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Equations of state in three centuries. Are we closer to arriving to a single model for allapplications?

Kontogeorgis, Georgios M.; Liang, Xiaodong; Arya, Alay; Tsivintzelis, Ioannis

Published in:Chemical Engineering Science: X

Link to article, DOI:10.1016/j.cesx.2020.100060

Publication date:2020

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Citation (APA):Kontogeorgis, G. M., Liang, X., Arya, A., & Tsivintzelis, I. (2020). Equations of state in three centuries. Are wecloser to arriving to a single model for all applications? Chemical Engineering Science: X, 7, [100060].https://doi.org/10.1016/j.cesx.2020.100060

https://doi.org/10.1016/j.cesx.2020.100060

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Chemical Engineering Science: X 7 (2020) 100060

Contents lists available at ScienceDirect

Chemical Engineering Science: X

journal homepage: www.elsevier .com/locate /cesx

Equations of state in three centuries. Are we closer to arriving to a singlemodel for all applications?

https://doi.org/10.1016/j.cesx.2020.1000602590-1400/� 2020 The Authors. Published by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Abbreviations: APACT, Associated Perturbed Anisotropic Chain Theory; BTEX, Benzene-Toluene-Ethylbenzene-Xylene.; COR, Chain-of-Rotators; CPA, CuAssociation; EoS, Equation of State; LCVM, Linear Combination of Vidal and Michelsen mixing rules; LLE, Liquid-Liquid Equilibrium; MEA, Ethanol Amine; MEG,Glycol; MHV2, Modified Huron-Vidal Second Order mixing rules; NRHB, Non Random Hydrogen Bonding; NRTL, Non-Random Two-Liquid model; PC-SAFT, PerturbeStatistical Associating Fluid Theory; PHCT, Perturbed Hard-Chain Theory; PR, Peng-Robinson Equation of State; PSRK, Predictive Soave–Redlich–Kwong modStatistical Associating Fluid Theory; SAFT-VR, Statistical Associating Fluid Theory – Variable Range; SGE, Solid-Gas Equilibrium; SLE, Solid-Liquid Equilibrium; SRKRedlich-Kwong Equation of State; UNIFAC, UNIQUAC Functional-Group Activity Coefficients; UNIQUAC, Universal Quasi-Chemical model; vdW1f, van der Waalsmixing rules; VLE, Vapor-Liquid-Liquid Equilibrium; VLLE, Vapor-Liquid Equilibrium; VTPR, Volume-Translated Peng–Robinson.⇑ Corresponding author.

E-mail address: [email protected] (G.M. Kontogeorgis).

Georgios M. Kontogeorgis a,⇑, Xiaodong Liang a, Alay Arya a, Ioannis Tsivintzelis b

aCenter for Energy Resources Engineering (CERE) & KT-Consortium, Department of Chemical and Biochemical Engineering, Building 229, Technical University of Denmark, DenmarkbDepartment of Chemical Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 November 2019Received in revised form 10 February 2020Accepted 15 February 2020

Keywords:Equations of stateCubic Equations of StateSAFTCPAMixing rulesPhase equilibriaWaterElectrolytes

Equations of state represent the cornerstone of thermodynamic models. They have the potential of beingused for a wide range of properties, systems and conditions and thus in many practical applications ofrelevance to chemical, petroleum, environmental and other types of engineering disciplines. Equationsof state with great potential appeared first in the 19th century in the form of the van der Waals (and othersimilar ones) and for almost 150 years, both these classical and more advanced equations of state haveshown great potential. This manuscript presents the opinion of the authors on the capabilities - limita-tions, current status and future challenges of equations of state. It is not a comprehensive review and onlyequations of state which, according to the authors’ views, have potential general applicability are dis-cussed. This work is an account of the authors’ experience based on their many years’ collaborationand involvement with a wide range of diverse general equations of state. We feel that, while the topicis still of great importance, it has stagnated a bit during the 21st century. We also believe that severalof the personal views expressed in this manuscript may be contested and diverse opinions are available,but we consider that, should this be the case, this is also very positive as this manuscript can contribute toa healthy debate on the actual status and future perspectives of this important field.� 2020 The Authors. Published by Elsevier Ltd. This is an open access articleunder the CCBY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Contents

1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1. Diversity of applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2. Diversity of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Cubic equations of state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1. Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2. Pure component parameters estimated in two ways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3. Mixing rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3. Beyond cubic equations of state – The advanced association theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74. Future challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125. Concluding remarks and some future expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

bic PlusEthylened Chain -el; SAFT,, Soave-

one-fluid

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Nomenclature

List of Symbolsa,b Parameters in the equation of stateg Gibbs Energyk12 Equation of state binary interaction parameterl12 Equation of state binary interaction parameterR Gas constantT TemperatureV VolumeVw van der Waals volumex Mole fraction

Greek Symbolsc Activity coefficient

d Parameter defined in Eq. (1), Solubility parameteru Volume fraction

Subscriptsi,j Component i or j

Superscripts1p Infinite pressure0p Zero pressurecomb Combinatorial termfv Free volumeE Excessres Residual contribution

2 G.M. Kontogeorgis et al. / Chemical Engineering Science: X 7 (2020) 100060

Declaration of Competing Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Appendix A. The MEA parameter estimation for the CPA equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1. Motivation

We start with the motivation of the current ‘‘opinion” manu-script as well as an explanation on what this manuscript is not. Itis not a comprehensive review of all possible thermodynamic mod-els (for this there are several textbooks available (e.g. Prausnitzet al., 1999; de Hemptinne et al., 2012; McCabe and Galindo,2010; Sandler, 2017; Kontogeorgis and Folas, 2010) but the aimis to present the authors’ opinion on what is today’s status withthermodynamic models and whether we are now closer than –say – 30 or 50 years ago to arriving to a single model for ‘‘all” appli-cations. Indeed, the manuscript is a quest for thermodynamic mod-els which have the potential of being widely used. Thus,specialized multiparameter equations of state (despite being oftenuseful) are not considered here. We do not contest the usefulnessof such multiparameter equations of state but, as we see it, theirapplication is limited to specific fields and they are not widespreadused in process and product design.

