Equation of state mixing rules from GE models.

Fluid Phase Equilibria, 29 (1986) 485-494

485

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

of state

Equation

mixing

rules from GE models.

P.A. Gupte, P. Rasmussen, Instituttet

Aa. Fredenslund

for Kemiteknik,

Danmarks Tekniske

Hrajskole, DK-2800,

Lyngby,

Denmark

ABSTRACT This paper shows that combining Modified

the van der Waals equation of state with a

UNIFAC model enables accurate

group-interaction

parameter matrix

high-pressure

VLE predictions

based on low-pressure

using a

VLE data.

INTRODUCTION Equations

of state are widely used in phase equilibrium

are applicable cription

to all phases of interest and thereby ensure a consistent

of phase equilibria

at low as well as high pressures.

lopment of the PFGC equation the GC-EOS

calculations.

(Cunningham,

(Skjold-Jorgensen,

as a predictive

highly nonideal

1984, 1986) the application

tool was restricted

carbon systems). Although systems,

1974; Moshfeghian

the GC-EOS

to relatively

They des-

Until the deve-

et al., 1979) and

of equations of state

simple systems

(e.g., hydro-

is able to predict the phase-equilibria

it is at present confined

of

to a limited range of com-

pounds. Models for the excess Gibbs function, sent accurately

highly nonideal

e.g., UNIFAC (Fredenslund they are powerful for

low

systems.

on the other hand, are able to repreIn their group-contribution

et al., 1977) and ASOG (Kojima and Tochiji,

predictive

tools. Nevertheless,

pressure applications

(typically

versions 1979),

these models are only suited

up to 10 bar) inasmuch as the vapor

phase is assumed to obey the ideal gas law or the virial equation of statetruncated after the second virial coefficient. The possibility models

of combining

is highly attractive.

of extending

equations

of state and excess Gibbs function

The equation of state approach

the large matrix of group-interaction

0378-3812/86/$03.50

has the potential

parameters,

0 1986 Elsevier Science Publishers B.V.

based on low-

486 VLE data, to apply to high pressures.

pressure

high-pressure pressure

In this work we show how a modi-

with van der Waals equation

fied UNIFAC model combined

using group-interaction

VLE predictions

of state yields parameters

accurate

based on low

VLE data.

Combination

of GE and EOS methods

Although forthcoming

the van der Waals equation development

is equally

of state was selected

applicable

The van der Waals model for the molar Ar(T,v, x)/RT

for this work,the

to any equation

residual

Helmholtz

of state. function

is:

=-an(l- i) - a(T)/RTv

The expressions

(1)

for the excess Gibbs function

and the pressure

resulting

from Eq. (1) _ are: E '

'EOS

=- &n(l-b/v)

+ T xiRn(l-bi/vi)-

&

xiai + y F + i xiRnvi/v

E + .$,-

(2)

and (3) The basic principle

in combining

te the excess Gibbs function

equation

from the equation

of state and GE models

is to equa-

of state to that from the GE mo-

del. E %

E

E (4)

lEOS = +iT lGEmodel = %T This equation

is taken to be valid at all system pressures.

The GE model clould modified

be any group-contribution

model

like ASOG or UNIFAC or

UNIFAC.

The modified

UNIFAC method

that the group-interaction parameters

have been estimated

UNIFAC method

is similar to UNIFAC. Tha main difference

parameters

is described

have been made temperature

from VLE and heats of mixing

and parameters

is

dependent.

The

data. The modified

are given in Larsen

(1986) and Lar-

sen et al. (1986). It is equally

possible

We note that the excess perature

and composition

function

of pressure

Combining

to use a molecular

Gibbs function only, whereas

also. The pressure

Eqs. (2) and (4) yields:

model

such as UNIQUAC

from the GE model

that from the equation dependence

or NRTL.

is a function

of tem-

of state is a

of gE/RT is small.

487

x.a.

&=fCOMB+f~R;6f-f~

(5)

ii

where f

=

v/b

(6)

fi = vi/hi

(7)

and COMB

=

r;;lf_ [an

P(i;b)

[

Eq. 5 is a mixing

1

_

i

Xi&n [

‘(“Rsbi) 11

rule for the a/b parameter.

