232372441 correlation-and-prediction-of-vle-and-lle-by-empirical-eos

Fluid PhaseEquilibria,27 (1986) 289-308
ElsevierSciencePublishersB.V., Amsterdam-Printed in TheNetherlands
289
CORRELATIONAND PREDICTIONOF VLE AND LLE BY EMPIRICAL EOS
GHAtYNAKOLASINSKA
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 EisevierSciencePublishersB.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-Robinson1976 v2+2cv-c8 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
RTe-, 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) has 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
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