1.1. Diversity of applications

Thermodynamic models are required in a wide range of indus-tries (oil and gas, chemical, petrochemical and biochemical, mate-rial, pharmaceutical and others) for design of processes andproducts and sometimes for applications in which diverse proper-ties, such as various types of phase equilibrium (i.e. VLE, LLE, VLLE,SLE, SGE. . .), thermal, derivative, interfacial properties should berepresented, or other properties, such as transport properties,should be used. The task of having a single or even just few modelsfor such wide range of needs is immensely difficult due to the largenumber and diversity of compounds and mixtures involved (hy-drocarbons/oil, alcohols, water, polymers, electrolytes, biochemi-cals, proteins, enzymes,. . . - in all possible combinations) overwide ranges of conditions (concentration, temperature, pressure),types of phase behavior, but often the ‘‘actual industrial needs”may be different. Whereas for some applications a precise calcula-tion of the phase behaviour may be needed, in other cases such asfor product design and certain polymer-related applications, sim-pler and sometimes qualitative estimations of solubility areadequate.

Another aspect is the diversity of applications, for examplewhether applied thermodynamics and the associated models canbe useful beyond the ‘‘classical applications”, which are typicallythose within the petroleum and chemical industries and whichtoday also include carbon dioxide (CO2) capture, transport andstorage. More recently, use of thermodynamic models in environ-mental engineering and mechanical engineering has gained inter-est (Sandler, 2017; Kontogeorgis and Folas, 2010), whereasprogress is much slower in the biotechnology field, despite excel-lent encouragement by some of the pioneers in the field(Prausnitz, 1989, 2003; von Stockar, 2003). Indeed, even thoughbiotech processes and products can be designed without detailedthermodynamics, its usefulness in the future may become evidentin a wide spectrum of applications, ranging from the prediction ofphysico-chemical properties of biomolecules and the prediction ofphase equilibria for downstream processing (bioseparations), tounderstand the structural and functional stability of proteins andother biomolecules, in biocatalysis (effect of temperature and pres-sure conditions, pH, solvents and solutes on activity and selectiv-ity), biotransformation (prediction of reaction equilibria),efficiency of bioprocesses (correct formulation of driving forcesand equilibrium position) and maybe cellular growth (thermody-namic effects including heat generation) and cellular metabolism– metabolic engineering e.g. optimizing biomass and productyields. Some consider the above ‘‘wishful thinking”, as essentiallyno modern or advanced thermodynamic models are today adoptedby top Biotech companies (NovoNordisk and Novozymes, 2017,personal communication), but the future may bring new conceptswhich can turn out to be useful.

Some of the aforementioned challenges for thermodynamicscan be considered a bit ‘‘exotic” but the trend of using thermody-namics in such fields is not new and can be traced several decadesago. While we see no problem in pursuing such tasks, we certainlyagree with the statement of S. Zeck from BASF more than 25 yearsago (Zeck, 1991):

‘‘The problems discussed very briefly and certainly not comprehen-sively show that there is still considerable potential for improve-ment for phase equilibrium thermodynamics even in long-established areas of the chemical industry. With all the enthusiasm

G.M. Kontogeorgis et al. / Chemical Engineering Science: X 7 (2020) 100060 3

about the possibilities of thermodynamics in new areas, it is neces-sary first to concentrate on performing the basic tasks”

Basic tasks mentioned by Zeck (1991) were referring to e.g. pre-dicting ternary LLE using binary parameters based on binary dataor having models which can handle both VLE and LLE needed incertain separations. We are not sure if his comments and sugges-tions were very much noticed, as his manuscript has received avery small number of citations. This is a manuscript where webelieve that the impact could and maybe should have been muchgreater.

1.2. Diversity of models

Inspiration for the current manuscript and its title can be alsotraced to John M. Prausnitz and an academia-industry discussionpublished in Fluid Phase Equilibria in 1983 (Prausnitz et al. 1983).In that study, more than 35 years ago, Professor Prausnitz antici-pated many new developments in the years to come:

� (Semi) theoretical equations of state (EoS) for complexmixtures.

� Abandonding quadratic mixing rules.� EoS for petroleum fractions and for polymers.� New techniques for incorporating in EoS scaling laws for thecritical region of pure fluids and mixtures.

� A comprehensive framework for multicomponent mixturescontaining aqueous electrolyte solutions with hydrocarbonsand weak electrolytes as well (CO2, ammonia, acetic acid).

And in addition to the above he was hoping for some type ofconsolidation of models, moving towards unique models in theyears to come. This is what he wrote in the 1983 manuscript(Prausnitz et al. 1983):

‘‘At present,. . . we use the Wilson equation for mixture of waterand ketone; the Redlich-Kwong EoS for a mixture of methaneand CO2; the Flory-Huggins for polymer solutions and someform for the Debye-Huckel theory for aqueous solutions of salts.I see a trend which will both extend and consolidate many ofthese theories and correlating equations. At present, appliedthermodynamics is a tool box with very many tools, eachdesigned for a particular job. I expect that, if current trends con-tinue, in a few years we will have not only better tools but alsofewer tools for covering a much wider range of problems”.

So, are we closer to arriving to a single model for all applica-tions? This is an underlying topic of the current manuscript.

Such a ‘‘single model”, if existed, might have the form of anequation of state (EoS), as from an EoS a wide range of thermody-namic properties can be derived, not limited to phase equilibria butalso enthalpies, densities and even activity coefficients, to mentionjust a few.

