Calculation of multiphase equilibria by a group contribution equation of state

Fluid Phase Equilibria, 52 ( 1989) 75-82 ElsevierSciencePublishersB.V.,Amsterdam-PrintedinTheNetherlands

CALCULATION BY

A GROUP

OF MULTIPHASE

CONTRIBUTION

EQUILIBRIA

EQUATION

SHING-YI SUEN and YAN-PING CHEN* Department of Chemical Engineering, Taipei, Taiwan, Republic of China

15

OF STATE

National

Taiwan

University,

and DAVID SHAN-HILL WONG Department of Chemical Engineering, National Tsing University, Hsinchu, Taiwan, Republic of China

Hua

ABSTRACT This work presented a generalized correlation for multiphase equilibrium computation. The Patel-Teja equation of state (EOS) was used, the mixture parameters of which were evaluated by equating the excess Gibbs free energy at an effective high pressure condition of 1000 atm to that predicted by the UNIFAC group contribution liquid model. Vapor-liquid (VLE), liquidliquid (LLE) and vapor-liquid-liquid (VLLE) calculation results over a wide pressure range were presented. Accuracy of predictions was found to be comparable to existing models.

INTRODUCTION Modeling of multiphase equilibrium had been important to many engineering calculations. Often excess Gibbs free energy methods were used for low pressure fugacity calculations and EOS methods were used at high pressures. Huron and Vidal (1979) proposed a method for using nonideal excess Gibbs free energy models to calculate EOS parameters for mixture fluids. Gupte et al. (1986) extended the method to VLE calculations for nonideal mixtures using the UNIFAC excess Gibbs free energy model. Sheng et al. (1989) modified this method and the combinatorial part of the UNIFAC model for calculating VLE of hydrocarbon mixtures containing light components at high pressures. The objective of this work was to investigate the applicability of this approach for LLE and VLLE calculations.

THEORY In this was used:

study,

the

EOS

suggested

by

Pate1

P = RT/(_V-b) - a/[y2+(b+c)y-bc] 0378-3812/89/%03.50

0 1989 Elsevier Science PublishersB.V.

and

Teja

(1982)

(I)

76

To calculate the mixture, van der Waals investigators: which mixtures.

fugacity coefficient of a component in a mixing model was commonly employed by most was not very satisfactory for nonideal

The relation between parameter am of the Patel-Teja equation of state and excess Gibbs free energy of a mixture fluid can be expressed as:

GE = PV -m -

C Z XiPvi i=l

-

RTln[P(ym-b,)/RT]

Wm+ P,+c,)/2+

C + Z XiRTln[P(vi-bi)/RT] i=l

(b,c,+

(b,+c,)

[bmcm+(bm+cm)2/4](-1/2)

t

2/4) (1’2)I)

-

C z xiailn[[~i+(bi+Ci)/2-(biCi+(bi+Ci)2/4)(l'2)]/ i=l

[biCi+(bi+Ci)2/4](-l'2)/2

(2)

At high pressures, the excess Gibbs free energy of mixture fluids would resemble that of a liquid. If we assumed that the excess volume of the system were negligible, the effect of pressure on excess Gibbs free energy of a liquid solution would be minimal. We could therefore choose to calculate the value of am by equating the excess Gibbs free energy at a high pressure limit to that calculated from a liquid solution by the UNIFAC model: even though this generalized correlation was obtained using low pressure data only:

~EE,S(P=lOOO

The following used:

atm, T, x)

mixing

rules

= GEUNIFAC(T,

x)

for the parameters

(3) b, and

cm

were

C

b, =

I: Xibi i=l

(4)

C cm =

C

i=l

XiCi

(5)

in order to be as consistent volume for the liquid solution

with the assumption at high pressure.

of

no

excess

The activity coefficients at infinite dilution for four binary liquid mixtures were presented in Figure 1 calculated by our method over a wide pressure range. At pressures less than 1000 atm, the calculated activity coefficients were nearly constants. Pressure effects were observed only at higher pressures. greater than IO4 atm, On the other hand, at pressures it was doubtful that the EOS could represent accurate PVT behavior. Extrapolation of the UNIFAC model to pressure of such magnitude would also be doubtful. It was thus concluded that an effective high pressure limit in this study could be defined at 1000 atm, within which, for practical purposes, our assumption that the pressure effect on the excess Gibbs free energy of liquid solutions was negligible, was valid.

