Fluid Phase Equilibria, 27 (1986) 289-308 Elsevier Science Publishers B.V., Amsterdam -Printed in The Netherlands

289

CORRELATION AND PREDICTION OF VLE AND LLE BY EMPIRICAL EOS

GHAtYNA KOLASINSKA

University of Ui’areaw, Department of Chemistry, Pasteura 1,

02-093 Warsaw (Poland)

ABSTRACT

This report presents a brief review of the most recent

C 1980 - 1985) advances in the field of empirical equations of

state for the correlation and prediction of fluid phase equilib-

ria. Topics included are the applications of the nonanalytical

and analytical (virisl- and van der \faals-type) equations of

state to description of PVT behaviour of pure fluids and their

mixtures (binary and multicomponent) , the use of equations of

state for ill-defined systems, parameter evaluation and computa-

tional techniques used. Refrences to the main sources of infor-

mation are given.

INTRODUCTION

Publications on equations of state are so numerous that, as

stated by R.C. Reid(1983), it is “a full time job just to mein-

tain familiarity with the new publications in this field”. In

1980/81 year alone, Reid(1983) counted 885 papers concerning the

use of EOS. In the limited time available to compile this report

the author has not had the time to count the number of publica-

tions in the last years. However. it is easy to predict that any

linear extrapolation based on the data of previous years will

grossly underestimate the number of papers published this year

on the subject.

Considering that the book on EOS published by Chao and Robin-

son (1979) covers most of the advances of the previous decade, in

this report the effort will be concentrated on the progress made

from 1980 and mostly during 1985 on the use of empirical EOS for

the correlation and prediction of fluid phase equilibria. Such

a report is obviously biased by the personal judgement of the

0378-3812/86/$03.50 0 1986 Eisevier Science Publishers B.V.

290

author. Some important contributions may have been left out of

consideration either because they were not available to the au-

thor or because they were considered to be beyond the scope of

this review.

Depending on the form selected to express the relationship

between EOS variables, EOS may be arbitrary classified in the

following types:

1. nonanalytical

2. analytical

a. inspired on a virial expansion

b. inspired on the two-term van der Waals form.

Even when cubic equations of state, pertaining to group 2b,seem

to be the most attractive due to their simplicity, low computa-

tional costs and reliability, we will also discuss briefly some

new achievements in the field of nonanalytical and virial-type

EOS.

NONANALYTICAL EOS

A3 stated by Levelt Sengers et a1.(1976): “analytical EOS

yield a critical isotherm that is asymptotically of the third or

fifth degree, a quadratic or quartic coexistance curve, a fini-

te constant volume specific heat Cv in the one phase region and

an analytical vapour pressure curve. On the other hand, real

fluids have a critical isotherm that is somewhat flatter than

the fourth degree curve but not as flat as a fifth degree curve,

a coexistance curve that is almost cubic, a weakly divergent

specific heat Cv and a nonanalytical vapour pressure curve”.Even

good quality multiparameter equations may lead to large errors

in the critical region,

Recent work of Levelt Sengers et al. (1983)and Fox(1983) have

concentrated upon the idea of combining classical and nonclassi-

cal treatments in order to cover the noncritical and critical

regions of the phase diagram. In the work of Fox(1983) a state

function measuring the effective dietance between the state in

question and the critical state is constructed. Near the criti-

cal point, the nonclassical scaling bshaviour is developed by

the parammetrised function while outside the critical region the

classical formulation is generated. In the work of Levelt Sen-

gers et a1.(1983) the two regions are separately treated and

stitched together by a “switching@’ function. The main problem

291

associated with this treatment is to assure that second deriva-

tive properties are free from discontinuities in the boundaries

between different regions.

So far, no predictive techniques for locating critical lines

in nonclassical models have been developed. The experimental

critical line is an input to the calculation.

The modeling of mixtures by nonclassical methods follow clo-

sely the treatment of pure compounde. Parametric representation

with appropriate choice of scaling variables are employed. To

the knowledge of the author, thermodynamic behaviour in liquid-

liquid mixtures have not been studied by systematic scaled anal-

ysis up to date.

VIRIAL-TYPE EOS

A historical development of the virial-type EOS up to 1979

has been presented by H8rmsns(l980). The main trend observed in

this field is the attempt to generalize the parameters so as to

reduce the amount of input data for a given compound.Experience

has shown that the multiplicity of pure compound parameters does

not give a unique set and frustrates the unambiguous definition

of mixing rules.

Harmens(1980) has shown that the generalized treatment pro-

posed by Lee and Kesler(1975)

xing rules (LKP treatment)

implemented with Pldcker(1977) mi-

may be extended to strongly esymetric

mixtures such as hydrogen/hydrocarbon systems. Briefly stated,

the Lee-Kesler(l975) treatment considers a generalized equation

of state of the form

z= z(O) ++q( zm - z(o) ) (1) where the superscript (0) stands for a simple fluid (0 = 0) and

(r) for a reference fluid (n-octane) in corresponding’states

with the fluid in question. For each of the pivot fluids, the

compressibility factor is obtained by

z= pR

zz= 1 + BSR + ce;

where B,C,D are polynomials in l/TR. PlUcker(1977)introduced 8

mixing rule for the pseudocritical temperature incorporating 8

binary parameter, In his treatment, Harmens (1980) used two dif-

ferent binary interaction coefficients, one for the vapour phase

292

and one for the liquid phase. Average absolute errors of the or-

der of 2 to 3% in the K values were obtained for hydrogen/hydro-

carbon systems. The other popular virial EOS is an extended BWR

equation (

Benedict et al. 1940, 1942 Nishiumi and Saito(1977)

and Nishiumi 1980, (

1985 >

).

have proposed a few modifications of

this equation to describe pure nonpolar, polar and some associa-

ted compounds and their mixtures.

