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Fluid Phase Equilibria 362 (2014) 35– 40

Contents lists available at ScienceDirect

Fluid Phase Equilibria

j ourna l ho me page: www.elsev ier .com/ locate / f lu id

orrelation of phase equilibria for the systems containing-butanol + water by concentration dependent surfacerea parameter model

oshio Iwai ∗, Issei Taniguchiepartment of Chemical Engineering, Faculty of Engineering, Kyushu University, Fukuoka 819-0395, Japan

r t i c l e i n f o

rticle history:eceived 4 July 2013eceived in revised form 29 August 2013ccepted 30 August 2013vailable online 9 September 2013

a b s t r a c t

The concentration dependent surface area parameter (CDSAP) model is applied to correlate thevapor–liquid equilibria for binary systems and liquid–liquid equilibria for ternary systems containing1-butanol + water. The model is based on the quasi-chemical theory. The surface area parameters in themodel depend on partner molecules and concentrations. The activity coefficients of multi-componentsystems can be calculated with the binary parameters of constituting binary systems. The advantage of

eywords:ctivity coefficient modeluasi chemicalocal fractiononcentration dependent surface areaarameter

the model is that the liquid–liquid for ternary systems and the vapor–liquid equilibria for constitutingbinary systems are calculated well with the same parameter set. The parameters in CDSAP model areexplained by functions of temperature to apply in wide range of temperature. The calculated results byCDSAP model are almost the same as those by NRTL and UNIQUAC models for the vapor–liquid equilibriaof binary systems, and better than those by NRTL and UNIQUAC models for the liquid–liquid equilibriaof ternary systems containing 1-butanol + water.

. Introduction

Activity coefficient is a fundamental physical-property valueequired for the design of separation equipment. Many mod-ls [1–12] have been proposed for the calculation of activityoefficients. NRTL [1] and UNIQUAC [2] models are widely used forhe correlation of activity coefficients. NRTL and UNIQUAC mod-ls have two or three adjustable parameters, and are expected forhe accurate correlation by fitting the adjustable parameters tohe experimental data. Furthermore, one can calculate the phasequilibria for multi-component systems with the binary parame-ers determined by the constituting binary systems by NRTL andNIQUAC models. However, UNIQUAC model fails in the descrip-

ion of the activity coefficient outside the finite concentration rangehen the parameters are fitted to activity coefficients of finite

oncentrations [8,13]. Furthermore, the estimation accuracy is notood for many systems when the binary parameters are deter-ined with the constituting binary systems, and the parameters

re applied for the calculation of the ternary liquid–liquid equilib-ia [13,14]. This means it is difficult to correlate well both binary

apor–liquid and ternary liquid–liquid equilibria with the samearameter set by UNIQUAC model. Islam et al. [15] applied UNI-UAC model for the partition coefficients of ternary liquid–liquid

∗ Corresponding author. Tel.: +81 92 802 2751; fax: +81 92 802 2751.E-mail address: iwai@chem-eng.kyushu-u.ac.jp (Y. Iwai).

378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.fluid.2013.08.038

© 2013 Elsevier B.V. All rights reserved.

equilibrium systems. They showed that the results are within 1order of magnitude for some systems (consistent systems) and thedeviations are even more than 8000% for other systems (in con-sistent systems) using both common and specific parameters byUNIQUAC model. Islam et al. [13] calculated the vapor–liquid andliquid–liquid equilibria in hexane + butanol + water system by twoparameter models (UNIQUAC and LSG [4]) and three parametermodels (NRTL and GEM-RS [6]). They showed that the two param-eter models fail to represent the data of vapor–liquid equilibriafor hexane + butanol binary system and liquid–liquid equilibria forhexane + butanol + water ternary system, whereas the three param-eter models represent the data well.

The authors proposed concentration dependent surface areaparameter (CDSAP) model [16]. The proposed model is basedon the quasi-chemical theory [17]. The surface area parametersin the model depend on partner molecules and concentrations.The vapor–liquid equilibria for binary systems and liquid–liquidequilibria for ternary systems are calculated well with the sameparameter set by the model. In the previous work [16], theparameters in CDSAP model were determined at each system andtemperature. In this work, the model is applied for the ternaryliquid–liquid equilibrium systems containing 1-butanol + watersystems with plate points. Those systems are difficult to correlate

by normally used models because the liquid–liquid equilib-rium regions are narrow. The parameters in CDSAP model areexplained by functions of temperature to apply in wide range oftemperature.