The practical

xing rule is complicated

by the fact that the RHS contains

v and the pure component

volumes

contribution

vi. The term

to the excess Gibbs function,

This contribution

does not contain

the mixture

'COMB' represents

stemming

the energy

use of this mi-

from the equation

parameter

volume

a combinatorial

a. The mixing

of state. rule, Eq.

( 5) is seen to contain the excess Gibbs function from the GE model. To analyse

the mixing

rule represented

ship to the infinite-pressure

mixing

der the limit of Eq. (5) as the pressure ven a detailed combinatorial

derivation expression,

As P -t m, v + b, vi + b.,,

+I?=

F

2

i

-

i

equation

of Vidal's

rule (Vidal, 1978; Huron and Vida1,1979).

to obtain the infinite pressure

this work. This rule is al-

limit given by Eq. (9). Note that

is at a hypothetical

relation

between

infinite

pressure

the excess Gibbs function

is given by Eq. (5). This equation

x.a.

can be rewritten

re-

and the

as follows:

E

11

mi

is

(9)

jTlp,,

state. A more general

c

in the

Eq. (8):

rule given by Eq. (9a) is used throughout

a/b parameter

&=

(1978) has gi-

f + 1, fi + 1 and COMB * 0. The resulting

in Eq. (9) the excess Gibbs function ference

Vidal

terms occuring

Pa)

Eq. (9) is a statement

so required

tends to infinity.

of the limits of the individual

(1978), we consi-

'ribi i

b=

The mixing

by Vidal

E

. . Lim p+_

by Eq. (5) and to show its relation-

rules developed

-

F '

'GEmode

(10)

488 where x.a.

c

11

-f!

iF

;&

+

%I

E - f(COMB)

ii

F=

GEmode

(11)

E '

'GEmode

Vidal's rule is recovered da1 (1979) and Soave

from Eq. (lo), since F + 1 as P + ~0. Huron and Vi-

(1984) have correlated

del for the excess Gibbs function. se equilibria function. unity

VLE using Eq. (9) with the NRTL mo-

Gupte and Daubert

(1985) have predicted

with Eq. (9),using the UNIFAC model to represent

Clearly,

the assumption

even at normal

Boutrouille

pha-

the excess Gibbs

in these efforts has been to set F eaual to

pressure conditions.

et al. (1985) have regressed

the value of F for several systems.

In each case, the excess Gibbs function in Eq. (10) was obtained from the UNIFAC model. A single value of F was obtained data set. The parameter temperature

for a given isothermal

or isobaric

F was for most systems found to be a strong function

and to depart significantly

from unity. These authors found tha:, F

was related to the v/b ratio of the mixture.

No direct function describing

relationship

(1985) has given an approximate

was obtained,

however. Mollerup

thod to obtain F at low reduced temperatures For the rigorous expression tion of temperature,

pressure,

of

this me-

(less than 0.7).

given by Eq. (ll), it is seen that F is a funcand composition.

Furthermore,

a pure fluid is vi/hi. For the van der Waals equation,

the value of F for

the ratio vi/hi can be

shown to be about 1.25 at the normal boiling point and 3.00 at the critical point. For the mixture, Gibbs function.

F is a more complex function

The value of F for mixtures

increase with temperature. for the mixture

depending

may typically

on the excess

be 1.25 at 330 K and

Since the vi/hi ratio is 3.00 at the critical

the value of F at the mixture

critical

point,

point would be expected

to be about the same. Table

I contains the optimum F values using Eq. (10) for the methanol-benze-

ne mixture.

A single value of F was obtained

and modified

I shows that the regressed decreases dependence

for each isotherm and the UNIFAC

UNIFAC models were used to obtain the excess Gibbs function.

Table

value of F for the UNIFAC model is 1.23 at 318 K and

to 0.98 at 493.15 K. This is contrary of F from Eq. 11 and is attributed

to the expected

temperature-

to the weak temperature

dependen-

ce of the UNIFAC model. We have obtained stems not containing

F for several systems with the UNIFAC model. For most sywater, F decreases

with temperature

benzene) and is close to unity at high temperatures.

