Artigo - The State Of The Cubic Equations Of State

REVIEWS The State of the Cubic Equations of State Jose´ O. Valderrama* Faculty of Engineering, University of La Serena, Casilla 554, and Center for Technological Information (CIT), Casilla 724, La Serena, Chile The development of van der Waals cubic equations of state and their application to the correlation and prediction of phase equilibrium properties is presented and analyzed. The discussion starts with a brief account of the contributions to equation of state development during the years before van der Waals. Then, the original equation proposed in the celebrated thesis of van der Waals in 1873 and its tremendous importance in describing fluid behavior are analyzed. A chronological critical walk through the most important contributions during the first part of the 1900s is made, to arrive at the proposal that I consider to be the most outstanding since van der Waals: the equation proposed by Redlich and Kwong in 1949. The contributions after Redlich and Kwong to the modern development of equations of state and the most recent equations proposed in the literature are analyzed. The application of cubic equations of state to mixtures and the development of mixing rules is put in a proper perspective, and the main applications of cubic equations of state to binary and multicomponent mixtures, to high-pressure phase equilibria, to supercritical fluids, to reservoir fluids, and to polymer mixtures are summarized. Finally, recommendations on which equations of state and which mixing rules to use for given applications are presented. Contents Introduction 1603 van der Waals’ Proposal 1605 Redlich-Kwong (RK) Proposal 1605 From Redlich-Kwong to Soave 1606 Modifications to the SRK and PR Equations 1606 Modifications to r(TR) 1606 Volume-Translated Equations 1607 Three-Parameter Equations of State 1608 Application to Mixtures 1608 Classical Mixing Rules 1609 Volume-Dependent Mixing Rules 1609 Nonquadratic Mixing Rules 1609 EoS + Gibbs Free Energy Models 1610 Mixing Rules of Mansoori and Co-workers 1611 Applications of Cubic EoS to Reservoir Fluids 1612 Mixtures Containing Supercritical Components 1612 Cubic EoS Applied to Polymer Solutions 1613 Recommendations 1614 Conclusions 1614 Introduction Since van der Waals proposed the first version of his celebrated equation of state (EoS) more than a century ago,1 many modifications have been proposed in the literature to improve the predictions of volumetric, thermodynamic, and phase equilibrium properties. Al- though the van der Waals (vdW) equation is not ac- curate for most applications, it can be considered the major contribution to this field since the first attempts to represent PVT behavior made by Boyle in the 17th century.2 The vdW equation and the many modifications that are now available are special cases of a generic cubic equation, which can be written as Here, a, b, c, and d can be constants or functions of temperature and some fluid properties (acentric factor, critical compressibility factor, normal boiling point, etc.). These parameters cannot be chosen arbitrarily, as certain theoretical and empirical restrictions must be imposed.3,4 Many complex and more accurate EoS have been proposed through the years and several theories have been devised to better represent PVT properties and vapor-liquid equilibrium. Complex molecular-based equations of state have received special attention, and the power and usefulness of this type of model have been noted many times. However, not many articles ques- tioning the results and conclusions obtained from these equations have been published. Recently, Nezbeda5 analyzed this type of EoS and summarized some basic rules that should be observed for developing molecular- based EoS that can be considered as truly molecular- based. The papers by Anderko6 and Wei and Sadus7 * E-mail: citchile@entelchile.net, jvalderr@userena.cl. P ) RT V - b - Patt(T,V) (1) Patt(T,V) ) a V(V + d) + c(V - d) (2) 1603 Ind. Eng. Chem. Res. 2003, 42, 1603-1618 10.1021/ie020447b CCC: $25.00 © 2003 American Chemical Society Published on Web 03/20/2003 reviewed several of these complex EoS, although they also pay some attention to cubic vdW equations of state. Empirical multiparameter EoS, all of them of noncubic type and with 15-60 or more parameters, became available during the 1970s. The advantages and disad- vantages of these equations and the development of such equations during the past 20 years are reviewed and analyzed in the book by Sengers et al.8 For pure substances, multiparameter EoS are divided into refer- ence and technical EoS. For mixtures, the introduction of Helmholtz free energy based multifluid mixture models enabled highly accurate descriptions of thermo- dynamic properties for the first time. Multiparameter EoS have recently been reviewed by Span et al.9 According to these authors, the unsolved problems in this field offer a multitude of scientific challenges with regard to the actual development of EoS and mixture models. Despite the many equations of different types and the several new applications, cubic equations similar to eq 1 are still used in semiquantitative predictions of equilibrium phenomena, in process design, and in simulations. There are several reasons for this popular- ity but also several disadvantages of cubic equations, which have encouraged many researchers to look for different approaches such as those mentioned above. Table 1 summarizes some of these advantages and disadvantages of cubic EoS. To clarify the limits and scope of this paper, a convenient classification is shown in Figure 1. This paper is limited to those aspects that the author finds relevant in the development of the group called “cubic empirical equations”. The paper starts with a short review of the history of cubic EoS, leading to an assessment of which EoS might be most useful for several applications. Because of the enormous amount of literature on the subject of EoS, one can always quarrel with which publications to include in a review like the present one. Thus, the publications included here represent what the author considers most relevant for the practitioners in the field of fluid properties and phase equilibria using cubic EoS. A recent book published by IUPAC8 reviews the main advances of EoS for fluids and fluid mixtures. The Table 1. Advantages and Disadvantages of van der Waals Cubic EoS advantages disadvantages a third degree in volume, which makes calculations relatively simple to perform a actual PVT data tend to follow a fourth-degree equation instead of a cubic equation b present correct limiting behavior: as V f b, P f ∞ in all van der Waals type equations b both the repulsive and attractive terms are inaccurate, as shown by molecular simulations c known inaccuracies of both the repulsive and attractive terms are canceled when the EoS are used to calculate fluid properties, in particular VLE c cubic equations cannot represent all properties of a fluid in all different ranges of P and T d for most applications, cubic EoS can be tuned to give accurate values for any volumetric or thermodynamic property d temperature dependency of the force constant a is not well established; co-volume b seems to be density-dependent, but the dependence is unknown e extension to mixtures is relatively easy using mixing and combining rules of any complexity e because interactions between unlike molecules are unknown, most mixing and combining rules are empirical, and interaction parameters are usually required f cubic equations are suitable for the application of modern mixing rules that include Gibbs free energy models or concentration-dependent parameters f in applications to complex mixtures, several interaction parameters might be required, even with the use of modern mixing rules Figure 1. Classification of various