A different approach to generalize virial-type EOS is based

on the shape factors method of corresponding states (Brandl,l981

Mentzer et al. ,198l )*

Although different EOS may be used with

this method, both analytical and nonanalytical, the Bender EOS

(Bender,1975) has received special attention. Mentzer et alG9B$

have performed a comparative study of four EOS, two analytic and

two nonanalytic ones, and have obtained good results for hydro-

carbons but poor results for higher ketones and alcohols. For

mixtures they have used the one-fluid model with two binary in-

teraction parameters. Fair results have been obtained for mix-

tures of light gases not containing hydrogen.

A third approach has been recently developed by Brulh and

Starling (1984). In this work a modified Benedict-Webb and Rubin

EOS (Benedict et al. 1940,1942) is used for simple hydrocarbons

and combined with s three parameter corresponding states and a

conformal solution model for the description of VLE of multicom-

ponent mixtures c

Brule/ et a1.,1982; Watanasiri et a1.,1982).

While Brul& and Corbett(l984) estimated the EOS characterization

parameters and ideal-gas thermodynamic properties for coal liq-

uids and crude oils using empirical correlations and pseudocom-

ponent method, Brul& and Starling(1984) extended the treatment

to molecules containing other functional groups such as N, NH

and OH, In the absence of critical properties for coal liquids

and crude oils, a multiproperty analysis, called the therm-trans

characterization procedure was used to obtain parameters able to

represent simultaneously thermodynamic and transport properties

(viscosity). However the viscosity correlation is limited to the

high reduced temperature region.

VAN DER WAALS’-TYPE EOS

Equations based on the van der Waals’ model may be arbitrari-

ly subdivided in cubic and noncubic depending on their volume

dependence. In general, they have the form

’ = P(repuleive) - ‘(attrective)

293

(3) Cubic equations of state have received much attention in the

last few years due to their simplicity snd practical success

even when it is well known (Abbott.1979) that a cubic equation

of state cannot reproduce simultaneously all thermodynamic func-

tions with desirable eccuracy. From the many publications on the

subject only those that represent major improvements, on the

judgement of the author of this review, will be considered. It

is virtually impossible to comment on all recent developments

and keep this report to a reasonable lenght.

In a recent publication, Adechi et al. . (1983b) evaluated six-

teen two-term three parameter EOS for the representation of set-

uration properties and the high liquid density region (eRs3) of pure compounds. The equations were obtained by combination of

the repulsive and artractive terms presented in Table 1.

TABLE 1

Different forms of the repulsive and attractive terms in van der Waale-type EOS

‘(repulsive) ‘(

attractive J

van der Waals 1873

Scott 1971

Guggenheim 1965

Carnahan-Starling 1969

RT Redlich-Kwong 8

z;:t$

1949 v (v+c)

Clausius 1881 *

:iJ,1: 4 22 ,,b3v-b 3 )_ Peng-Robinson 1976 v2+2cv-c 8 2

Among the four repulsive terms, the van der Waals type was found

to perform better and among the four attractive terms the

Redlich-Kwong type appeared to be the best. Although open to

discussion, this study seems to point out the potential adven-

tages of cubic EOS.

Following the studies of Martin 1979 ( ) and Abbott (1979) ,

Vera et al l ( 1’ 1984 discussed the flexibility and limitations of

cubic equations of state written in their most general form

294

KT P

a(v - k3b) 5--

v-b (v-b)(v2+kibv+k2b2) C4) It was observed that successful cubic equation of stare either

use k3=1 or, if the value of kg is relaxed, they use k2=0 in or-

der to obtain the equality of fugacities of pure compounds at

low reduced temperature. According to the opinion of the authors

the use of kg=1 and zc=Pc;jc/RTc and Qb=bPc/RTc as independent

variables is better than fixing “a priori” the values of kj and

k2. The Schmidt-Wenzel EOS (Schmidt and Wenzel, 1980) was dis-

cussed in detail. For this equation k,_=i+3w, k2=-3w(k3=1), and

thus it reduces to the Redlich-Kwong equation (Redlich and Kwong

1949) for W=O, to the Peng-Robinson equation (

Peng and Robinson

1976) for W=1/3 and the Harmens equation (Harmens,l977) for

0=2/3. Alternatively, the selection of zc and Q,, as independ-

ent variables simplifies the solution of the dependent variables

and preserves all the advantages of the Schmidt-Wenzel EOS.

On the aame line, Adachi et al

ance of e,.(4) with k3-

.(1985a) studied the perform-

-1. In their nomenclature kl=u and k2=w.

They concluded that R, is substance dependent and determinant

for liquid volume estimations; saturated vapour volumes are not

sensitive to the values of the parameters u and w; a value w=-2,

as in the Peng-Robinson EOS, is suitable for representing satu-

rated liquid volumes. A representation of different equations on

the “u-w plane” indicated that the Schmidt-Wenzel EOS(1980) and

the four parameter EOS of Adachi et a1.(1983) give good repro-

duction of saturated vapour volumes and, thus, the P-V-T behav-

iour in the gas phase.

Since the early studies of Abbott(1979), it is known that CU-

bit EOS cannot describe with good accuracy the critical isotherm

of fluids. Recently Michels and Meijer(1984) have presented a

mathematical discussion showing that the critical point and the

high density region are better described by equation for which

Qb is less sensitive to the necessary adjustment of t. The

Schmidt-Wenzel EOS was found to be the best in this respect. It

was also found that improved predictions around and above the

critical density unavoidably entrain a large error in the re-

duced critical second virial coefficient and oonsequently lead

to less accurate predictions at elevated gas densities. The op-

timal four-parameter equation is especially accurate at very

295

high pressures and the root-mean-square deviation in pressure

cannot be reduced any further by introduction of a fifth parame-

ter. The errors in volume become minimal only with the general

five-parameter cubic equation.