While the beginning of EoS can be traced back to the excellentideal gas EoS (PV = nRT), excellent in terms of its simplicity andlack of adjustable parameters, this is naturally a model of limitedapplicability. One division of equations of state (other categoriesor ways of presentation can be perceived) includes the virial equa-tion, the specialized multiparameter BWR-type or Span-Wagnerand other related EoS, the cubic equations of state, and theadvanced molecular EoS (Statistic Associating Fluid Theory(SAFT)-type and like) (Kontogeorgis and Folas, 2010). Only the lasttwo EoS categories will be considered here, again using the crite-rion of potential greater generality compared to other types ofEoS (which can be very useful and very accurate, but for a morelimited range of applications). The virial equation can only be used,

in its usual form, for vapor phase applications. However, a specialnote should be made for the multiparameter EoS (often expressedin forms of reduced Helmholtz energy functions) (Span andWagner, 1996; Span, 2000; Wilhelmsen et al. 2017; Kunz andWagner, 2012). These models are very useful for a number of sys-tems (pure compounds andmixtures) and for a wide range of prop-erties but often limited to certain type of systems, e.g. natural gas,CO2, refrigerants, etc. They have rarely been used for a wider rangeof polar and associating compounds. Span and Wagner (1996) pre-sent a well-established model for CO2 (an extremely well-citedwork) and Kunz and Wagner (2012) present one extension of thesemodels for mixtures. Span (2000) and Wilhelmsen et al. (2017)present reviews of such models, while Wilhelmsen et al. (2017)focus also on various computational challenges.

All these various models (and more e.g. explicit activity coeffi-cient models both general and specialized ones such as those forelectrolytes and other modeling theories such as COSMO-RS(Klamt et al., 1998)) are in use by industry and often even withinthe same company, as shown recently by Dr. Gerard Krooshof fromDSM (Krooshof, 2016).

Based on the above, the questions we will address in this workcan be summarized as follows:

� What are the capabilities and limitations of equations of state aswe know them today?

� How is the performance of cubic and non-cubic EoS comparedto each other?

� What is the true application spectrum of cubic and non-cubicEoS (mainly based on association theory)? Both in terms of sys-tems and properties.

� Where do we stand today? What have we learnt from almost150 years of work with equations of state?

� What are the future challenges and what can we expect that thefuture will bring?

2. Cubic equations of state

Sometimes we read in articles, even in established thermody-namic textbooks, statements about the cubic equations of state(EoS) which are not very precise and, in some cases, simply untrue.Some of these ‘‘myths” about cubic EoS include that they are:

� Completely empirical models.� Not providing good results for mixtures containing complex(polar, hydrogen bonding) fluids.

� Not performing well for size-asymmetric systems e.g. gas/heavyhydrocarbons or mixtures of hydrocarbons of different size.

� Not applicable to polymer systems.

The origin of the cubic EoS can be traced back to the van derWaals equation of state and van der Waals himself who, in hisNobel prize speech, was rather more positive and optimistic aboutthe theoretical origin of his equation of state, both in terms of itsmolecular origin and the nature of the EoS parameters. He had sta-ted (van der Waals, 1910):

”It will be perfectly clear that in all my studies, I was quite con-vinced of the real existence of molecules, that I never regardedthem as a figment of my imagination. . . When I began my stud-ies I had the feeling that I was almost alone in holding thatview. . . Many of those who opposed it most have ultimatelybeen won over, and my theory may have been a contributoryfactor”.”The two constants that appear in the theory have a real phys-ical meaning, namely that of molecular volumes and attraction


and no one will deny that the theory will influence progress inthis field”.

He also did not consider that the parameters of the EoS shouldbe temperature-independent (van der Waals, 1910):

”I have never been able to consider that the last word had beensaid about the equation of state and I have continually returnedto it during other studies. As early as 1873 I recognized the pos-sibility that a and b might vary with temperature, and it is wellknown that Clausius even assumed the value of a to be inverselyproportional to the absolute temperature”.

We agree with these comments and the general optimism aboutcubic EoS. In reality, cubic EoS can accommodate much of theabove ‘‘so-called limitations” and they exhibit many positive char-acteristics e.g. they are representations of the Corresponding StatesPrinciple and contain terms which account (largely accurate inmany cases) for both free-volumes (entropic effects) and energeticinteractions. The simple van der Waals one-fluid mixing (vdW1f)rules satisfy the theoretically-correct quadratic dependency ofthe second virial coefficient and the geometric mean combiningrule often used for cross-energetic interactions is justified basedon the theory of intermolecular interactions (London theory).Finally, cubic EoS can predict 5 out of 6 types of phase diagrams(van Konynenburg and Scott classification) types I-V (vanKonynenburg and Scott, 1980).

What is sometimes forgotten is, when evaluating the capabili-ties and limitations of cubic EoS (and any other EoS for that mat-ter), that we should take a careful look at the ‘‘details” (as ‘‘thedevil is indeed in the details” even in this context). In addition tothe form of the EoS itself (its functionality), the crucial ‘‘details”include the pure component and mixture parameters and morespecifically, how the pure compound parameters are estimated(based on which data) and how the model is extended to mixturesi.e. which mixing and combining rules are used. In many cases, andwe will show examples, the type of data used in parameter estima-tion and the way of estimating the model parameters (for purecompounds and mixtures) can have such a dramatic influence onthe results which may overshadow the functional form of theEoS itself.

Let us look at some of the details, which help appreciating thetrue value of cubic EoS.

