Liquid Equilibria of Oxygenate Fuel Additives with ... - Springer Link

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Journal of Solution Chemistry [josc]

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Style file version Nov. 19th, 1999

Journal of Solution Chemistry, Vol. 30, No. 3, 2001

Liquid–Liquid Equilibria of Oxygenate Fuel Additives with Water at 25◦ C: Ternary and Quaternary Aqueous Systems of Methyl tert-Butyl Ether and tert-Amyl Methyl Ether with Methanol or Ethanol Kazuhiro Tamura,* Yao Chen, and Toshiro Yamada Received April 25, 2000; revised November 29, 2000 Experimental tie-line data have been determined for the ternary system water + methyl tert-butyl ether + tert-amyl methyl ether and the quaternary systems water + methanol + methyl tert-butyl ether + tert-amyl methyl ether, and water + ethanol + methyl tertbutyl ether + tert-amyl methyl ether at 25◦ C and ambient pressure. The experimental results have been satisfactorily correlated using the modified UNIQUAC and extended UNIQUAC models with ternary and quaternary, in addition to binary parameters. KEY WORDS: Multicomponent; liquid–liquid equilibria; oxygenate fuel additives; modified and extended UNIQUAC models.

1. INTRODUCTION The use of ethers such as methyl tert-butyl ether (MTBE) and tert-amyl methyl ether (TAME) as fuel additives has been increasing. The multicomponent phase equilibria of the oxygenate ether and alcohol mixtures with water are useful and considerably focused not only on process design for reformulated gasoline production but also on the problem of contamination of groundwater. Here we report liquid–liquid equilibrium (LLE) measurements on two quaternary systems water + methanol + MTBE + TAME and water + ethanol + MTBE + TAME and one relevant ternary system water + MTBE + TAME, all at 25◦ C. The experimental LLE data were correlated by means of the modified UNIQUAC and extended UNIQUAC models(1−3) including both ternary and quaternary parameters coming Department of Chemistry and Chemical Engineering, Division of Physical Sciences, Kanazawa University, 40-20, Kodatsuno 2-chome, Kanazawa, Ishikawa 920-8667, Japan. 291 C 2001 Plenum Publishing Corporation 0095-9782/01/0300-291$19.50/0

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from multicomponent intermolecular interactions, in addition to binary parameters. The binary parameters of miscible binary mixtures of constituents of the ternary and quaternary systems were obtained from vapor–liquid equilibrium data and those of immiscible mixtures were obtained from mutual solubility data. These binary phase equilibrium data have been published.(4−8) The results for the ternary water + methanol + MTBE or TAME and water + ethanol + MTBE or TAME systems, available from Refs. 8, 9, were used to obtain ternary parameters for accurate representation of the two quaternary LLE systems studied in this work. 2. EXPERIMENTAL 2.1. Materials TAME, MTBE, and methanol used were purchased from Aldrich Chemical Company, Inc. with nominal minimum purities of 97, 99.8, and 99.8 mass %, respectively. Ethanol and water were obtained from Wako Pure Chemical Industries, Ltd. with purities of 99.5 and 99.9 mass %. All chemicals were used without further purification. GC analysis did not detect appreciable peaks and gave the purities of 98.9 mass % for TAME, and better than 99.9 mass % for water, MTBE, methanol, and ethanol. Densities of the chemicals determined using a densimeter (DMA58, Anton Paar) at 25◦ C are presented in Table I, along with published values.(6,10) 2.2. Apparatus and Procedures Ternary and quaternary LLE measurements were carried out at 25 ± 0.01◦ C. The experimental apparatus and procedure are described in Ref. 2. About 70 cm3 of each mixture was loaded into the equilibrium glass cell placed in a thermostated water bath. The mixture was then stirred vigorously by magnetic stirrer for 5 h and allowed to settle 5 h, which was sufficient for separation into two phases. Dry nitrogen gas was used to prevent contamination with moisture in the headspace of Table I. Densities of Pure Components at 25◦ C Densitya Component

Ib

IIc

Methanol Ethanol MTBE (methyl tert-butyl ether) TAME (tert-amyl methyl ether) Water

0.78652 0.78524 0.73540 0.76587 0.99692

0.78637 0.78493 0.73528 0.76577 0.99705

g-cm−3 . work. c Refs. 6 and 10.

