Prediction of phase equilibria for CO2–C2H5OH–H2O system using the SAFT equation of state

Fluid Phase Equilibria 169 Ž2000. 1–18 www.elsevier.nlrlocaterfluid

Prediction of phase equilibria for CO 2 –C 2 H 5 OH–H 2 O system using the SAFT equation of state Zhi-Yu Zhang, Ji-Chu Yang ) , Yi-Gui Li Department of Chemical Engineering, Tsinghua UniÕersity, Beijing 100084, People’s Republic of China Received 15 July 1999; accepted 23 November 1999

Abstract The SAFT equation of state is applied to the correlation of thermodynamic properties of the binary systems consisting of water–ethanol, water–carbon dioxide and ethanol–carbon dioxide. Based on this, the liquid phase compositions of ternary system are calculated from vapor phase compositions using SAFT equation in the range of 308.15–338.15 K and 10.1–17.0 MPa, in which carbon dioxide acts as a supercritical extraction solvent. The phase equilibria predicted by SAFT agree with the experimental data from literatures accurately. q 2000 Elsevier Science B.V. All rights reserved. Keywords: SAFT; Equation of state; Vapor–liquid equilibria; Supercritical carbon dioxide; Ethanol

1. Introduction Dehydration of fermented crude ethanol from fermentation broths, in which the ethanol concentration is about 5–40 wt.%, requires a large amount of energy. Supercritical carbon dioxide can be used as a potential solvent for the extraction of alcohol from dilute aqueous solutions produced in biochemical processes because the supercritical fluid extraction is a less energy-consuming method than the traditional distillation method w1–3x. There is high ethanol–water selectivity in this process, even though it is not possible to break the azeotropic composition by means of a simple CO 2 extraction cycle w4x. In the last decade, the phase equilibria for CO 2 –C 2 H 5 OH–H 2 O ternary system have been studied w5–10x. Takishima et al. w5x used the Patel–Teja equation of state with Wilson’s local composition model to correlate the binary systems CO 2 –C 2 H 5 OH and CO 2 –H 2 O. But the predictions of phase behavior for ternary system were not satisfactory. Ossa et al. w8x applied a

)

Corresponding author. Tel.: q86-10-6278-5514; fax: q86-10-6277-0304.

0378-3812r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 Ž 9 9 . 0 0 3 4 1 - 6

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

2

perturbed-dipolar-hard-spheres equation of state, which has hard spheres with dipole moment as its reference term and the van der Waals attractive term as its perturbation term, to correlate the binary phase equilibria and to predict ternary phase behavior under 313K, 3.86–20MPa with good results. Yao et al. w10x proposed the modified Peng–Robinson equation of state to correlate binary systems of CO 2 Ž1. –C 2 H 5 OHŽ2. and CO 2 Ž1. –H 2 OŽ 3. under 31, 35, 40 and 508C and get adjustable parameters k 12 , kX12 , k 13 and kX13 respectively. They correlated VLE data for ternary system by adjusting parameters of k 23 and kX23 with good accuracy. In this work we use the statistical associating fluid theory Ž SAFT. to correlate and predict the phase behavior of the system. The equation is based on the Wertheim’s first-order perturbation theory for associating fluids w11–14x, and was first proposed by Chapman et al. w15x. Huang and Radosz w16,17x converted it into a useful engineering equation. Fu and Sandler w18x used the single dispersion term of Lee et al. for the square-well fluids to simplify the SAFT equation of state ŽSSAFT.. Gil-Villegas et al. w19x modified the SAFT to deal with different potential functions Ž SAFT-VR. . SAFT EOS has parameters with explicit physical meaning and few in number. It can describe vapor–liquid equilibria of fluids accurately and can make extrapolations or predictions at other thermodynamic conditions. Using SAFT Huang and Radosz w17x correlated phase equilbria of CO 2 –C 2 H 5 OH system, and Fu and Sandler w18x correlated C 2 H 5 OH–H 2 O system. Blas and Vega w20x measured the systems and then used SAFT to predict binary and ternary diagrams and got excellent agreements. Wolbach and Sandler w21x described phase behavior for cross-associating mixtures. But all of them have no report for the CO 2-C 2 H 5 OH–H 2 O ternary system.