2.1. Functionality

One way to understand (cubic) EoS and their mixing rules is byobserving and analyzing the excess Gibbs (GE) or Helmholtz (AE)energies and activity coefficient expressions derived from them.In the case of GE or AE expressions, the expressions are generaland no mixing/combining rules are involved, while the activitycoefficient expressions depend on mixing and combining rules.For example, the van der Waals EoS using the classical vdW1f mix-ing and combining rules (with all interaction parameters set tozero) can be written as (under the assumption that the excess vol-ume is zero, see Kontogeorgis and Economou (2010):

lnci ¼ lnccomb�fvi þ lncresi

¼ lnufv

i

xiþ 1�ufv

i

xi

!þ Vi

RTdi � dj� �2u2

j

� �ð1Þ

ufvi ¼ xi Vi � bið ÞP

jxj Vj � bj� � di ¼

ffiffiffiffiai

pVi

where c is the activity coefficient, u the volume fraction, x themolar fraction, V the volume, while b and a are the equation of state

parameters (superscripts comb-fv and res represent combinatorialfree volume and residual, respectively).

The similarities between the van der Waals EoS written in thisform and classical models used for polymer solutions, shown inEqs. (2a) and (2b) are evident (Kontogeorgis and Economou2010). These are the Flory-Huggins (2a) or Free-Volume combina-torial (size) terms (2b) combined with the Hildebrand solubilityparameter theory:


¼ lnui

xiþ 1�ui

xi

� �þ Vi


j

� �ð2aÞ


¼ lnufv

i

xiþ 1�ufv

i

xi

!þ Vi


j

� �ð2bÞ

where the volume and free-volume fractions are defined as (Vw isthe van der Waals volume and d is the usual solubility parameter):

ui ¼ xiViPjxjV j

ufvi ¼ xi Vi � Vwið ÞP

jxj V j � Vwj� �

It should thus not be so surprising that even the vdW EoS can beused for polymer solutions and it has indeed been used for bothpolymer-solvent VLE and LLE and actually many more cubic EoSe.g. Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) have beenapplied successfully to polymer solutions (using diverse mixingrules), see Sako et al. (1989) for a review and pertinent references.The first cubic EoS for polymers was, to our knowledge, proposedby Prausnitz and co-workers (Sako et al. 1989) and it has beenextensively used for polymer-solvent VLE including high pressures.Also, such model is mentioned in the DSM review (Krooshof, 2016)and is still used today in industrial polymer-related applications(Moebus and Greenhalgh, 2018).

Having said the above, both for the van der Waals and the Sako-Wu-Prausnitz (Sako et al. 1989) equations of state lie their valuelargely to historical reasons and/or for specific applications. Today,cubic EoS are used in practice mostly in the form of the SRK and PRequations of state. Hundreds other cubic EoS have been proposedand forgotten as their functionality has not offered any advantagecompared to SRK and PR. In itself, SRK and PR are overall rathersimilar (with the former a bit better for some pure componentfugacities and the latter for densities) and any of them could havedominated, but –maybe again for historical reasons– both are usedtoday in different parts of the globe in engineering practice.

2.2. Pure component parameters estimated in two ways

The pure component parameters of cubic EoS are typically esti-mated from the critical point constraints in combination withvapor pressure data (ignoring liquid densities and other properties;although a volume translation can easily correct the liquid densi-ties without affecting the phase behavior performance). Such esti-mated pure compound parameters follow non-linear trends withcarbon number or molecular weight for families of organic com-pounds e.g. alkanes, thus making difficult to extrapolate to heaviercompounds. When, however, the same pure component parame-ters are estimated simultaneously from vapor pressure and liquiddensity data (in the same way as done in SAFT and othertheoretically-based EoS), then the pure compound parameters fol-low smooth trends with carbon number or molecular weight. Thishas been shown by several researchers (Ting et al., 2003;Tsivintzelis et al., 2015, 2016). Thus, a characteristic originallythought to be present only in theoretically-based models is alsoshared by cubic EoS when the parameter estimation is carried


out in the same way, as described, for the advanced equations ofstate.

2.3. Mixing rules

Mixing and combining rules can have a profound effect on theperformance of cubic EoS. For example, it has been shown(Kontogeorgis and Folas, 2010; Kontogeorgis and Coutsikos,2012) that when the so-called a/b-mixing rule (ab ¼

Pixi

aibi) is used

in cubic EoS like SRK and PR, very good predictions of activity coef-ficients in asymmetric alkane mixtures e.g butane or hexane withdiverse n-alkanes are obtained. If the same EoS is used with theclassical vdW1f mixing rules, then the predicted activity coeffi-cients are entirely wrong (even qualitatively; predicting positiveinstead of negative deviations from Raoult’s law). This is illustratedfor one example in Fig. 1.

Indeed, improved mixing rules are obtained when the a/b mix-ing rule is used, preferably combined with an explicit externalactivity coefficient model, as done in the so-called EoS/GE modelswhich dominated in the relevant scientific literature in the 1980sand 1990s. Our own preferred model and what we consider as‘‘benchmark” for all new models is the Huron-Vidal mixing rule(Huron and Vidal, 1979):

gE;1P

RT¼ 1

RT

Xi

xiaibi

� ab

!¼ gE;res;M

RTð3Þ

where the superscripts 1, res and M in the excess Gibbs energy, g,indicate infinite pressure and the ‘‘residual” contribution of theexternal activity coefficient model, denoted as M.

As discussed elsewhere (Kontogeorgis and Folas, 2010;Kontogeorgis and Coutsikos, 2012; Kontogeorgis and Economou2010) using the a/b mixing rule, the ‘‘size or entropic” term of acubic (SRK or PR) EoS appears to be (at least qualitatively) correct.Moreover, when such a mixing rule is combined with a local-composition based ‘‘residual” (energy term) such as NRTL or theenergetic part of UNIQUAC, an overall very powerful equation ofstate is obtained. The resulting model (SRK or PR using Huron-Vidal mixing rule) is clearly superior to both SRK when classicalmixing rules are used (both in terms of asymmetry and represen-tation of energetic effects) and to NRTL which alone cannot be used

Fig. 1. Activity coefficients at infinite dilution of n-butane in alkane solvents at 373 K andatmospheric pressure as a function of the alkane carbon number using the Peng-Robinsonequation of state. Results are shown using the vdw1f mixing rules (with kij = 0) and the a/bmixing rule (which is essentially theHuron-Vidal mixing rulewithout the term for ‘‘residual”interactions): ab ¼

Pixi

aibi. The results in Fig. 1are obtained using the linearmixing rule for the

co-volume. For comparison are shown the results with the modified UNIFAC activitycoefficient model. From Kontogeorgis and Folas (2010). Reprrinted with permission.

at high pressures and does not contain a ‘‘clear” size (entropic)term either. This is partially due to the possibility of incorporatingmore adjustable energetic parameters (in the NRTL part of theseadvanced mixing rules) compared to the classical mixing rulesand partially due to the local composition nature of NRTL (andother similar models) which permits accounting for polar effects(to some extent).