a Units: b This

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Table II. Experimental Ternary Liquid–Liquid Equilibrium Compositions for Water (1) + MTBE (2) + TAME (3) at 25◦ C Organic phase

Aqueous phase

x1

x2

x3

x1

x2

x3

0.0614 0.0651 0.0743 0.0801 0.0656 0.0500 0.0479

0.9386 0.8022 0.6605 0.4386 0.1956 0.0573 0.0000

0.0000 0.1327 0.2652 0.4813 0.7388 0.8927 0.9521

0.9675 0.9743 0.9841 0.9884 0.9932 0.9949 0.9966

0.0325 0.0256 0.0155 0.0112 0.0063 0.0030 0.0000

0.0000 0.0001 0.0004 0.0004 0.0005 0.0021 0.0034

the equilibrium cell. Samples, withdrawn from upper and lower phases in the cell by a microsyringe, were analyzed by a gas chromatograph (GC-8A, Shimadzu) equipped with a thermal conductivity detector. Each component of the ternary and quaternary mixtures was separated clearly using a 2-m long stainless steel column packed with Porapak SQ. The temperatures of the injection port and detector were set at 190◦ C. The initial temperature of the oven, initially at 150◦ C, was increased up to 240◦ C at a rate of 32◦ C min−1 . The helium flow rates for both the separation and reference columns were set at 0.5 ml-s−1 . The peak areas of the components, detected with a chromatopac (C-R6A, Shimadzu), were calibrated with weighed mixtures. The mass of each component of the mixture was determined from the calibration and converted to mole fraction. Three analyses were done for each sample to obtain a mean value. The accuracy of the measurements was estimated to be within ±0.001 in mole fraction. Table II lists the experimental LLE results for the ternary mixtures for the water + MTBE + TAME system. The mutual solubilities of water with MTBE or TAME measured in this work are comparable to values reported previously.(9) The schematic in Fig. 1 shows a tetrahedron to depict three planes of the quaternary LLE for the water + methanol or ethanol + MTBE + TAME systems. The quaternary mixtures were prepared by mixing stepwise the binary MTBE + TAME mixtures whose compositions areM1 , M2 , and M3 with water, and then methanol or ethanol to cover the two-phase regions. The values of M1 , M2 , and M3 are 0.25, 0.50, and 0.75, respectively, indicating the mole fraction of TAME in the binary MTBE + TAME mixtures. Tables III and IV present the experimental LLEs of the quaternary mixtures for the water + methanol + MTBE + TAME and water + ethanol + MTBE + TAME systems at 25◦ C. 3. ANALYSIS OF EXPERIMENTAL DATA The experimental results were correlated with the modified(1) UNIQUAC and extended UNIQUAC models described in the Appendix. Our modified model

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Fig. 1. Phase equilibria in quaternary systems water + methanol or ethanol + MTBE + TAME. M1 , M2 , and M3 denote quaternary sectional planes.

assumes that the combinatorial term can be expressed by a modification of the treatment of Gmehling et al.(11) The residual term is introduced by a third parameter C. The adjustable binary parameter τji obtained from binary experimental phase equilibrium data, is defined by the binary energy parameter aji aji τji = exp − (1) CT where the third parameter C is set equal to 0.65 in the modified UNIQUAC model and T is the Kelvin temperature. The additional ternary parameters, τ231 , τ132 , and τ123 , are obtained from the experimental ternary LLE results and the quaternary parameters, τ2341 , τ1342 , τ1243 , and τ1234 , are obtained from the correlation of the quaternary results. The extended UNIQUAC model has been described in detail elsewhere.(3) The binary parameter in the extended model is expressed by aji (2) τji = exp − T Table V shows the molecular-structural volume and area parameters, r and q, for MTBE and TAME taken from Arce et al.,(9) while the others are taken from

