Universal Liquid Mixture Models for Vapor−Liquid and Liquid−Liquid

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Ind. Eng. Chem. Res. 2011, 50, 1034–1045

Universal Liquid Mixture Models for Vapor-Liquid and Liquid-Liquid Equilibria in the Hexane-Butanol-Water System Akand W. Islam,* Anand Javvadi, and Vinayak N. Kabadi Department of Chemical Engineering, North Carolina A&T State UniVersity, 1601 East Market Street, Greensboro, North Carolina 27411, United States

The research conducted here was an attempt to use the currently available activity coefficient methods with universal sets of parameters to simultaneously predict binary and ternary vapor-liquid and liquid-liquid equilibrium data. Literature studies available with such correlations based on two-parameter models (UNIQUAC and LSG) and three-parameter models (NRTL and GEM-RS) used different binary interaction parameters to calculate vapor-liquid and liquid-liquid equilibrium data. The focus of this research was to predict phase equilibrium data (binary mutual solubility (BMS), infinite dilution activity coefficient (γ∞), binary vapor-liquid equilibrium (VLE), distribution coefficient (Dsw), and ternary liquid-liquid equilibrium (LLE)) within fair error using only a single set of parameters obtained from the above-mentioned models regardless of vapor-liquid or liquid-liquid equilibrium. This was proven by an investigation of the hexane-butanol-water ternary system, in which hexane-butanol, hexane-water, and water-butanol binary LLE data, binary VLE data, and γ∞ data were used to analyze the ternary system. Ternary LLE data of the hexane-butanol-water system were also analyzed. In each of the mentioned binary systems and the ternary system, the calculated and experimental data were compared. The results of this analysis predicted binary mutual solubility data, binary VLE data, ternary LLE data, γ∞, and Dsw concurrently within reasonable error (e15%). Introduction There are different models available for correlating phase equilibria. The simplest and effective models among them are Margules, Van Laar, Redlich-Kister, and so on.1 They often give good results, but extrapolation to concentrations beyond the range of data or the prediction of ternary phase diagrams from only binary information should not be carried out with these models because the results are often not qualitatively correct. Local composition models, like LSG, LCG, GEM-QC, GEM-RS,2,3 NRTL,4 and UNIQUAC,5,6 have proven superior to the simple models, both for correlating binary and ternary LLE and for predicting ternary phase diagrams from binary data. For phase equilibrium calculations, all models require binary interaction parameters. These parameters can be obtained in several ways. If the two components are completely miscible, then parameters can be regressed by taking VLE data only. However, for partial miscible pairs, these can be obtained either by mutual solubility data or by vapor-liquid equilibrium data. Binary interaction parameters can also be fitted from specific ternary systems. These show good results for that particular ternary system; however, they cannot be used to calculate data like binary VLE, binary LLE, γ∞, and so on. These parameters cannot represent the data of both very dilute regions and finite concentrations. On the other hand, the parameters that are regressed from binary data cannot predict ternary data. Hence, a single set of parameters are unable to represent different types of phase equilibrium data. Here, new kinds of interaction parameters, termed as universal parameters, will be introduced. Universal binary interaction parameters are the parameters that are able to predict BMS data, binary VLE data, ternary LLE data, γ∞, and Dsw correctly with reasonable error. These parameters are fitted using all possible phase equilibrium data simultaneously. In this work, a process to obtain said universal parameters was devised. * To whom correspondence should be addressed. E-mail: [email protected].

In Dechema,7 a list of common and specific parameters is reported. A number of aqueous ternary systems have been studied, and using those common parameters, Dsw values have been calculated. For most of the systems, the Dsw value was predicted within acceptable error. For example, literature8 Dsw of benzene-ethanol-water is 0.19, whereas the calculated value is 0.21; Dsw of hexane-ethanol-water is 0.04, and the calculated value is 0.03. Similarly, benzene-pyridine-water, toluenemethanol-water, diethyl ether-acetic acid-water, etc., systems show good agreement between literature and calculated Dsw. However, some systems were also found, which show an order of magnitude difference between literature and calculated Dsw, for instance, hexane-butanol-water (literature 1.21/calculated 10.0), CCl4-propionic acid-water (0.068/3.53), heptanepropanol-water (0.25/2.345), and so on. From these systems, hexane-butanol-water was chosen for thorough analysis, and phase equilibrium data (BMS, binary VLE, ternary LLE, γ∞, and Dsw) of the pairs (hexane-water, butanol-water, and hexane-butanol) were investigated. Table 1 shows the overview of this monograph. It shows the anomalies between experimental and calculated data by twoparameter models UNIQUAC and LSG, and three-parameter models NRTL and GEM-RS. In this case, parameters of these models were estimated regressing a particular set of data. BMS data were used for hexane-water and butanol-water pairs, and VLE data for hexane-butanol pair. From this table, it can be clearly observed that two-parameter models can represent the data only which were used for parameters estimation, and for other calculations they show an order of magnitude difference (for example, experimental γ∞ of butanol in hexane is 38.6, whereas calculated result by UNIQUAC is 5.89 and by LSG is 6.03). However, for three-parameter models, this is not observed. For all four models (UNIQUAC, LSG, NRTL, GEM-RS), a universal set of parameters were attempted to obtain using every set of data (BMS, binary VLE, ternary LLE, γ∞, and Dsw). Upon obtaining these parameters, experimental and calculated data were compared. In Table 1, all the experimental data were