The SRK with Huron-Vidal mixing rules can be, thus, used forboth low and high pressure phase equilibria, both simple and com-plex polar mixtures, both VLE and LLE, both binary system correla-tions and predictions of multicomponent VLLE of e.g. water-glycol-alkanes and some illustrative results are shown in various literaturestudies (Kontogeorgis and Folas, 2010; Kontogeorgis and Coutsikos,2012; Kontogeorgis and Economou, 2010; Folas et al., 2007).

Some have considered such so-called EoS/GEmodels to be essen-tially a ‘‘clever mathematical trick” but it is a very important one, asthe ‘‘best of two worlds” (cubic equations of state and the localcomposition activity coefficient models) are combined. The latter(models like Wilson, NRTL and UNIQUAC) were, prior to the adventof association theories, our best choices for representing phaseequilibria of complex fluids (polar, associating ones). According tode Hemptinne et al. (2012), the availability of EoS/GE modelsconsiderably widens the applications range of cubic equations ofstate, since, if models are well parameterized, virtually all complexsystems can be accurately described.

Some have considered it a problem that in the Huron-Vidalmethodology (as the infinite reference pressure is used), existing(previously estimated from low pressures) parameters of the activ-ity coefficient models cannot be used. Thus, one should re-fit theinteraction parameters of the incorporated local-compositionactivity coefficient model. This has led to the development of thezero or approximate zero reference pressure models like MHV2and PSRK with the purpose of using existing tables of interactionparameters including those of UNIFAC (more information is pro-vided in Kontogeorgis and Folas (2010), Kontogeorgis andCoutsikos (2012), Michelsen and Mollerup (2007), Mollerup(1986), Boukouvalas et al. (1994), Vidal (1997). In our view, thisoverall discussion is a rather minor issue as, for accurate design,the binary interaction parameters should best be adjusted basedon binary data (which often are available) and then multicompo-nent predictions can be performed. Anyhow, significant efforts todevelop zero reference pressure models have not been very suc-cessful when such models should be expressed in the form ofexplicit mixing rules. Such explicit zero reference pressure modelslike MHV2 and PSRK are only ‘‘approximate” zero reference mod-els, as shown in literature (Kontogeorgis and Coutsikos, 2012;Kontogeorgis and Economou, 2010; Kalospiros et al. 1995).

This can be further clarified from the comparison of the follow-ing equations 4a and 4b, which express the difference betweenexcess Gibbs energies at infinite (gE;1P) and zero (gE;0P) or approx-imately zero reference pressure.

General equation and result for cubic EoS (derivation inMichelsen and Mollerup (2007)):

gE;P

RT� gE;0P

RT¼Z P

0

VE

RTdP ) gE;1P

RT� gE;0P

RT

¼Xi

xilnbbi

�Xi

xilna

bRT

� �ai

biRT

0@

1A ð4aÞ

Difference between infinite and approximate zero referencepressure models (as in MHV1 and PSRK):

gE;1P

RT� gE;approx:0P

RT�Xi

xilnbbi

ð4bÞ


This ‘‘approximate zero reference pressure” character meansthat thesemixing rules (MHV1,MHV2, PSRK and other similar ones)do not reproduce the activity coefficient model they are combinedwith, and this poor reproduction becomesworse at higher asymme-tries. Moreover, the resulting model contains a so-called ‘‘doublecombinatorial term difference” (difference between the combinato-rial term of the external activity coefficient e.g. UNIFAC and a Flory-Huggins term stemming from the ‘‘approximate” zero referencepressure approximation) (Kontogeorgis and Coutsikos, 2012;Kontogeorgis and Economou, 2010). This ‘‘combinatorial term dif-ference” should be zero, as Mollerup (1986) had once stated, but itis not (!) and it increases with size asymmetry. This has the conse-quence of affecting negatively the overall successful combinatorial(entropic) term of the cubic EoS and at the same time makes it dif-ficult to estimate unique group parameters e.g. for the CO2/alkanegroup and similar ones. While models like LCVM (Boukouvalaset al., 1994) have appeared which empirically eliminate this doublecombinatorial difference, the best approach is possibly to entirely‘‘forget” the zero reference pressure approximation and go back tothe Huron-Vidal mixing rules, which is essentially what is done inmostmodern successfulmixing rules approaches. Possibly themostpromising approach is that by Gmehling and co-workers with theVTPR EoS (Schmid and Gmehling, 2016), as these authors are refit-ting the group energetic interaction parameter by correctly usingonly the UNIFAC ‘‘residual” contribution and in the context of theEoS, thus aiming at a group contribution-based EoS which retainsall the positive characteristics mentioned above.