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Table III. Experimental Quaternary Liquid–Liquid Equilibrium Compositions for Water (1) + Methanol (2) + MTBE (3) + TAME (4) at 25◦ Ca Organic phase x1

x2

x3

0.0658 0.0801 0.1137 0.1230 0.1695 0.2145 0.2202 0.2333 0.3252

0.0251 0.0610 0.1269 0.1401 0.1834 0.2349 0.2511 0.2679 0.3448

0.2130 0.1851 0.1697 0.1552 0.1488 0.1231 0.1123 0.1032 0.0688

0.1019 0.1039 0.1255 0.1290 0.1326 0.1556 0.1701 0.2051 0.2598 0.3243

0.0395 0.0434 0.0833 0.1080 0.1189 0.1675 0.1933 0.2189 0.2671 0.3257

0.0959 0.1302 0.1455 0.1672 0.1970

0.0490 0.1384 0.1689 0.2079 0.2212

Aqueous phase x4

x1

x2

x3

x4

x30 = 0.25 0.6961 0.9554 0.6738 0.9053 0.5897 0.8244 0.5817 0.8014 0.4983 0.7462 0.4275 0.6911 0.4164 0.6621 0.3956 0.4580 0.2612 0.4487

0.0412 0.0892 0.1676 0.1829 0.2294 0.2717 0.2920 0.2901 0.2845

0.0021 0.0028 0.0041 0.0036 0.0059 0.0085 0.0103 0.0523 0.0612

0.0013 0.0027 0.0039 0.0121 0.0185 0.0287 0.0356 0.1996 0.2056

0.4026 0.3977 0.3709 0.3535 0.3485 0.3191 0.2901 0.2582 0.1978 0.1536

x30 = 0.50 0.4560 0.9351 0.4550 0.9343 0.4203 0.8814 0.4095 0.8635 0.4000 0.8278 0.3578 0.7686 0.3465 0.7249 0.3178 0.6722 0.2753 0.6477 0.1964 0.5902

0.0589 0.0590 0.1105 0.1275 0.1616 0.2145 0.2457 0.2890 0.3035 0.3377

0.0047 0.0051 0.0060 0.0065 0.0076 0.0113 0.0150 0.0221 0.0264 0.0335

0.0013 0.0016 0.0021 0.0025 0.0030 0.0056 0.0144 0.0167 0.0224 0.0386

0.6282 0.5302 0.4926 0.4454 0.4206

x30 = 0.75 0.2269 0.9264 0.2012 0.8115 0.1930 0.7788 0.1795 0.7323 0.1612 0.6840

0.0639 0.1713 0.1983 0.2354 0.2667

0.0087 0.0150 0.0195 0.0267 0.0391

0.0010 0.0022 0.0034 0.0056 0.0102

prepared by mixing pure water and methanol with x30 methyl tert-butyl ether + (1 − x30 ) tert-amyl methyl ether.

a Mixtures

Prausnitz et al.,(12) together with the interaction correction factor, q 0 , for which the value for self-associating components was taken from the literature,(1−3) while that for nonassociating components was set to q 0 = q 0.75 in the modified UNIQUAC model and q 0 = q 0.20 in the extended UNIQUAC model. The expressions for ln γ1 of the modified and extended UNIQUAC models are shown in the Appendix. The parameters for completely miscible binary mixtures were obtained from experimental vapor–liquid equilibrium data. The binary data reduction was performed by using a computer program described by Prausnitz et al.(12) according

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Table IV. Experimental Quaternary Liquid–Liquid Equilibrium Results for Water (1) + Ethanol (2) + MTBE (3) + TAME (4) at 25◦ Ca Organic phase x1

x2

x3

0.0573 0.1190 0.2130 0.3249 0.3451 0.4486

0.0191 0.0844 0.1605 0.2835 0.2927 0.2951

0.2182 0.1886 0.1480 0.0915 0.0845 0.0600

Aqueous phase x4

x1

x2

x3

x4

x30 = 0.25 0.7054 0.9772 0.6080 0.9407 0.4785 0.9100 0.3001 0.8570 0.2777 0.8492 0.1963 0.8233

