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Fluid Phase Equilibria 309 (2011) 97– 101

Contents lists available at ScienceDirect

Fluid Phase Equilibria

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

rediction of vapor–liquid equilibrium data of the systemTBE + methanol + ethanol by PRSV2 EOS

.R. Moradi ∗, M. Rahmanzadehhemical Engineering Department, Razi University, Kermanshah, Iran

r t i c l e i n f o

rticle history:eceived 12 March 2011eceived in revised form 11 June 2011ccepted 6 July 2011

a b s t r a c t

The Stryjek and Vera (1986) [9] modification of Peng–Robinson (PRSV2) equation of state has beenapplied for modeling vapor–liquid equilibrium of the systems MTBE + methanol, MTBE + ethanol andmethanol + ethanol. Binary interaction parameters for mixing rules have been estimated by using exper-imental data at the atmospheric pressure. The calculated binary interaction parameters were used for

vailable online 14 July 2011

eywords:RSV2quation of stateapor–liquid equilibriumixing rules

predicting azeotropic behavior at high pressure and also for isobaric equilibrium points which showedan excellent agreement with experimental data. In addition, estimated binary interaction parametersfor binary systems were used for ternary system (MTBE + methanol + ethanol). The predictions deviatedonly slightly from the experimental data. The results show PRSV2 can be used for VLE prediction of polarsystems.

© 2011 Elsevier B.V. All rights reserved.

. Introduction

Tertiary-alkyl ethers are low toxic and low polluting oxygenatedetrochemical compounds, used as an octane booster for lead-freer low-leaded gasoline and also increasingly valued as solvents ands chemical reactants [1,2].

In recent years, increasing use of ethers as oxygenated addi-ives for gasoline has necessitated large increases in worldwidether production, with important implications for the hydrocar-on processing industry. The ether most widely added to gasoline

s 2-methoxy-2-methylpropane (more commonly MTBE, methylert-butyl ether). These ethers are used in combination with

ethanol and ethanol co-solvents as octane-enhancing agentsnd anti-pollutants in gasoline blends [1]. The thermodynam-cs of ether + alcohol mixtures are thus of some interest [3]. Inhe literature vapor–liquid equilibrium (VLE) measurements for

TBE + methanol, MTBE + ethanol and methanol + ethanol mix-ures were correlated using the Wilson [23], NRTL [24], and

NIQUAC [25] models for the liquid phase and the equation of ideal

tate for the vapor phase [4–8]. The equation of the ideal state forhe vapor phase in high pressure cannot predict the behavior of theapor phase accurately.

∗ Corresponding author. Tel.: +98 9123895988; fax: +98 8314274542.E-mail addresses: gmoradi@razi.ac.ir, moradi m@yahoo.com (G.R. Moradi).

378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2011.07.003

For system of strongly non-ideal at high pressures, the ideal con-ditions for vapor phase or simple cubic equations of state (EOS)cannot be applied for correlating vapor liquid equilibrium data. But,using more appropriate and somewhat complicated mixing rulesfor different types of mixture in simple cubic EOS provide high capa-bility for these EOS in vapor–liquid equilibrium calculations. So,‘fugacity coefficient–fugacity coefficient’ approach for vapor andliquid could be used instead of the ‘activity coefficient–fugacitycoefficient’ approach, i.e., using an excess Gibbs energy model forthe liquid phase and an equation of state for the vapor phase.

A modification of the Peng–Robinson equation of state [12] hasbeen discussed by Stryjek and Vera [9]. This modification, called thePRSV equation of state, contains one adjustable parameter per purecompound and represents vapor pressure data with high accuracy.Stryjek and Vera have used their modified equation to correlatevapor–liquid equilibrium data of hydrochloric acid solutions [10].Vera has added two additional parameters for higher accuracy inPVT prediction for pure component and has called it PRSV2 [11].

In this article the behavior of the vapor–liquid equilibrium ofMTBE + methanol, MTBE + ethanol and methanol + ethanol systemshas been modeled by the PRSV2 EOS. Then, by using the six obtainedbinary interaction coefficients and suitable mixing rules the poten-tial of the PRSV2 equation of state for correlating the vapor–liquid

equilibrium of ternary system (MTBE + methanol + ethanol) hasbeen illustrated. The PRSV2 equation, using six adjustable param-eters, gives an excellent representation of this system containing astrong polar component.

9 uid Phase Equilibria 309 (2011) 97– 101

2

2

P

P

w

a

a

b

w

˛

ta

k

w

k

Tb

otip

2

b

a

a

m

l

w

a

fitf

Table 1Binary parameters for the system MTBE(1) + methanol(2) + ethanol(3).