At this stage we should also mention the highly-cited and verywell-known andwidely usedmixing rule proposed byWong and San-dler in 1992 (Wong and Sandler, 1992). This model is special inmanyways. It uses the infinite reference pressure model and the finalexpression for a/b is given by Eq. (3), i.e. the Huron-Vidal mixing rule,existing activity coefficientmodel parameters can be used in themix-ing rulee.g. UNIFACparameters and themixing rule for the co-volumeparameter is chosen so that the quadratic mixing rule for the secondvirial coefficient is obeyed. The Wong-Sandler mixing rule is indeedvery interesting and it has been extensively discussed in anotherreview by the authors (Kontogeorgis and Coutsikos, 2012). As men-tioned, in some respects, while using the same interaction parametersas inUNIFACmodels, theWong-Sandlermixing rulesbearsmany sim-ilarities with the Huron-Vidal mixing rules, as essentially the sameformfor thea/b rule isusedaswellas the infinitepressure limit isused.The ‘‘trick” with the Wong-Sandler mixing rules is the use of anon-linear mixing rule for the co-volumewhich permits maintainingthe same UNIFAC parameters even if the infinite zero referencepressure condition is used. However, this is done at a cost of an adjus-table parameter in the mixing rule for the co-volume and there aresome issues as extensively discussed by Kontogeorgis and Coutsikos,2012.

While all these recent models return to the Huron-Vidal mixingrule, several of them modify the linear mixing rule for the co-volume parameter, using either other functionalities of the com-bining rule for the co-volume parameter or a correction parameterin the classical arithmetic mean cross co-volume parameter (lij). Ithas not as yet been fully evaluated what is the influence of thesealternative combining rules in the context of cubic EoS e.g. foractivity coefficients, but there is little doubt that small changesin the co-volume combining rule may result to big changes inthe results, as in the activity coefficient expression, the combiningrule of the co-volume appears in several parts of the cubic EoS(both in the size and energetic terms). This is shown in the equa-tion below (for the infinite dilution activity coefficient of SRK usingthe classical vdW1f mixing rules and for a binary system):

lnc11 ¼ lnccomb�FV1 þ lncres1

¼ lnV1 � b1

V2 � b2

� �þ 1� V1 � 2b12 � b2ð Þ

V2 � b2

� �

þ a1b1RT

lnV1 þ b1

V1

� �þ a2b2RT

V1b2 þ V2b2 � 2V2b12

V22 þ V2b2

" #

� a2b2RT

lnV2 þ b2

V2

� �2a12a2

þ 1� 2b12

b2

� �ð5aÞ

where:

a12 ¼ ffiffiffiffiffiffiffiffiffiffia1a2

p1� k12ð Þ ð5bÞ

b12 ¼ b1 þ b2

21� l12ð Þ ð5cÞ

In all the above equations, a, b, and V have their usual meaningsas the energy, co-volume parameters and the molar volume,whereas 1,2 indicate the two components of a binary mixture.

Previous studies (Kontogeorgis et al., 1998) have shown that asingle lij parameter around 0.02 (and without use of kij in thecross-energy term) is adequate for the correlation with PR of VLEof the whole ethane/alkane series, possibly providing an additionalsmall improvement in an already good size term of the EoS.

In conclusion, the power of cubic EoS is indeed amazing, maybedue to partial cancellation of errors (Gibbs statement that ‘‘thewhole is simpler than the sum of its parts”), but we believe thatthe examples above indicate more than that.

In a review on Industrial Needs in Physical Properties, Guptaand Olson (2003) state that:

‘‘Most past work focused on extensions of cubic vdW-type EoS . . .

This has occurred despite the now famous recommendation of Hen-derson. No matter how sophisticated a mixing rule, the use of vander Waals-type cubic equations of state forces their inherent limi-tations on the users.”

While we sympathize with these and other statements pointingout some of the deficiencies of cubic vdW-type EoS, we believe thatthe advantages clearly outperform the limitations and this explainsthe widespread use and acceptance of cubic equations of state byindustry.

Tsonopoulos and Heidman (1986) stated that ‘‘We, in the petro-leum industry, continue to find such simple EoS are very useful high-pressure VLE models, and we found as yet no reason to use complexnon-cubic equations of state”. Ten years later (1998) during a con-ference in Denmark, Jack Heidman repeated ‘‘cubic EoS are hereto stay” and Hendriks (2011) stated that ‘‘the general philosophy isto use standard and proven methods such as the PR equation ofstate. . ..” – and more advanced methods ‘‘whenever and as long as atrue gap exists”.

Maybe for all of these reasons, expert colleagues with profoundand broad knowledge of equations of state have expressed similarthoughts. A colleague at DTU (professor Michael L. Michelsen) hasoften stated that cubic equations of state are ‘‘A technology thatrefuses to die”.

And Marco Satyro in a personal communication shortly afterPPEPPD conference in 2016 has written to one of the co-authorsabout cubic EoS:

� ”Qualitative predictions are not good in some cases but correlativepowers are formidable”

� ”There are simple fixes we can do to the cubic estimation frame-work that still allow us to use it reliably to model asymmetricsystems”


� ”I have not seen yet a detailed study showing an objective compar-ison between a cubic and SAFT if both equations of state are param-eterized in the same way”

Since the development of the well-known cubic equations ofstate (SRK, PR) and the clarification of the capabilities (and limita-tions) of the EoS/GE mixing rules, not many new developments offundamental nature in the field of these classical models appearedin literature. The area seems to be rather mature and these cubicEoS are now standard approaches in both process and reservoirsimulators. One notable exception with new interesting develop-ments is the work by J.N-Jaubert, R. Privat and co-workers on thedevelopment of the predictive group contribution-based methodsto estimate the interaction parameter (inspired by the EoS/GE

approach) (Vitu et al., 2008) and the recent work by the sameauthors on the analysis and criteria to determine the correct tem-perature dependency of the energy parameter of cubic EoS(Guennec et al., 2016). This may be very significant as several prop-erties from the cubic EoS such as excess enthalpies and entropiesare sensitive on the temperature dependency of the energy param-eter (via its temperature derivative da/dT; this derivative does notappear in e.g. fugacity and excess Gibbs or Helmholtz energies).