10.1021/ie902028y  2011 American Chemical Society Published on Web 12/15/2010

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

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Table 1. Comparison between Experimental and Calculated Results Using Two-Parameter Models (UNIQUAN and LSG) and Three-Parameter Models (NRTL and LSG) solubility i in j

j in i cal

cal

pair

exp

Uniquac

LSG

NRTL

GEM-RS

exp

Uniquac

LSG

NRTL

GEM-RS

hexane-water butanol-water hexane-butanol

2.57 × 10-6 0.0187

2.57 × 10-6 0.0187

2.57 × 10-6 0.0187

2.57 × 10-6 0.02

2.57 × 10-6 0.021

4.7 × 10-4 0.505

4.7 × 10-4 0.505

4.7 × 10-4 0.505

4.7 × 10-4 0.49

4.7 × 10-4 0.48

γ∞ i in j

j in i cal

pair

exp

Uniquac

hexane-water butanol-water hexane-butanol

3.89 × 10 51.37 5.18

5

LSG

3.89 × 10 56.9 5.3

3.89 × 10 58.1 5.26

5

cal NRTL

5

3.89 × 10 55.8 5.42

5

GEM-RS

exp

3.89 × 10 56.2 5.32

2.13 × 10 5.06 38.6

5

Uniquac 3

LSG

2.13 × 10 3.04 5.89

3

binary VLE (average error in %) P

2.13 × 10 5.02 33.91

cal

cal

Uniquac

LSG

NRTL

GEM-RS

Uniquac

LSG

NRTL

GEM-RS

hexane-water butanol-water hexane-butanol

12.5 12.0

13.1 11.9

1.29 2.68

1.41 2.88

7.3 1.64

7.9 1.87

3.24 1.95

3.32 2.05

component

Bii (cm3/mol)

ref

ViL (cm3/mol)

ref

pisat (kPa)

ref

hexane butanol water

-1749.3 -3475.9 -1162.1

39 39 40

131.6 91.96 18.064

41 41 42

20.167 0.906 3.165

43-46 43, 45, 47, 48 42

selected after a thorough data selection process. Parameters estimation is discussed in the Results and Discussion. Because, in the literature,9 only finite concentration ternary LLE data are available, the data were measured in the lab over the concentration range, and from the most dilute concentrations, Dsw was predicted. Data Selection A thorough literature search was carried out to collect BMS, γ∞, and VLE data of hexane-water, butanol-water, and butanol-hexane pairs, Dsw and LLE data of hexanebutanol-water, and vapor pressure of hexane, butanol, and water. From this search, the final data were carefully selected and were used to regress the parameter. Hence, after careful evaluation, a list was made for each kind of data, and final selection was done by averaging the values that were in close proximity, and outliers were ignored. Here, all the data are at 25 °C. The selected final data are shown in Tables 2 and 3. Pair-wise descriptions are as follows. Hexane-Water. This is an immiscible pair. Solubility of hexane in water is very low and vice versa. Solubility and γ∞ data of hexane in water and water in hexane were collected from Dechema Chemistry data series7,10-12 and from other literature.13-19 From these references, closely matched data of γ∞ of hexane in water were found as 2.592 × 10-6, 2.571 × 10-6, and 2.568 × 10-6. The final selected datum was 2.577 × 10-6, which was the average of these values. Similarly, data of water in hexane were 5.1 × 10-4 and 4.3 × 10-4; the average of these, 4.7 × 10-4, was the selected datum. After finalizing solubility data, γ∞ values of hexane in water and water in hexane were taken as the inverse of corresponding solubility data.

2.13 × 103 4.98 31.72

hexane-butanol-water

pair

Table 2. Second Virial Coefficients (Bii), Molar Volumes (ViL), and Vapor Pressure (pisat) Data

GEM-RS 3

Dsw y

cal

NRTL

2.13 × 10 3.07 6.03

3

exp

Uniquac

LSG

NRTL

GEM-RS

1.21, 1.45

10.0

9.76

1.64

1.65

Because of the fact that the solubility is very low, the infinite dilution activity coefficient can be taken as the reciprocal of solubility data.20 Butanol-Water. Butanol-water is partially miscible. Solubility of butanol in water is low, whereas the solubility of water in butanol is high. Available solubility data of butanol in water and that of water in butanol in literature21-24 lie very closely. For instance, from these references, solubility data of butanol in water were observed to be 0.01851, 0.01869, 0.01880, and 0.01892. The average of these, 0.01877, was selected. However, γ∞ data of butanol in water differ much from one source to another. For example, Djerki et al.25 reported the γ∞ value of butanol in water as 205.6, whereas in other sources26-31 γ∞ values were 49.02, 50.2, 50.5, 50.9, 51.6, 52.24, 52.8, and 53.7. The selected value was 51.37 (average of these values). Only one γ∞ value of water in butanol was available at 25 °C,32 which was 3.8. On the other hand, when other γ∞ values were observed at different temperatures from other sources,11 it was found to be more than 3.8 at 25 °C. Even when x-γ values were extrapolated, it was found to be greater than 3.8. Hence, after analyzing x-γ values, the final value of 5.06 was selected from the extrapolation. The x-γ value refers to activity coefficient (γi) values for corresponding compositions (xi). These values were generated from VLE data by the following thermodynamic relation: yipφi

γi ) sat xipsat i φi

[

VLi (p - psat i ) exp RT

]

(1)

φi have been calculated from the virial equation as follows: m

ln φi ) 2(

∑yB

j ij

j)1

- Bmix)

p RT

(2)

where B ) y2i Bii + 2yiyjBij + y2j Bjj

(3)

