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Liquid–liquid equilibria for the ternary system water

+ diethylene

glycol monohexyl ether

+ 2-methyl-2-butanol

Da-Ren Chiou, Li-Jen Chen

∗

Department of Chemical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan Received 27 May 2003; received in revised form 26 November 2003; accepted 10 January 2004

Abstract

Liquid–liquid equilibrium compositions of the ternary system water+ diethylene glycol monohexyl ether + 2-methyl-2-butanol were measured at 293.15, 303.15, and 313.15 K under atmospheric pressure. The experimental equilibrium data were successfully correlated with the UNIQUAC model.

Keywords: Liquid-liquid equilibria; Phase equilibria; Ternary system; UNIQUAC model

1. Introduction

Non-ionic surfactants poly(oxyethylene glycol)

mo-noethers CH3(CH2)i−1O(CH2CH2O)jH (symbolized by

C_iE_j hereafter) are widely used in the industrial processes

as liquid–liquid extraction emulsifying agents [1], tertiary

oil recovery detergents[2], and herbicides. Hence the

ther-modynamic liquid–liquid equilibrium data for such systems are therefore essential. Recently, liquid–liquid equilibrium

data of systems containing C_iE_j have been measured for

binary and ternary mixtures in our laboratory[3–9]. There

are, to the best of our knowledge, no liquid–liquid equi-librium experimental data for the ternary systems water

+ diethylene glycol monohexyl ether + 2-methyl-2-butanol

available in the literature. In this study, liquid–liquid equi-librium measurements were performed for two binary

systems water + diethylene glycol monohexyl ether and

water + 2-methyl-2-butanol, and a ternary system, water

+ diethylene glycol monohexyl ether + 2-methyl-2-butanol,

at 293.15, 303.15, and 313.15 K under atmospheric pres-sure. The experimental data were then correlated with the

UNIQUAC model of Abrams and Prausnitz[10]. The phase

behavior of the system was successfully described by the UNIQUAC model.

∗_{Corresponding author. Tel.:}_{+886-2-23623296;} fax:+886-2-23623040.

E-mail address: ljchen@ccms.ntu.edu.tw (L.-J. Chen).

2. Experimental

2-Methyl-2-butanol was obtained from Merck with a pu-rity of 99% and was used as received. The non-ionic

sur-factant diethylene glycol monohexyl ether (C6E2) was an

Aldrich chemical product with a purity of 98% and was fractionally distilled under reduced pressure until a purity of >99% was obtained, as determined by gas chromatog-raphy. Water was purified by Millipore Milli-RO PLUS 10 followed by Milli-Q system with the resistivity better than

18.2 M cm.

To analyze the composition of the samples, a gas chro-matograph (China Chrochro-matography 9800) equipped with thermal conductivity detector was used. An 1 m long by 3.175×10−3m diameter stainless steel column stuffed with Poropak P 80/100 mesh was used. The signal was transferred to an integrator (Shimadzu, Chromatopac C-R6A) to accom-plish data recording. The temperatures of the injection port and of the detector were held at 553.15 and 573.15 K, respec-tively. The oven temperature was initially held at 398.15 K. Five minutes after injection, the oven temperature was raised to a final temperature of 518.15 K at a speed of 37 K min−1.

Helium was the carrier gas with a flow rate of 60 ml min−1.

Each analysis took about 18 min.

Single phase binary mixtures of C6E2 +

2-methyl-2-butanol and water+ 2-methyl-2-butanol with known

com-positions were used to calibrate the instrument in the composition range of interest. For the calibration of gas

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Table 1

The calibration composition ranges and the corresponding errors of gas chromatograph

Water+ 2-methyl-2-butanol C6E2 + 2-methyl-2-butanol

Mass fraction range of 2-methyl-2-butanol 0.00496–0.0858 0.203–0.955 0.0475–0.995

Average errorsa _0.000176 _0.000751 _0.000284

a _{Average errors}₌ 1

np

j=1|wexp− wcalb|j, where np is the number of calibration points and w is the mass fraction of 2-methyl-2-butanol. The

subscript “exp” and “calb” represent the prescribed experimental values and calibrated values, respectively.

chromatography, each prescribed composition was analyzed at least three times. The calibration results were further fitted to a third order polynomial function. The devia-tions of the calibration curves from the actual values are

tabulated in Table 1. These two polynomial functions of

calibration curves are combined to simultaneously solve

the compositions of the ternary system water + C6E2

+ 2-methyl-2-butanol. For each tie-line, the water + C6E2 + 2-methyl-2-butanol mixtures were prepared in three test

tubes with the same total composition. These samples were vigorously shaken and then put into a water-thermostat, whose temperature stability was controlled uncertainty in

temperature within±0.05 K, for at least 24 h to reach

equi-librium. During the equilibration process, these samples were shaken several times to ensure thorough mixing. After equilibrium was reached, both liquid phases of each sample were analyzed at least three times by gas chromatography to determine the compositions. The compositions for each tie-line were determined by averaging over three samples. The experimental uncertainty among three samples was

within±0.0009 mass fraction.