For technical calculations it is important to have a good

representation of the vapour pressure curve and, to some extent.

of the saturated liquid volumes of pure compounds at temperatur-

es below the critical. At higher temperatures a good PVT repre-

sentation is desirable. To reach these goals with cubic EOS two

major lines of work have been developed. For two-parameter cubic

EOS the method proposed by Zudkevitch and Joffe(1970) has re-

ceived special attention. In this method both parameters “a” and

“b” are allowed to vary with temperature. Following the line of

work of yarborough (1979) ,Morris and Turek (1985) have recently

presented an extension of the method to cover the supercritical

region as well the subcritical region.

An alternative method is to improve the vapour pressure curve

representation by introducing specific adjustable parameters to

the temperature dependence of “a” (attractive term) and to im-

prove the saturated liquid molar volume representation by the

translation concept introduced by Martin(1979) and elaborated by

Peneloux and Rauzy (1982). Many publications improving the vapour

pressure representation by introducing of adjustable pure com-

pound parameters in cubic EOS have appeared in the recent liter-

ature. Pate1 and Teja(1983) using their own EOS and Mathias(l983)

using the Redlich-Kwong-Soave EOS have introduced one adjustable

parameter. Soave (1984) with the van der Waals EOS and Gibbons and

Laughton (1984) with the Redlich-Kwong-Soave EOS have proposed

the introduction of two adjustable parameters. Mathias and Cope-

man(1984) introduced three adjustable parameters in the Peng -

Robinson EOS. The latest development on this line of work has

been presented by Stryjek and Vera(1985 a,b,c). Stryjek and Vera

(1985a) have modified the Peng-Robinson EOS to extend its use to

low reduced temperatures. The modified equation, PRSV, with one

adjustable parameter is able to reproduce pure compound vapour

pressure down to 10 mmHg with accuracy comparable with Antoine’s

equation for nonpolar, polar and associated compounds. With

three adjustable parameters its accuracy is symilar to Chebyshev

polynomials (Stryjek and Vera, 1985c).

The volume translation to improve saturated liquid molar vol-

296

ume representation has been discussed by Peneloux and Rauzy

(1982). Vidal(1983) examined several modifications that improve

density calculations while retaining the accuracy in computing

vapour-liquid equilibrium. It should be observed that the volume

correction proposed by Chung-Tang Li and Daubert(l980) does not

preserve this property. From Vidal(1983) study it is interesting

to observe that even the van der Waals ECS after translation in

volume becomes comparable to more complex cubic EOS. Good re-

sults are obtained with the simple EOS proposed by Joffe(1981)

and then by Kubic(1982)

P =“-&

v-b (5)

In this equation the translation parameter is temperature de-

pendent.

Even when the volume translation improve the representation

of liquid volumes at low reduced temperatures the error in-

creases in the proximity of the critical temperature end affects

the high pressure region. Heyen (

1930, 1981) has proposed to al-

low for a temperature variation of the covolume to account for

this ef feet .

An alternative path is to search for noncubic EOS. Behar and

Ja in (1981) and Behar et al .(1985) have proposed to keep van der

Waals repulsion term and to expand the the attraction term as a

function of density.

P RT e-, a(T)

v-b(T) v(v+b(T)) {l - ‘d(T)bo + r( T)q}

V V (61

This equation produces results comparable with 42 parameter

viriel-type form.

A different approach, strongly promoted by Henderson(l979))

is to use as repulsion term the form proposed by Carnahan and

Starling (1972) and an attractive term of Redlich-Uwong form

?y+$ _ * b(T) with rj =4v

This equation seems attractive Vida1,(1983), but it appears

necessary to further modify the attractive term.

297

MIXTURES

Two theoretical studies of mixing rules have recently appea-

red in the literature. Freydank(1985) has studied the effect of

binary parameters and different fluid models in the calculation

of excess functions of binary mixtures with EOS. As repulsive

term he used van der Waals form and also different forms based

on radial distribution function. As attractive term he used van

der Waals and Redlich-Kwong forms. The absolute value of excess

function calculated using either the repulsive or the attractive

van der Waals terms was lower comparing with those obtained by

other expressions. There were no significant differences between

the results obtained with the other forms of the repulsive term.

The effect of the following two binary parameters were studied

ai j = ( aiaj)Oo5(1-ki j )

b.. iJ

= 0.125(bi1’3 + bj1’3)(i-lij)

While a higher value of kij increases an excess function and 1. ICi

has the opposite effect, even allowing to obtain S-shaped forms,

the values of both parameters are highly correlated. The one -

fluid model with two parameters allows to reach all regions of

the g E

- TsE plane but other fluid models cannot reproduce the

region gE< 0, TsE< 0.

Mansoori (1985) on the other hand, discussed a statistical me-

chanical conformal solution technique to derive mixing rules of

EOS. He pointed out that it is generaly incorrect to use classi-

cal mixing rules without attention to the algebraic form of the

cubic EOS and reported mixing rules for the van der Wasls,Red-

lich-Kwong and Peng-Robinson &OS. The capabilities of the new

expressions for the Peng-Robinson EOS have been tested for high-

ly polar and hydrogen bonding mixtures giving excellent correla-

tions and predictions of these complex systems (

Benmekki and

Flansoori, 1985). Some conformal mixing rules previously tested

by Radosz et a1.(1982) produced elso good results for esymmetric

mixtures.

On the more applied side, we may distinguish four main ap-

proaches to mixing rules for EOS. As it will be discussed with

more detail below, Wenzel et al . (1982), and also Pate1 and Teja

(1982) , have retained a one parameter mixing rule and incorpo-

298

rated additional correction terms. The second approach considers

volume dependent mixing rules so as to preserve the quadratic

dependence on composition of the second virial coefficient of

mixtures. Plollerup (1981) and also Whitting and Prausnitz(l982)

have suggested one-parameter local composition treatments which

at low densities reduce to the quadratic dependence on composi-

tion for the attractive term. Methias and Capemen (19S3) end

Luedecke end Prausnitz(l985) have conserved the one parameter

mixing rule end tlltroduced a vo1ur.1~ dependent correction term,

including two more adjustable parameters, to account noncentral

forces. Recently Sandle r (1985) h as discussed a theoretical basis

for density dependent mixing rules coming to the conclusion,that

the models currently in use do not properly account for nonren-

dom mixing due to attractive energy effects.