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Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

Table 3. Data Selection for Hexane-Water, Butanol-Water, and Butanol-Hexane Binary Pairs binary pair hexane-water

type of data mutualsolubility ∞

water-butanol

hexane-butanol

pressure range

xh,w ) 2.577 × 10 xw,h ) 4.7 × 10-4 ∞ γh,w ) 3.88 × 105 ∞ γw,h ) 2127.659 xb,w ) 0.01875 xw,b ) 0.5056 ∞ γb,w ) 51.37 ∞ γw,b ) 5.06

ambient ambient

mutualsolubility

ambient

γ∞

ambient

VLE γ∞

1.022-3.51kPa ambient

VLE

2.7-23.5

The second virial coefficients (Bii), saturated liquid volume and saturated vapor pressure (psat i ) data were carefully selected. Emphasis was made on obtaining the best empirical functions from the literature.33-38 On the basis of these selections of Bii of hexane and butanol, the mixed virial coefficient (Bij) of hexane-butanol/water-butanol pairs has been calculated by the Tsonopoulos correlation.39 Bii of water was taken from Dymond and Smith.40 Because the VLE calculations were at 25 °C and at very low pressures Bii or Bij does not vary much using any method and does not affect x-γ generation significantly, VLi values of hexane and butanol are calculated from temperature-dependent correlation according to Daubert and Danner,41 and for water, from ASME.42 psat i data are selected from experimental values in the literature.43-49 Two VLE data sets, one in Dechema43 and the other in Butler et al.,50 for the butanol-water pair at 25 °C were obtained. These data sets were checked by analyzing x-γ values. Both of these two data sets were combined and were used for parameters estimation. Hexane-Butanol. This pair is completely miscible. There is no experimental data of γ∞ of butanol in hexane available exactly at 25 °C. The nearest temperature at which these data ∞ is 33.0. are available is at 27.85 °C,51 and the value of γb,h However, the Dsw value of hexane-butanol-water is available ∞ ∞ ∞ )/(γb,h ) relation, γb,h was at 25 °C.8 Hence, from the Dsw ) (γb,w ∞ shown to be 38.6. For γh,b, very closely reported data were

Figure 1. Schematic diagram of the experimental setup.

ref -6

γ

(VLi ),

data values

kPa

γ∞h,b ) 5.18 γ∞b,h ) 38.6

10, 13-17 9, 10, 15, 17, 19 11, 18 9, 12, 15, 17, 19 10, 21-23 10, 21, 24 11, 26-31 12, 32 43, 50 12, 52, 53 8, 12, 53 44, 45

available in the literature12,52,53 (such as 5.12 and 5.25, and the average of these, 5.18, was the selected value). Two sources were found for VLE data of butanol-hexane at 25 °C from Rodriguez et al.44 and Gracia et al.45 The x-γ values of these two data sets were analyzed, and both were consistent with each other. Data from Gracia are in isothermal P-x form. Hence, Barker’s54 method was applied to get y (vapor composition) data and to generate the corresponding γ. Experimental The ternary LLE data of this system were measured in the lab, and details of these experiments are discussed in this section. The purpose of these experiments is 2-fold, to get experimental ternary LLE data of the hexane-butanol-water system from very dilute region to higher concentration and to get the correct Dsw value. For the experiment, the slow stirring method was used, which was the most effective method. Figure 1 shows the setup for the experiment. The experiment was mounted on a bench in the laboratory. The details of the measurement technique are described in the following subsections. Equilibrium Cell. A programmable water bath was purchased from Cole Parmer; magnetic stirrer and diaphragm pump were purchased from Fisher Scientific Co. The equilibrium cell was designed in-house and fabricated at the Lorillard Tobacco Co. in Greensboro, NC.

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

Gas Chromatograph. A Perkin-Elmer Autosystem gas chromatograph was used to determine the concentrations of solute, solvent, and water in each of the two phases. Because water is one of the components in the system, a thermal conductivity detector (TCD) has to be used. The column was chosen on the basis of the retention times of all three components. The rapid elution of water through PORAPAK materials makes these GC column packings ideal for analyzing aqueous systems. PORAPAK N was chosen to be the packing in the column because of the distinct differences in the retention times. The column was obtained from Alltech Associates, Inc. Chemicals. For our experiments, solvents (hexane) and solutes (1-butanol) were purchased from Aldrich, and solvents were 99+% purity and 1-butanol was 99.8+% purity. Ultrapure water was obtained from a Milli-Q water system in the laboratory. Setup. The experimental setup basically consists of four simple components. It consists of a programmable water bath complete with a pump, an equilibrium cell with a magnetic stirrer, and a gas chromatograph for analysis. The pump takes water at a constant temperature from the bath through the tubing and annulus of the equilibrium cell and back to the water bath. In this way, the contents in the equilibrium cell are maintained at a constant temperature for the experiment. Procedure. The equilibrium cell is a straight, double-walled Pyrex glass flask with a capacity of approximately 1 L. Water samples are taken from a Teflon tap that is connected to the interior of the vessel at approximately 2 cm from the bottom. Water of a constant temperature, 25 °C, is pumped through the outside ring (annulus) of the cell from the water bath. The water runs through rubber tubing, which connects the whole setup. In each run, 80 mol of water and 20 mol of hexane are brought into the equilibrium cell together with a Teflon-coated magnetic stirring bar. The mixture is thoroughly mixed using the magnetic stirrer. The required amount of 1-butanol is added, and the stirring process is continued for 6-8 h. Samples from both water-rich and hexane-rich phases are taken at regular intervals and analyzed using the gas chromatograph until the two phases reach the equilibrium. Results and Discussion The area of chromatogram corresponding to a particular component gives the amount of that component present in the sample injected, and the corresponding volume is calculated from the calibration curves. From the component volume and the sample volume and assuming that the density of pure component and density of component in the mixture is the same, moles of each component present in the sample are calculated. Mole percentages for each component in two phases are calculated and are tabulated in Table 4. To determine the distribution coefficient, the ratio of lowest butanol concentrations in the hexane-rich phase to the water-rich phase is taken. This is 1.23. This Dsw is in very good agreement with the literature.8 Literature values are 1.21 and 1.45. Now to verify model, first, the variations between calculated and experimental results of BMS data, γ∞, Dsw, and binary VLE data using the modified UNIQUAC6 model are shown. These parameters have been obtained from BMS data for hexane-water/ water-butanol pairs and for hexane-butanol x-γ data generated from VLE data. The results are reported in Table 5. The second column of this table shows the types of data from which parameters have been estimated. Here, symbols w1, w2, w3, and w3′ represent the weighted BMS, γ∞, VLE data of all concentration ranges, and VLE data of only finite concentration