3. Results and discussion

The experimental results of liquid–liquid

equilib-rium of two binary systems water + C6E2 and water

+ 2-methyl-2-butanol at 293.15, 303.15, and 313.15 K are

listed inTable 2, also compare with the literature[3,4,6,11]. The data are expressed in units of mass fraction and the superscripts u and l stand for upper and lower liquid phases, respectively. The experimental compositions of tie-lines

for the ternary system water+ C6E2+ 2-methyl-2-butanol

at 293.15, 303.15, and 313.15 K are given inTables 3–5.

Figs. 1–3 show the corresponding triangular phase

dia-Table 2

Experimental and literature mass fraction of water of equilibrium liquid phases for the binary systems water (1) + C6E2 (2) and water (1)

+ 2-methyl-2-butanol (3)

T (K) Water (1)+ C6E2 (2) Water (1)+ 2-methyl-2-butanol (3)

This study Literature[4] Literature[6] This study Literature[3] Literature[11]

wl

1 wu1 wl1 wu1 wl1 wu1 wl1 wu1 wl1 wu1 wl1 wu1

293.15 0.981 0.569 0.976 0.566 0.981 0.558 0.884 0.246 0.894 0.238 0.879 0.243 303.15 0.986 0.432 0.980 0.429 0.980 0.430 0.906 0.228 0.907 0.226 0.899 0.227

313.15 0.988 0.348 0.981 0.341 0.982 0.341 0.921 0.207 0.921 0.218 – –

grams of the system water+ C6E2+ 2-methyl-2-butanol at

293.15, 303.15, and 313.15 K. Note that in the temperature range of our experiments, the two-phase region is belt-like and it enlarges with increasing temperature. Total compo-sitions of samples prepared for tie-line measurements are

also given in Figs. 1–3 by filled diamonds. The fit of a

linear expression to each tie-line and its corresponding total composition data point is always better than 0.999.

The UNIQUAC model was used to correlate the experi-mental data. In this work, the relative van der Waals volume

r_iand surface area q_iwere adopted from the UNIFAC group

contribution of Hansen et al.[12], listed inTable 6. The ef-fective binary interaction parameter a_ij, is defined byaij=

(uij−uji)/R, where uijis the UNIQUAC interaction

param-eter between molecules i and j and R is the gas constant.

The effective binary interaction parameters a_ij and a_ji

of the binary systems water (1) + C6E2 (2) and water (1)

+ 2-methyl-2-butanol (3) can be numerically solved for each

temperature by using experimental binary compositions as input data according to the iso-activity criterion:

xl

iγil= xuiγiu, i = 1–3 (1)

where x_i is the mole fraction of component i, γ_i is the

activity coefficient of component i.

For a ternary system, there are six group-interaction pa-rameters, a_ij. In this work, four effective binary interaction parameters, a12, a21, a13, and a31can be directly determined

from the experimental data of the binary systems. Therefore,

there are two parameters a23 and a32 left to be determined

by a numerical method.

First, we defined an objective function[13]:

Fa= m j=1 3 i=1 (xl ijγ l ij− x u ijγ u ij) 2 ₍₂₎

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Table 3

Experimental and calculated mass fractions of equilibrium liquid phases for the ternary system water (1)+ C6E2(2)+ 2-methyl-2-butanol (3) at 293.15 K

Experimental results Calculated results

wu 1 wu2 wl1 wl2 wu1 wu2 wl1 wl2 0.242 0.078 0.884 0.002 0.246 0.073 0.910 0.004 0.246 0.155 0.890 0.004 0.253 0.144 0.928 0.007 0.267 0.236 0.900 0.006 0.268 0.220 0.942 0.009 0.277 0.278 0.911 0.007 0.279 0.262 0.949 0.010 0.289 0.328 0.911 0.009 0.294 0.308 0.956 0.011 0.308 0.366 0.921 0.010 0.311 0.346 0.960 0.012 0.337 0.401 0.930 0.011 0.334 0.387 0.965 0.014 0.380 0.426 0.943 0.012 0.360 0.420 0.969 0.015 0.425 0.451 0.952 0.015 0.404 0.452 0.973 0.016 0.497 0.450 0.965 0.018 0.472 0.462 0.977 0.018

Average absolute deviationa 0.009 0.012 0.032 0.002

a _{Average absolute deviation}₌N

i=1|w

exp i −wcalci |

N , where N is the number of tie-lines.