The third approach is specially designed to keep the cubic in

volume form of a cubic EOS for mixtures. This approach sacri-

fices the quadratic in composition dependence of the second vi-

riel coefficient on the basis that cubic ECS give e poor repre-

sentation of pure compound second viriel coefficients. Huron and

Vidal(1979) , Heyen (1981),

Stryjek and Vera (

Panegiotopoulos and Reid(1985) and

1985 e,b,c >

have followed this path. Stryjek

and Vera have tested many different mixing rules including those

arising from the general relation between “a” parameter and g E

(P-o6)(Huron end Videl , 1979). They have found(Stryjek end Vera,

1985c) that the composition dependent mixing rule

aij = (eiaj)Os5 (1 - x, :ijrii, )

= ij J jl

(10) gives consistently better results than other methods for complex

systems such es water/elcohols end elkene/alcohols. This is pro-

bably the first time that a simple cubic EOS proves to produce

better than methods using excess Gibbs energy functions for sys-

tems of such complexity,

Finally the fourth approach has been recently proposed by

Skjold-Jfirgensen (1984) and Tochigi et e1.(1985 e,b). It uses t

group-contribution version of an equation of state. Skjold -

Jfirgensen proposed e new equation based on the generalized van

der Waals partition function for polar as well nonpolar compo-

nents. The GC-EOS provides particularly good predictions of mu

ticomponent high-pressure vapour-liquid equilibria and fairly

he

l-

299

good predictions of Henry’s constant in mixed solvents. In the

method, presented by Tochigi et al.{1985 a,b), the energy pera-

meter “a” in the Redlich-Kwong-Soave, the Peng-Robinson and the

three paramerer Martin EOS has been expressed by an originated

from Huron and Vidal’s(1979) concept mixing rule based on ASOG

group contribution method. Satisfectory results have been ob-

tained for VLE mixtures containing n-paraffins,nitrogen,hydrogen

carbon dioxide, carbon monoxide, hydrogen sulfide, alcohols and

acetone in the range 60-600 K.

While most of the above mixing rules, some with more limita-

tions than others, may be satisfactory for VLE calculations of

many systems of industrial interest, they have not all been ful-

ly tested for simultaneous correlation of VLE and LLE in systems

containing strongly polar or associating compounds. The combina-

tion of e cubic EUS with chemical self association models, on

the other hand, has been specifically designed for this purpose.

This method, as described by Wenzel et a1.(1982) and Kolesinska

et a1.(1983), requies an initial choice in the stoichiometry of

associations, criticel parameters and acentric factors of the

associated forms and association constants to be determined with

the help of data of the pure substances. In addition, interac-

tion parameters are adjusted from binary VLE or LLE data. The

method, although complex, gives a self-consistent description of

pure substance properties and simultaneous correlation of VLE

and LLE (Kolasinska et al.,

by Kolasiriska et a1.(1983)

1983). As shown by Moorwood(l982),

and by Peschel and Wenzel(1984) the

method allows a good prediction of ternary VLE and LLE from bin-

ary information only. In an interesting study, Peschel and Wen-

zel (1984) have shown that several association models fit the

binary data equally well for methanol containing systems. Addi-

tional information as enthalpy of formation of the hydrogen bond

and IR spectroscopic determination of the OH groups are in a-

gresment with results.

EOS FOR ILL-OEFINEU MIXTURES

Many mixtures of industrial interest contain far too many

compound to allow a detailed chemical analysis of their identity

and concentrations. Petroleum, coal derived liquids and vegeta-

ble oils are typical examples. Three methods of applying equa-

tions of state to the calculation of phase equilibria for such

300

mixtures have received special attention in the recent litera-

ture.

Ilajeed and irlagner(1985) have used PFGC equation (Parameters

From Group Contribution) proposed by Cunningham and wilson(197d$

In its final form the PFGC EOS has five adjustable parameters

and, in addit ion, one interaction parameter per pair of groups

present. The functional groups may be identified, for example,

by NMR spectroscopy. Volumetric properties and VLE of light hy-

drocarbons and hydrocarbon/water systems have been predicted by

this method.

Vogel et a1.(1983)have used the pseudocomponent method to

calculate reservoir fluid properties. In this method, mixtures

are separated into fractions which, once characterised, are con-

sidered to be some equivalent pure compound. The authors discuss

the effect of important variables for EOS use such as character-

ization and optional lumping of the pseudocomponents. They warn

the users against possible errors due to maximum lumping and ad-

vice to use EOS parameters only with the EOS for which they were

obtained.

Pedersen et al .(1983) used the Redlich-Kwong-Soave EOS for

calculations of cruide oils. G more detailed description of the

computational method used was preaented in additional publica-

tions- Pedersen et al., (1984 a,b). It is interesting to observe

that in this work no “matching” or “tunning” of the parameters

to experimental data was required.

Although not directly related with EOS direct calculations it

is worth to observe here that Watanasiri et al . (1985) have re-

cently developed correlations to estimate parameters used in EOS

from data normally obtained for undefined petroleum and coal

fluids.

Cotterman et a1.(1985) have recently tested the use of an EOS

of the ven der Waals form for dew and bubble point calculations

of continuous and semicontinuous mixtures. In the continuous

thermodynamic approach an ill defined mixture is considered to

be formed by an infinite number of compounds with compositions

described by a continuous distribution function C

Kehlen and

RBtzsch, 1980, 1984; RBtzsch and Kehlen, 1983). Cotterman and

Prausnitz(l985) have considered the more complex problem of

flash calculations. Notably the results obtained with continuous

thermodynamics, even using an approximate method of solution,

301

were found to be in good agreement with the exact 24-component

system results while values obtained using the pseudocomponent

method were found dependent on the choice of pseudocomponents.