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Table 4. Experimental Ternary Data for Hexane-Butanol-Water (in mol %) hexane-rich phase

water-rich phase

hexane

butanol

water

hexane

butanol

water

99.791 99.544 99.057 98.444 97.826 97.225 96.964 96.607 95.694 95.280 93.030 88.656 83.966 80.071 75.459 71.569 65.068 24.740 16.978 11.762

0.209 0.456 0.943 1.556 2.174 2.775 3.036 3.393 4.306 4.720 6.410 10.301 14.224 17.338 21.029 23.895 28.635 52.850 55.206 56.548

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.560 1.043 1.810 2.591 3.512 4.536 16.296 22.410 27.816 31.690

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.170 0.343 0.550 0.695 0.792 0.819 0.871 0.887 0.955 0.997 1.037 1.144 1.216 1.228 1.279 1.326 1.348 1.411 1.451 1.480

99.830 99.657 99.450 99.305 99.208 99.181 99.129 99.113 99.045 99.003 98.963 98.856 98.784 98.772 98.721 98.674 98.652 98.589 98.449 98.520

ranges, respectively. Corresponding parameters are also reported. Combinations of weighted functions are shown as regression no. in the third column of this table. All computational work was carried out using FORTRAN 77 and Absoft 10.1 as the compiler. Regression calculations were done by using IMSL FORTRAN Numerical Library 5.1. In all cases, the objective function was set as Nk

Fi(obj) )

Ni

∑ ∑ N1 w

k

k

i

Fi,k(exp) - Fi,k(cal) Fi,k(exp)

(4)

where N ) Nk*Ni

(5)

Here, Ni ) number of data points, Nk ) total number of property, and wk ) weightage. In this table, it is observed that regression using only the solubility data for the water-butanol pair presented an error (err) of around 40% in the γ∞ calculation. Using the γ∞ data to calculate solubility, a variation of 40% is observed as compared to the experimental data. Regression using VLE data results in the reduction of error in all calculations by 21%. Subsequently, using every data type in conjunction with the necessary weighted objective functions resulted in a further reduced error of 15%. Hence, for this pair, the UNIQUAC model performs well. For the hexane-butanol pair, using only the VLE data for parameter regression calculations of γ∞ shows 68.4% error. In fact, for some data points of VLE (x-γ), error is 100%. This pair was found to be more challenging and was thus studied extensively to better understand the problem. The other twoparameter model, LSG,2 has also been checked. The LSG model presents the same kind of behavior observed when using the UNIQUAC model. In this pair, γ of butanol increases abruptly as the butanol concentration is reduced, and these two-parameter models are unable to take into account this steep behavior. This is clearly shown in Figure 2. In Figure 2, it can be seen that the regression of the parameters using finite concentration data to calculate γ∞ of butanol in hexane shows an order of magnitude difference from the experimental value. To solve this problem, extra weight is given to the data in very dilute regions. This allows the models to predict γ∞ suitably; however, γ values in the finite concentration

3.96a 1.53a 10.0a 1.54a

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

1.21,1.45 3.7 6.6 3.34 7.3 5.8 2.94 3.0 1.64 2.98 2.91 11.5 2.59 12.5 6.01 28.9 39.4 12.0 35.8

y

erri )

N

i)1

average err )

N

∑ err

i

abs(expi - cali) × 100 expi

(N ) number of data points)

Figure 2. Variation of experimental and calculated γ’s in the binary hexane-butanol liquid mixture by two-parameter models. The γh scale is given on the y-axis to the left and γb on the y-axis to the right of the figure (*parameters used are according to the regression no. in Tables 5 and 6).

where erri is the relative percentage of error of data point i.