0.0184 0.0544 0.0845 0.1311 0.1375 0.1570

0.0023 0.0025 0.0027 0.0047 0.0050 0.0066

0.0021 0.0024 0.0028 0.0072 0.0083 0.0131

0.0386 0.0484 0.0713 0.0956 0.1139 0.1395 0.1491 0.1639 0.1705

0.0049 0.0050 0.0055 0.0071 0.0084 0.0123 0.0132 0.0177 0.0201

0.0014 0.0016 0.0020 0.0030 0.0040 0.0075 0.0096 0.0141 0.0159

0.0198 0.0366 0.0688 0.0919 0.0965 0.1141 0.1244 0.1273 0.1412

0.0073 0.0082 0.0097 0.0111 0.0117 0.0154 0.0177 0.0188 0.0234

0.0007 0.0009 0.0012 0.0016 0.0019 0.0026 0.0035 0.0039 0.0056

0.0893 0.1129 0.1919 0.2876 0.3493 0.4247 0.4521 0.5111 0.5436

0.0586 0.0793 0.1324 0.1975 0.2405 0.2790 0.2847 0.2804 0.2703

0.4120 0.3840 0.3246 0.2471 0.1957 0.1411 0.1194 0.0971 0.0891

x30 = 0.50 0.4401 0.9551 0.4238 0.9450 0.3511 0.9212 0.2678 0.8943 0.2145 0.8737 0.1552 0.8407 0.1438 0.8281 0.1114 0.8043 0.0970 0.7935

0.1032 0.1782 0.2029 0.3189 0.3418 0.3811 0.4239 0.4352 0.4924

0.0318 0.0579 0.1379 0.1988 0.2049 0.2456 0.2570 0.2600 0.2616

0.6346 0.5617 0.4867 0.3538 0.3311 0.2748 0.2329 0.2223 0.1795

x30 = 0.75 0.2304 0.9722 0.2022 0.9543 0.1725 0.9203 0.1285 0.8954 0.1222 0.8899 0.0985 0.8679 0.0862 0.8544 0.0825 0.8500 0.0665 0.8298

prepared by mixing pure water and ethanol with x30 methyl tert-butyl ether + (1 − x30 ) tert-amyl methyl ether.

a Mixtures

to the following thermodynamic equations P yi 8i = xi γi Pis 8si exp ViL P − Pis RT X XX ln 8i = 2 yj Bij − yi yj Bij P RT j

i

(3) (4)

j

where P, x, y, and γ are the total pressure, liquid-phase mole fraction, vapor-phase mole fraction, and activity coefficient, respectively. The pure-component vapor pressure P s was calculated by using the Antoine equation with coefficients taken from the literature.(6,10) The liquid molar volume V L was obtained by a modified

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Table V. Structural Parameters for Pure Componentsa q0 Component Methanol Ethanol MTBE TAME Water

r

q

Ib

1.43 2.11 4.07 4.74 0.92

1.43 1.97 3.63 4.17 1.40

1.48 1.40 q 0.75 q 0.75 1.28

IIc 1.00 0.92 q 0.20 q 0.20 0.96

a Refs.

9 and 12. UNIQUAC model; Refs. 1 and 2. c Extended UNIQUAC model; Ref. 3. b Modified

Rackett equation.(13) The fugacity coefficient 8 calculated by the virial equation of state with the second virial coefficient B was expressed by Eq. (4). The subscript s stands for a value at saturation vapor pressure. The pure and cross-second virial coefficients were estimated by the method of Hayden and O’Connell.(14) An optimum set of binary parameters was obtained by minimizing the following objective function " # exp 2 exp 2 exp 2 X Pical − Piexp 2 Tical − Ti xical − xi yical − yi F= + + + σx2 σy2 σP2 σT2 i (5) where the subscripts cal and exp indicate, respectively, the most probable calculated value corresponding to each measured point and the experimental value. The standard deviations in the experimental values(12) used in Eq. (5) were taken as: σP = 1 mmHg for pressure, σT = 0.05 K for temperature, σx = 0.001 for liquidphase mole fraction, and σy = 0.003 for vapor-phase mole fraction. Table VI lists the binary parameters of the modified UNIQUAC model and the extended UNIQUAC models for the constituent binary mixtures, along with the root-mean-square deviations between experimental and calculated values: σP for pressure, σT for temperature, for σx liquid-phase mole fraction, and σy for vapor-phase mole fraction. Good agreement between experimental results and those calculated by both models was obtained, except for binary vapor-liquid equilibria of the MTBE + TAME system. The calculated results, can be improved by adjusting the interaction correction factor q 0 of MTBE and TAME to the experimental values. However, further calculations were not performed in the absence of the experimental results sufficient to correlate the models. The binary parameters for partially miscible mixtures were obtained by solving the thermodynamic relation given by Eqs. (6, 7), simultaneously. (xi γi )I = (xi γi )II

(6)

40 60 24.99 54.91 ∼78.19 73.66 ∼85.73 25 55.69 ∼85.16 25 25

Methanol + MTBE Methanol + TAME Methanol + water Ethanol + MTBE

MTBE + TAME

TAME + water

number of data points. deviation. c Units: mmHg. d Units: K. e I, modified UNIQUAC model; II, extended UNIQUAC model. f MS, mutual solubility.