PARAFIETER EVALUATION AND CALCULATION I>!ETHODS

Recent improvements on EOS have motivated parallel studies on

parameter evaluation and calculation methods. Recently Skjold -

Jdrgensen(l983) discussed the applicetion of the Maximum Likeli-

hood Method (FILM) to obtain binary interaction parameters for

EDS using binary VLE data. Heidemann(1983) has presented an ex-

cellent review of some recent advances in the computation of

phase equilibria covering most references up to 1982. Some addi-

tional publications appearing after 1982 are discussed below.

Mehra et a1.(1983) have presented an accelerated succesive

substitution method, Nghiem (1983 a,b) has worked on a new ap-

proach to quasi-Newton methods with application to compositional

modeling and a robust iterative method for flash calculations

using the RKS and PR EOS. Ziervogel and Poling(1983) have pro-

posed a simple method for constructing phase envelopes for mul-

ticomponent mixtures. Nghiem and Li(1984) presented a general

approach to multiphase equilibrium with particular applicetion

to L1L2V systems. The PH EOS was used to predict LlL2V equili-

bria for reservoir oil-CO2 mixture. Novak et a1.(1985)have pres-

ented a Double Application of the Newton-Raphson DAN for VLE

bubble and dew point calculations.

Some additional particular problems originated when computing

phase equilibria with EOS have also received special attention.

Jovanovic and Paunovic(l984) proposed a method to avoid the

trivial solution in phase equilibria calculations for a cubic

EOS. The failure is prevented by generating artificial density

values leading to a correct solution, Nathias et al . (1984) con-

sidered the computation of the density root at conditions where

the appropriate density root does not exist. A strategy of return

ning suitable pssudoproperties under such conditions was propo -

sed.

Saville and Szczepanski (1983) considered the problem arising

due to the use of slightly thermodynamically inconsistent fun-

ct ions (

shape factor correlation 1

in multiparameter EOS,Notably,

convergence is impaired when functions end gradients are incon-

sistent. In contrast, these problems do not occur when using

302

thermodynamically consistent properties from a cubic equation of

state. Vetere(1984)discussed the main obstacles which impair s

useful application of complex equations to fluid phase equili-

bria. An increase in the number of parameters makes difficult to

define reasonable mixing rules, increases the computation time

and does not improve significantly the accuracy of the calcula-

tions.

CONCLUSIONS

When discussing the applicability of an equation of state it

is always necessary to keep in mind the purpose for which the

equation was constructed. In this review we have concentrated on

recent advances in the field of phase equilibria of pure com-

pounds and mixtures. No attempt has been msde to cover enthalpy

related properties.

Within the scope of this work, we may briefly summarize the

state of the art in the practical use of EOS.

Multiparameter virial-type equations of state are able to re-

produce with good accuracy pure compound properties. However,

they loose much of their advantages when generalized or extended

to mixtures.

Regarding the correlation of binary VLE we can remind the

results of Knapp et a1.(1982) who compared the performance of

two multiparameter and two cubic EOS. No single equation was ob-

viously better than others. The cubic EOS were simpler to use

and required less computation time. New empirical mixing rules

have recently allowed to correlate even data for complex systems

like HCl/H20 with cubic EOS (Stryjek and Vera, 1985b).

Correlation of LLE can be done with cubic EOS using extended

mixing rules (

Luedecke and Prausnitz, 1985) or using a combina-

tion with a chemical theory (Kolasinska et al., 1983).

The treatment of multicomponent ill-defined mixtures with EOS

has edvanced greatly in the last few years. Pseudocomponent me-

thods, continuous thermodynamics and even group contribution

methods (PFGC) have been shown to produce satisfactory results.

Numerical techniques for evaluation of binary parameters and

for dew and bubble point and flash calculations have been per-

fected and received new impulse by the introduction of more re-

fined mathematical techniques.

The outstanding challenging problems for equation of state

303

application are the prediction of VLE for binary mixtures from

pure component data only, the prediction of VLE for multicompo-

nent mixtures in the presence of associated or highly polar com-

pounds and the simultaneous correlation and prediction of LLE

and VLE for binary mixtures starting from VLE data.

ACKNUWLEDGEMENTS

I am indebted to Prof .A.Sylicki(Institute of Physical Chemis-

try, Polish Academy of Sciences for inspiring me to write this

review and to Prof.J.H.Vera McGill (

> University, Montreal) for

helpful correspondence.

LIST OF SYMBOLS

a

b

C

9

k

1

P

R

S

T

V

X

Z

w

e,

attraction parameter in cubic equation of state

covolume

translation parameter

molar Gibbs energy

interaction term to the attraction parameter

interaction term to the covolume

pressure

perfect gas constant

molar entropy

thermodynamic tsmpereture

molar volume

molar fraction

compressibility factor

scentric factor

density

Subscripts

C critical

ij component of the mixture

R reduced value

Superscripts

E excess function

0 simple fluid

r reference fluid

304

REFERENCES

Abbott, M.M., 1979. Cubic equations of state: an interpretative review,

Adechi, Y., Advances in Chemistry Series, 182: 47 - 70,

B.C.-Y. Lu and H. Sugie, 1983 a. A four - parameter equation of state,

Adechi, Y., Fluid Phase Equilibria, 11: 29 - 48.

B.C.-Y. Lu and H. Sugie, 1983 b. Three parameter equations of state,

Adachi, Y., Fluid Phase Equilibria, 13: 133 - 142.

H. Sugie and B.C.-Y. Lu, 1985. Development of van der Waals type of equation of state, J.Chem.Eng.Japan. 18: 20 - 29.

Behar, E. end C. Jain, 1981. Application d’une equation d’etat a la simulation du comportement thermodynamique des fluides, Revue de 1’Institut Francais du Petrole, 36: 173 -181.