a

3.37 13.37 3.0 14.44 6.95 4.89 0.98 2.73 0.89 3.89 × 10 54.74 51.37 54.53 56.9 59.05 4.75 5.18 5.30 4.98 3.89 × 10 51.37 51.37 51.37 51.37 51.37 5.18 5.18 5.18 5.18 4.7 × 10 0.42 0.30 0.38 0.505 0.47 2.57 × 10 0.021 0.026 0.022 0.0187 0.0186 2.57 × 10 0.0187 0.0187 0.0187 0.0187 0.0187 1 1 2 3 4 5 1 2 3 4 1545.48,845.14 w1 ) 1, w2 ) 1, w3 ) 1 274.45,144.7 w1 ) 0, w2 ) 1, w3 ) 0 214.9,209.7 w1 ) 0, w2 ) 0, w3 ) 1 286.16,142.28 w1 ) 1, w2 ) 0, w3 ) 0 362.73,77.4 w1 ) 5, w2 ) 10, w3 ) 1 355.56,82.72 hexane-butanol w2 ) 1, w3 ) 1 520.25,60.71 w2 ) 1, w3 ) 0 723.17,67.51 w2 ) 0, w3′ ) 1 295.41,106.48 w2 ) 5, w3 ) 1 703.45,64.2 hexane-water butanol-water

pair i-j

weightage parameter (Aij,Aji)

reg no.

exp

-6

i in j

cal

-6

4.7 × 10 0.505 0.505 0.505 0.505 0.505

exp

-4

j in i

cal

-4

exp

5

i in j

cal

5

2.13 × 10 5.06 5.06 5.06 5.06 5.06 38.6 38.6 38.6 38.6

exp

3

2.13 × 10 4.58 5.06 3.97 3.04 4.43 14.9 38.6 5.89 38.14

3

cal j in i γ∞ solubility

Table 5. Comparison between Experimental and Calculated Results Using UNIQUAC Model

Using γ∞ value of butanol in water of 59.05 and that of butanol in hexane in the corresponding row. b In all cases, average err values have been calculated as follows:

5.61 5.16 4.47 5.67 4.52 22.43 39.76 9.34 38.97

P j in i i in j

average errb in γ

average err

exp

Dsw

cal

1038

range are inconsistent. Regression using all of the concentration data slightly improves γ∞ calculation, yet this is unacceptable as compared to experimental data. Despite using the best parameters, still the results were not within an error of 15-20%. Because no weighted objective functions were able to produce acceptable results, the two-parameter models failed to represent the data for this pair. Complete results of the LSG model are shown in Table 6. Parameters shown in the last row of each pair of Tables 4 and 5 are the final parameters of the UNIQUAC and LSG models. After this, the NRTL4 model was used, because it contributes an additional parameter, that is, the nonrandomness factor. The NRTL model provided acceptable results for the hexane-butanol pair unlike the previous two-parameter models. Using the NRTL model, the data were fit with an overall error of 5%. To judge the consistency of this result, another three-parameter model (GEM-RS2) was used with the hexane-butanol pair. GEM-RS also showed similar behavior as compared to NRTL. Both NRTL and GEM-RS are able to represent steep functions. The third parameter of these models helps to adjust functionality changes. This fact is shown more clearly in Figure 3. After the success of using NRTL and GEM-RS models on the hexane-butanol pair, these said models were used for the entire computational work. For all three pairs, various combinations of weightage, mentioned in earlier sections, were used to obtain the best fitted parameters. For the hexane-water pair, there is no flexibility to regress all three parameters because there are only two data sets (BMS of hexane in water and that of water in hexane). Hence, a third parameter was fixed (R, λ for NRTL, GEM-RS, respectively), while the other two parameters were regressed. In the case of NRTL, different combinations of binary interaction parameters (g12, g21) and nonrandom factor (R) are possible, and choosing the best R was a challenge. The effects of different non random factor (R) on ternary LLE calculation, especially in the hexanerich phase, were investigated. In the water-rich phase, compositions of all three components do not vary much throughout the concentration range (as seen in Table 4). However, in the hexane-rich phase, addition of more butanol amounts to considerable changes in the concentration of hexane and water. Hence, different R values of the hexane-water pair will have substantial effects in ternary LLE calculations, particularly in the hexane-rich phase. The value of R should be between 0 and 1. The calculations were checked at every 0.05 increment of R, starting at R )

cal

3.19 12.57 2.98 15.05 6.15 4.14 0.87 2.53 0.78

5.0 4.98 4.17 5.17 4.19 20.98 38.06 8.95 38.69

3.01 11.2 2.78 13.1 6.34 27.7 38.6 11.9 34.1

3.92 6.2 3.81 7.9 5.91 2.98 2.92 1.87 2.32

1.21,1.45

Using γ∞ value of butanol in water of 58.95 and that of butanol in hexane in the corresponding row. a

exp y P i in j j in i cal exp

3.89 × 105 3.89 × 105 5.06 4.27 5.06 5.06 5.06 3.81 5.06 3.07 5.06 4.72 38.6 17.17 38.6 38.6 38.6 6.03 38.6 38.32

Table 7. Change of Activity Coefficient Values with the Change of Hexane-Water Concentrations in Different r’s

2.13 × 103 53.88 51.37 53.63 58.1 58.95 4.83 5.18 5.26 5.01

cal exp cal

4.7 × 10-4 0.44 0.28 0.40 0.505 0.48

exp

4.7 × 10-4 0.505 0.505 0.505 0.505 0.505

cal

2.57 × 10-6 0.019 0.027 0.02 0.0187 0.0185

exp

2.57 × 10-6 0.0187 0.0187 0.0187 0.0187 0.0187

reg no.