type of equations of state, with a selection of equations for each group. In this classification, van der Waals EoS are those cubic and noncubic equations that consider the compressibility factor as Z ) Zrep + Zatt. 1604 Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 theoretical basis and practical use of each type of equation is discussed, and the strengths and weak- nesses of each are analyzed. Topics addressed include the virial equation of state, cubic equations and gener- alized van der Waals equations, multiparameter EoS, perturbation theory, integral equations, corresponding states, and mixing rules. Special attention is also devoted to associating fluids, polydisperse fluids, poly- mer systems, self-assembled systems, ionic fluids, and fluids near critical points. For those interested in cubic EoS, the chapters by Anderko on cubic and generalized van der Waals equations and by Sandler and Orbey on mixing and combining rules, included in this IUPAC book, should be read. For those interested in more theoretical aspects of EoS, the review by Sarry10 and the recently published book by Eliezer et al.11 present detailed pedagogical accounts of EoS and their applica- tions in several important and fast-growing topics in theoretical physics, chemistry, and engineering. van der Waals’ Proposal Before van der Waals, some attempts to represent the real behavior of gases were made. The main drawback of the proposals presented before van der Waals was that they did not consider the finite volume occupied by the molecules, which is also the case for the ideal gas model. The idea of including the volume of the molecules into the repulsive term was suggested by Bernoulli at the end of the 18th century,12 but it was ignored for a long time. The works of Hirn13 and Dupre14 revived this idea. The volume V was replaced by (V - b), where b represents the volume occupied by the molecules, which Dupre named the “covolume”. How- ever, none of these contributions were of general use, and none was able to answer the many questions related to fluid behavior remaining at that time. It was van der Waals with his celebrated doctoral thesis “The Continu- ity of the Liquid and Gaseous States” and other works derived from it who gave rise to what seems to be the most longed-for but still unattainable goal of many researchers: finding the most simple and generalized EoS. van der Waals1 proposed in his thesis the following equation In this equation, P is the external pressure, V is the molar volume, b is a multiple of the molecular volume, a is the “specific attraction”, and R is a constant related to the kinetic energy of the molecules. This equation later became what is now known as the van der Waals equation of state The parameters, or so-called “equation of state con- stants”, a and b were first calculated using PVT data but were later related to critical properties by applying the condition of continuity of the critical isotherm at the critical point. This requirement allows the constants to be related to the critical pressure and temperature, Pc and Tc, respectively, and also gives a constant value for the critical compressibility factor (zc ) 0.375). van der Waals’ equation of state and his ideas on intermolecular forces have been the subjects of many studies through the years. Andrews’ discovery of the existence of the critical point helped van der Waals in formulating a theory that accounts for the behavior of fluids both above and below the critical point. van der Waals unified most of the experimental knowledge on fluid properties up to the 1870s in a single equation, which not only accounted for deviations from the ideal gas but also predicted the existence of a critical point. The equation also simultaneously considered the vapor and liquid phases, phase equilibrium below the critical point, and even the separation of phases above the critical point, a phenomenon that was experimentally verified several years later by Krichevskii.15 The con- cepts developed by van der Waals on the separation of repulsion forces caused by molecular size from cohesive forces caused by molecular attraction still remain as the basis of several theories concerning the prediction of fluid properties and of computer simulations based on statistical mechanics. Also, his ideas on the existence of a continuous equation connecting the liquid and vapor states, on the corresponding state principle, and on the extension of the pure-component equation of state to mixtures through the use of mixing rules have contrib- uted greatly to the present developments in this area. A good study on the van der Waals equation is given in the book by Vukalovich and Novikov.16 Clausius,17 contrary to what van der Waals claimed in his thesis, recognized that the attractive term should be temperature-dependent. Clausius also modified the volume dependency of the attractive term. His proposal was2 Clausius arrived at his equation by considering that molecules at low temperature do not move freely but form clusters of molecules in which stronger attractions occur (the van der Waals term a/V2 is too small at low temperatures). This equation has been the subject of several studies through the years, and some contradic- tory arguments about its accuracy can be found in the literature.4,12,18 It can be seen that Clausius’ equation allows for the use of an additional empirical parameter, because the critical compressibility factor is no longer constant, as occurs with the van der Waals equation. This fact has given rise to a group of important EoS known as “three-parameter equations”. Modifications to the repulsive term were also considered soon after van der Waals’ proposal. However, most modifications to the repulsive term give rise to noncubic equations. The Redlich-Kwong (RK) Proposal By the time of the proposal of Redlich and Kwong,19 there were about 200 equations of state. Otto20 gave a list of 56 equations, most of them modifications to vdW equation. Also, the book by Vukalovich and Novikov16 contains a list of 150 EoS published by 1944, and the book by Walas21 gives a list of about 60 equations published before 1949. The renewed interest in van der Waals type equations came years after Redlich and Kwong’s contribution, but these authors certainly showed the way on how to improve upon van der Waals’ ideas. As Prausnitz22 wrote, “Redlich’s great contribution was to revive the spirit of van der Waals.” Redlich and Kwong were very much concerned about the limiting behavior of the EoS. They wanted correct (P + a V 2)(V - b) ) R(1 + Rt) (3) P ) RT (V - b) - a V 2 (4) P ) RT (V - b) - a/T (V + c)2 (5) Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1605 representations at low density and at high density and proposed the following equation This equation did not have a strong theoretical back- ground but proved to give good results for many gaseous systems. It should also be mentioned that, when Redlich and Kwong proposed their celebrated equation of state, they were interested in developing a good equation for gases only. Not a single application to liquids can be found in Redlich and Kwong’s original paper. During the period 1960-1980, the interest in Redlich- Kwong-type equations was so high that the RK equation can be considered the most modified EoS ever. A complete account of all such equations is not easy to provide; however, following some good review articles and our own findings, it is not adventurous to estimate that there must be about 150 RK-type equations and a total of 400 cubic EoS proposed to date in the literature. During the 15 years after Redlich and Kwong’s proposal, the equation seemed to be one more of the already many modifications to the vdW equation, with the Chao- Seader correlation being the only major application of this EoS. The works of Wilson,23 Barner et al.,24 