Behar, E., R. Simonet and E.Rauzy., tion of state,

1985. A new non-cubic equa- Fluid Phase Equilibria, 21: 237 - 255.

Bender, E. Equations of state for ethylene and propylene, 1975. Cryogenics, 15 : 667 - 673.

Benedict, PI., G.R. Webb and L.C. Rubin, 1940. An empirical equa- tion for thermodynamic properties of light hydrocarbons and their mixtures.I.CH4,C2H6,propane and n-butane, J.Chem.Phys., 8: 334 - 345.

Benedict , Id., G.R. Webb and L.C. Rubin, 1942. An empirical equa- tion for thermodynamic properties of light hydrocarbons and their mixtures.II.Mixtures of methane,ethane,propane and bu- tane, J.Chem.Phys., 10: 747 - 758.

Benmekki,E.M. and A.G. Mansoori, 1985. Phase equilibrium calcu- lations of highly polar systems, 2nd CDDATA Symposium, Paris, Sept .ll-13,

Brandl, H., 1981. Eine Methodik zur Gewinnung generalisierter Thermischer Zustandsgleichung reiner Fluide, Z.phys.Chemie, Leipzig, 262: 415 - 427.

Brule, M.R., C.T. Lin, L.L. Lee and K.E. Starling, 1982. Multi- parameter corresponding states correlation of coal-fluid thermodynamic properties, AIChE 3.. 28: 616 - 625.

Brulh, M.R. and R.W. Corbett, 1984. What makes critical-solvent process work?, Hydrocarbon Process, June:73.

Bruld, M.R. and K.E. Starling, of complex systems:

1984. Thermophysical properties applications of multiproperty analysis,

Ind.Eng.Chem.Process Des.Dev., 23: 833 - 845. Carnahan, N.F. and K.E. Starling. 1969. Equation of state for

non-attracting rigid spheres, J.Chem.Phys., 51: 635 -636. Chao, K.C. and R.L. Robinson,Jr. ed. ,1979. Equations of State

in Engineering and Research. Advances in Chemistry Seriss,No. 182, American Chemical Society, Washington.

Chung-Tong Li and T.E. Daubert, 1980. Estimation of partial mol- ar volume and fugacity coefficient of components in mixtures from the Soave and Peng-Robinson EOS, Ind.Eng.Chem.Process Des.Dev., 19: 51 - 59.

Clausius, R., 1881. Ann. Phys. Chem., IX: 337. Cotterman, R., R. Benden and J. Prausnitz, 1985. Phase equilib-

ria for mixtures containing very many components.Development and application of continuous thermodynamics for chemical process design, Ind.Eng.Chem.Process DesJev., 24: 194 - 203.

Cotterman, R. and 2. Prausnitz, 1986. Flash calculations for continuous and semicontinuous mixtures using en equation of stats, Ind.Eng.Chem.Process Des.Dev., 24: 434 - 444.

305

Cunningham, J.R. and G.M. Wilson, 1974. Equation of state anal- ogy to an activity coefficient equation.Calculation of par- smeters from group contributions, Proc.,Annu.Conv.,Procees. Assoc .,Tech.Pap., 53: 77 - 86.

Fox, J.R., 1983. Method for construction of nonclassical equa- tions of state, Fluid Phase Equilibria, 14: 45 - 53.

Freydank, H., 1965. Modellierung thermodynamischer Mischungs- funktionen mit thermischen Zust~ndegl~ichungen.I.Eigenschaft- en der Zustandsgleichungen in G - l-9 - Feld., Z.phys.Chemie Leipzig, 266: 81 - 93.

Gibbons, R.N. and A.P. Laughton, 1984. An EOS for hydrochloric acid solutions, Fluid Phase Equilibria, 18: 61 - 68.

Guggenheim, E.A., 1965. Variations on van der Waals equation of state for high densities, Mol.Phye., 9: 199 - 200.

Harmens, A., 1977. A cubic equation of state for the prediction of N2-Ar-0 phase equilibria, Cryogenics, 17: 519 - 523.

Harmens, A., P 980. A three-parameter cubic equation of state for normal fluids, Thesis, Technical University, Berlin.

Harmens, A. and H. Knapp, 1980. Three parameter cubic equation of state for normal substances, Ind.Eng.Chem.Fundam., 19: 291 - 294.

Heidemann, R.A., 1983. Computstion of high pressure phase equi- libria, Fluid Phase Equilibria, 14: 55 - 78.

Henderson, D., 1979. Practical calculations of equation of state of fluids and fluid mixtures using perturbation theory and related theories, Advances in Chemistry Series, 182: 1 - 30.

Heyen, G., 1980. Liquid and vapour properties from a cubic equa- tion of state. 2nd International Conference on Phase Equilib- ria and Fluid Properties in the Chemical Industry, Berlin.

Heyen. G,, 1981.A cubic equation of state with extended range of application. 2nd World Congress of Chemical Engineering, Montreal.

Huron, M.-J. and J. Vidal, 1979. New mixing rules in simple equa- tions of state for representing VLE of strongly non-ideal mixtures, Fluid Phase Equilibria, 3: 255 - 271.

Doffe, J., 1981. Vapour-liquid equilibria and densities with the Martin equation of state, Ind.Eng.Chem.Process Des.Dev., 20: 158 - 172.

Jovanovic, S. and Paunovic R., 1984. Generating appropriate den- sity values from EI cubic state equation to avoid false unit I< values. Application to distillation problems, Ind.Eng.Chem. Process Oee.Dev., 23: 801 - 805.

Kehlen, H. and M. RBtzsch, 1980. Continuous thermodynamics of multicomponent mixtures. Proc.Gth Int.Conf.Thermodyn.,Merse- burg: 41 - 51.

Kehlen, H. and M. Rltzech, 1984. Separate treatment of paraffins and aromatics in complex hydrocarbon mixtures by continuous thermodynamics. 8th International CHISA Conference, Praha.

