimated. The first one needs only one parameter (Flory parameter), while in the second model the Flory parameter involves five parameters to be adjusted. Close results are obtained by the models M1 and M2, with a little improvement to the first one, demonstrating that the addition of a great number of parameters does not improve the result of the simulation. In M3 and M4 models, the residual contribution can be described using the NRTL equation. The main difference is that for the M3 model two binary interaction parameters are estimated, while M4 model is predictive. To any solvent type, results demonstrated the M3 model is superior. This model was the best, not only in relation to the M4 model but also in relation to all the analyzed models, for all systems used. M6 and M7 models are very similar. Their structures are expressed as a sum of fixed terms: a combinatorial free-volume and a residual term. The difference is on the model used to calculate the residual contribution. The first one uses UNIQUAC (with two parameters estimated) and the second one uses UNIFAC (predictive), with the new group parameter table developed by Hansen el. al. (1992). Normally, M6 and M7 models are called in the literature as entropic-free volume models and are considered as good precision models. The results in Table 3 demonstrate that the M7 model predictive capability is not so good, principally with strongly polar solvents. M6 model shows very good results, but it requires the same number of adjusted parameter than M3 model (two parameters). Model M3 however, demonstrates an explicit advantage over M6 model, with all solvent classes. Models M5, M8 and M9 are predictive ones. Moreover, group parameters for many compounds are not available. In M5 model, systems with strongly polar solvents show errors greater than 100% for the great majority of systems. Thus, these errors were not considered for this model. The comparison of M5 and M7 models, both predictive, demonstrated an improvement in the second one, when used with non-polar and strongly polar solvents. Summarizing, the results in the Table 3 demonstrate that M3 model (it depends on 2 parameters estimation) is the best to all analyzed systems. Considering predictive models, M7 model is better to non-polar and strongly polar solvents and M4 model for weakly polar solvents. Analyzing solvents, it can be explained by the fact that 6

2nd Mercosur Congress on Chemical Engineering 4th Mercosur Congress on Process Systems Engineering the attractive/residual contribution that modeling molecular interactions is not so good in M4 model, which was expected, since these phenomena are more difficult to modeling. Moreover, other models based on group contribution calculations present good results, such as those obtained with M5 model. This result can not be attributed to group contribution calculation, since density calculations are made with the equation of state itself, with group contribution methods been used solely to calculate equation parameters. Figures 1 and 2 show graphical comparisons of the results predicted by the nine different models. The ∞

predicted values of the weight fraction activity coefficient of component i at the infinite dilution ( Ω i ) are plotted against the experimental ones. In these figures, a point above the diagonal line indicates over prediction, whereas a point below the diagonal lines shows under prediction. Figure 2 shows the predictive methods and Figure 1 the others one. 3

predicted lnΩ1 at infinite dilution

predicted lnΩ1 at infinite dilution

3

M1 Model

2

1

0

M2 Model

2

1

0 0

1

2

3

0

experimental lnΩ1 at infinite dilution

2

3

3

predicted lnΩ1 at infinite dilution

3

predicted lnΩ1 at infinite dilution

1

experimental lnΩ1 at infinite dilution

M3 Model

2

1

0

M6 Model

2

1

0 0

1

2

experimental lnΩ1 at infinite dilution

3

0

1

2

3

experimental lnΩ1 at infinite dilution

Fig. 1. Prediction of infinite dilution activity coefficients versus experimental values

The results for the models that use adjusted parameters (M1, M2, M3 and M6 models) are shown in Figure 1. There are no significant differences between them. For the majority of cases, these models overestimate the predictions.

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3

predicted lnΩ1 at infinite dilution

predicted lnΩ1 at infinite dilution

3

M4 Model

2

1

1

0

0 0

1 2 experimental lnΩ1 at infinite dilution

0

3

1

2

3

experimental lnΩ1 at infinite dilution

3

predicted lnΩ1 at infinite dilution

3

predicted lnΩ1 at infinite dilution

M5 Model

2

M7 Model

2

1

0

M8 Model

2

1

0 0

1

2

3

0

experimental lnΩ1 at infinite dilution

1

2

3

experimental lnΩ1 at infinite dilution

predicted lnΩ1 at infinite dilution

3

M9 Model

2

1

0 0

1 2 experimental lnΩ1 at infinite dilution

3

Fig. 2. Prediction of infinite dilution activity coefficients versus experimental values – predictive models

As shown in Figure 2, no biased scatter is obtained with models M4, M7 and M8, showing that these models do not present systematic errors, nor under predict neither over predict. Models M5 and M9 show under predictions. Table 4 summarizes the analyses of the results obtained by all the models. It contains the number of polymer-solvent systems tested in each model. The best results for M5 model are unrealistic, since it was 8

2nd Mercosur Congress on Chemical Engineering 4th Mercosur Congress on Process Systems Engineering impossible to simulate the infinite dilution activity coefficients for all 40 selected systems. M5 model is predictive and some groups necessary to the simulation were unavailable. Table 4. Summary of results of infinite dilution activity coefficients calculated by different models for polymer-solvent systems.