1 1 2 3 4 5 1 2 3 4 -9.04,384.06 w1 ) 1, w2 ) 1, w3 ) 1 81.18, -16.62 w1 ) 0, w2 ) 1, w3 ) 0 39.22, 25.13 w1 ) 0, w2 ) 0, w3 ) 1 103.65,-36.3 w1 ) 1, w2 ) 0, w3 ) 0 198.31, -111.2 w1 ) 5, w2 ) 10, w3 ) 1 192.33, -104.15 hexane-butanol w2 ) 1, w3 ) 1 -150.45, 270.08 w2 ) 1, w3 ) 0 -160.36,307.58 w2 ) 0, w3′ ) 1 -102.36, 176.68 w2 ) 5, w3 ) 1 -161.31, 308.7

weightage

Figure 3. Variation of experimental and calculated γ’s in the binary hexane-butanol liquid mixture by three-parameter models. The γh scale is given on the y-axis to the left and γb on the y-axis to the right of the figure.

2.13 × 103 51.37 51.37 51.37 51.37 51.37 5.18 5.18 5.18 5.18

i in j j in i i in j pair i-j

hexane-water butanol-water

Dsw average err in γ average err j in i γ∞ solubility

Table 6. Comparison between Experimental and Calculated Results Using LSG Model

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3.43a 1.54a 9.76a 1.54a

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

R

average err of butanol composition in hexane phase

average err of water composition in hexane phase

0.05 0.1 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.35

27.83 20.13 13.2 12.04 11.3 10.68 10.1 9.91 9.33 9.02 9.53 10.01 10.78 21.38

27.96 21.32 15.18 14.13 13.40 12.12 11.61 10.91 10.23 10.41 11.03 12.21 13.32 24.62

0.05. In each case, the overall percentage of error of the butanol and water composition in the hexane-rich phase was compared. From this comparison, it was observed that the appropriate R value should be between 0.15 and 0.25. Subsequently, starting at R ) 0.15, the error percentage was observed at every 0.01 increment. This resulted in overall minimum error at R ) 0.22. Hence, the final selection was R ) 0.22. Comparison of results for different R’s is shown in Table 7. Graphical representation of R ) 0.15, 0.22, and 0.25 is shown in Figures 4 and 5. L1 and L2 indicate hexane-rich phase and water-rich phase, respectively. The same investigation that was conducted on the R parameter was also used to investigate the effect of λ in the GEM-RS model. From NRTL, variations of x-γ based on the appropriate third parameter are known. In the case of the GEM-RS model, attempts to obtain a third parameter, which would show similar x-γ transformations, were made. Starting at λ ) -50 and increasing by increments of 25, changes of x-γ seem to be more similar with NRTL’s R variation. From this observation, the optimal λ value was found to be between 250 and 260. Again, starting at λ ) 250 and increasing by increments of 1, the optimal λ value was found to be 254, after checking the error percentage. Thus, λ ) 254 was selected as the final parameter for our ternary calculation. Comparative results for different λ values are shown in Table 8. Differences between calculated results of butanol and water composition in the hexane-rich phase and experimental values for different λ’s are shown in Figures 6 and 7.

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Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

Figure 4. Comparison of experimental data and results of butanol compositions calculated by NRTL for different R values of the hexane-water pair.

Figure 6. Comparison of experimental data and results of butanol compositions calculated by GEM-RS for different λ values of the hexane-water pair.

Figure 5. Comparison of experimental data and results of water compositions calculated by NRTL for different R values of the hexane-water pair.

Figure 7. Comparison of experimental data and results of water compositions calculated by GEM-RS for different λ values of the hexane-water pair.

Table 8. Average Percentage of Error of Butanol and Water Composition in Hexane Phase for Different λ’s λ

average err of butanol composition in hexane phase

average err of water composition in hexane phase

250 251 252 253 254 255 256 257 258 259 260

13.68 12.83 11.21 10.12 9.13 9.82 10.10 11.2 12.41 13.39 14.08

14.92 13.38 12.91 11.18 10.73 11.08 12.13 13.23 14.51 15.68 16.71

After establishing the third parameter for the hexane-water pair, R and λ were fitted as independent parameters for the water-butanol and hexane-butanol pairs. Complete results for NRTL and GEM-RS models are shown in Tables 9 and 10, respectively. For fair comparison, all four models used the same weights when estimating the final parameters. In these tables, parameters of regression numbers (reg no.) 1, 3, and 2 were the best fitted parameters for hexane-water, butanol-water, and hexane-butanol pairs, respectively. Ternary results using the best fitted parameters of all four models were compared. Results of L1 and L2 phase are shown

in Figures 8 and 9, respectively. According to Figure 8, both the UNIQUAC and the LSG models fail to represent the ternary data at a higher concentration range, whereas three-parameter models represent the data well. As in Figure 9, these twoparameter models show a peculiar behavior in L2 phase. In dilute concentration ranges, the calculated data reach a peak, which is not observed in the experimental data (as seen in Table 4). Three-parameter models do not show this peak. Hence, in this regard, three-parameter models are better than two-parameter models. Two-set experimental data themselves do not match with each other, and therefore none of the models show good prediction after the feed fraction of 0.02. Comparison of water in the hexane-rich phase is shown in Figure 10. Here, also threeparameter models show better performance than two-parameter models. Comparisons between calculated and experimental data for all four models in very dilute regions are shown in Figures 11 and 12. One very important observation is that the three-parameter models behave similar when the third parameter is kept constant and the two interaction parameters are estimated. As noticed earlier, the main difficulty was fitting the data of the hexane-butanol pair. All three parameters were estimated for this pair; however, keeping the third parameter fixed and estimating the other two parameters in both the NRTL and the GEM-RS models results in problems similar to the UNIQUAC

weightage parameter (Aij,Aji,R)