2. Background In this paper we consider a molecule of the chain or polar fluid to be composed of spherical segments of equal-size and equal-interaction parameters with Lennard–Jones potential. Therefore, the residual Helmholtz free energy of molecules Žfor pure components or mixtures. can be expressed as the sum of hard sphere repulsion, hard chain formation, and dispersion and association terms: Ares s Ahs q Adisp q Achain q Aassoc

Ž1.

2.1. Hard sphere term The hard sphere term can be calculated with the residual part of the Carnahan–Starling equation multiplied by the averaged chain parameter m x w15x: Ah s NkT

smx

4h y 3h 2

Ž1 y h .

2

Ž2.

where h is a reduced density, defined as follows

hs

p NA 6

r d 3x m x

Ž3.

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

3

In Eq. Ž3. , the effective chain length m x s Si x i m i where x i is the mole fraction of component i. The effective hard sphere diameter of the conformal fluid is given by w22x: dx 1 q 0.2977Tr x s Ž4. sx 1 q 0.33163Tr x q 0.0010477Tr2x where reduced temperature Tr x s kTr´ x , and sx and ´ xrk are given as follows: 1r3

sx s

m2r3 x

žÝ Ýx x m m s / i

i

´x s

3 ij

j

i

j

Ý Ý x i x j m i m j si 3j ´ i j



Ž5.

j

i

j

Ý Ý x i x j m i m j si 3j i

j

0

Ž6.

where sii is the soft sphere diameter and ´ iirk is the Lennard–Jones energy parameters between segments Žof pure fluid.. For mixtures, we have the following mixing rules: sii q sj j si j s Ž7. 2 disp d i j is needed in the calculation of mhs in mixtures. Based on the Barker–Henderson i q mi perturbation theory, d i j can not be calculated from si j by use of Eq. Ž4.. We have to use a mixing rule that is independent of Eq. Ž 7. as follow: d ii q d j j di j s Ž8. 2

´ i j s Ž 1 y k i j . Ž ´ ii ´ j j .

1r2

Ž9.

where k i j is an adjustable parameter. 2.2. Hard chain formation term The contribution to the Helmholtz free energy from the formation of the covalent bond between hard-sphere segments is given by w15x: Achain NkT

s Ý x i Ž 1 y m i . ln g ii Ž d ii .

ž

hs

Ž 10.

/

i

Table 1 Regressed segment parameters and correlation deviations for pure fluids

H 2O C 2 H 5 OH CO 2

Data points

T range ŽK.

´ r k ŽK.

s Ž10y10 m.

m

´ AB r k ŽK.

k AB

D P Ž%.

D r L Ž%.

56 9 17

283.15–643.15 302–503 225–300

236.53 192.90 114.26

2.0863 3.0183 2.2018

2.756 2.7042 3.7249

681.23 2114.7

0.49572 0.028255

0.897 0.441 0.677

2.73 2.02 1.49

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

4

Fig. 1. Comparison between experimental and correlated values of density for pure carbon dioxide.

The contact value of radial distribution function is expressed as follows w15x: gi j Ž di j .

seg

hs

f gi j Ž di j . s

1 1yj3

q

3d i i d j j

j2

d i i q d j j Ž1 y j 3 . 2

q2

ž

dii d j j dii q d j j

2

/

j 22

Ž1 y j 3 .

3

Ž 11.

where

p NA r Ý x i m i d iki jks

i

6

,

k s 0,1,2,3

Ž 12 .

Fig. 2. Comparison between experimental and correlated values of saturated vapor pressure for pure carbon dioxide.

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

5

Fig. 3. Comparison between experimental and correlated values of density for pure water.

2.3. Dispersion term In most of the SAFT EOS, the square well potential is used to calculate the dispersion term. For real fluids the Lennard–Jones potential is more reasonable. So in this work the LJ potential is used. The dispersion contribution for Lennard–Jones fluids to Helmholtz free energy is as follows w22x: Adisp NkT

smx

1 Tr x

disp ž Adisp 01 q A 02 Tr x /

Fig. 4. Comparison between experimental and correlated values of saturated vapor pressure for pure water.

Ž 13.

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

6

Fig. 5. Comparison between experimental and correlated values of density for pure ethanol.

with 2 3 Adisp 01 s rt Ž y8.5959 y 4.5424 rt y 2.1268 rt q 10.285 rt .