3 hase Equilibria 362 (2014) 35– 40

2

2

wa

e

G

wasfa

�

wsaf

�

wc

∑

c

wp

mcf

q

wqa

l

T

l

qd

Table 1Surface area parameters for pure compounds [18].

Compound qo

1-Butanol 3.052Water 1.400Methanol 1.432

6 Y. Iwai, I. Taniguchi / Fluid P

. CDSAP model

.1. Activity coefficient

The detailed derivation of activity coefficient of CDSAP modelas shown in the previous work [16]. The model is briefly explained

s follows.The excess Gibbs free energy GE is explained by the following

quation from the concept of quasi-chemical theory [17].

E = RT∑

jqjnj ln

�jj

�j(1)

here R is the gas constant, T is the temperature, q is the surfacerea parameter, n is the amount of substance, �j is the averageurface area fraction of component j, �jj is the local surface arearaction of component j around component j. The average surfacerea fraction of component i, �i, is defined as

i = qixi∑j

qjxj

(2)

here x is the mole fraction. The local area fractions �ji and �ijhould meet the following conservation equation, because inter-ction between components i and j is the same when it is countedrom components i and j.

i�ji = �j�ij (3)

here �ji is the local surface area fraction of component j aroundomponent i.

In addition, the following equation should be valid.

j

�ji = 1 (4)

Further, the following equation is obtained from the quasi-hemical theory [17].

�ij�ji

�ii�jj= �ij = exp

[−2uij

kT

]= exp

[−2aij

T

](5)

here �ij is the interaction parameter, uij is the interaction energyarameter, k is the Boltzmann constant, aij is the binary parameter.

The surface area parameters in CDSAP model depend on partnerolecules and concentrations [16]. The surface area parameter of

omponent i, qi, is given by the following linear function of moleraction.

i = qoi xi +

∑j=1,j /= i

q∞ji xj (6)

here qio is the surface area parameter of pure component i and

ji∞ is the surface area parameter of component i in component j

t infinite dilution.The activity coefficient � i can be obtained by the next equation.

n �i =[

∂GE

RT∂ni

]T,p,nj /= i

(7)

he activity coefficient is obtained as follow.

n �i ={

qi +(

qoi − qi

)xi

}ln

�ii

�i+

∑j=1,j /= i

(q∞

ij − qj

)xj ln

�jj

�j(8)

The local mole fractions �ii and �jj can be obtained by solvinguadratic equations for binary system, and by an iterative proce-ure for multi-component system from Eqs. (3)–(5).

2-Propanol 2.508Acetone 2.3361-Propanol 2.512

2.2. Parameters

The surface area parameters for pure components qio were cited

from the surface area parameters of UNIQUAC model in order toreduce the number of parameters. The values of qo are listed inTable 1. The fitting parameters for binary system are aij, qji

∞, andq∞

ij. The activity coefficients for multi-component systems can be

calculated by using the three fitting parameters of consisting binarysystems.

The parameters in CDSAP model for vapor–liquid equilibriawere determined to fit the experimental data of activity coefficientsat each temperature. Table 2 lists the values of parameters. Then,the parameters were explained by linear functions of temperatureas follows.

a12 [K] = at12 (t[◦C] − 25) + a25

12 (9)

q∞21 = q∞,t

21 (t[◦C] − 25) + q∞,2521 (10)

q∞12 = q∞,t

12 (t[◦C] − 25) + q∞,2512 (11)

The coefficients of the linear functions are listed in Table 3.The parameters in CDSAP model for the liquid–liquid equilibria

of 1-butanol + water system were determined as follows. In general,two fitting parameters are required to fit the experimental dataof liquid–liquid equilibria for a binary system, because two fittingparameters are required to solve x1