We share the excitement about the capabilities of cubic equa-tions of state and conclude that they are very powerful models,which can be used for both size-symmetric and asymmetric sys-tems, even for polymers with very good results and for polar sys-tems using the EoS/GE mixing rules. We believe that thefunctional form of cubic equations of state provides the right mag-nitude of the size (combinatorial-free volume or entropic) effects,at least with respect to activity coefficient and in a qualitativelycorrect manner. The best approach in using cubic equations of statewith the advanced EoS/GE mixing rules is by ‘‘eliminating” the dou-ble combinatorial difference, or in other words by using the origi-nal Huron-Vidal formulation. We believe that cubic equations ofstate combined with the Huron-Vidal mixing rule and a suitablelocal composition activity coefficient model like NRTL is a state-of-the-art approach, possibly one of the best we have today inapplied thermodynamics and certainly a ‘‘benchmark” modelagainst which modern SAFT EoS and the like should be comparedto. A few such comparisons are already available in the literature,as further discussed below (Kontogeorgis and Folas, 2010;Kontogeorgis and Coutsikos, 2012; Kontogeorgis and Economou,2010; Folas et al. 2007; Tsivintzelis et al., 2014a, 2014b).

At this stage, we should mention that much of the success of theEoS/GE approach/mixing rules depends on a careful parameteriza-tion of the local-composition models. This is not always easy. AsPrausnitz et al. (1980) wrote as early as in 1980 for UNIQUAC itis sometimes necessary to fit the interaction parameters simulta-neously for ‘‘providing effectively constraints on the binary param-eters, preventing them from attaining values of little physicalsignificance”. Or in other words, both types of data, VLE and LLE,may be needed in the parameter estimation so that the local com-position model can be used for both VLE and LLE. Still, when suchparameter estimation is done carefully, even the simplest cubicEoS, van der Waals, has been shown by Soave (1984) to predictcomplex multicomponent VLE when coupled with the Huron-Vidal mixing rule and NRTL. And in a number of studies we havecarried out (Kontogeorgis and Folas, 2010; Kontogeorgis andCoutsikos, 2012; Folas et al., 2007; Tsivintzelis et al., 2014a,2014b) we have demonstrated a great capability of equations ofstate with the NRTL/Huron-Vidal mixing rules for a wide range ofsystems from binary LLE of aniline-water and water-hydrocarbons to multicomponent VLE, VLLE and LLE of water/alcohols, glycols/hydrocarbons and CO2/water/methane. Whilethese results are obtained with two or three adjustable parametersper binary system, especially the predictive capability of the

SRK/Huron-Vidal/NRTL model for multicomponent systems isimpressive, and possibly improved compared to what the local-composition models alone are capable of (many of the aforemen-tioned systems are at very high pressures). So, do we need more?This is discussed in the next section.

3. Beyond cubic equations of state – The advanced associationtheories

Jean Vidal stated 25 years ago in connection with a PhD coursehe taught (see also Vidal (1997)) after the developments of EoS/GE

models, in which he played a major role, that ‘‘time for easy successhas passed by. . .”. He explained what he had in mind by referring, asan example Vidal (1997), to two types of systems where, accordingto him, cubic EoS have reached their limitations; mixtures contain-ing molecules of very different sizes (e.g. methane/ higher alkanes)and methanol-ethylene VLLE. For the latter he commented thatsuch systems could not be (then) described by any model and ingeneral the prediction of LLE or three-phase equilibria brings usto the limits of our models as, in addition to complex interactions,such three-phase equilibrium calculations also require speciallyrobust algorithms.

We do not agree on the first example (cubic EoS with advancedmixing rules can perform very well with size-asymmetric systems)but we agree on the second example. And indeed there may be cir-cumstances where more advanced models than cubic EoS would beneeded. In many cases, phase equilibrium calculations with cubicEoS can be particularly sensitive to interaction parameters e.g.for CO2-alkanes using vdW1f mixing rules (zero adjustable interac-tion parameters [kij = 0] is not possible for these systems) and thereare cases e.g. solid-gas equilibria where two interaction parame-ters may be needed again with the vdW1f mixing rules. This ishighly unfortunate as such interaction parameters are very muchcorrelated and cannot be generalized. In some cases, even for sim-ple systems like CO2/methane, it is difficult to obtain an excellentcorrelation of various properties (phase equilibria and densities)using the same interaction parameter (Li and Yan, 2009).

While some of the aforementioned problems are attributed tomixing rules and can be addressed with EoS/GE mixing rules, thereare some important characteristics of complex mixtures which arenot sufficiently captured by cubic EoS, thus indicating a need formore advanced equations of state.

Opinions on what may be missing from cubic EoS differ, but wedo not consider as crucial limitations for cubic EoS their repulsiveor attractive terms or the ‘‘apparent” lack of a chain term, sincesuch limitations seem to be effectively overcome in various ways(and there is also an element of cancellation of errors betweenthese terms). Non-cubic EoS like Perturbed hard-chain theory(PHCT) (Beret and Prausnitz, 1975), Associated Perturbed Anisotro-pic Chain Theory (APACT) (Ikonomou and Donohue, 1986), Chain-of-Rotators (COR) (Chien et al., 1983) and many others which wereproposed in the 1970 s and 1980 s as replacements of cubic EoS byintroducing ‘‘better” theoretically more correct attractive, repul-sive and chain terms (compared to those of cubic EoS) failed (inthe sense that no significant improvement over cubic EoS wasobtained) and they are today forgotten. They are not available incommercial simulators and are even abandoned by the academiccommunity.