10. T = 298.15 K 8-

6-

-2

-

C

O-

-2

I

d

1

I 10

I

I f02

lo”

10’

P (arm)

Figure

1

Plot of Activity Coefficients at Infinite Dilution Calculated in a: 1-Pentanol(1) and Water(Z); b: This Study Against Pressure. c: Toluene(1) and Acetic Acid(Z) Methanol(l) and n-Octane(2); d: Propane(l) and n-Eicosane(2)

RESULTS AND DISCUSSIONS The particular form and parametric values of the modified UNIFAC model proposed by Larsen et a1.(1987) were used in this study, since we would like to calculate VLE, LLE and VLLE. The thermodynamic model presented in the above section would be denoted as the EOS+UNIFAC model in the following discussions, to be distinguished from the UNIFAC activity coefficient model and the EOS+kij model which employed the van der Waals mixing rule:

c

am=

C

c Z xixj

11/2k. 13.

(aiaj

i=lj=l

(6)

instead of equation (3). Figure 2 illustrated T-x-y diagram of 1-pentanol and water of the at 1 atm. Calculated result of the temperature heterogeneous VLLE azeotrope of 365K was found to be close to the There were appreciable experimental value of about 369K. deviations in composition of the 1-pentanol rich phase of LLE with respect to the experimental data for both EOS+UNIFAC and the UNIFAC methods. Figure 3 presented y-x diagram of the binary The predicted mixture of methanol and n-hexane at 1 atm. temperature of the heterogeneous VLLE azeotrope of 322K was also close to the experimental value of about 323K. In Figure 4 were the LLE calculation results of the binary mixture of methanol and n-octane at 1 atm. Predictions using modified UNIFAC were also

_ 360 Y + 340 l

320

2801

0

. 0

’ 0.2

Expt.

0 Expt. ___ Il.IIC

I 0.4

I 0.6

I 0.8

1.0

x2 or !h

Figure 2 Calculated T-x-y Diagram of 1-Pentanol(1) and Water(a) at P=l atm Experimental Data were Taken from Gmehling et a1.(1981) and Sorensen and Arlt(1979)

79

0

0.2

0.4

0.6

0.8

1.0

XI

Calculated y-x Diagram Experimental Data Were

Figure 3 of Methanol(l) and n-Hexane(2) at P=l atm Taken from Gmehling et a1.(1977)

340

330

Y I-

320

310

300 290

I

I

I

02

0.4

0.6

I

0.8

5, Figure Calculated T-x Diagram P=l atm. Experimental a1.(1987)

4

of LLE of Methanol( Data are Obtained

,) and n-Octane(2) from Higashiuchi

at et

80

TABLE

1

Comparison of Calculated Three-Phase Region Mole Fractions Methane(l)-n-Butane(2)-Water(3) Ternary Mixture at 310.93K