Knapp, H., R. Wring, L. Oellrich, U. Plucker and J.M. Prauenitz 1982. Vapor-Liquid Equilibria for Mixtures of Low Boiling Substances. Institue of Thermodynamics and Plant Deeiun,Tech- nical University, Berlin, DECHEMA.

Kolaeinska, G., K.A.S. Moorwood and H. Wenzel, 1983. Calculation of vapour-liquid and liquid-liquid equilibria by an equation of state. Fluid Phase Equilibria, 13: 121 - 132.

Kubic, W.L., 1982. A modification of the Martin eauation of sta- te-for calculating vapour-liquid equilibria. Fluid Phase Equi libria, 9: 79 - 97.

Lee, 6.1. and M.G. Kesler, 1975, Generalized thermodynamic cor- relation based on three-parameter corresponding states, AIChE J., 21: 510 - 527.

306

Levelt Sengers, J.M.H., W.L. Greer, J.V. Senger, 1976. Scaled equation of state parameters for gases in the critical region J.Phys.Chem.Ref.Data, 5 1 : 1 - 51.

Levelt Sengers, J.M.H., G. Morrison and R.F. Chang, 1983. Criti- cal behavior in fluids and fluid mixtures, Fluid Phase Equi- libria, 14: 19 - 44.

Luedecke, D. and J.M. Prausnitz, 1985. Phase equilibria for strongly nonideal mixtures from an equation of state with density dependent mixing rules, Fluid Phase Equilibria, 22: 1 - 19.

Majeed, A.I. and 3. Wagner, 1985. The PFGC equation and phase equilibria in light hydrocerbon systems. 189 ACS National Meeting, Miami, April 28 - May 3.

Mansoori, A.G., 1985, Mixing rules for cubic equations of stats. 189 ACS National Meeting, Miami, April 28 - May 3.

Martin, J.J., 1979. Cubic equations of state - which? Ind.Eng. Chem.Fundam., 18: 81 - 97.

Hathias, P.M., 1983. A versatile phase equilibrium equation of state. Ind.Eng.Chem.Process Des.Dev., 22: 385 - 391.

Mathias, P.M. and T.W. Copeman, 1984. Extension of the Peng - Robinson EOS to complex mixtures: evaluation of the various forms of the locel composition concept, Fluid Phase Equilib- ria, 13: 91 - 108.

Mathias, P.M., J.F. Boston and S. Wetanasiri, 1984. Effective utilization of equations of state for thermodynamic properties in process simulation, AIChE J., 30: 182 - 187.

Mehra, R.K., R.A. Heidemann and I<. Aziz, 1983. An accelerated successive substitution algorithm, Can.J.Chem.Eng., 61: 590 - 596.

Mentzer, R.A., R.A. Greenkorn and Kweng-Chu Chao, 1981. Princi- ple of corresponding states and VLE of molecular fluids, and their mixtures with light gases, Ind.Eng.Chem.Process Des.Dev. 20: 240 - 253.

Michels, M.A.J. and Meijer H., 1984. The performance of cubic equetiona of state on the critical isotherms of pure fluids, Fluid Phase Equilibrie, 17: 57 - 75.

Mollerup, J., 1981. A note on excess Gibbs energy models, equa- tions of state end the local composition concept, Fluid Phese Equilibria, 7: 121 - 138.

Moorwood, R .A.S., 1982. Die Berechnung von Phasengleichgewichten in verschiedenen aesoziierten Systemen mit Hilfe einer Zu- standsgleichung. Thesis, University Erlangen-NUrnberg.

Morris, R.W. and E.A. Turek, 1985. Optimal temperature-dependent “e” and “b” parameters for the Redlich-Kwong equation of sta- te. 189 ACS National Meeting, Miami, April 28 - May 3.

Nghiem, L.X., 1983. A new approach to quasi-Newton methods with application to compositional modeling. 7th Soc.Petrol.Eng. Reservoir Simulation Symp.,Nov. 15 - 18, San Francisco, CA, Pap .SPE 12242.

Nghiem, L.X., K. Azia and Y.-K. Li, 1983. A robust iterative me- thod for flesh calculations using the Soave-Redlich-Kwong or Peng-Robinson equation of state. Soc.Petrol.Eng.J., 23: 521 - 530.

Nishiumi, H. and S, Saito, 1977, Correlation of the binary inter action parameter of the modified generalized BWR equation of state. J.Chem.Eng.Japan, 10: 176 - 180.

Nishiumi, H., 1980. An improved generalized BWR equation of state with three polar parameters applicable to polar substances. J.Chem.Eng.aapan, 13: 178 - 183.

307

Nishiumi, H., 1985. Prediction of phase equilibria for systems containing hydrogen from an extended BUR equation of state, 2nd COOATA Symposium, Paris, Sept. 11 - 13.

Novak, J.P., V. Ruricka and A. Mslijevsky, 1985. The DAN method for calculation of vapour-liquid equilibria using an equation of state.l.Bubble and dew point calculations.2.Flash calcula- tion. CCCC, 50: 1 - 22, 23 - 32.

Panagiotopoulos, A.Z. and R.C. Reid, 1985. A new mixing rule for cubic equations of state for highly polar, asymmetric systems 189 ACS National Meeting, Miami, April 28 - May 3.

Patel, N.C. and A.S. Teja, 1982. A new cubic equation of state for fluids and fluids mixtures, Chem.Eng.Sci., 37: 463 - 473.

Pedersen, K.S., P. Thomassen and AA. Fredenslund, 1983. SRK-EOS calculation for crude oils, Fluid Phase Equilibria, 14: 209 - 218.

Pedersen, K.S., P. Thomassen and AA. Fredenslund, 1984e.Thermo- dynamics of petroleum mixtures containing heavy hydrocarbons l.Phaae envelope calculations by use the Soave-Redlich-Kwong equation of state. Ind.Eng.Chem.Process Oes.Dev., 23: 163 - 170.