(similar to methanol-

F may even decrease

below

AyxlOO

AP% P

JP?$

RMS

= /-'

Data from Gmehling

*

(1977).

nx

100

/ n x 100

and Onken

0.89

22.32 -56.89

493.15

1.71 1.32

13.01 -27.32 16.71 -40.28

1.54

2.42

473.15

433.15

453.15

6.65 -11.68 9.49 -18.20

413.15

- 7.03

4.80

393.15

7.47

5.45

3.07

0.98

1.03

1.04

1.13 1.10

3.14

1.15

1.22

1.23

F

VLE data.

2.45

2.02

2.00

2.24

3.08 - 4.12 1.81

4.81

2.90

0.31 - 0.60

(abs)

RMS

RMS

UNIFAC

373.15

P (atm)

of F from experimental

- Benzene*

I

318.15

TK

Regression

Methanol

Table

9.17 10.84

3.27

2-28

1.98

1.68

1.56 4.38

1.43 1.33

2.17

1.43

1.33

1.36

l-34

1.24

F

3-39

1.64

0.37 0.78

1.19

8.11

labs)

Aq’x100

RMS

UNIFAC

0.90

2.65

AP% P

RMS

MODIFIED

490 unity for some systems. ted good results more,

This explains

for some systems

their low temperature

bias. This happens

bubble pressure

because

the optimum

ratures

and setting

systems

(see Eq. 9). For systems

gressed

F is approximately

The regressed increases

why Gupte and Daubert

at high temperatures

temperatures.

We have studied systems

dependence in addition

to obtain

gE

with temperature. UNIFAC model

are extremely

substitution

very poor predictions

UNIFAC model.

This is directly

sensi-

of the modified-

of F with respect

of the modified-UNIFAC

at high

to temperature

cases, the value

In all

attributable

model obtained

to the correct by fitting

excess

'to VLE.

F-UNIWAALS

A new method pressure

the behaviour

using the modified

enthalpies

Method

that direct

rule) will yield

with temperature.

temperature

for positive

with the modified

from Eq. (11). Since VLE predictions

in Eq. 9 (Vidal's

negative

water, we have found that the re-

1.6 and does not vary significantly

UNIFAC model

for several

the a parameter

containing

value of F for methanol-benzene

as expected

reveal a consistent

value of F is about 1.25 at these tempe-

F to unity underpredicts

tive to the value of F, it is evident

of F increases

results

(1985) have repor-

with Eq. 9 (F=l). Further-

Equation

has been developed

and composition.

of state to calculate

This is carried

F as a function

out by simultaneously

and (10) (along with Eq. (9a) for the b parameter).

of temperature, solving

F is merely

Eqs. (3)

an intermediate

variable

and is not needed explicitly.

For the liquid phase, the simultaneous

solution

of Eqs. (3) and (10) directly

yields

the mixture

compressibility

factor

Z. For the vapor phase the ratios f and fi are calculated

at liquid-phase

con-

ditions

to yield a/RTb. This value of a/RTb is then used with the van der Waals

equation

(Eq.3) to obtain

dure requires

the correct

the calculation

Supercritical extrapolation,

compounds

are descriged

the product of F and

(11) for subcritical by the modified

conditions.

NRTL equation

then used at conditions pound. The details

vapor compressibility

factor.

The proce-

of liquid roots for all pure components.

g-

by an extrapolation is calculated

The FFoduct

(Prausnitz

from equations

temperature

procedure

In this (10) and

is then represented

g

et al., 1980).Rihe

beyond the critical

of the calculation

of F and

scheme.

NRTL equation

of the lightest

are explained

is

com-

by Gupte et al.

(1986). Table

II contains

and modified

sed on low-pressure progressively

the prediction

UNIFAC models

VLE data. As expected,

deteriorate

FAC model given

results

in Table

for the new method

with their original

with temperature.

II are exellent.

group-interaction

the results

using the UNIFAC parameters

The results with the modified Figures

ba-

using the UNIFAC model

1 through

UNI-

3 show the results

P (atm)

-

-

16.71 -40.28 22.32

473.15 493.15

*

-

Data from Gmehling

and Onken

-56.89

(1977).