and Chueh and Prausnitz25 attracted renewed interest in exploring Redlich and Kwong’s equation. From Redlich-Kwong to Soave Wilson23 made a major contribution to the attempts at generalizing the RK equation. He was able to consider the variations in behavior of different fluids at the same reduced pressure and reduced temperature by introduc- ing Pitzer’s acentric factor (ω) into the attractive term. Wilson’s equation was largely ignored during the rest of the 1960s and until 1972, the time of Soave’s proposal. Soave26 proposed a new improved version of Wilson’s idea. He redefined the R(TR,ω) function and kept the RK volume functionality, giving what became, in a short period, one of the most popular EoS in the hydrocarbon industry. Computer simulation packages became popu- lar during the late 1960s, and there was great need for a simple, generalized, and reasonably accurate EoS for the many repetitive calculations required in process simulations. Soave’s EoS, commonly known as the SRK equation, satisfied the need at that time. Also, in the mid 1970s, optimization of industrial processes became extremely important, because of the so-called oil crisis, and Soave’s equation was again of great help for the type of calculations required in optimization algorithms. The SRK equation can be summarized as follows After Soave’s proposal, many modifications were presented in the literature for improving predictions of one or another property. These works were not limited to proposing new temperature models for R(TR,ω), but also considered modifications of the volume dependence of the attractive pressure term. The most popular of all of these modifications is the one proposed by Peng and Robinson.27 Peng and Robinson improved upon Soave’s equation by recalculating the R(TR,ω) function and by modifying the volume dependency of the attractive term. These changes allowed them to obtain better results for liquid volumes and better representations of vapor- liquid equilibrium (VLE) for many mixtures. Peng and Robinson’s equation is The SRK and the PR equations are the most popular cubic equations used currently in research, simulations, and optimizations in which thermodynamic and VLE properties are required. These two equations have been considered for all types of calculations, from simple estimations of pure-fluid volumetric properties and vapor pressures to descriptions of complex multicom- ponent systems. New models for the equation of state parameters and for the mixing rules are the most common modifications. Overall, all of these modifica- tions did not contribute to any major development until the early 1980s, with the so-called three-parameter EoS and the development of new mixing rules. Nevertheless, most modern computer process simulation packages (ChemCAD, AspenPlus, Hysim, PRO/II) include the SRK and PR equations among the thermodynamic options. Modifications to the SRK and PR Equations The trends in research on cubic EoS after Soave’s and Peng and Robinson’s contributions have followed three main routes: (i) modifications to R(TR) in the SRK and PR equations, (ii) modifications of the volume depen- dence of the attractive pressure term, and (iii) use of a third substance-dependent parameter. The first ap- proach has focused on looking for more accurate predic- tions of vapor pressure and vapor-liquid equilibrium. The second route has given rise to the “volume-transla- tion” concept. The third line of development constitutes the so-called group of “three-parameter equations of state”, of which some promising generalized three- parameter equations have been proposed. Modifications to r(TR) Modifications of the temperature-dependent function R(TR) in the attractive term of the SRK and PR equa- tions have been mainly proposed to improve correlations and predictions of vapor pressure for polar fluids. Table 2 shows some selected expression proposed in the P ) RT V - b - acR(TR,ω) V(V + b) + b(V - b) (9) ac) 0.45724 RTc 2.5 Pc R(TR,ω) ) [1 + m(1 - TR 0.5)]2 (10) b ) 0.07780 RTc Pc m ) 0.37464 - 1.54226ω - 0.26992ω2 P ) RT V - b - acR(T) V(V + b) (6) R(T) ) a/T0.5 ac ) ΩaR2Tc 2.5/Pc Ωa ) 0.4278 b ) ΩbRTc/Pc Ωb ) 0.0867 P ) RT V - b - acR(TR,ω) V(V + b) (7) ac ) 0.427 47 RTc 2.5 Pc R(TR,ω) ) [1 + m(1 - TR 0.5)]2 b ) 0.086 64 RTc Pc m ) 0.480 + 1.574ω - 0.176ω2 (8) 1606 Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1607 Table 2. Selected Models for the Temperature Dependence of the Attractive Term a(7) in Cubic Equations of State expression for a(T) ref 1/ JT, Redlich and Kwong!? TA + mT!) Wilson?? mIT, + nIT,2 Barner et al.24 [1+ md — /7)P Soave? 1+ mT, - 120 - JT. Usdin and McAuliffe>® a(Ty) = 1+ (1 — 7,0 + n/T;) Soave?® exp[C(1 — T;")] Heyen*? [1+md — fT) +n /1 = 70.77 Raimondi'® m, + md/T, + m3/T 2 Ishikawa et al.!©! exp[C( — T;4] Boston and Mathias!® (1 +m — JT) — pd — 70.7 — TP Mathias!®° l+m(1— JT) +m(1 - JTY + m3 - JT» Mathias and Copeman®? (+m — /T PT; Bazua!4 m, + m/T, + m3/T2 + ma/T 3 Adachi and Lu!® 1+m(T.— 1) + m/T, —1) Gibbons and Laughton!© {1 +m + T.)/7 Kabadl and Danner!® [+m —./7) +nd — 70.7 -T)P Stryjek and Vera® [1 +m —- JT)? + n(T, — 0.6)2 Adachi and Sugie°? [1+ md — JT, — pd — Tq - THP Du and Guo!®* {fl +7 exp(-A)V/[1 + n-exp(-kT?") 3} T2""! exp[(1 — 72”) Twu!©? exp[m(1 — T;) +n — JT] Melhem et al.!7° exp[p(1 — 7;)|1 — Tr! + q(T! — 1) Almeida et al.?9 [l+m(i —- JOP, 0 =(T — Ty)MTe — Tp) Nasrifar and Moshfeghian!7! 2 and Mansoori*? have mentioned that when, EoS are applied to mixtures, the classical vdW mixing rules are woe Twat? for constants of an equation of state and not for any wuaeeeee Boston and Mathias? thermodynamic state function that might appear in the —_———— Soave equation of state. Therefore, they parametrized the PR ———~ Soave" 0 equation by introducing an additional parameter, re- —— — Aznar and Siva-Telles sulting in an EoS with three temperature-independent E, constants. This point is further discussed later in this 3s paper in the section on mixtures. i Volume-Translated Equations ee The volume-translation concept first suggested by KAS Martin* and developed by Peneloux et al.*4 has received AQT some attention.*>~*° Peneloux et al.*4 proposed a con- 0 +20 SEs a sistent volume correction in the SRK equation of state, ° 2 4 6 8 10 which improves volume predictions without changing Reduced Temperature,Tr the VLE conditions. The method consists of using a Figure 2. Selected temperature functions in the attractive term corrected volume V* = V + ¢, where ¢ is a small of cubic EoS for carbon dioxide. component-dependent molar volume correction factor. . . Thus, for the vdW equation the following expression is literature for the a(ZR) function. The most popular of obtained all of the generalized models is that of Soave.*® Of the models containing component-dependent parameters, RT a those of Soave?® and Almeida et al.?? have been exten- P= Vt+t-b wap (12) . ae . t—b (v+n sively studied; these expressions are _ _ Peneloux et al.*+ pointed out that other volume a= 1 ~ Tn + nit) corrections such as the correction factor of Lin and Daubert*! do not preserve this characteristic of improv- and ing a given property while keeping others unchanged. 1 1 The method has been highly recommended for calculat- a(T,) = exp[pU — 7,)|1 — T,] + qT —1)) (1) ing phase and volumetric behavior of hydrocarbon mixtures and reservoir fluids. An extension of the respectively. The parameters m and n for the Soave volume-translation concept has been proposed by Math- model and p, g, and T for the Almeida et al. model are ias et al.,4? consisting of adding another term to the available for about 500 substances for the SRK, PR, and corrected volume (V + s) proposed by Peneloux et al.*4 Patel—Teja—Valderrama (PTV) equations.