I�1I = x1

II�1II andx2

I�2I = x2

II�2II

two equations. Where the superscripts I and II mean phases I and II,respectively. The CDSAP model has three fitting parameters for eachbinary system. So, many sets of values for the fitting parameters canbe obtained to fit the experimental data for liquid–liquid equilibriaof 1-butanol + water binary system. In this study, the parameters for1-butanol + water binary system were determined to exactly sat-isfy the liquid–liquid equilibrium data of 1-butanol + water binarysystem, and to reduce the deviations from the liquid–liquid equilib-rium data for 1-butanol (1) + methanol (2) + water (3) [35,36] and1-butanol (1) + 2-propanol (2) + water (3) [37,38] ternary systemsat 25, 60, and 80 ◦C. The values of parameters are listed in Table 2.Then, the three values of a12 for 1-butanol + water binary systemwere averaged. The a12 was fixed to 167.336 K, and the parame-ters q21

∞ and q12∞ were determined to fit the experimental data

of liquid–liquid equilibria for 1-butanol + water binary system [34]in the temperature range from 20 to 120 ◦C. The parameters wereexplained by Langmuir type equations. The equations are shown inTable 3.

3. Results and discussion

3.1. Vapor–liquid equilibria for binary systems

The parameters of binary vapor–liquid equilibria were obtainedto fit the experimental data of each system for both NRTL and UNI-

QUAC models. The values of parameters are listed in Table 4. Table 5shows the errors of activity coefficients calculated by CDSAP, NRTL,and UNIQUAC models for the binary systems. The errors by CDSAPmodel are slightly less than those by NRTL and UNIQUAC models

Y. Iwai, I. Taniguchi / Fluid Phase Equilibria 362 (2014) 35– 40 37

Table 2Parameter values of CDSAP model determined at each temperature.

System t (◦C) Refs. of data CDSAP

Comp. (1) Comp. (2) q21∞ q12

∞ �12 a12(K)

Vapor–liquid equilibria1-Butanol Methanol 25 [19] 2.687 0.779 0.901 15.5771-Butanol 2-Propanol 40 [20]a 3.306 1.871 0.977 3.4691-Butanol Acetone 25 [21] 3.673 2.334 0.625 70.000Methanol Water 25 [22] 1.034 0.736 0.625 70.059

45 [23] 1.285 0.956 0.595 82.52460 [24] 1.228 0.759 0.568 94.35580 [25] 1.201 1.209 0.507 119.788

2-Propanol Water 25 [26] 1.979 1.152 0.328 166.09860 [27] 2.571 1.234 0.342 178.63980 [28] 2.508 1.084 0.351 185.108

Acetone Water 20 [29] 1.406 1.229 0.253 201.17125 [30] 1.376 1.224 0.243 210.66035 [31] 1.595 1.396 0.292 189.80350 [29] 1.774 1.316 0.280 205.53360 [32] 2.011 1.721 0.327 186.057

100 [33] 1.723 1.001 0.279 238.310Liquid–liquid equilibria1-Butanol Water 25 [34] 3.862 1.611 0.326 165.274

TC

TP

60 [34]

80 [34]

a Data of 1-butanol + 1-propanol were used because accurate data for 1-butanol + 2-pro

able 3oefficients of CDSAP parameters in Eqs. (9)–(11).

System CDSAP

Comp. (1) Comp. (2) q21∞,t×103 (◦C−1) q21

∞,25

1-Butanol Methanol 0 2.687

1-Butanol 2-Propanol 0 3.306

1-Butanol Acetone 0 3.673

Methanol Water 0 1.187

2-Propanol Water 9.615 1.976

Acetone Water 8.606 1.426

1-Butanol Water a

a q∞21 = 3.516×10−2(t[◦C]−25)

1+2.266×10−2(t[◦C]−25)+ 3.862

b q∞12 = 3.787×10−2(t[◦C]−25)

1+2.304×10−2(t[◦C]−25)+ 1.610

able 4arameter values of NRTL and UNIQUAC models.