−4 −4

8 G.R. Moradi, M. Rahmanzadeh / Fl

. Theory

.1. The PRSV2 equation of state

The PRSV2 equation of state retains the basic form of theeng–Robinson equation [12]

= RT

v − b

a

v2 + 2bv − b2(1)

ith

=(

0.45724R2T2c

Pc

) (2)

nd

= 0.0778RTc

Pc(3)

here

= [1 + k(1 − T0.5R )]

2(4)

Peng and Robinson considered k to be a function of the acen-ric factor only. The PRSV2 equation considers k to be function ofcentric factor and reduced temperature of the form:

= k0 + [k1 + k2(k3 − TR)(1 − T0.5R ) × (1 + T0.5

R )(0.7 − TR)] (5)

here k0 is given by

0 = 0.378893 + 1.4897153ω − 0.1731848ω2 + 0.0196544ω3 (6)

And k1, k2 and k3 are pure compound adjustable parameters.he values of k1, k2 and k3 for MTBE, methanol and ethanol haveeen reported by Vera [11,13].

As observed before, the Peng–Robinson equation is applicablenly for hydrocarbons and slightly polar compounds at reducedemperatures of about 0.7 and above, while the PRSV2 equations applicable for all compounds independent of their size, shape,olarity or degree association [11].

.2. The mixing rules for vapor–liquid equilibrium calculations

In this work we use the conventional mixing rules,

=∑

i

xibi (7)

nd

=∑

i

∑j

xixjaij (8)

The fugacity coefficient of a component i in a multi componentixture takes the form [11]:

n ϕi = bi

b(z − 1) − ln(z − B) − A

2√

2B

×(

ai

a+ 1 − bi

b

)ln

z +(

1 +√

2)

B

z +(

1 +√

2)

B(9)

ith

¯ i =(

∂na

∂ni

)nj /= i

(10)

For convenience, ai has been left unspecified in Eq. (8) since itsnal form depends on the assumption made regarding the crosserm aij. The following expression for the cross term aij gives satis-actory results.

k12 = 6.385 × 10 T (K) − 0.175 k21 = 5.479 × 10 T (K) − 0.235k13 = 5.803 × 10−4T (K) − 0.149 k31 = 4.17 × 10−4T (K) − 0.152k23 = 9.446 × 10−5T (K) − 0.03 k32 = −0.00638

For a binary mixture of components i and j, Margules-type two-binary-parameter has the following form [11],

aij(aiiajj)0.5(1 − xikij − xjkji) (11)

where aij = aji, but kji /= kji.Using Eqs. (9)–(11) as mixing rules, the fugacity coefficient for

component i in a binary mixture of components i and j, takes theform

ln ϕi = bi

b(z − 1) − ln(z − B) − A

2√

2B

×[

2a

(xiaii + xjaij + xix

2j

√aiiaji(kji − kji)

)−b1

bln

z+(

1+√

2)

B

z+(

1+√

2)

B

]

(12)

In addition, for component i in a ternary mixture including i, jand k, fugacity coefficient has been obtained as follows

ln ϕib

bi(z − 1) − ln(z − B) − A

2√

2B

{2a

[xiaii + xjaij + xkaik

+ xjxk

√ajjakk (xjkjk + xkkkj) + xixj

√aiiajj(xikij + xjkji − kij)

+ xixk

√aiiakk(xikik + xkkki − kik)

]− bi

b

}ln

z +(

1 +√

2)

B

z +(

1 −√

2)

B(13)

As Eq. (13) depicts for binary system there are two interactionparameters and for ternary system there are six interaction param-eters in the fugacity coefficient equation exists. Eq. (13) reduced toEq. (12) when xk = 0.

3. Results and discussion

3.1. Estimation of binary interaction parameters

For vapor and liquid phases in equilibrium, the criterion of equi-librium is as follows:

yi = xiϕLi

ϕvi

(14)

where xi and yi are the mole fraction of component i in the liquidand vapor phases respectively.

The experimental isobaric VLE compositions ofMTBE + methanol, MTBE + ethanol and methanol + ethanol sys-tems at atmospheric pressure were extracted from literature[14,15], then by using these experimental data and the methodof algorithm genetic the best fitted interaction parameters havebeen found in different temperatures in such a way that theminimum deviations from experimental data have been raised.In the next step, these interaction parameters have been fittedin terms of temperature, the results are shown in Table 1. Theseexpressions for binary interaction parameters that are functionof temperature were used in Eq. (11) for calculating aij. All theexperimental VLE data were thermodynamically consistent by

the PRSV2 EOS as have been shown in Fig. 1. In this figure theexperimental isobaric VLE compositions for MTBE + methanol,MTBE + ethanol and methanol + ethanol systems have been com-pared with the predicted values of PRSV2 EOS with the obtained

G.R. Moradi, M. Rahmanzadeh / Fluid Phase Equilibria 309 (2011) 97– 101 99

F anol al

iMt

hwM

3

3

wVr

Fp

ig. 1. Isobaric VLE data at 101.33 kPa for the systems MTBE + methanol, MTBE + ethiterature experimental data ([14,15]) and lines to the model results.

nteraction parameters which showed negligible error [14,15]. TheTBE + methanol system has a minimum boiling azeotrope, while

he other systems do not show azeotropic behavior.The non-ideality of MTBE + alcohol is greater than alco-

ol + alcohol, also the non-ideality of MTBE + alcohol was decreasedith increasing carbon number of alcohols where is theTBE + methanol system shows azeotropic behavior.