*°~ 72 Other works on translated equations include applica- Figure 2 shows how some of these functions behave tions to the calculation of saturated densities of binary at low and high temperature. The main conclusions from mixtures,** to the modeling of solid phases,*+ and to all of these works is that, indeed, two-parameter cubic phase equilibrium in mixtures containing supercritical EoS can be adjusted to give good representations of PVT components.* Tsai and Chen*® developed a volume- properties of pure polar fluids by modifying the tem- translated Peng—Robinson (VTPR) equation in which perature functionality of the attractive term. Benmekki the temperature dependence of the EoS energy param- 1608 Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 Table 3. PTV Equation of State and Fugacity Coefficients p= RT _ a V-b VV+b)+c(V—b) a= a,(Tr) Q. = 0.661 21 — 0.761 057z. a= Q(R?T.7/P.7) Qy = 0.022 07 — 0.208 68z. a(Tr) =[1 + FU - JT)? Q. = 0.577 65 — 1.870 80z- b = Q(RTSP.) F = 0.462 83 + 3.582 30wz, + 8.194 17(wz,.) c = QA(RT./P.) B+(C- 127+ [A—-2BC— BP - (B+ O]Z+ (B°C + BC — AB) =0 z= 4, = 2, a A=eP pa PP cae RT *' pep 7) RTE RT Rr RT RT an __|A wa _),|22 Eas) Veer (Eo) ign ne A 1 “se Rh In — Bta. foo BO? + fact) Bex SY] [z+ P55 + fact (F55) ” " B. | “ ‘fa ZB) +73 BC+ ( 5 ) a=(@4) . w=(™4) B= (MQ) ‘ an; TV.n, ‘ an, TV.n, ‘ an; TV.n, a” = 3BC; + 3CB; + CC; + BB; B” =(—-B + C\CB; — BC) “ Expressions for the partial derivatives included in the above equations are obtained for each mixing rule to be employed. eter was regressed by an improved expression that McAuliffe>° revived the interest in this approach. Patel yields better correlations of pure-fluid vapor pressures. and Teja*! reworked the equation previously proposed The VTPR equation is comparable to other EoS in VLE by Heyen>’to obtain calculations with various mixing rules, but it yields better predictions for the molar volumes of liquid P=RTKV — b)— aacDi[WV + b) + c(V — b)] mixtures. De Sant'Ana et al.*° evaluated an improved volume-translated EoS for the prediction of volumetric The constants a,, b, and c are determined as functions properties of fluids. According to these authors, the of two substance-dependent parameters, ¢, and F. proposed equation does not present some inconsistencies Valderrama>’ generalized this equation using the acen- found by several authors for this type of EoS, an tric factor (w) and the critical compressibility factor (zc) argument that has been refuted in the literature.*° More as generalizing parameters. In another work, the author recently, Cabral et al.4° used molecular simulation had justified the use of these two properties, w and Z., results to study the performance of a translated Peng— for generalizing the parameters of a cubic EoS.** The Robinson EoS. The authors emphasized the special generalized Patel—Teja EoS, known as the PTV equa- effects of the combining rules for better predicting tion, has been successfully applied to correlate vapor— vapor—liquid equilibrium and excess properties using liquid equilibrium in mixtures.°°~°® Valderrama and the proposed translated PR—Lennard-Jones EoS. De- Alfaro*? evaluated the PTV and other generalized EoS spite the progress made on this type of modification, equations for the prediction of saturated liquid densities, translated cubic EoS have not become popular for concluding that, for this type of calculation, cubic EoS practical users. should be used with care. Xu et al.°> found that the PTV equation was the best equation for predicting VLE in Three-Parameter Equations of State CO> reservoir fluids without using interaction param- . . eters. Table 3 summarizes the PTV equation of state. It has been mentioned several times that one of the major drawbacks of vdW-type equations, is that the Application to Mixtures critical compressibility factor z. takes on fixed values, regardless of the substance (z, = 0.375 for vdW, 0.333 Until recent years, most of the applications of EoS to for RK and SRK, and 0.307 for PR). To overcome this mixtures considered the use of the classical mixing deficiency, it has been suggested that the fixed value of rules. An interaction parameter has been introduced Z, Should be replaced by a substance-dependent adjust- into the force parameter a in vdW-type equations to able critical parameter.* This approach has been mainly improve predictions of mixture properties.© It has been applied by introducing a third parameter into the recognized, however, that, even with the use of inter- equation of state. Many three-parameter equations have action parameters, the classical vdW one-fluid mixing been proposed during the past 25 years, although the rules do not provide accurate results for complex idea is not new and can be traced back to the end of the systems.°!~® During the past 20 years, efforts have been 19th century. As mentioned earlier here, Clausius!’ made to extend the applicability of cubic EoS to obtain proposed a three-parameter equation that allowed for accurate representations of phase equilibria in highly variations of the critical compressibility factor, although polar mixtures, associated mixtures, and other very the concept of z, was not well established at that time. complex systems. The different approaches presented Himpan’*’ provided a major contribution in this direc- in the literature include the use of multiple interaction tion, but the idea did not gain popularity, especially parameters in the quadratic mixing rules,°+® the because of the success of other simpler equations. The introduction of the local-composition concept,>? the studies of Elshayal and Lu,** Fuller,4? and Usdin and connection between excess Gibbs free energy models and Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1609 EoS,® and the use of nonquadratic mixing rules.°’~”? correlation for the interaction parameter in the PR Solorzano et al.”° presented a comparative study of equation in terms of the critical volumes and acentric mixing rules for cubic EoS in the prediction of multi- factors of the pure components. Gao et al.”> correlated component vapor—liquid equilibria. They concluded that the interaction parameters in the PR equation as a true evaluation of the accuracy of mixing rules is their functions of the critical temperatures and critical com- application to multicomponent mixtures. pressibility factosr of the pure components. Coutinho et al.’”° used combining rules for molecular parameters Classical Mixing Rules to propose correlations for the interaction parameters . . .. in the SRK equation as functions of the pure-component In the past, simple, classical mixing rules of the van parameters (critical properties and ionization poten- der Waals type were used in most applications tials). None of these proposals, however, have proven to be of general applicability, and at present, an accurate a— Vyxxa; predictive or correlating method for evaluating the pod interaction parameters does not exist. The available correlations and estimation methods are not always b= VY xx; (13) suitable for extrapolation and, in many cases, are only ij applicable to particular mixtures.**.7>-7° Therefore, re- gression analysis of experimental phase equilibrium c= VY xx, data is the preferred way to obtain the required inter- ij action parameters. Care must be taken, however, in . . complex systems in which multiple optimum interaction Customarily, the geometric mean was used for the force parameters could be obtained. parameter a,j, and the arithmetic mean was used for Also, combining rules for intermolecular parameters the volume parameters bj and cj. Concentration- such as the energy (€) and size (0) parameters in the independent interaction parameters have been intro- Lennard-Jones potential have been used in connection duced into aj, bj, and cj to improve the correlation of with cubic EoS. Several combining rules have been phase equilibrium. This has been done as follows proposed in the literature and different expressions for the EoS parameters (for instance, a and b in the SRK a, = faa — k;) equation) have been derived from the combining rules for « and o.334+.77-79 The results obtained using this type b.