0.0187 0.0187

2 3

-6

cal -6

0.02

0.023

2.57 × 10 0.021

i in j

0.505

0.505

weightage parameter (Aij,Aji,λ)

a

0.49

0.39

1 1 2 3 1 2

reg no. 2.57 × 10 0.0187 0.0187 0.0187

exp -6

cal 2.57 × 10 0.022 0.024 0.021

i in j -6

-4

exp

cal -4

5.18 5.18

51.37

51.37

cal 3

5.06

5.06

5

γ∞ exp

cal

3

j in i

33.91 38.5

5.02

4.32

3

cal

2.13 × 10 4.48

j in i 2.13 × 10 5.06

exp

38.6 38.6

cal

5

γ∞

3

3.33 4.48

3.8

2.87

3.37

j in i

2.68 3.0

1.29

1.16

1.23

P

1.95 2.41

3.24

2.47

2.91

y

average err

5.42 5.21 5.78 3.15 3.23

i in j

3.51 2.93 3.93 3.63 4.83

j in i

1.31 1.23 1.41 2.88 3.26

P

3.01 2.52 3.32 2.05 2.74

y

Dsw exp

1.21 1.45

exp

Dsw

1.21, 1.45

average err in γ average err

2.99 3.03

5.6

5.06

5.34

i in j

average err in γ

3.89 × 10 3.89 × 10 2.13 × 10 2.13 × 10 51.37 55.14 5.06 4.93 51.37 59.67 5.06 4.57 51.37 56.2 5.06 4.98 5.18 5.32 38.6 31.72 5.18 5.25 38.6 38.2 5

i in j

5.42 5.23

55.8

58.65

3.89 × 10 53.74

exp

5

i in j 3.89 × 10 51.37

4.7 × 10 0.41 0.37 0.48

j in i -4

4.7 × 10 0.505 0.505 0.505

exp

Using γ∞ value of butanol in water of 56.2 and that of butanol in hexane in the corresponding row.

hexane-water butanol-water

-438.05, 1380.45, 254.0 w1 ) 1, w2 ) 1, w3 ) 1, -198.86, 271.76, 75.87 w1 ) 0, w2 ) 0, w3 ) 1, -234.34, 339.57, 84.73 w1 ) 5, w2 ) 10, w3 ) 1, -85.68, 104.79, 58.58 hexane-butanol w2 ) 1, w3 ) 1, -272.87, 464.36, 70.56 w2 ) 5, w3 ) 1, -278.75, 485.91, 70.56

pair i-j

solubility

cal 4.7 × 10 0.42

j in i -4

4.7 × 10 0.505

exp

Using γ∞ value of butanol in water of 55.8 and that of butanol in hexane in the corresponding row.

1 2

2.57 × 10 0.0187

exp

1 1

reg no.

Table 10. Comparison between Experimental and Calculated Results Using GEM-RS Model

a

3381.1, 2013.86, 0.22 w1 ) 1, w2 ) 1, w3 ) 1 1070.01, 146.32, 0.35 w1 ) 0, w2 ) 0, w3 ) 1 1099.3, 133.41, 0.34 w1 ) 5, w2 ) 10, w3 ) 1 1031.46, 229.43, 0.41 hexane-butanol w2 ) 1, w3 ) 1 871.16, 329.66, 0.55 w2 ) 5, w3 ) 1 910.21, 323.42, 0.56

hexane-water butanol-water

pair i-j

solubility

Table 9. Comparison between Experimental and Calculated Results Using NRTL Model

1.65a 1.47a

cal

1.64a 1.45a

cal

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Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

Figure 8. Comparison of experimental data and results calculated by the UNIQUAC, NRTL, LSG, and GEM-RS models over the concentration range in the L1 phase.

Figure 9. Comparison of experimental data and results calculated by the UNIQUAC, NRTL, LSG, and GEM-RS models over the concentration range in the L2 phase.

Figure 11. Comparison of experimental data and results of butanol compositions calculated by the UNIQUAC, NRTL, LSG, and GEM-RS models in the very dilute region in the L1 phase.

Figure 12. Comparison of experimental data and results of butanol compositions calculated by the UNIQUAC, NRTL, LSG, and GEM-RS models in the very dilute region in the L2 phase.

Figure 10. Comparison of experimental data and results of water compositions calculated by the UNIQUAC, NRTL, LSG, and GEM-RS models over the concentration range in the L2 phase.

Figure 13. Variation of experimental and calculated γ’s in the binary hexane-butanol liquid mixture by three-parameter models if the third parameter is kept constant. The γh scale is given on the y-axis to the left and γb on the y-axis to the right of the figure (parameters used are according to the regression no. in Tables 9 and 10).

and LSG models. Without the third parameter, binary interaction parameters of three-parameter models are unable to represent sharp function. In Figure 13, R and λ values are kept fixed to 0.2 and 200, which results in poor calculations as in twoparameter models.