System t (◦C) NRTL

Comp. (1) Comp. (2) g12–g22 (J mol−1) g21–g11 (J mol−1)

Vapor–liquid equilibria1-Butanol Methanol 25 −2034.804 3321.332 0.21-Butanol 2-Propanol 25 41.780 44.992 0.31-Butanol Acetone 25 −823.911 4815.021 0.2Methanol Water 25 −192.190 1564.346 0.3

45 −246.156 2100.828 0.360 129.163 1787.931 0.380 1430.548 1134.031 0.2

2-Propanol Water 25 372.906 4925.962 0.260 2017.138 6142.336 0.580 1026.925 6735.739 0.4

Acetone Water 20 2083.183 3007.085 0.225 1806.965 3296.983 0.235 2611.699 3395.652 0.450 2416.806 4183.421 0.460 1514.109 4342.519 0.2

100 569.416 6207.749 0.2

Liquid–liquid equilibria1-Butanol Water 20 −2676.572 12,723.272 0.2

25 4016.160 8273.121 0.440 −4114.073 15,291.033 0.160 2919.518 10,150.470 0.480 −275.996 11,788.816 0.3

4.491 2.025 0.366 161.1754.895 2.848 0.388 175.560

panol are not available.

q12∞,t × 103 (◦C−1) q12

∞,25 a12t (K ◦C−1) a12

25 (K)

0 0.779 0 15.5770 1.871 0 3.4690 2.334 0 70.0005.660 0.932 0.896 67.0400 1.115 0.347 166.2030 1.298 0 205.256

b 0 167.336

UNIQUAC

Ref. ofparameter

u12–u22 (J mol−1) u21–u11 (J mol−1) Ref. ofParameter

483 [39] 508.377 168.991 [39]022 [40] 362.149 −291.468 [40]870 [41] −470.272 1902.348 [41]022 [42] −514.836 514.919 [42]002 [43] −122.015 290.160 [43]009 [44] 965.997 −643.609 [44]991 [45] 2160.415 −1080.793 [45]884 [46] 2041.376 −336.380 [46]228 [47] 983.325 841.538 [47]524 [48] 540.685 1167.424 [48]920 [49] 3535.712 −874.658 [49]915 [50] 4293.194 −1089.461 [50]878 [51] 3500.116 −894.153 [51]456 [52] 3095.137 −583.587 [52]908 [53] 3802.650 −881.081 [53]862 [54] 2151.091 −88.825 [54]

000 (This work) −0.550 2079.037 [34]500 (This work) −76.322 2220.799 [34]700 (This work) −282.892 2610.499 [34]500 (This work) −591.185 3146.617 [34]700 (This work) −942.946 3657.460 [34]

38 Y. Iwai, I. Taniguchi / Fluid Phase Equilibria 362 (2014) 35– 40

Table 5Calculated results of vapor–liquid equilibria for binary systems.

System t (◦C) Refs. of data Errora

Comp. (1) Comp. (2) CDSAPb CDSAPc NRTLd UNIUQACd

Completely miscible system1-Butanol Methanol 25 [19] 0.019 0.019 0.026 0.0261-Butanol Acetone 25 [21] 0.170 0.170 0.172 0.170Methanol Water 25 [22] 0.006 0.011 0.014 0.014

45 [23] 0.005 0.010 0.007 0.00760 [24] 0.012 0.024 0.017 0.01780 [25] 0.026 0.028 0.026 0.027

2-Propanol Water 25 [26] 0.009 0.012 0.012 0.01260 [27] 0.062 0.072 0.061 0.07580 [28] 0.017 0.018 0.014 0.019

Acetone Water 20 [29] 0.014 0.021 0.015 0.02625 [30] 0.051 0.051 0.053 0.05735 [31] 0.034 0.040 0.033 0.03450 [29] 0.008 0.018 0.009 0.00960 [32] 0.031 0.038 0.038 0.029

100 [33] 0.027 0.036 0.024 0.026Averagee 0.033 0.038 0.035 0.037

Partly miscible system1-Butanol Water 25 [55] 0.035 0.036 0.072 0.095

60 [56] 0.081 0.137 0.044 0.114Averagee 0.058 0.087 0.058 0.105

a Error = (∑

N

∑i(|�exp

i− �calc

i|/�exp

i))/2N where i is the component and N is the number of data.

b Parameters are determined at each temperature as shown in Table 2.le 3.

wTa

3

ciowtciMsqfx�i(qpnabssm

3

1rt

and ˛) for 1-butanol + water were determined to exactly satisfy theliquid–liquid equilibrium data of 1-butanol + water binary system,and to reduce the deviations from the liquid–liquid equilibrium

c Parameters are obtained by temperature dependent equations as shown in Tabd Parameters are determined at each temperature as shown in Table 4.e Average = summation of errors/number of systems.

hen the parameter values are determined at each temperature.he errors by CDSAP model slightly increase when the parametersre obtained from the linear functions of temperature.