= 1 _ of combining rule are not very impressive. If accurate gf 58; + BA B;) . La . 2 results are required, one cannot avoid introducing binary interaction parameters into the mixing rules. Cy = sc, +o) — 6,) (14) Volume-Dependent Mixing Rules Volume-dependent mixing rules have also been pro- These modifications retain the quadratic concentration posed in the literature. The local-composition concept”? dependence of the EoS parameters and the quadratic has been used in this approach,®°8! and applications of concentration dependence of the second virial coefficient. cubic EoS to highly nonideal mixtures have been made However, the introduction of such parameters does not with relative success. Mathias and Copeman®? intro- improve correlations in some complex cases such as duced some modifications to the original ideas of Mol- those found in supercritical fluid processes and wine lerup, extended the Peng—Robinson EoS to complex distillation processes, among others. mixtures, and evaluated various forms of the local- Interaction parameters such as k;, jj, or 6; in eq 14 composition concept. Zheng et al.8? developed a unified are usually calculated by regression analysis of experi- density-dependent local-composition model combined mental phase equilibrium data, although some predic- with an EoS to be applied to strongly polar and tive correlations have been proposed for some mixtures. asymmetric mixtures. Conventional quadratic mixing The basic idea in this regression analysis is to apply rules (vdW one-fluid rules) were used, and the approach the EoS to the calculation of a particular property and was successfully extended to ternary mixtures. The then minimize the differences between predicted and model failed to give accurate representations of polar— experimental values. The values of the interaction polar systems such as methanol—water and acetone— parameters that minimize these differences correspond water. Inaccuracies were also found in the critical to the optimum interaction parameters. The differences region. between calculated and experimental values are ex- Sandler* presented a theoretical analysis of the van pressed through an objective function that is arbitrarily der Waals partition function to conclude that several but conveniently defined. Several objective functions models based on the local-composition concept, including have been presented in the literature,7!~7? with the those of Mollerup®° and of Whiting and Prausnitz,*! do most popular being those that include deviations in the not satisfy necessary boundary conditions required by bubble pressure. statistical mechanical analysis. Sandler recognized, Some authors have attempted to obtain binary inter- however, the importance of Mollerup’s works in the action parameters from pure-component property data. development of better mixing rules for EoS. The main Chueh and Prausnitz*> related the binary interaction contribution was to recognize that the application of EoS parameter in the RK equation to the critical volumes to mixture is not restricted to the use of quadratic of the pure components. Graboski and Daubert” cor- mixing rules. related the binary interaction parameter in the SRK . . equation in terms of the solubility parameter difference Nonquadratic Mixing Rules between the hydrocarbon and non-hydrocarbon compo- Quadratic mixing rules are usually sufficient for the nents. Arai and Nishiumi’* developed a semiempirical correlation of phase equilibrium in simple systems. To 1610 Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 Table 4. Selected Mixing Rules and Combining Rules Used in Two-Constant Cubic Equations of State mixing/combining rule formulas van der Waals a= xX: b= x.x,D;, one parameter: ki Ly or Ly oe two parameters: kj, lj 1 ay = Jaa — kj) by = xi + bj — U,) Panagiotopoulos—Reid (PR) ay = Jaall — ki + (ki — kjaxi) two parameters: kj, kji 1 . three parameters: kjj, kji, lj b= 30; + bj) — 1) general nonduadratic (GNQ) ay = [aja 1 — ki) ki = Oxi + Ox; two parameters: 0j, 0; _ _ _ three parameters: 0;, 0), Bi by = 0.5[b(1 — 6) + 51 B)I fi ~ 0 for all solutes and §; = 0 for all solvents Kwak—Mansoori (KM) aj = ./aat1 — kj) by = 0.5(b)'? + B37. — By) three parameters: ki, Bij, Lij di = 0.5(d,"3 + d)® (1 — ly) Kwak—Mansoori modification 1 (KM-1) aj = o/aa,1 — ky) bj = 9.5[b(1 — Bi) + b(1 — B)I three parameters: kj, li, 8; (one solute) dj = 0.5(d;!7 + dP (1 — Ly) /; = 0 for all solutes and f; = 0 for all solvents Kwak—Mansoori modification 2(KM-2) ay = J a,a,1 — ki) dj = 0.5(d;!? + dj)? three parameters: 6,;, 6;, 8; (one solute) kij = Oix; + Ox; bi = 0.5[bi1 — Bi) + b(1 — B)) fi ~ 0 for all solutes and §; = 0 for all solvents Kurihara et al. (KTK) a= VY xxlaa)” — (t— Pgeps/In[(b — db — 1) three parameters: 11, 72, 3 ij : 1 E 2. b= VY xxi by = 3 +B) res = RTx)x2[M) + My% ~ Xz) + 30% — 2)" ij Wong-Sandler b = LYxix(b — alRT);i/[1 — Xxjalbj;RT — AE(@/QRT] one parameter: ki a= b[Lxjia/b; — AE(x)/Q] two parameters: kj, /; (one solute) (b — alRT) jj = 0.5[bK1 — 1) + bj] — (aiaj)°°C. — kij/RT 1; # 0 for all solutes and J; = 0 for all solvents treat more complex systems, Panagiotopoulos and Reid®’ Wong and Sandler (WS)*® proposed a mixing rule for introduced a second interaction parameter by making two-parameter cubic EoS consistent with statistical the k,; parameter concentration-dependent, thus trans- mechanical requirements. In particular, the model gives forming the mixing rule in a nonquadratic form the quadratic concentration dependence of the second virial coefficient. The WS mixing rule is a,j = faall — ky + (ky — kx) with k; ~ kj; (15) _ by = VY xx(b — alRT),/ Expressions similar to that of Panagiotopoulos and Reid E have been presented by Adachi and Sugie® and by [1 — )\x,a/b,RT — A, (XVQRT] (16) Sandoval et al.8> The classical and nonquadratic mixing rules can be summarized in one general form, called the an = byl), x,a,/b,+ A*( xQ “general nonquadratic mixing rule”,®° in terms of two parameters 6; and 6;. That is, kj = 6x; + 6)x; This _ _ _ 0.574 mixing rule, although it suffers from the so-called (b — alRT),, = (8; + b)l2 — (ajay) 1 — Ky RT Michelsen—Kistenmacher syndrome, has been success- . _. . fully applied to binary mixtures containing a supercriti- This mixing rule has been the focus of sever al studies cal component.%6 during the past several years.87°°-° Verotti and Costa®’ presented an extensive study on the use of the WS EoS + Gibbs Free Energy Models mixing rule to correlate liquid—liquid equilibrium in 47 polar binary liquid mixtures. The authors also consid- Among the modern approaches presented in the ered the mixing rules of Huron and Vidal,°° Heidemann literature to describe phase equilibria in mixtures, and Kokal,?! and Dahl and Michelsen®? and the NRTL methods of the type “EoS + Gibbs free energy” seem to and UNIQUAC models for the excess Gibbs free energy. be the most appropriate for modeling mixtures with The study shows that, for strongly polar—nonpolar highly asymmetric components. The basic concepts mixtures, the combination WS + NRTL gives the best related to this type of model can be found in Orbey and results and, for strongly polar + strongly polar mixtures, Sandler®’ and Sengers et al.® Since the first proposals the combination WS + UNIQUAC gives the best results. of Vidal8’ and Huron and Vidal,°®* these models have Yang et al.! extended the application of the WS been extensively used and applied to low- and high- mixing rule to three-parameter equations. They used pressure vapor—liquid mixtures, to liquid—liquid equi- the Patel—Teja equation of state and tested the model libria, and gas—solid equilibria. Important contributions using VLE data for binary and ternary mixtures. A to this area are those of Mollerup,®° Kurihara et al.,°° comparison with results obtained using classical vdaW Michelsen,” Heidemann and Kokal,°! Dahl and Mich- mixing rules showed much better results for highly elsen,°* Holderbaum and Gmehling,®? Soave,?4 and asymmetric mixtures and for conditions near the critical Wong and Sandler.