Ž 14.

2 3 Adisp 02 s rt Ž y1.9075 q 9.9724 rt y 22.216 rt q 15.904 rt .

Ž 15.

where

rt s

6

p'2

h

Fig. 6. Comparison between experimental and correlated values of saturated vapor pressure for pure ethanol.

Ž 16.

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

7

Table 2 Binary interaction parameters and errors in liquid phase mole fraction and vapor phase mole fraction for mixtures < D x < s < x exp y x calc <, < D y < s < yexp y ycalc < Data T range points ŽK. H 2 O-C 2 H 5 OH k 12 sy0.15738q0.0004308 T AB k 12 s 0.55682–0.0010178 T CO 2-C 2 H 5 OH k 23 s 0.61893–0.0019401 T CO 2 –H 2 O

32

24 33 k 13 sy0.37442q0.00084 95 T 24

P range ŽMPa.

10 2 < D x < 10 2 < D y < Data source

313.15–347.94 0.0119–0.0872 0.65

0.86

w24x

304.2–308.2 3.75–7.67 304.65–323.15 2.22–9.17 298.15–348.15 5.066–70.93

0.46 0.368 0.392

w5x w25x w26x

2.50 3.74 1.046

2.4. Association term The Helmhotz energy change due to association is calculated as w15x: Aassoc NkT

s Ý xi

Ý

i

Ai

ž

Ai

InX y

X Ai 2

/

1 q 2

Mi

Ž 17.

In this paper, we use the association scheme of 4C for H 2 O moleculesŽ M s 4. and 3B for C 2 H 5 OH molecules Ž M s 3. w21x, while Chapman et al. w15x did not report association scheme used, and Huang and Radosz w16x used 3B for H 2 O and 2B for C 2 H 5 OH molecules. X A i is defined as the mole fraction of molecules i not bonded at site A: y1

X A i s 1 q NA Ý j

Ý x i r X B DA B j

i

j

Ž 18.

Bj

where associating strength is:

D A i B j s d i3j g i j Ž d i j .

seg

k A i B j exp Ž ´ A i B jrkT . y 1

Ž 19.

Fig. 7. Comparison between experimental and correlated values of VLE for the system water–ethanol at 313.15 K.

8

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

Fig. 8. Comparison between experimental and correlated values of VLE for the system water–carbon dioxide at 304.65 K.

In Eq. Ž19. , k AB is noted as bonding volume and ´ ABrk as associating energy. For cross-associating mixtures, we have the mixing rules w15x: k A i Bi q k A j B j k A j Bi s k A i B j s Ž 20. 2 A i Bi A j B j ´ A j B i s ´ A i B j s ž 1 y k iAB ´ . j /Ž´

1 2

Ž 21.

3. Fitting procedure of pure fluids According to the thermodynamic relation, in phase equilibria, the Gibbs free energy of the liquid phase equals to that of the vapor phase. GV s GL Ž 22.

Fig. 9. Comparison between experimental and correlated values of VLE for the system ethanol–carbon dioxide at 313.15 K.

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

9

Table 3 Comparison between experimental and predicted values of phase equilibria for the waterŽ1. –ethanolŽ2. –carbon dioxideŽ3. system in the range of 308.15–338.15 K and 10.1–17.0 MPa. < D x < Ž10y2 . s < x exp y x calc < T ŽK.

P ŽMPa.

308.15 308.15 308.15 323.15 338.15 308.2 313.2 323.2 333.2

10.2 13.6 17.0 10.2–17.0 10.2–17.0 10.07–10.31 10.1 10.1 10.1

Data points 6 5 6 6 6 5 4 6 7

Average absolute deviation < D x1 <

< D x2 <

< D x3<

1.32 3.06 3.32 2.57 3.29 1.67 2.79 1.95 4.56

1.17 2.60 3.18 2.73 3.23 2.07 3.32 2.72 5.20

0.46 0.75 0.56 0.80 1.49 1.33 0.63 2.04 1.47

Data source w6x w6x w6x w6x w6x w5x w7x w7x w7x

Corresponding work by Yao et al. w10x < D x1 <

< D x2 <

< D x3<

1.62 2.06 1.65

1.10 1.52 1.40

0.72 0.98 0.83

1.09

1.19

0.68

then A V q PV V s A L q PVL

Ž 23.