P

EOS+UNIFAC

atm

Cl

C4

EOS+kij W

Cl

C4

Experiment W

Cl

C4

of

Data W

13.75

V LH LW

.6991 .0738 .0014

.2964 -9260 .0009

.0045 .0002 -9977

.6963 .0534 .oooo

-2982 -0055 .9446 -0020 .oooo 1.0000

-7810 .0694 .oooo

.2139 .0051 .9299 .0007 .oooo 1.0000

32.25

V LH LW

.8388 -1968 .0037

.1590 .8030 .0006

.0002 .0002 -9957

.8362 -1475 .OOOO

-1609 .0029 .8504 .0021 .OOOO 1.0000

.8450 .1478 -0007

.1527 .8514 .OOOl

0023 :0008 .9992

V LH LW

-8610 .2619 -0049

.1373 .7379 -0005

.0017 .0002 .9946

.8589 .2000 .OOOl

. 1392

.0019 0003 :9999

.8694 . 2075

. 1288

42.97

.0006

.7917 . 0001

.0018 0008 :9993

V .8747 LH .4846 LW -0069

.1239 .5148 .OOOl

.0014 .0006 .9930

.8729 -3968 .OOOl

. 1250 6007

. 0021 . 0025

: 0000

-9999

.8781 -3857 -0013

. 1208 . 6135 . 0001

.0011 .0008 -9986

V LH LW

.1324 -4482 .0002

.OOlO .0003 .9910

.8648 .4570 .OOOl

-1331 -5404

. 0021 . 0026 .9999

.1419 .5490 .OOOl

. 0010

.oooo

.8571 4501 :0015

83.02

95.67

.8666 .5515 .0088

.7997

.oooo

.0009 .9984

V: Vapor phase, LH: Hydrocarbon-rich Liquid Phase, LW: Water-rich Liquid Phase, W:Agueous Phase, Data from McKetta and Katz (1948), Binary ki' parameters used in EOS+k.. method were: k12=1.0, k 13=l.035 and k23=0.629, obtained by da+t% regression

illustrated. The EOS+UNIFAC model was found to be eguilvalent or slightly better than the modified UNIFAC method in these cases. Partial miscibility behavior of Methane-n-Butane-Water VLLE results of ternary system was also examined in this study. calculated by EOS+UNIFAC and the equilibrium mole fractions Direct EOS+k* * were compared with experimental data in Table 1. use 0 r the activity coefficient method was difficult due to the existence of the light component methane. The vapor compositions comparable in accuracy. The predicted by the two methods were would be EOS+k*. method predicted that the hydrocarbons The EOS+UNIFAC neglig ‘1 ble in the water-rich liquid phase. method, however, indicated that certain amount of alkanes should which was qualitatively be found in the water-rich phase, consistent with but quantitatively greater than the experimental data. Deviations from data of composition of the hydrocarbonThese rich liquid phase were found for the EOS+UNIFAC model. deviations could be improved by modifying the combinatorial part of the UNIFAC model.

81 CONCLUSIONS This study demonstrated that it was possible to calculate LLE, VLLE as well as VLE by combining the Patel-Teja equation of state and the UNIFAC model. Accuracy of predictions of VLE, LLE, and VLLE of this model was comparable to the existing UNIFAC activity coefficient model. It was also applicable at both low and high pressures. However, further research would be necessary to obtain a quantitatively useful generalized correlation.

NOMENCLATURE a,b,c C G P T P X

Equation of state parameters Number of species Molar excess Gibbs free energy Pressure (atm) Temperature (K) Molar Volume (liter) Mole fraction

Subscript i,j m

Components i and j, respectively Mixture property

Superscript E

Excess properties

REFERENCES Gmehling, J., Onken, U., Arlt, W., Grenzheuser, P., Weidlich, U. and Kolbe, B 1981. Vapor-Liquid Equilibrium Data Collection, part la, DE&&A, Frankfurt Gmehling, J., Onken, U., Arlt, W., Grenzheuser, P., Weidlich, U. and Kolbe, B., 1977. Vapor-Liquid Equilibrium Data Collection, part 2a, DECHEMA, Frankfurt Gupte, P. A., Rasmussen, P. and Fredenslund, A., 1986. Ind. Eng. Chem. Fundam., 25:636-645 Higashiuchi, H., Sakuragi,_Y.,Iwai, Y., Arai, Y. and Nagatani, M, 1987. Fluid Phase Eguilibria,36:35-47 Huron, M. J. and Vidal, J., 1979. Fluid Phase Equilibria, 3:255271 Larsen, B. L., Rasmussen, P. and Fredenslund, A., 1987. Ind. Eng. Chem. Res., 26:2274-2286 McKetta, J. J. and Katz, D. L., 1948. Ind. Eng. Chem., 40:853-863 Patel, N. C. and Teja, A. S., 1982. Chem. Eng. Sci., 37:463-473

82 Sheng, Y.J., Chen, Y. P. and Wong, D. S. H., Fluid Phase E q u i l i b r i a 46:197-210.

1989.

Sorensen, J. M. and Arlt, W., 1979. L i q u i d - L i q u i d Data Collection, Part i, DECHEMA, Frankfurt

Equilib]