*° regions of the mixtures. Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1611 Table 5. Mean Deviations in Vapor Mole Fraction for Table 5 gives some selected results for five solid—gas EoS + Wong—Sandler Mixing Rule (WS) and Three systems. It should be noted that deviations in the gas Equations of State? solute mole fraction, y2, are presented. The solvent y2 (%) concentration is correlated with deviations below 0.5% SRK PR... PTV. PTV in all cases. Also, we have developed a new model using no. system T(K) WS/KTK WS/KTK WS/KTK MRS the PTV equation of state with a residual contribution “| naphthalene 308 1.3/1.1 -1.1/23 1.03.9 12_ determined by a modified regular solution model that 313. «15/12 13/724 11/41 2.1 considers the polar and hydrogen-bonding contributions. 318 2.4/2.6 1.9/5.3 1.5/5.3 2.6 For the nonpolar part, the concepts of regular solutions 2 phenanthrene 318 7.9/7.6 7.3/5.7 6.3/7.1 5.7 (for which the excess volume and excess entropy are 0) 328 8.3/7.3 5.5/7.0 3.4/7.8 2.4 are used to derive a mixing rule for the force parameter 338 8.3/8.4 7.1/6.5 6.3/8.4 6.4 a in the equation of state.!°! The last column in Table 3 anthracene 303. -6.6/10.7. 5.9/19.3. 5.2/8.1 2.6 . : 323 15.7/72 162/186 191/95 64 5 (PTV—MRS) also reports the results obtained with 343. 9.2/9.9 8.4/21.3. 7.1/11.2 9.5 this new model. * *"Vaphthalene 318 7126 S388 43/45 28 Mixing Rules of Mansoori and Co-workers 328 «2.7/4.3 4.3/6.7 3.7/1.8 5.2 A new concept for the development of mixing rules 5 2,6-dimethyl- 308 6.6/4.9 5.5/2.4 5.4/6.4 4.2 for cubic EoS consistent with statistical mechanical naphthalene 318 6.4/5.1 4.0/3.5 3.1/7.2 2.8 theory of the van der Waals mixing rules was introduced ; 328 IS.7IS.S 5.6/5.2 4.2/8.8 5.0 by Benmekki and Mansoori.*? This concept is based on 6 caffeine 313) 4.4/15.3 2.2/15.6 2.0/15.5 2.4 y pen : P 333 -4.3/22.8 2.5/24.2 28/229 23 statistical mechanical arguments and the fact that rules 353. -—s-6.3/20.1. 4.1/20.9 4.1/19.9 7.7 are for constants of an equation of state and not for any % average 6.5/7.7 5.0/10.4 4.6/8.7 4.1 thermodynamic state function that might appear in an deviation equation of state. For these mixing rules, the cubic EoS “Results obtained using a modified regular solution (MRS) must be rewritten. For instance, the Peng—Robinson model are also included. ” In the mixing rule the UNIQUAC model EoS is reformulated as for AE was used. . _ . _ RT _ Am + RT dy — 2d gRT Kurihara, Tochigi, and Kojima®® proposed a mixing PE Fo ——_ (19) rule (KTK) based on a convenient separation of the V— by VV + by) + OV — Bp) excess Gibbs free energy as This form suggests three independent EoS parameters E_ ,E E Am, Dm, and d,), which are expressed using the classical § Bas + 8res ay) ae der Waal mixing rules hs . where Shs is the excess Gibbs free energy for a regular non solution and oRES is the residual excess Gibbs free a, = y’yix Xai energy. Kurihara et al.8? derived an expression for the i=1j=1 regular solution contribution using a general cubic non equation of state, and for the residual contribution, the _ used a Redlich—Kister expansion. For the force param. Pm = yy Pi (20) eter, the mixing rule is given by Sul non am = VV xa(aia;” ~ d= Ldreidy ij i=1j= E (T — )8res/Inlb — ob — 1)] (18) The combining rules for aj, bj, and dj given by Kwak EB 5 and Mansoori (KM) are Sees = RTX X99) + My (1 — Xp) + 3 Hy — X2)'] . a, = (aa) — k;) and the mixing rules for the other parameters are detailed in Table 4. b)? +b)? A recent paper by Wyczesany®® presents a critical b= >. dad — Bis) (21) analysis of several EoS + excess Gibbs free energy models for the correlation of vapor—liquid equilibria of di? + ap’) several mixtures at high pressures. The author indicates di = (gai) d — 6) that, for mixtures containing a supercritical component, this type of model is not sufficiently accurate. In our a,=a(Tc)( + my group in La Serena, we have extensively explored both ' ' ' WS and KTK mixing rules for describing solid—gas and 0.077 80RTc; liquid—gas mixtures containing supercritical carbon b,= Pa (22) dioxide. We have been successful in correlating phase “i equilibrium properties in liquid—gas and solid—gas a(Tc,)m 2 binary systems by introducing an additional parameter d- a (1;) into the solute volume constant only. This modified : RTc; mixing rule is shown in Table 4. For solubility calcula- tions, this modified WS mixing rule has an acceptable Figure 3 shows results for the vapor-phase concentra- physical meaning and does not suffer from the so-called tion for the system 2-methyl-1-pentanol + CO» at high Michelsen—Kistenmacher syndrome. pressure and 453 K using the PR equation with the above mixing rules. Our results show that, as for other complex mixing rules, correlation of VLE is not im- proved as expected by the complexity and the alleged foundations of the mixing and combining rules. For complex systems, more than one interaction parameter must be used if accurate results for the solute concen- tration in the gas phase are needed. The clear advantage of the KM reformulation is that an EoS that includes temperature-independent parameters only is obtained. Applications of Cubic EoS to Reservoir Fluids The application of EoS to correlate VLE and proper- ties of reservoir fluids has received special attention. Important contributions include those of Yarborough,102 Firoozabadi et al.,103,104 Vogel et al.,105 Willman and Teja,106-108 Lira-Galeana et al.,109 and Skjold-Jorgen- sen.110 Cubic equations of state have shown surprisingly good capabilities for correlating VLE and volumetric properties of complex reservoir fluids.103,111-115 Two approaches are usually used in these applications. One of these approaches is the use of pseudocomponents, that is, grouping the mixture into a limited number of fractions, with each fraction having specific critical properties and acentric factor. These properties are calculated using well-known standard correlations. The system then becomes a defined mixture of a given number of pseudocomponents. The other approach is the representation of the properties of a naturally occurring reservoir mixture through a continuous distribution, using some characteristic property such as the molec- ular weight or the normal boiling temperature. This method is known as “continuous thermodynamics”, a concept that was already used by Bowman and Edmis- ter more than 50 years ago116,117 and reformulated later by Ratzsch and Khelen118 and by Cotterman et al.119 using the equations of state. Other works on continuous thermodynamics include those by Cotterman and Praus- nitz,120 Du and Mansoori,121 Haynes and Mathews,122 Zuo and Zhang,123 and Vakili-Nezhaad et al.124 Xu et al.55 and Danesh et al.57 evaluated the perfor- mance of several cubic EoS for predicting phase behav- ior and volumetric properties of reservoir fluids. Satu- ration pressures, liquid and gas densities, and equilib- rium ratios for several multicomponent mixtures were correlated using the selected EoS. The main conclusions obtained from these works are as follows: (i) the modified Patel-Teja equation (PTV)53 and the Zud- kevitch and Joffe modified Redlich-Kwong equation60 are, overall, superior to all of the other EoS; (ii) the abilities of the SRK and the PR equations to predict liquid density were improved by the inclusion of the volume translation concept; (iii) the phase volumes obtained by flash calculations were unsatisfactory with all of the equations tested; and (iv) phase concentrations were reasonably well predicted by all of the equations at all conditions. Mixtures Containing Supercritical Components Equations of state are usually employed for high- pressure phase equilibria such as systems containing supercritical components. The first efforts to model the phase behavior in systems containing supercritical fluids were made using the virial EoS,125 but these attempts were not successful. The best results have been obtained using cubic EoS such as SRK and PR.126,127 Several combinations between cubic EoS and mixing rules have been employed and have been presented in the literature. These included different applications and modifications of the PR and SRK equations with mixing rules such as van der Waals, Panagiotopoulos-Reid, Kwak-Mansoori, Huron-Vidal, Kurihara et al., and Wong-Sandler, among others.128-139 The book by Sa- dus140 includes a chapter dedicated to cubic equations applied to phase equilibria in multicomponent mixtures. However, none of these works represents a thorough study on the problem of phase equilibrium in systems containing a supercritical fluid. Thus, the problem is not yet exhausted, and there is ample room for research on different aspects of phase equilibrium modeling. Literature information and our own research indicate that, with the present state of knowledge, it is necessary to include more than one interaction parameter, even in complex models such as the Kurihara et al. and Wong-Sandler mixing rules. A common practice in several applications of cubic EoS to mixtures containing supercritical fluids has been to analyze the results in terms of the concentration of the supercritical solvent in the gas phase, y1. Some optimistic statements and conclusions found in the literature are usually drawn by analyzing the vapor- phase concentration of the solvent only (usually carbon dioxide) and not the concentration of the solute, y2, as should be done to accurately test the capabilities of an Figure 3. Gas-phase concentration vs pressure in the system 2-methyl-1-pentanol + CO2 using the Peng-Robinson EoS and Kwak-Mansoori mixing rules and some modifications. 1612 Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 1613 Table 6. Best Combination of EoS + Mixing Rules for in this area in our group in La Serena. For all of the Several Gas~ Liquid and Gas—Solid Systems Containing cases shown in Table 6, the pressure is predicted with Supercritical Carbon Dioxide deviations below 10% and the solvent concentration system T(K) P(MPa) EoS mixingrule* Ay (%) with deviations below 1%. The solute concentration in J-octanol --403.—~—s«6.5-18.4 PTV. GNQ 5.2. the gas phase is predicted with variable deviations. 453 +6.5-19.0 PTV GNQ 81 However, these deviations are below those reported in 1-decanol 348 = 7.0-19.0 PTV GNQ 12.4 the literature for similar systems. 403. 6.0-19.0 PR KM-2 10.5 453 6.5-19.0 SRK GN' 3.4 . . . 2-methyl-1- 348 65-119 PR ONO 13 Cubic EoS Applied to Polymer Solutions pentanol 403, 65-154 PTV — KTK 7.3 For polymer—solvent and polymer—polymer mixtures, i, 453, 6.5-17.9 PTV GNQ Ll several polymer-specific EoS have been proposed that lauric acid 423 9.1-5.1 PR KM-2 15.5 +o : . : 473. 91-51 PR KM2 Bi can be classified into two broad groups: lattice models palmitic acid 423.» «9.1-5.1 PR KM.2 143 and continuum models. "4° Applications of cubic EoS to 473. 9.1-5.1. SRK ws 178 describe VLE in this type of mixture have been under- oleic acid 313 7.2-28.2 PR KM-1 15.1 taken with some success.!47~!°° 333» 7.1-28.6 PTV KTK 17.1 Sako, Wu, and Prausnitz!*© (SWP) used the van der limonene 313, 5.9-7.9 PTV vdW-2 11.6 Waals theory to propose a cubic EoS applicable to large 323, 4.9-10.3, PR KM-2 19.3 molecules and polymers and extended it to mixtures o-pinene 313, 3.3-7.9 PR KM 6.1 using simple mixing rules. The main advantage of the 323 4,579.6 PTV GNQ 9.6 proposed equation is that it can be applied to polymer— 328 4.8-9.5 PR KM-1 15.3 : . : phenanthrene 318 11.9-27.7 PTV GNQ 45 solvent systems with a minimum of experimental 328 11.9-27.7 PTV GNO 30 information. The SWP equation has been used with 338 11.9-27.7 PTV GNQ 6.7 relative success by Tork et al.!>4+1!55 Using a different benzoic acid 318 11.9-27.7 SRK KTK 71 approach, Orbey and Sandler!*’? combined the SRK 328 11.9-27.7 SRK KTK 4.0 equation with the Flory—Huggins activity coefficient 338 11.9-27.7 PTV MRS 0.9 model in a Huron—Vidal EoS + G* mixing rule. They caffeine 313° 19.7-29.7 PTV MRS Is analyzed binary polystyrene—hydrocarbon solvent mix- 333 19.7~29.6 PTV MRS 2.7 tures, obtaining acceptable results. The authors con- 353 19.7-—29.6 PR vdW-2 3.5 . anthracene 303. 103-41.0 PTV MRS 26 cluded that, unless extensive VLE data for accurately 323 +-9.0-41.0 SRK MRS 6.4 correlating the data and determining the model param- 343 11.7-20.6 PTV GNQ 78 eters are available, the use of multiparameter phase naphthalene 308 10.4-29.7 PTV KTK 1d equilibrium models for the moderately polar polymer— 313. 9.9-34.6 PTV MRS 2.1 solvent mixtures studied is not justifiable. Orbey et al.!#° 318 10.1-31.1 PTV MRS 2.6 used a polymer SRK equation to correlate polyethylene— 2,6-dimethyl- 308 9.6—27.7 PTV MRS 4.2 ethylene mixtures and compared the results with the naphthalene 318 9.6277 PTV — KTK 2.4 Sanchez—Lacombe and SAFT calculations. Other EoS ; 328 9.5~27-7 PIV ws 43 + G®* models have been used with some success.!*° 2,3-dimethyl- 308 9.8-27.7 PTV WS 2.4 K : - 149 : : naphthalene 318 9.8-27.7 PTV GNQ 32 alospiros and Tassios found satisfactory results with 328 9.8-27.7 SRK KTK 1.8 the use of a simplified Wong—Sandler mixing rule B-cholesterol 313: 9.9-24.8 PR KM 4.4 proposed by Zhong and Masuoka (ZM).'°° 323. -9.9-24.8 PR KM 4.5 Louli and Tassios!** applied the PR equation to 333 12.9—24.8 PR vdW-2 5.5 polymers including a single set of energy and co-volume @ Notation for mixing rules as indicated in Table 4. parameters per polymer (a and b) fitted to experimental volume data. Excellent results for the volumetric be- EoS. In the studies presented in the literature, although havior of the polymer up to very high pressures were the deviations in the calculated solvent concentration obtained. Correlations of VLE data for a variety of in the gas phase (which is usually on the order of 0.999) nonpolar and polar polymer solutions, including hydro- are lower than 1%, the deviations in the calculated gen-bonding ones, were carried out by using three solute concentration (which is close to 0) can be as high mixing rules. The best results were found with the ZM as 200% when expressed as percent deviations: Ay (%) mixing rule.!°° Kang et al.!°” used the PR equation with = 100[(vexp — Yeat)Vexp].4°-4!- 4 These high deviations the Wong-—Sandler mixing rule to calculate bubble-point in the gas-phase solute concentration (y2) are not usually pressures and vapor-phase mole fractions for several reported and discussed in papers related to phase polymer mixtures. This model was found to give gener- equilibrium modeling of mixtures containing a super- ally good results away from critical regions and except critical component using cubic EoS. Not reporting these for nonpolar polymers in polar, nonassociating solvents. high deviations is at least a misleading way of analyzing These studies show that mixing rules for polymer— the accuracy of a proposed model. solvent mixtures need to be further investigated. Although an accurate general conclusion cannot be Future developments of EoS for polymer mixtures are drawn at present, results indicate that, for mixtures not clear, and some contradictory statements can be including a supercritical component, the use of Gibbs found in the literature. Some authors indicate that cubic free energy models in the EoS parameters and nonqua- equations can be extended to accurately correlate and dratic mixing rules with interaction parameters in the predict VLE in polymer mixtures. !47:!5?