1.69

3.76

2.23 -

23.61

6.78

13.01 -27.32

2.12

1'62

5.15 5.64

453.15

-11.68 17.92

9.49 -18.20

433.15

14'95

6.65

413.15

1.09 2.01

3.79

6.24

3.08 - 4.12 4.80

373.15 - 7.03

1.79

/ I

i

Modified

2.59

1.86

2.37

tabs)

RMS Ayxl 00

RMS AP/P%

UNIFAC

10.48

T

parameters

data only

UNIFY

393.15

0'31 - 0.60

1

318.15

TK

and modified

F

- benzene*

to calculate

based on low-oressure

and UNIFAC

II

for methanol

using new approach

VLE predictions

Table

3.50

3-10

1.67

1.34

o-49

O-86

O-72

1.43

UNIFAC

492

Oo-0 MOLE

FRACTION,

X(l),Y(l)

Fig. 1. Comparison of UNIWAALS-EOS predictions with experimental data for methanol-benzene. Data at 318.15 K and 373.15 K.

OOOMO

MOLE

Fig. 2. Data at 393.15

FRACTION,

X(l),

Y(1)

of UNIWAALS-EOS predictions with experimental data for methanol-benzene. K, 413.15 K and 433.15 k.

1noeoN40,‘ MOLE

FRACTION,

’ OH X(l),

I

Y(1)

Fig. 3. Comparison of UNIWAALS-EOS predictions with experimental data for methanol-benzene. Data at 453.15 K,473.15 K and 493.15 K.

493 obtained.

The modified

to establish

UNIFAC model was used in conjuction

a new "UNIWAALS"

was tested for a large yield excellent

Ideal gas method

vapor phase mole-fraction absolute

deviation

quantities

superior

from UNIWAALS

At higher

pressures

to about 4% avg and 0.02 absolute

al. (1986) have obtained

superior

of state

The UNIWAALS

of the modified predictions

(less than 10 atm) the bubble

predictions

respectively.

predictions

and yields

at low pressures

equation

(Gupte et al., 1986) and found to

at both high and low pressures.

the high quality

at low pressures

sures. Typically,

with the new approach

of state. The UNIWAALS

number of systems

predictions

tion of state preserves

equation

at high prespressure

and

are about 2% avg and 0.01 the model

deviation

predictions

equa-

UNIFAC-

predicts

respectively.

from UNIWAALS

these

Gupte et

over GC-EOS.

ACKNOWLEDGEMENT The authors Gmehling

thank Jsrgen Mollerup

of the University

Bank. Thanks

for financial

nical and Scientific

of Dortmund support

Research

for inspiring

has kindly provided

are directed

(STVF).

Helmholtz

function

a

energy

b

size parameter

f

v/b ratio

parameter

F

adjustable

gE

molar

parameter

excess Gibbs function

n

number of moles

P

pressure

R

gas constant

T

temperature

V

molar volume

X

mole fraction

Subscripts C

critical

value

EOS stemming

from an equation

GE

stemming

from an excess

i

component

r

reduced

cc

pertaining

of state

Gibbs function

model

number

property to the infinite

pressure

state

Jiirgen

the Dortmund

to the Danish

LIST OF SYMBOLS A

discussions.'Dr.

Council

Data

for Tech-

494 Superscripts E res

excess property residual

property

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A., Moysan,

J., Paradowski,

red at the 8th Seminar Cunningham,

A., Gmehling,

Using UNIFAC.

Elsevier,

P., 1977. Vapor-Liquid

Equilibria

New York.

Data Series,

P.A. and Daubert,

Presen-

Trieste.

Provo, UT.

J. and Rasmussen,

3. and Onken, U., 1977. Vapor-Liquid

DECHEMA Chemistry Gupte,

H., and Vidal, J., May 1985.

Thermodynamics

J.R., 1974. M.Sc. Thesis,

Fredenslund,

Gmehling,

Applied

T.E.,

Equilibrium

Data Collection.

Frankfurt.

1985. Fluid Phase Equilibria,

accepted

for publi-

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P. and Fredenslund,

Aa., 1986. Ind.Eng.Chem.

Fundam.,

for publication.

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for Kemiteknik,

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Univer-

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J.M., Anderson,

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K. 1979. Prediction

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Equilibria

by the