.2. Model evaluation

.2.1. VLE compositions of MTBE + methanol systems

For evaluation of the model, the predicted results of model

ere compared with the isothermal and isobaric experimentalLE compositions of MTBE + methanol systems at 298 K and 50 kPaespectively [17,18]. The results are shown in Fig. 2.

ig. 2. (a) Isothermal VLE data at 298.15 K for the system MTBE + methanol and (b) isobhase). Symbols correspond to the literature experimental data ([17,18]) and lines to the

nd methanol + ethanol (x: liquid phase, y: vapor phase). Symbols correspond to the

In addition, azeotropic behavior in the system ofMTBE + methanol at the different pressure (115–1000 kPa)has been investigated by using the PRSV2 EOS with the fittedkij for mixing rule. The azeotropic temperature and compositionpredicted by the model was compared with the experimen-tal temperature and composition. The results are shown inTable 2. In all cases, azeotropic compositions were obtained byfixing the value x1 (mole fraction of MTBE) and calculating y1in such a way that absolute difference of x1 and y1 approacheszero. Then the corresponding temperature was computed. Theresults are shown in Table 2. As it illustrates, there is good

agreement between experimental and predicted values. Thepercent of average absolute relative deviation (AARD%) for pre-diction of azeotropic temperature is 0.07% and 1% for azeotropiccomposition.

aric VLE data at 50 kPa for the systems MTBE + methanol (x: liquid phase, y: vapor model results.

100 G.R. Moradi, M. Rahmanzadeh / Fluid Phase Equilibria 309 (2011) 97– 101

Table 2Predicted azeotropic points for the system MTBE + methanol using PRSV2 EOS.

Pressure/kPa x1 (model) x1 (experimental) T/K (model) T/K (experimental) Ref.

115.85 0.6698 0.681 328.22 328.15 [18]138.09 0.6517 0.662 333.28 333.15 [19]162.23 0.6347 0.637 338.06 338.15 [18]164.4 0.6333 0.639 338.46 338.15 [20]354.6 0.5464 0.548 363.34 363.54 [21]500 0.5047 0.496 375.67 374.93 [22]

1000 0.4160 0.412 403.11 403.42 [22]

F mposio ntal d

3

tGbttctc

4

rmioivppbf

ig. 3. Comparison of experimental and predicted equilibrium temperatures and cobtained using the PRSV2 equation. Symbols correspond to the literature experime

.2.2. VLE for the system ethanol + methanol + MTBEIsobaric vapor–liquid equilibrium data at 101.32 kPa for the

ernary system MTBE + methanol + ethanol have been reported byonzalez et al. [16]. The binary interaction parameters for theinary systems which have been obtained in Section 3.1 were usedo predict the VLE of the ternary systems. In Fig. 3, the equilibriumemperature data obtained by using the PRSV2 EOS (Eq. (13)) wereompared with the experimental temperature data [16]; in addi-ion, the corresponding predicted and experimental vapor-phaseomposition data are compared in Fig. 3.

. Conclusion

In this study, behaviors of the binary vapor–liquid equilib-ium systems including MTBE + methanol, MTBE + ethanol andethanol + ethanol were considered using the PRSV2 EOS. Binary

nteraction parameters in the related mixing rules have beenbtained as a function of temperature using atmospheric exper-mental binary vapor–liquid equilibrium data. To investigatealidity of the model (PRSV2 EOS with estimated interaction

arameters) the equilibrium points at the 50 kPa and azeotropicoints between 115 and 1000 kPa for MTBE + methanol haveeen predicted and compared with experimental data. AARD%or prediction of azeotropic temperature is 0.07% and 1% for

tions for the system MTBE + methanol + ethanol at 101.32 kPa; the predictions wereata ([16]) and lines to the model results.

azeotrpic composition. The predicted results are in a good agree-ment with experimental values at an extensive range of pressure(50–1000 kPa).

Also, vapor–liquid equilibrium data at 101.32 kPa for ternarysystem MTBE + methanol + ethanol were satisfactorily predictedusing the PRSV2 EOS and corresponding binary interaction param-eters for constituent binary subsystem. The predictions deviatedonly slightly from the experimental data. The results showed thecapability of PRSV2 EOS for using in VLE calculation in polar sys-tems.

List of symbolsa parameter of the equation of stateA Pa/(RT)2

b parameter of the equation of stateB Pb/RTkij binary interaction parameterP total pressureR universal gas constantT absolute temperature

v molar volumexi liquid phase mole fraction of component iyi vapor phase mole fraction of component iz compressibility factor

uid Ph

G˛kkkϕω

SciR

R

[[[[[[

[[[

[

[[21] K. Fischer, S.J. Park, J. Gmehling, J. Phys.-Chem. Data 2 (1996) 135.[22] W. Yong, T. Anyu, S. Yuguang, Shiyou Huagong 18 (1989) 442.

G.R. Moradi, M. Rahmanzadeh / Fl

reek letters function of reduced temperature and acentric factor

function of reduced temperature and acentric factor0 function of acentric factor1, k2, k3 pure compound parameteri fugacity coefficient of component i

acentric factor

ubscripts at critical point, j components

Reduced variable

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31 (1999) 1231.

[[[

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