-!>3 Other authors volume constants of the EoS give the best results. state that, considering the complexity of this type of However, one cannot avoid including more than one mixture, it seems that simplicity is not a necessary interaction parameter in the mixing rules if accurate requirement for an EoS, with the calculation of param- correlations of the solute concentration in the gas phase eters for the mixture components being more impor- are needed. Table 6 summarizes the work we have done tant.!°8 There is agreement, however, on the fact that future developments of EoS for polymer mixtures must emphasize the study of mixing rules and that the EoS input parameters should be related to common mea- sured properties of the polymers. Recommendations On the basis of the abundant information available in the literature and our own findings, some general recommendations on which cubic EoS to use for different applications are summarized in Tables 7 and 8. The recommendations must be considered as general guide- line, and they do not mean that other EoS or mixing rules could not be used for the applications listed in the tables. For instance, in several cases for pure fluids (Table 7), the SRK and PR equations are recommended, although other cubic equations could also be used. However, these two EoS are of common use, have been widely studied, and are incorporated in most commercial and academic software. Thus, for the practical user, there is no reason to use other equations. Also, in some cases in Table 7, empirical correlations are recom- mended, although some cubic EoS could give equally good results. However, the good results found for some applications cannot be generalized. This is the case for liquid volume, for which empirical correlations and generalized models have been demonstrated to be accurate and of general applicability.59,159 Similar comments can be made for the recommenda- tions given in Table 8. Although several other mixing rules can be used in some applications, the list in this table includes those commonly used and frequently studied in the literature (classical vdW-2, P&R, Wong- Sandler) This fact does not mean that for some particu- lar cases, models such as Kwak-Mansoori, Kurihara- Tochigi-Kojima, Zhong-Masuoka, or other mixing rules could not give similar or better results. The recommendations given in Table 8 mean that those rules give reasonable results for the wide group of substances indicated in the first column of that table. Conclusions Through the years, many researchers have discovered and taken advantage of the tremendous capabilities of cubic EoS. These equations have been modified and Table 7. Recommendations on Generalized Equations of State to Use for Several Pure-Fluid Properties property comments recommendations gas volume at moderate/ high pressure most cubic EoS with two or three parameters SRK, PR, PTV gas volume at low temperature, moderate/high pressure most cubic EoS with two or three parameters adjusted using low-temperature data SRK, PR saturated vapor volume most cubic equations SRK, PR, PT saturated liquid volume for nonpolar fluids three-parameter equations seem to be better PTV, PT, but empirical correlations should be preferred saturated liquid volume for polar fluids two- or three-parameter equations with parameters adjusted for polar fluids PT, but empirical correlations should be preferred compressed liquid volume none of the cubic equations empirical correlations or specific equations for a given fluid should be used volume near the critical point most cubic EoS fail in this region, but three- parameter equations should be preferred PTV or other three-parameter EoS with parameters adjusted using near-critical data vapor pressure for nonpolar fluids most cubic EoS, although those that use more involved a((T) functions give better results SRK, PR, PTV, but prefer R(T) with specific parameters for polar fluids, such as Soave-polar28 vapor pressure for polar fluids most cubic equations, although those that use specific parameters for polar fluids should be preferred PR, PTV with complex R(T) function, such as that of Twu169 vapor pressure for associating fluids none of the cubic equations noncubic equations specially developed for this type of fluid should be preferred enthalpy and entropy of liquids none of the cubic equations give accurate results specific equations, usually polynomial, should be preferred enthalpy and entropy of gases at low pressure most cubic equations, although those that use specific parameters for polar fluids should be preferred SRK, PR, PTV, but prefer R(T) with specific parameters enthalpy and entropy of gases at moderate/ high pressure most cubic equations, although those that use specific parameters for polar fluids should be preferred PR and PTV with R(T) with specific parameters; noncubic equations are also good Table 8. Recommendations on Generalized EoS and Mixing Rules for Different Types of Liquid-Vapor Mixturesa type of mixture EoS temp function mixing rule low pressure (<10 atm) nonpolar + nonpolar SRK, PR, PTV Soave26 P&R, WS nonpolar + polar SRK, PR, PTV Soave-polar28 P&R, WS polar + polar SRK, PR, PTV Soave-polar,28 Mathias163 WS asymmetric mixtures SRK, PR, PTV Twu P&R, MWS-1P polymer solutions SRK, PR, SWP Soave,26 Mathias163 HV, MWS-1P, ZM moderate and high pressures (>10 atm.) nonpolar + nonpolar SRK, PR Soave26 vdW-1 or -2 nonpolar + polar SRK, PR, PTV Soave-polar28 vdW-1 or -2 polar + polar SRK, PR Soave-polar,28 Mathias163 vdW-2, P&R nonpolar + nonpolar SRK, PR Soave26 vdW-1 or -2 reservoir fluids SRK, PR, PTV Soave26 vdW-1 or -2 polymer solutions SRK, PR, SWP Soave,26 Mathias163 WS, ZM one supercritical component PR, PTV Soave,26 Twu169 P&R, WS-2P, WS-3P a Notation for mixing rules as indicated in Table 4. Also ZM is Zhong-Masuoka, HV is Huron-Vidal, and SWP is Sako-Wu-Prausnitz. 1614 Ind. Eng. Chem. Res., Vol. 42, No. 8, 2003 applied to almost any situation in which they have been needed. When applied to pure components, the main modifications include changes in the EoS volume de- pendency and in the temperature dependency of the attractive term. For mixtures, several theories have been devised to propose new mixing and combining rules. One of the most successful to date has involved the use of Gibbs free energy models in the EoS param- eters. Theoretical approaches have not been practically successful, and those that have been more accurate and useful employ noncubic EoS. Also, we know that a single cubic EoS cannot provide reliable predictions for all volumetric, thermodynamic, and phase equilibrium properties, for all type of fluids and mixtures. Therefore, we cannot expect any major developments in EoS until we better understand how molecules interact. However, considering their over- whelming advantages, cubic EoS can be adjusted in many ways to find acceptable results for most practical applications. Thus, it is my impression that, for the time being and for the near future, cubic equations of state are here to stay. Acknowledgment The author acknowledges the support of the National Commission for Scientific and Technological Research (CONICYT, Chile) through Research Grant FOND- ECYT 1000031; the Direction of Research of the Uni- versity of La Serena, Chile, for permanent support through several research grants; and the Center for Technological Information (CIT, La Serena, Chile) for computer and library support. Literature Cited (1) van der Waals, J. D. On the Continuity of the Gaseous and Liquid States. Ph.D. Dissertation, Universiteit Leiden, Leiden, The Netherlands, 1873 (English Translation by Threlfall and Adair172). (2) Boyle, R. New Experiments Physico-Mechanical. Touching the Spring of the Air (1660). 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