.2. Test of Michelsen–Kistenmacher syndrome

Michelsen and Kistenmacher [57] pointed out that someomposition-dependent mixing rules were not invariant to divid-ng a component into a number of identical subcomponents. Ifne calculates a binary mixture (x1, x2) as a ternary (x1, x2, x3)here the ternary is formed by dividing component (2) into

wo “new” components in identical amounts and with identi-al properties, a different total molar properties will result withnadequate mixing rules. We performed the numerical test of

ichelsen–Kistenmacher syndrome for CDSAP model. The testedystem is acetone (1) + water (2) at 20 ◦C. The parameters are1

o = 2.336, q2o = 1.400, q21

∞ = 1.406, q12∞ = 1.229, and �12 = 0.253

rom Tables 1 and 2. When the mole fractions are x1 = 0.3 and2 = 0.7, the activity coefficients are calculated as �1 = 2.3549 and2 = 1.19361 for the binary system. When water (2) is divided

nto two new components, the parameters related component3) become q3

o = q2o = q23

∞ = q32∞ = 1.400, q31

∞ = q21∞ = 1.406,

13∞ = q12

∞ = 1.229, �13 = �12 = 0.253, and �23 = 1.0, because theroperties of component (3) are the same as those of compo-ent (2). When the mole fractions of two water componentsre x2 = 0.2, x3 = 0.5 (x1 = 0.3), the calculated activity coefficientsecome �1 = 2.3549, �2 = 1.19361, and �3 = 1.19361 for the ternaryystem. The calculated results of the binary system and the ternaryystem give exactly the same properties. It is shown that CDSAPodel does not suffer from Michelsen–Kistenmacher syndrome.

.3. Liquid–liquid equilibria for 1-butanol + water binary system

Fig. 1 shows the calculated results of liquid–liquid equilibria for-butanol + water binary system by CDSAP model. The calculatedesults are in good agreement with the experimental data in theemperature range from 20 to 120 ◦C.

3.4. Liquid–liquid equilibria for ternary systems

Eqs. (9)–(11) and the coefficients in Table 3 were used forthe calculations of liquid–liquid equilibria for ternary systemsby CDSAP model. In the case of UNIQUAC model, the binaryparameters for 1-butanol + water system were obtained to fit theliquid–liquid equilibrium data for 1-butanol + water binary system,because UNIQUAC model has only two fitting parameters. In thecase of NRTL model, three binary parameters (g12–g22, g21–g11,

Fig. 1. Liquid–liquid equilibria for 1-butanol (1) + water (2) binary system. �: Exper-imental data [34]; ( ): CDSAP. The parameter values are listed in Table 3.

Y. Iwai, I. Taniguchi / Fluid Phase Equilibria 362 (2014) 35– 40 39

Table 6Calculated results of liquid–liquid equilibria for ternary systems.

System t (◦C) Refs. of data Errora

Comp. (1) Comp. (2) Comp. (3) CDSAPb NRTLc UNIQUACc

1-Butanol Methanol Water 25 [35] 2.609 3.092 4.61160 [36] 2.715 4.911 5.27275 [36] 2.359 4.305 4.141

1-Butanol 2-Propanol Water 25 [37] 2.358 2.570 3.02060 [37] 2.413 4.186 5.49180 [38] 1.888 4.381 4.496

1-Butanol Acetone Water 20 [58] 6.273 8.797 7.57840 [58] 4.372 6.702 5.882

Averaged 3.123 4.868 5.061

a Error = 100

√√√√ N∑k=1

2∑j=1

3∑i=1

(Xj,exp

i,k−Xj,calc

i,k

)2

5N where k is the tie line, j is the phase, i is the component, and N is the number of tie lines. The mole fractions of component 2

in 1-butanol rich phase were fixed to those of the experimental values.b Parameters are obtained by temperature dependent equations as shown in Table 3c Parameters determined at each temperature as shown in Table 4 were used except for the following cases. The parameters determined at 25 ◦C were used for comp.