Model M1 M2 M3 M4 M5 M6 M7 M8 M9

AAD%

No. of systems

26.64 40 26.77 40 29.44 40 32.44 34 22.43 35 27.15 40 35.27 40 36.39 29 31.47 31 * Average absolute deviation

The success of the M3 model behavior regard to the finite concentration was not the same in relation to the infinite dilution, as shown in Figure 1 and Table 3. M1 and M2 models provide more accurate results than the others. It is important to emphasize that the same parameters estimated with finite concentration are used to calculate the infinite dilution. Therefore, for the systems selected to study, we can conclude that there is not a relation between the successfully prediction for one case and the other case.

4. Conclusions The polymer is present only in the liquid phase, and thus, the VLE problem reduces to calculating the activity coefficient of the solvent in liquid phase. A comparison of nine models for the correlation of VLE data for polymer systems was performed. The database in this study contains 63 systems at finite concentrations and 40 systems at infinite dilution. The comparison performed indicates the great advantage of the Chen (1993) model (M3 model) to represent the activity coefficient at finite concentration, which is able to produce the best correlation for the VLE experimental data using a pair of interaction parameters. The same nine models were applied to calculate the solvent activity coefficients at infinite dilution of solvents, with the same parameters estimated for no-predictive models, with finite dilution. Analyses for the currently available experimental data showed that the Flory (1953) model (M1 model) and Qian et al. (1991) model (M2 model) gave the best predictions. Both models are not predictive. M1 model uses one estimated parameter and M2 model uses five estimated parameters. The similarity in behavior between the M1 and M2 models, regarding to the infinite dilution, was also observed with finite concentration.

References Bondi, A. (1968). Physical Properties of Molecular Crystals, Liquids and Glasses. John Wiley & Sons: New York. Chen, C. (1993). A Segment-Based Local Composition Model for the Gibbs Energy of Polymer Solutions. Fluid Phase Equilibria, 83, 301.

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2nd Mercosur Congress on Chemical Engineering 4th Mercosur Congress on Process Systems Engineering Chen, F., Fredenslund, Aa., Rasmussen, P. (1990). Group-Contribution Flory Equation of State for Vapor-Liquid Equilibria en Mixtures with Polymers. Ind. Eng. Chem. Res. 29, 875. Danner, R. P., High, M. S. (1993). Handbook of Polymer Solution Thermodynamics. Design Institute for Physical Property Data (DIPPR), Elbro, H.S., Fredenslund, Aa., Rasmussen, P. (1990). A New Simple Equation for the Prediction of Solvent Activities in Polymer Solutions. Macromolecules, 23 (21), 4707. Elbro, H.S., Fredenslund, Aa, Rasmussen, P. (1991). Group Contribution Method for the Prediction of Liquid Densities as a Function of Temperature for Solvents, Oligomers, and Polymers. Ind. Eng. Res., nº12, 30, 2576. Fedors, R.F.(1974). A Method for Estimating Both the Solubility Parameters and Molar Volumes of Liquids. Polym.Eng.Sci. 14, 147. Flory, P.J. (1953). Principles of Polymer Chemistry. Cornell University Press, New York, NY. Hansen, H.K., Coto, B., Kerhlmann, B. (1992). UNIFAC with Linearly Temperature-Dependent Group-Interaction Parameters. Technical Report (nº.9212), IVC – SEP Research Engineering Center, Institute for Kemiteknik, The Technical University of Denmark: Lyngby. High, M.S., Danner, R.P. (1990). Application of the Group Contribution Lattice – Fluids EOS to Polymer Solutions. AICHE J., 36, 1625. Kontogeorgis, G.M., Fredenslund, Aa., Tassios, D. (1993). Simple Activity Coefficient Model for the Prediction of Solvent Activities in Polymer Solutions. Ind. Eng. Chem. Res., nº 2, 32, 362. Qian, C., Mermby, S.J., Eichinger, B.F. (1991). Phase Diagrams of Binary Polymer Solutions and Blends. Macromolecule 24, 1991, 1655. Oishi, T., Prausnitz, J.M. (1978). Estimation of Solvent Activities in Polymer Solutions Using a Group-Contribution Method. Ind. Eng. Chem. Process Des. Dev., nº 3, 17, 333. Vetere, A. (1994), Rules for Predicting Vapor-Liquid Equilibria of Amorphous Polymer Solutions Using a Modified FloryHuggins Equation. Fluid Phase Equlibria, 97, 43. Wen, H., Elbro, H.S., Alessi, P. (1991). Polymer Solution Data Collection. DECHEMA, Chemistry Data Series XIV. Spencer, C.F.; Danner, R.P. (1972).Improved Equation for Prediction of Saturated Liquid Density. J.Chem.Eng.Data, 17, 236.

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