Pedersen, K.S., P. Thomassen and AA.Fredenslund, 1984b. Thermo- dynamics of petroleum mixtures containing heavy hydrocarbons P.Flash and PVT calculations with the SRK equation of state, Ind.Eng.Cham.Process Oes.Dev., 23: 566 - 573.

Peneloux, A. and E. Rauzy, 1982. A consistent correction for Redlich-Kwong-Soave volumes, Fluid Phase Equilibria, 8: 7 -23

Peng, O.Y. and D.B. Robinson, 1976. A new two-constant equation of state, Ind.Eng.Chem.Fundam., 15: 59 - 64.

Peschel, W. and H. Wenzel, 1984. Equation-of-state predictions of phase equilibria at elevated pressures in mixtures con- taining methanol, Ber.Bunsenges.Phys.Chem., 88: 807 - 812.

Pldcker, U., 1977. Berechnung von Hochdruck-Phasengleichgewich- ten mit einer Korrespondenzmethode unter besonder Berdcksich- tigung asymmetrischer Gemische. Thesis, Technical University, Berlin.

Radosz, M. Lin Ho-Mu, Chao Kwang-Chu, 1982. High-pressure VLE in asymmetric mixtures using new mixing rules, Ind.Eng.Chem. Process Des .Oev., 21: 653 - 658.

RLltzsch, M.T. and H. Kehlen, 1983. Continuous thermodynamics of complex mixtures, Fluid Phase Equilibria, 14: 225 - 234.

Redlich, 0. and 3.N.S. Kwong, 1949. On the thermodynamics of so1utions.V. Chem.Rev., 44: 233 - 244.

Reid, R.C., 1983. Retrospective comments on physical property correlations, Fluid Phase Equilibria, 13: 1 - 15.

Sandler, S.J., 1985. A basis for equation of state mixing rules, 2nd CODATA Symposium, Paris, Sept. 11 - 13.

Saville, G. and R. Szczepanski, 1983. Calculation of phase equi- libria by minimization techniques - problems of thermodynamic consistency, Fluid Phase Equilibria, 14: 246 - 264.

Skjold - Jdrgensen, S., 1983. On statistical principles in re- duction of thermodynamic data, Fluid Phase Equilibria, 14: 273 - 288.

Skjold - Jdrgensen, S.. 1984. Gas solubility calculations.11. Application of a new group-contribution equation of state, Fluid Phase Equilibris, 16: 317 - 351.

Schmidt, G. and H. Wenzel, 1980. A modified Van der Weals equa- tion of state, Chem.Eng.Sci., 35: 1503 - 1512.

Scott, R.L., 1971. Physical Chemistry,An Advanced Treatise.0. Henderson, ed . ,Vol.8A,Chapter 1, Academic Press ,New York.

308

Stt-yjek, R. and J.H. Vera, 1985 a. An improved cubic equation of state.

Stryjek, 189 ACS National Meeting, Miami, April 26 - May 3.

R. and J.H. Vera, 1985 b. Vapor-liquid equilibrium of hydrochloric acid solutions with the PRSV equation of state, Fluid Phase Equilibria, in press.

Stryjek, R. and J.H. Vera, 1985 c. n-alcohols -

Vapor-liquid equilibrium of n-alkane systems with the PRSV equation of ste

te. IUPAC Conference, Paris, Sept. 5 - 7. Tochigi, K., K. Kurihara and K. Kojima, 1985 a. Prediction of

high pressure vapor-liquid equilibria with mixing rule using ASDG group contribution method, J.Chem.Eng.Japan, 18: 60 - 66

Tochigi, K., K. Kurihara and K. Kojima, 1985 b. Prediction of high pressure vapor-liquid equilibria by SRK-ASDG equation of state. 2nd CODATA Symposium, Paris, Sept. 11 - 13.

van der Waals, J.D., 1873. Over de Continuitet van den Gas-en Vloeistoftoestand. Doctoral dissertation, Leiden.

Vera, J.H., M.D. Huron and J. Vidal, 1984. On the flexibility and limitations of cubic equations of state, Chem.Eng.Commun. 26: 311 - 318.

Vetere, A., 1984. Algorithm for fluid phase equilibria calcula- tions by means of a multiconstant equation of state, Chem. Eng.Sci., 39: 1779 - 1784.

Vidal; J., 1983. Equations of state - reworking the old forms, Fluid Phase Eauilibria. 13: 15 - 33.

Vogel, J.L., E.A: Turek, R.S. Metcalfe D.F. Bergman and R.W. Morris, 1983. Applications of equations of state to calculate reservoir fluid properties, Fluid Phase Equilibria, 14: 103 - 116.

Watanasiri, S., M.R. Bruld and K.E. Starling, 1982. Correlation of phase-separation data for coal conversion systems, AIChE J 28: 626 - 637.

Watanasiri, S., V.H. Owens and K.E. Starling, 1985. Correlations for estimating critical constants, acentric factor, and di- pole moment for undefined coal-fluid fractions, Ind.Eng.Chem. Process Des.Oev., 24: 294 - 297.

Wenzel, H., R.A.S. Moorwood and M.Baumgaertner, 1982, The calcu- lation of vapour-liquid equilibrium of associated systems by equation of state, Fluid Phase Equilibria, 9: 225 - 266.

Whiting, W.B. and 3.1~. Prausnitz, 1982. Equations of state for strongly nonideal fluid mixtures: application of local compo- sitions toward density-dependent mixing rules, Fluid Phase Equilibria, 9: 119-147.

Ziervogel, R.G. and B.E. Poling, 1983. A simple method for con- structing phase envelopes for multicomponent mixtures, Fluid Phase Equilibria, 11: 127 - 135.

Zudkevitch, D. and J. Joffe, 1970. Correlation and prediction of vapor-liquid equilibrium with the Redlich-Kwong equation of state, AIChE 3., 16: 112 - 119.

Yarborough, L., 1979. Application of a generalized equation of state to petroleum reservoir fluids. Advances in Chemistry Series, 182: 385 - 439.