0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.1 0.1 0.0 0.0 0.3 0.3

δT b,d 0.1 0.6 1.9 1.6 0.6 0.6 1.9 1.9 1.0 4.3 1.5 0.9 2.6 2.6

103 δx b 0.5 4.7 9.2 9.4 4.0 4.1 7.7 8.1 7.3 7.3 6.0 4.8 20.8 21.0

103 δy b

8

8

6

7

6

6

5

4

4

Ref.

298

b Root-mean-square

0.8 0.9 1.5 1.7 0.6 0.6 0.6 0.5 1.6 1.6 0.6 0.6 4.9 4.5

δ P b,c

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MS

MS f

28

10

569.52 540.64 657.78 626.20 158.59 70.15 523.26 513.31 496.31 496.53 −46.98 37.08 −274.50 −270.93 173.24 399.09 171.30 465.96

a21 (K)

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TAME + water

Ethanol + water

−107.03 −63.71 −89.93 −8.96 −160.39 −71.81 −56.88 −41.96 −19.50 23.55 212.17 157.12 475.66 517.21 1196.10 1023.70 1692.00 1341.90

a12 (K)

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I II I II I II I II I II I II I II I II I II

e

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Ethanol + TAME

◦C

System (1 + 2)

Table VI. Calculated Results of Binary Phase Equilibrium Data Reduction

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Fig. 2. Experimental and calculated LLEs of three ternary mixtures making up the quaternary system water + methanol + MTBE + TAME at 25◦ C. •–·–·–·–•, experimental tie-lines; −−−−, predicted by the modified UNIQUAC model using only binary parameters; ——, correlated by the modified UNIQUAC model with binary and ternary parameters taken from Tables VI and VII.

X

xiI = 1

i

and

X

xiII = 1

(I, II denote two liquid phases)

(7)

i

Figures 2 and 3 compare the experimental results for the ternary LLE mixtures making up the quaternary water + methanol + MTBE + TAME and water + ethanol + MTBE + TAME systems at 25◦ C with those predicted using only the binary parameters given in Table VI. As is shown in Figs. 2 and 3, as well as in Table VII, it seems that a good representation cannot always be obtained using only the binary parameters. The ternary and quaternary parameters, τjki and τjkil , resulting from three- and four-body interactions are necessary to accurately correlate experimental multicomponent LLEs. The ternary and quaternary parameters can be obtained by fitting the model to the experimental ternary and quaternary LLEs using the simplex method(15) by minimizing the objective function #0.5 " X X exp X 2 cal 2 F = 10 min (8) xijk − xijk M k

i

j

where min denotes minimum values, i = 1 to 3 for ternary mixtures or i = 1 to 4 for quaternary mixtures, j = 1, 2 (phases), k = 1, 2, . . . , M (number of tielines), M = 2ni, and x is the liquid-phase mole fraction. Table VII presents the

6 8 10 7

Water + methanol + MTBE Water + methanol + TAME Water + ethanol + MTBE Water + ethanol + TAME Water + MTBE + TAME

I I I I II

0.64 0.83 1.73 2.74 3.56 3.56 4.30 3.81 1.44 1.56

0.34 0.32 1.11 1.16 0.58 0.77 0.97 1.08 0.66 0.68

rmse,g −0.95210 −0.13910 0.01919 0.13680 −0.06410 0.03636 0.00840 −0.16997 0.28584 0.27848

τ231 0.95866 0.25021 0.17874 0.16510 0.32754 0.56749 0.15693 0.90885 0.47302 0.09221

τ132

This work

8

8

9

9

Ref.

300

b Number

τ123 −0.81427 −0.64896 −0.08545 −0.19029 0.95089 0.40938 0.78578 −0.00434 −1.12780 −1.13470

Ternary parameters

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I, only one binary system shows phase separation; type II, two binary systems show phase separation. of tie-lines. c I, modified UNIQUAC model. d II, extended UNIQUAC model. e Root-mean-square deviation (mol %). f Predicted using only binary parameters. g Correlated using binary and ternary parameters.