The segment parameters needed in our equation of state for pure fluids can be obtained by simultaneously fitting the experimental saturated vapor pressures and liquid densities w23x, and they are shown in Table 1. We use vapor–liquid equilibrium data of pure CO 2 with temperature up to 300 K instead of density data of fluids with temperature and pressure above its critical point to fit parameters because the former is proved to be more accurate in the calculation of phase equilibria of

Fig. 10. Comparison between experimental and predicted values of VLE for the system water–ethanol–carbon dioxide at 308.15 K and 10.2 MPa.

10

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

Fig. 11. Comparison between experimental and predicted values of VLE for the system water–ethanol–carbon dioxide at 308.15 K and 13.6 MPa.

binary and ternary systems. Correlation results of the saturated density and pressure data of pure CO 2 , H 2 O and C 2 H 5 OH are shown in Figs. 1–6, respectively. 4. Correlation of binary systems The parameters of carbon dioxide in Table 1 were extrapolated to higher pressure and higher temperature to correlate phase behaviors of binary systems. Using the mixing rules of Eqs. Ž 7. – Ž 9. and Eqs. Ž20. – Ž21. , we obtain the adjustable parameters, as shown in Table 2.

Fig. 12. Comparison between experimental and predicted values of VLE for the system water–ethanol–carbon dioxide at 308.15 K and 17.0 MPa.

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

11

Fig. 13. Comparison between experimental and predicted values of VLE for the system water–ethanol–carbon dioxide at 323.2 K and 10.1 MPa.

As examples, Figs. 7–9 show a comparison between calculated isothermal P–xy curves and experimental data for the three binary systems C 2 H 5 OH–H 2 O, CO 2 –H 2 O and CO 2 –C 2 H 5 OH, respectively.

Fig. 14. Comparison between experimental and predicted values of VLE for the system water–ethanol–carbon dioxide at 338.15 K, 13.6 MPa and 17.0 MPa.

12

Model

Correlated binary parameters AB k 23 , k 13 , k 12

This work Yao et al. w10x

SAFT EOS Modified PR EOS

k 12 , k 12 , k 23 , k 13 , k 12’, k 23’, k 13’

Ossa et al. w8x

Perturbed–dipolarhard-spheres EOS

0 1 0 1 l 12 , k 12 , k 12 , l13 , k 13 , k 13 , l 23 , 0 k 23 , k 123

T range ŽK.

Predictionrcorrelation

308.15–338.15 304.15, 308.15, 313.15, 323.15 312.9–313.2

Prediction with given y 1, y 2 and y 3 Correlation with given y 1, y 2 and y 3 Prediction Only y1 was given

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

Table 4 Comparison between different models in prediction or correlation of the ternary system waterŽ1. –ethanolŽ2. –carbon dioxideŽ3.

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

13

5. Prediction of phase equilibria of H 2 O–C 2 H 5 OH–CO 2 system The adjustable binary parameters thus obtained from these binary systems can be used to predict the phase equilibria of H 2 O–C 2 H 5 OH–CO 2 ternary system. To verify the predictive capabilities of the model at supercritical conditions of carbon dioxide, we calculated the vapor–liquid equilibria of the ternary system without refitting the binary parameters. The average absolute deviations between the predicted and experimental liquid compositions Ž if the vapor compositions are known. under different temperatures and pressures were shown in Table 3. We found that SAFT EOS can be used to describe the phase behavior of this system with acceptable accuracy and does not need additional parameters. Results under different temperatures and different pressures are shown in figures from Figs. 10–14. Compared with this work, Ossa’s model w8x can predict well in ternary system, but compositions of vapor fluid are not completely given, and there are too many parameters. In their model every pure fluid has nine parameters. Results of Yao et al. w10x can only be used to calculate the phase behavior at 318C, 358C, 408C and 508C, respectively, with no report of predictions. A comparison of the models is shown in Table 4. The solvent selectivity of ethanol between the liquid and the vapor phase is defined by the ratio of the distribution coefficients of ethanol to water w8x: yC H OH y H 2 O Ss 2 5 Ž 24. x C 2 H 5OH x H 2 O The ethanol loading is defined by the ratio of ethanol to carbon dioxide in the vapor phase w8x: yC H OH L s 104.55 2 5 Ž 25. yCO2 In this equation 104.55 s Ž MC 2 H 5OHrM CO2 . = 100. Here M is the molecular weight, so that the mole fraction is changed into weight percentage.