Because the final parameters of the NRTL and GEM-RS models are used for calculating binary phase equilibrium data (γ∞, BMS, VLE), these parameters can be named universal parameters because the calculations of the data types mentioned

Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

above can be solely obtained using these parameters. Although these parameters have been obtained from binary data, they can also predict ternary LLE data well for all concentration ranges. These can predict correct Dsw as well. Throughout this work, it was observed that careful selection of data and estimation of parameters using x versus γ data generated from VLE, all γ∞, BMS data made it possible to obtain a universal liquid state model as well as a universal set of parameters. These parameters are not only able to predict binary data but also ternary data very well. For the hexane-water and water-butanol pairs, either two-parameter or three-parameter models provide good results. The hexane-water pair does not show any problems due to the fact that the solubility itself is in infinite dilution and appropriate γ∞ values can be obtained using any model. Also, problems do not arise for the water-butanol pair because they are partially miscible and the x-γ functionality change within the miscibility gap is not known and cannot be described by any of the models. For hexane-water and water-butanol pairs, any of the models can be used to fit the data at infinite dilution and finite concentration simultaneously. These two pairs did not show large deviations between calculated and experimental data. However, it was not possible to fit the data for all concentration ranges for the hexane-butanol pair. From x-γ analysis, it was found that at dilute concentrations γ values of butanol increases very sharply. So any model that can represent sharp functionality changes can only be able to fit the data of this pair. The two-parameter models were found unable to show rapid changes of x-γ values. For the other two pairs, because γ changes smoothly with concentration, the two-parameter models perform well. Three-parameter models are capable of showing steep γ changes. The third additional parameter (in terms of nonrandomness) helps to adjust the fit. It is also very important that the three-parameter models can only fit the data when the third parameter is regressed independently. Otherwise, there would be no difference between the threeand two-parameter models. This additional parameter has the flexibility to fit the data for all concentration ranges. As seen in Figure 13, using the three-parameter models to fit the data showed the same behavior as in the UNIQUAC or LSG model. This is due to the fact that the third parameters were kept fixed and other two interaction parameters were estimated. For ternary systems, using the three-parameter model is not straightforward. Because of the immiscible pair, hexane-water, many combinations of third parameters and binary parameters are possible. Although every combination will give reasonable solubility values for this binary pair, they will not necessarily give good ternary results. The water-rich phase calculations can be done using any set of parameters, unlike the hexane-rich phase. In the water-rich phase, the amount of hexane and water remain nearly fixed throughout the entire concentration range. In the hexane-rich phase, as the butanol concentration increases, the concentration of hexane and butanol changes substantially. Hence, different third parameters will give different results for this phase. The most accurate way of selecting suitable third parameters is to compare the calculated results with the experimental data, which is very tedious work. As seen from the results, optimum ternary data can be achieved at R ) 0.22

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(NRTL) and λ ) 254 (GEM-RS) after a thorough investigation with this third parameter. Conclusion From detailed observations, the following conclusions can be drawn to obtain universal parameters: Data selections should be done very cautiously. The hexane-butanol pair was the main problem for fitting data because x-γ shows steep function. Two-parameter models cannot fit this steep function; however, three-parameter models can represent. Regression has to be done using the binary phase equilibrium data (LLE, VLE, γ∞). The third parameter has to be regressed independently. Selecting the appropriate third parameter of three-parameter model for an immiscible pair (hexane-water) is very important because only a fixed R or λ of NRTL and GEM-RS is able to predict ternary data well. From the results, it is observed that NRTL/GEM-RS (threeparameter models) are the universal models for the hexanebutanol-water system because these models can fit the data for the hexane-butanol pair, which show a sharp x-γ change, predict binary phase equilibrium data (LLE, VLE, γ∞), represent ternary data from very dilute region to finite concentration simultaneously, and calculate Dsw correctly. Hence, it is possible to obtain universal liquid mixture models for VLE and LLE, meaning that separate sets of parameters are not needed to calculate different types of phase equilibrium data. The work based on a typical nonideal ternary system that is hexane-butanol-water was satisfactory. However, more work is needed to be done to show that this process provides acceptable universal parameters for other nonideal ternary systems like benzene-methanol-water, hexane-propanol-water, and so on. The ternary systems reported in ref 55 display orderof-magnitude differences between experimental and calculated Dsw and will be good candidates for checking the proposed methods of obtaining universal parameters. This process can be further validated if used on quaternary systems. Reference 56 will be one of the good sources of quaternary liquid-liquid equilibrium data. Notation xw ) liquid molar composition of water yw ) vapor molar composition of water xh ) liquid molar composition of hexane yh ) vapor molar composition of hexane xh,w ) solubility of hexane in water xw,h ) solubility of water in hexane xb,w ) solubility of butanol in water xw,b ) solubility of water in butanol γw ) activity coefficient of water γb ) activity coefficient of butanol γh ) activity coefficient of hexane ∞ γh,w ) infinite dilution activity coefficient of hexane in water ∞ γw,h ) infinite dilution activity coefficient of water in hexane ∞ γb,w ) infinite dilution activity coefficient of butanol in water ∞ γw,b ) infinite dilution activity coefficient of water in butanol ∞ γh,b ) infinite dilution activity coefficient of hexane in butanol ∞ γb,h ) infinite dilution activity coefficient of butanol in hexane obj ) objective function

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Ind. Eng. Chem. Res., Vol. 50, No. 2, 2011

exp ) experimental cal ) calculated

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(55) Islam, A. W. Universal Liquid Mixture Models for Vapor-Liquid and Liquid-Liquid Equilibria in Hexane-Butanol-Water System. M.S. Thesis, North Carolina A&T State University, Greensboro, NC, 2009. (56) Macedo, E. A.; Rasmussen, P. Liquid-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt am Main, Chemistry Data Series; 1987; Vol. 5, Part 4.

ReceiVed for reView December 21, 2009 ReVised manuscript receiVed October 18, 2010 Accepted October 19, 2010 IE902028Y