I II I II I II I II I II

rmse, f

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Table VII. Calculated Results for Ternary Liquid–Liquid Equilibria at 25◦ C

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Fig. 3. Experimental and calculated LLEs of three ternary mixtures making up the quaternary system water + ethanol + MTBE + TAME at 25◦ C. •–·–·–·–•, experimental tie-lines; −−−−−−, predicted by the modified UNIQUAC model using only binary parameters; ——, correlated by the modified UNIQUAC model with binary and ternary parameters taken from Tables VI and VII.

ternary parametersobtained in fitting the modified UNIQUAC model and the extended UNIQUAC model to the experimental ternary LLE systems constituting the quaternary LLEs, together with the root-mean-square deviation between the experimental and calculated tie-lines for the ternary LLE systems. Figures 2 and 3 show good agreement between the experimental values and those correlated using the additional ternary parameters. The quaternary mixtures exhibit type 2 quaternary LLE behavior,(16) which are composed of two ternary LLEs for the water + methanol + MTBE and water + ethanol + TAME or the water + meth anol + MTBE and water + ethanol + TAME systems classified as type 1, and one ternary LLE for the water + MTBE + TAME system as type 2, illustrated in Figs. 2 and 3. The two-phase regions in the constituent aqueous systems, including TAME, are larger than those of MTBE because of the presence of an extra methyl group in TAME compared with MTBE. The solubilities of water in the MTBE + TAME mixture and of the ethers in water are increased synergistically in comparison with the solubilities in MTBE or TAME alone. Furthermore, ethanol is more soluble than methanol in the organic-rich phase of the biphase mixture. It seems that the alkyl group in ethanol acts to hydrophobically interact with water in comparison to methanol.This leads to increasing the solubilities of ethanol more in the organic phase than in the aqueous phase. Table VIII summarizes the correlated results for the quaternary mixtures obtained in fitting the modified UNIQUAC model and the extended UNIQUAC model with binary, ternary, and quaternary parameters to the experimental quaternary LLE

24

Water + methanol + MTBE + TAME Water + ethanol + MTBE + TAME I II I II

3.26 2.81 1.67 2.44

rmse, f 1.87 1.79 0.98 1.02

rmse,g −0.6422 −0.0298 −0.7563 0.0365

τ2341 0.2784 −0.5534 1.1168 1.3230

τ1342

−2.7059 −1.6024 0.6930 −0.6736

τ1243

Quaternary parameters

−0.5268 −0.4797 2.5669 4.2630

τ1234

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c I,

of tie-lines. modified UNIQUAC model. d II, extended UNIQUAC model. e Root-mean-square deviation (mol %). f Predicted using binary and ternary parameters. g Correlated using binary, ternary, and quaternary parameters.

b Number

a Type.

II

24

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Table VIII. Calculated Results for Quaternary Liquid–Liquid Equilibria at 25◦ C

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data for the water + methanol + MTBE + TAME and the water + ethanol + MTBE + TAME systems, together with the predicted results by these models with the binary and ternary parameters listed in Tables VI and VII. The correlated results obtained from the both models are better than the predicted ones in representing the quaternary LLEs measured in this work and are in good agreement with the experimental ternary and quaternary LLE results.

4. CONCLUSION Ternary LLE data for the water + MTBE + TAME system and quaternary results for the water + methanol + MTBE + TAME and water + ethanol + MTBE + TAME systems were measured at 25◦ C. The experimental ternary and quaternary LLE data were successfully correlated by using the modified UNIQUAC and extended UNIQUAC models, in both cases, including binary, ternary, and quaternary parameters.