Fig. 15. Ethanol loading in supercritical carbon dioxide at 308.15 K.

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

14

Fig. 16. Solvent selectivity for the system water–ethanol–carbon dioxide at 308.15 K.

Both selectivity and loading can be predicted acceptably, as Figs. 15 and 16.

6. Conclusion The phase equilibria of H 2 O–C 2 H 5 OH–CO 2 system under 308.15–338.15 K and 10.1–17.0 MPa can be satisfactorily predicted by SAFT EOS without any additional adjustable parameters or refitting the binary parameters. Compared with the earlier works by different authors, for the ternary system H 2 O–C 2 H 5 OH–CO 2 the SAFT EOS seems to be the best both for correlation and for prediction. List of symbols A d g k m Mi N NA P R T x xi X iA

Helmholtz energy Hard-sphere diameter Žm. Distribution function Boltzmann’s constant Effective number of segments Number of associate sites on molecule i Number of molecules Avogadro’s number, 6.02217 = 10 23 moly1 Pressure Gas constant Absolute temperature ŽK. Mole fraction in the liquid phase Mole fraction of component i Mole fraction of molecule i not bonded at site A

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

y Z

15

Mole fraction in the vapor phase Compressibility factor

Greek letters h Reduced density m Chemical potential r Number density Žmy3 . s Lennard–Jones segment diameter, m D AB Association strength between sites A and B k AB Bonding volume ´rk Energy parameter of dispersion ŽK. AB ´ rk Energy parameter of association between sites A and B ŽK. Subscripts i, j, k Component x Pure fluid or mixture Superscripts assoc Contribution due to association chain Contribution due to covalent chain-formation disp Contribution due to dispersion hs Hard-sphere A,B Association site

Appendix A. The Compressibility Factor The compressibility factor obtained from the thermodynamic equation E Zsr

A

ž / NkT Er

 0

s 1 q Z hs q Z disp q Z chain q Z assoc

Ž A1.

/

Ž A2.

T,N

with Z hs s m x

ž

Z disp s m x

Z

chain

4h y 2 h 2

Ž1 y h .

ž

Z01 Tr x

q

3

Z02 Tr2x

/

s Ý x i Ž1 y m i . r i

Ž A3.

ž

Eln g ii Ž d ii . Er

hs

/

Ž A4. T, xj

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

16

Calculation methods of Z01, Z02 and r Ž E g i i Ž d i i . hsrE r . T,x i can be refered to Chapman et al. w15x Z assoc s

1 RT

y Aassoc Ý x i massoc i

Ž A5.

i

The massoc is given by Eq. ŽA9. . With suitable optimizing algorithm and starting value of r 0 i chosen, an output of r with physical meaning can be obtained.

Appendix B. Chemical potential With r obtained we can also calculate chemical potential m i , which is essential in phase equilibrium calculation. hs disp mres q mchain q massoc i s mi q mi i i

mi

disp mhs s i q mi

mx

q

ž

mx

Adisp 01

q

Tr x

Tr2x

2 Ý x j m j si 3j

Ž Z hs q Z disp .

hs

y 3Ž Z q Z

disp

 .

j

1 q 0.2977Tr x

ž

Ž 1 y m i . RT ln ž g it Ž d it .

m

Ý RT Žln X

Ai

0

1 y 0.0010477Tr2x 1 q 0.33163Tr x q 0.0010477Tr2x

//  hs

j

0

j 3 x mx

´x s

/qÝx

j

r RT Ž 1 y m j .

j

assoc s i

y1

sx3 m x

2 Ý x j m j ´ i j si 3j y 2 ´ x Ý x j m j si 3j

1

m

mx

2 Adisp 02

y

chain s i

m i RT

Ž Ahs q Adisp . q

mi

Ž A6.



E ln g j j Ž d j j .

Ai

y X r2 . q RTMir2 q Ý x j r RT Ý

Ai

j

Ž A7.