(1) + comp. (2) at all temperatures. The parameters determined at 80 and 35 ◦C were used for the calculations of methanol (2) + water (3) and acetone (2) + water (3) at 75a ◦

dapcusttisdemos(lptTbt

Ft

N

nd 40 C, respectively.d Average = summation of errors/number of systems.

ata for 1-butanol + methanol + water, 1-butanol + acetone + water,nd 1-butanol + 2-propanol + water ternary systems at each tem-erature. The parameters are listed in Table 4. The parameters ofonstituting binary vapor–liquid equilibria listed in Table 4 weresed for the calculations of liquid–liquid equilibria for ternaryystems for both NRTL and UNIQUAC models. Table 6 showshe summary of calculated results of liquid–liquid equilibria forernary systems. The examples of the calculated results are shownn Figs. 2–5. The calculated results by NRTL model become onlylightly better compare with those of UNIQUAC model even if theata of liquid–liquid equilibria for ternary systems are consid-red to determine the parameters for 1-butanol + water by NRTLodel. The calculated results by NRTL and UNIQUAC models are

ver estimated for the liquid–liquid equilibrium regions. Fig. 3hows the calculated results in two-dimension graph for 1-butanol1) + methanol (2) + water (3) ternary system at 60 ◦C. The calcu-ated results by three models are almost the same in water richhase, however, the calculated results by CDSAP model are betterhan those by NRTL and UNIQUAC models in 1-butanol rich phase.

he calculated results by CDSAP model are much better than thosey NRTL and UNIQUAC models for all ternary systems studied inhis work.

ig. 2. Liquid–liquid equilibria for 1-butanol (1) + methanol (2) + water (3)ernary system at 60 ◦C. : Experimental data [36];

:CDSAP; : UNIQUAC; Δ Δ:RTL. The parameter values are listed in Tables 3 and 4.

Fig. 3. Liquid–liquid equilibria for 1-butanol (1) + methanol (2) + water (3) ternarysystem at 60 ◦C. : Experimental data [36]; : CDSAP; : UNI-QUAC; : NRTL. The parameter values are listed in Tables 3 and 4.Superscripts I and II mean 1-butanol and water rich phases, respectively.

40 Y. Iwai, I. Taniguchi / Fluid Phase E

Fig. 4. Liquid–liquid equilibria for 1-butanol(1) + 2-propanol(2) + water(3) ternarysystem at 80 ◦C. : Experimental data [38]: :

CDSAP; : UNIQUAC; Δ Δ: NRTL. The parametervalues are listed in Tables 3 and 4.

Fig. 5. Liquid–liquid equilibria for 1-butanol(1) + acetone(2) + water(3)ternary system at 40 ◦C. : Experimental data [58];

N

4

mbcefcbbtelspla

R

[[[

[[

[[[[

[

[[[[[

[[[

[[

[[[[[

[

[[[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

:CDSAP; : UNIQUAC; Δ Δ:RTL. The parameter values are listed in Tables 3 and 4.

. Conclusion

The concentration dependent surface area parameter (CDSAP)odel was applied to correlate the vapor–liquid equilibria for

inary systems and liquid–liquid equilibria for ternary systemsontaining 1-butanol + water. The parameters in CDSAP model arexplained by functions of temperature. The calculations can be per-ormed in the wide range of temperature by CDSAP model. Thealculated results by CDSAP model are almost the same as thosey NRTL and UNIQUAC models for the vapor–liquid equilibria ofinary systems. The calculated results by CDSAP model are bet-er than those by NRTL and UNIQUAC models for the liquid–liquidquilibria of ternary systems containing 1-butanol + water. Theiquid–liquid equilibria for 1-butanol + acetone + water ternaryystem are well calculated by CDSAP model with the binaryarameters of 1-butanol + water which are determined with the

iquid–liquid equilibrium data for 1-butanol + methanol + waternd 1-butanol + 2-propanol + water ternary systems.

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