APPENDIX The activity coefficient of component 1 in quaternary mixtures is given by the modified UNIQUAC model is as follows φ10 φ10 Z φ1 φ1 − q1 ln + 1 − lnγ1 = ln + 1 − x1 x1 2 θ1 θ1 " # 4 X 0 θj τj1 + θ2 θ3 τ231 + θ2 θ4 τ241 + θ3 θ4 τ341 + θ2 θ3 θ4 τ2341 − Cq1 ln j

+ Cq1

4 X

qj0

qj

j

θj

 0   q1 θ (τ − θ θ τ − θ θ τ − θ θ τ − 2θ θ θ τ )  1 11 2 3 231 2 4 241 3 4 341 2 3 4 2341  q − Cq1 1 4  P   θj τj1 + θ2 θ3 τ231 + θ2 θ4 τ241 + θ3 θ4 τ341 + θ2 θ3 θ4 τ2341 

(A1)

j

q20 +

q2

θ2 [τ12 + (1 − θ1 )θ3 τ132 + (1 − θ1 )θ4 τ142 − θ3 θ4 τ342 + (1 − 2θ1 )θ3 θ4 τ1342 ] 4 P j

θj τj2 + θ1 θ3 τ132 + θ1 θ4 τ142 + θ3 θ4 τ342 + θ1 θ3 θ4 τ1342

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q30 θ3 [τ13 + (1 − θ1 )θ2 τ123 + (1 − θ1 )θ4 τ143 − θ2 θ4 τ243 + (1 − 2θ1 )θ2 θ4 τ1243 ] q + 3 4 P θj τj3 + θ1 θ2 τ123 + θ1 θ3 τ143 + θ2 θ4 τ243 + θ1 θ2 θ4 τ1243 j

  q40  θ4 [τ14 + (1 − θ1 )θ2 τ124 + (1 − θ1 )θ3 τ134 − θ2 θ3 τ234 + (1 − 2θ1 )θ2 θ3 τ1234 ]  q4 + 4  P   θj τj4 + θ1 θ2 τ124 + θ1 θ3 τ134 + θ2 θ3 τ234 + θ1 θ2 θ3 τ1234  j

P where the coordination number Z = 10, the P segment fraction φi = xiri / j xjrj , 3/4 3/4 0 the corrected P segment fraction φi = xiri / j xjrj , and the surface fraction φi = xi qi / j xj qj . The expressions for ln γ2 , ln γ3 and ln γ4 are obtained successively by cyclic advancement of the subscripts in Eq. (2), by changing 1 to 2, 2 to 3, 3 to 4, and 4 to 1. The activity coefficient of the component 1 derived by the extended UNIQUAC model is expressed as: φ1 φ1 Z φ1 φ1 − q1 ln + 1 − lnγ1 = ln + 1 − x1 x1 2 θ1 θ1 # " 4 X 0 θj τj1 + θ2 θ3 τ231 + θ2 θ4 τ241 + θ3 θ4 τ341 + θ2 θ3 θ4 τ2341 − q1 ln j

+ q1

4 q0 X j

θj

qj

j

  q10    θ1 (τ11 − θ2 θ3 τ231 − θ2 θ4 τ241 − θ3 θ4 τ341 − 2θ2 θ3 θ4 τ2341 ) q − q1 1 4  P   θj τj1 + θ2 θ3 τ231 + θ2 θ4 τ241 + θ3 θ4 τ341 + θ2 θ3 θ4 τ2341 

(A2)

j

q20 +

q2

θ2 [τ12 + (1 − θ1 )θ3 τ132 + (1 − θ1 )θ4 τ142 − θ3 θ4 τ342 + (1 − 2θ1 )θ3 θ4 τ1342 ] 4 P

θj τj2 + θ1 θ3 τ132 + θ1 θ4 τ142 + θ3 θ4 τ342 + θ1 θ3 θ4 τ1342

j

q30 θ3 [τ13 + (1 − θ1 )θ2 τ123 + (1 − θ1 )θ4 τ143 − θ2 θ4 τ243 + (1 − 2θ1 )θ2 θ4 τ1243 ] q + 3 4 P θj τj3 + θ1 θ2 τ123 + θ1 θ3 τ143 + θ2 θ4 τ243 + θ1 θ2 θ4 τ1243 j

P1: GAD/FPW

P2: FYK/FPW


QC: FYK PP101-298034


March 14, 2001

17:34


305

  q40  θ4 [τ14 + (1 − θ1 )θ2 τ124 + (1 − θ1 )θ3 τ134 − θ2 θ3 τ234 + (1 − 2θ1 )θ2 θ3 τ1234 ]  q4 + 4  P   θj τj4 + θ1 θ2 τ124 + θ1 θ3 τ134 + θ2 θ3 τ234 + θ1 θ2 θ3 τ1234  j

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16.

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