Aj

ž

hs

/

Eri

ž

1 XAj

1 y 2

/ž

0

Ž A8. T , r k/i

EX Aj Eri

/

Ž A9. T , rk/ i

where EX Aj

ž / Erj

T , r 1/ i

s yX

A2j

NA

ž

ÝX Bi

Bi

D

A jBt

qr Ý k

Ý xk Bk

E X Bk

ž ž / D

A jBk

Eri

qX T , r 1/ i

Bk

ž

ED A j Bk Eri

/ //

Ž A10.

T , r1/ i

The expressions Ž E g jk Ž d jk . hs .rE r i . T, r 1/ i and Ž E D A j B k .rE r i . T , r 1 / i can be referred to Chapman et al. w15x. In order to obtain X Ai, nonlinear equations of Eq. Ž18. must be solved. For pure self-associating fluids and binary systems of one non-self-associating and another self-associating fluid, Eq. Ž 18. can

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

17

be reduced to a quadratic equation with only one variance. For example, for water of 4C association scheme w21x: H

O

X sX s

(

y1 q 1 q 8 NA x H 2 O rD HO

Ž A11.

4 NA x H 2 O rD HO

And for ethanol of 3B association scheme:

ž

1qX H 2

/

sX Os

y1 q NA x C 2 H 5OH rD HO q

(ž

2

1 q NA x C 2 H 5OH rD HO q 4 NA x C 2 H 5OH rD HO

/

4 NA x C 2 H 5OH rD HO

Ž A12. But the situation is quite different for cross-associating mixtures. Take waterŽ 1. –ethanolŽ 2. as an example: X H 1 s 1r Ž 2 NA x 1 X O1rD H 1O1 q 2 NA x 2 X O2rD H 1O2 q 1 . X H 2 s 1r Ž 2 NA x 1 X O1rD H 2O1 q 2 NA x 2 X O2rD H 2O2 q 1 . X O1 s 1r Ž 2 NA x 1 X H 1rD H 1O1 q NA x 2 X H 2rD H 2O1 q 1 .

Ž A13.

X O2 s 1r Ž 2 NA x 1 X H 1rD H 1O2 q NA x 2 X H 2rD H 2O2 q 1 . The nonlinear equations can be changed into a optimization problem as Eq. Ž A14. where z is defined as the object function of optimization and k as the times of iteration. It is fortunate to find that on the X H 1 X H 2 plate only one local minimum point of z exists and it proved to be the global minimum point. The initial value of X H 1 and X H 2 are 0.35 in this work. 2

H1 H2 min z s Ž X kq1 y X kH 1 . q Ž X kq1 y X kH 2 .

2

H1 X kq1 s 1r Ž 2 NA x 1 X kO1rD H 1O1 q 2 NA x 2 X kO2rD H 1O2 q 1 . H2 X kq1 s 1r Ž 2 NA x 1 X kO 1rD H 2O 1 q 2 NA x 2 X kO 2rD H 2O 2 q 1 . O1 X kq1 s 1r Ž 2 NA x 1 X kH 1rD H 1O1 q NA x 2 X kH 2rD H 2O1 q 1 .

Ž A14.

O2 X kq1 s 1r Ž 2 NA x 1 X kH 1rD H 1O 2 q NA x 2 X kH 2rD H 2O 2 q 1 .

k s 0,1,2, . . . With the numerical value of X Ai, the value of Ž E X A i .rŽ E r i . T, r 1/ i can be obtained. For system waterŽ 1. –ethanolŽ 2., Eq. ŽA10. can be transferred into linear equations as below and solved: 1 2