Table 4

Average absolute deviationa _0.002 _0.011 _0.027 _0.001

i=1|w

exp i −wcalci |

N , where N is the number of tie-lines. wherexl_ijandxu_ijstand for the experimental mole fraction of

component i of lower and upper phase, respectively, along a tie-line j,γ_ijl, andγ_iju are the corresponding activity coeffi-cient calculated from the UNIQUAC model and m is the

to-Table 5

Average absolute deviationa _0.005 _0.009 _0.021 _0.001

i=1|w

exp i −wcalci |

N , where N is the number of tie-lines.

tal number of tie-lines. Then the effective binary interaction

parameters a23 and a32 of the UNIQUAC model were

de-termined numerically by minimizing the objective function

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H2O (1) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 C6E2 (1) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2-methyl-2-butanol (3) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 1. Ternary liquid–liquid equilibria (mass fraction) for the system water (1)+ C6E2 (2)+ 2-methyl-2-butanol (3) at 293.15 K: experimental tie-lines

(䊊, dashed line); calculated tie-lines (solid line); calculated binodal curve (solid curve); and experimental total compositions (䉬).

Next, we used the effective binary interaction parameters obtained above as the initial guesses of the following objec-tive function[14]: Fx= k j i xexpl ijk − xijk calc xijkexpl 2 , (3) H2O (1) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 C6E2 (2) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2-methyl-2-butanol (3) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 2. Ternary liquid–liquid equilibria (mass fraction) for the system water (1)+ C6E2 (2)+ 2-methyl-2-butanol (3) at 303.15 K: experimental tie-lines

(䊊, dashed line), calculated tie-lines (solid line); calculated binodal curve (solid curve); and experimental total compositions (䉬). Table 6

The relative van der Waals volume r and van der Waals surface area q

Compound r q

Water 0.9200 1.4000

C6E2 7.9900 6.7700

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H2O (1) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 C6E2 (2) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2-methyl-2-butanol (3) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fig. 3. Ternary liquid–liquid equilibria (mass fraction) for the system water (1)+ C6E2(2)+ 2-methyl-2-butanol (3) at 313.15 K: experimental tie-lines

(䊊, dashed line); calculated tie-lines (solid line); calculated binodal curve (solid curve); and experimental total compositions (䉬).

Table 7

UNIQUAC binary interaction parameters for the system water (1)+ C6E2

(2)+ 2-methyl-2-butanol (3) ij a_ij(K) 293.15 K 303.15 K 313.15 K 12 599.58 −298.27 558.67 −270.26 519.77 −244.13 13 111.51 112.19 132.80 106.81 145.47 114.73 23 −419.83 701.25 −441.24 785.60 −452.68 791.99

where xexpl_ijk and x_ijkcalc are the experimental and calculated composition of component i in phase j along a tie-line k,

respectively. The minimization of the objective function F_x

was accomplished by using the subroutine DUMPOL of IMSL library. Then the regressed binary interaction param-eters were applied to the liquid–liquid equilibrium flash

cal-culation [15] to evaluate the calculated tie-lines, listed in

Table 8

Comparison of UNIQUAC binary interaction parameters for this work and literatures

T (K) Water (1)+ C6E2 (2) Water (1)+ 2-methyl-2-butanol (3)

This work Literature[4] This work Literature[3]

a12(K) a21(K) a12 (K) a21(K) a13(K) a31(K) a13 (K) a31(K)

293.15 599.58 −298.27 569.45 −290.05 111.51 112.19 118.78 108.28

303.15 558.67 −270.26 518.70 −257.23 132.80 106.81 132.25 108.44

313.15 519.77 −244.13 464.23 −223.50 145.47 114.73 155.50 99.95

Tables 3–5, by using the experimental total compositions as input data.

The regression results of the UNIQUAC interaction

parameters are shown in Table 7 and note that these

pa-rameters are temperature-dependent.Table 8compares the

UNIQUAC effective binary interaction parameters of this work and literatures, and there is a good agreement. The

calculated results for each tie-line are given inTables 3–5

to make a comparison with experimental data. The average absolute deviation between experimental data and

calcu-lated results are also listed in the bottom of Tables 3–5.

As one can see, the phase behavior of the system water

+ C6E2 + 2-methyl-2-butanol is successfully described by

the UNIQUAC model.

List of symbols

a group-interaction parameters of the

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F_a first objective function

F_x second objective function

m the total number of tie-lines

q the relative van der Waals surface area

r the relative van der Waals volume

R the gas constant

T temperature

w water

x mole fraction

x_ij the experimental mole fraction of

component i along a tie-line j

x_ijk composition of component i in phase j

along a tie-line k

Greek letters

γij activity coefficient of component i

along a tie-line j Subscripts 1 water 2 C6E2 3 2-methyl-2-butanol i component i

ij interactions of i–j pair

ji interactions of j–i pair

Superscripts

calc calculated values

expl experimental values

l lower liquid phase

u upper liquid phase

Acknowledgements

This work was supported by the Chinese Petroleum Com-pany and the National Science Council of Taiwan, Republic of China.

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