X H 1 NA 0,

,0,2 x 1 rD

1 2

X H 2 NA

H 1O1

,2 x 2 rD

,2 x 1 rD H 2 O1 ,2 x 2 rD H 2 O 2

2 x 1 rD H 1O1 , x 2 rD H 2 O1 , 2 x 1 rD

H 1O 2

H 1O 2

, x 2 rD

H 2O2

1 2

X O1 NA

,0,

,0

1 2

X O 2 NA

ž ž ž ž

E X H1 Er 1 EX

H2

Er 1 E X O1 Er 1 E X O2 Er 1

/ / / /

T,r2

T ,r2

T ,r2

T ,r2

Z.-Y. Zhang et al.r Fluid Phase Equilibria 169 (2000) 1–18

18

ž ž ž ž

y 2 X O1D H 1O1 q2 x 1 r X O1 y 2 X O1D H 2 O1 q2 x 1 r X O1 s

y 2X

H1

D

H 1O

q2 x 1 r X

H1

ž ž ž ž

y 2 X H 1D H 1O 2 q2 x 1 r X H 1

E D H 1O1 Er 1 ED

H 2 O1

Er 1 ED

H 1O1

Er 1 ED

H 1O 2

Er 1

/ / / /

q2 x 2 r X O 2

T ,r2

q2 x 2 r X O 2

T ,r2

q x2 r X

H2

T ,r2

q x2 r X H2 T ,r2

ž ž

ž ž

E D H 1O 2 Er 1 ED

Er 1

ED

/ / / /

H 2 O1

Er 1 ED H 2O2 Er 1

/ / / /

T ,r2

H 2O2

T ,r2

T ,r2

T ,r2

Ž A15.

The value Ž E X A jrE r 2 . T , r 1 can be obtained in the same way. References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x

R.P. De Dilippi, J.M. Moses, Biotechnol. Bioeng. Symp. 12 Ž1982. 205–219. G. Brunner, K. Kreim, Ger. Chem. Eng. 9 Ž1986. 246–250. A. Serra, M. Poch, C. Sola, ` Process Biochem. 10 Ž1987. 154–158. E.A. Brignole, P.M. Andersen, A. Fredenslund, Ind. Eng. Chem. Res. 26 Ž1987. 254–261. S. Takishima, K. Saiki, K. Arai, S. Saito, J. Chem. Eng. Jpn. 19 Ž1986. 48–56. M.L. Gilbert, M.E. Paulaitis, J. Chem. Eng. Data 31 Ž1986. 296–302. S. Furuta, N. Ikawa, R. Fukuzato, N. Imanishi, Kagaku Kogaku Ronbunshu 15 Ž1989. 519–525, Žin Japanese.. E.M. de la Ossa, V. Brandani, G. Del Re, G. Di Giacomo, E. Ferri, Fluid Phase Equilib. 56 Ž1990. 325–340. H. Horizoe, T. Tanimoto, I. Yamamoto, Y. Kano, Fluid Phase Equilib. 84 Ž1993. 297–320. S. Yao, Y. Guan, Z. Zhu, Fluid Phase Equilib. 99 Ž1994. 249–259. M.S. Wertheim, J. Stat. Phys. 35 Ž1984. 19–34. M.S. Wertheim, J. Stat. Phys. 35 Ž1984. 35–47. M.S. Wertheim, J. Stat. Phys. 42 Ž1986. 459–476. M.S. Wertheim, J. Stat. Phys. 42 Ž1986. 477–492. W.G. Chapman, K.E. Gubbins, G. Jackson, M. Radosz, Ind. Eng. Chem. Res. 29 Ž1990. 1709–1721. S.H. Huang, M. Radosz, Ind. Eng. Chem. Res. 29 Ž1990. 2284–2294. S.H. Huang, M. Radosz, Ind. Eng. Chem. Res. 30 Ž1991. 1994–2005. Y.-H. Fu, S.L. Sandler, Ind. Eng. Chem. Res. 34 Ž1995.. A. Gil-Villegas, A. Galindo, P.J. Whitehead, S.J. Mills, G. Jackson, J. Chem. Phys. 106 Ž1997. 4168–4186. F.J. Blas, L.F. Vega, Ind. Eng. Chem. Res. 37 Ž1998. 660–674. J.P. Wolbach, S.I. Sandler, Ind. Eng. Chem. Res. 37 Ž1998. 2917–2928. R.L. Cotterman, B.J. Schwarz, J.M. Prausnitz, AIChE J. 32 Ž1986. 1787–1798. C.F. Beaton, G.F. Hewitts, Physical Property Data for the Design Engineer, Hemi Sphere Pub., New York, 1989. J. Gmehling, U. Onken, Vapor–Liquid equilibrium Data Collection: Aqueous–Organic Systems, DECHEMA, 1977. S.-J. Yao, F.-H. Liu, Z.-X. Han, Z.-Q. Zhu, J.of Chem. Eng. of Chinese Universities 3 Ž1989. 9–15, Žin Chinese.. R. Wiebe, Chem. Rev. 29 Ž1941. 475–481.