But there is one effect which is so important and in our view itsdescription with cubic EoS even with the EoS/GE mixing rules andwith the best local composition models we have today is not opti-mal. This effect, which is indeed missing from cubic EoS, is anexplicit account for the hydrogen bonding or, in general, the Lewisacid-Lewis base interactions. Cubic EoS with EoS/GE mixing rulesaccount for such effects indirectly via the local composition con-

Fig. 2. DEA (Diethanolamine)-n-Hexadecane LLE with CPA at 1.013 bar with anoptimal value of kij . The lines are results with two different sets of DEA parameters.The CPA parameters are (in units, respectively, L/mol, bar (L/mol)2, –, bar (L/mol)and – �1000) equal to 0.0956, 20.3587, 1.9880, 89.7961, 149 (set-1) and 0.0942,17.4953, 2.1510, 109.6677 and 136 (set-2). The 4C scheme is used for DEA. Theerrors in vapor pressure, liquid density and heat of vaporization in the Tr-range0.45. �0.9 is, respectively, 2.6, 2.3, 1.03 (set-1) and 2.9, 1.9 and 1.2 (set-2).


cept but an explicit direct treatment (can be hoped that it) mayyield better results and with fewer interaction parameters. Thisbecame clear also based on input from industry. For example, thedevelopment of the Cubic Plus Association (CPA) model startedwhen Shell considered that the in-house cubic EoS could notdescribe satisfactorily VLLE for multicomponent water-alcohol orglycol-hydrocarbon systems (Hendriks and Mejer, 1995).

Even van der Waals in his Nobel-prize speech was aware of thisserious limitation of cubic equations of state, as he stated (van derWaals, 1910): ‘‘In fact, bluntly speaking, the result would be an equa-tion of state compatible with experimental data is totally impossible.No such equation is possible, unless something is added, namely thatthe molecules associate to form larger complexes” ”I have termed itpseudo-association to differentiate it from the association which is ofchemical origin” ”What is the origin of this complex formation, thispseudo association?” He does not clearly conclude that this pseudoassociation is due to hydrogen bonding but we would like tobelieve that this is what he meant!

To address some of the limitations of cubic EoS, especially thecomplex interactions, many advanced non-cubic EoS have beendeveloped, based on chemical, quasi-chemical (lattice) or perturba-tion theories. Thesemodels have different expressions for the phys-ical interactions (attractive, repulsive, chain terms), but all of themhaving in common that an association term is included, accountingfor hydrogen bonding and related strong interactions. Anothercommon feature for these models is their rather time consumingdevelopment as these models contain at least five or six pure com-ponent parameters for hydrogen bonding compounds. We will getback to this point later. The perturbation theory (Wertheim,1984a, 1984b, 1986a, 1986b), which resulted in the developmentof the SAFT type models (McCabe and Galindo (2010), is possiblythe most well-known trend among these theories. However, asEconomou and Donohue (1991) have discussed in detail, there areclear similarities in the hydrogen bonding terms of such models(based on quasi-chemical (lattice) or perturbation theories) for anumber of specific cases. This was also shown by Tsivintzeliset al. (2008), who compared the hydrogen bonding term of SAFTwith the corresponding term of a quasi-chemical (lattice) theoryfor alkanol mixtures, and from Hendriks et al. (1997) who presentthe equivalence between such theories in a more general way.

SAFT is, as mentioned, the most popular one of these theories,originated from the work of Wertheim (1984a, 1984b, 1986a,1986b) for chain and association effects, but, in reality, put in theform of an equation of state by Chapman et al. (1990). SAFT con-tains additive contributions from various effects (attraction, repul-sion, chain, association), is based on the segment concept and theparameters have a clear physical meaning, following smoothtrends for families of compounds against the molecular weight.These characteristics are sometimes used for interpretations andextrapolations but the parameters are typically obtained fromregressing vapor pressure and liquid density over extensive tem-perature range (typically up to a reduced temperature of 0.9 or0.95), thus excluding critical point, which is overestimated. Todaywhen we use the term ‘‘SAFT”, we are essentially talking about afamily of many different SAFT EoS, of which PC-SAFT (Gross andSadowski, 2001, 2002) is the most well-known and widely usedversion (with over 1500 citations and about 80 citations/year forthe original publication of the model from 2001).

Such SAFT models differ in many respects but mostly in how thephysical interactions are accounted for. Common in all is the use ofthe association term. In our view, the association term may well bethe most significant contribution in SAFT, which remains, to a largeextent, unchanged since the very first developments (Chapmanet al., 1990). The number and type of association sites (associationschemes) can be freely defined according to the type (nature, geom-etry, functional groups) ofmolecules, thus permitting the incorpora-

tion in the same framework of different types of schemeswhich candescribe different association phenomena e.g. linear oligomers inalcohols and 3D-networks for water. Despite the great importanceof the association term, until the development of CPA in 1996(Kontogeorgis et al., 1996) (combination of cubic EoS physical termwith the association term of SAFT), there have been only few appli-cations of SAFT for associating fluids, in most cases with unsatisfac-tory results e.g. as shown by Economou and Tsonopoulos (1997) forwater-hydrocarbons. Thus, theworkwithCPAwhich started in 1995entailed some risks as it was uncertain whether the cubic function-ality and the cubic molecular approach could be combined with thesegment-type association term of SAFT, whether the associationterm was any good and, last but not least, whether the significantcomplexity of the association term could be managed, especiallyfor practical applications and industrial use.

Today we know that the association term does work very welland it has been shown to be equally successful for both CPA andvarious SAFT variants, for a wide range of associating compoundsand also that it can be represented in a way, as shown byMichelsen and Hendriks (2001), which permits easy use andimplementations in computational tools.

Today, almost 30 years from SAFT’s advent (and 25 years fromCPA), we have a rather clear picture of the capabilities (but alsothe limitations) of these models. We have seen that the SAFT/CPAmodels yield impressive phase equilibria results for a very largenumber and wide range of binary and multicomponent systems(associating mixtures, oil systems, polymers, pharmaceuticals,. . .),as shown in numerous publications (see for some reviews inPrausnitz et al., 1999; de Hemptinne et al., 2012; McCabe andGalindo, 2010; Kontogeorgis and Folas, 2010, Kontogeorgis andEconomou, 2010; Folas et al., 2007; Tsivintzelis et al., 2008, 2018;Grenner et al., 2008).

The most important successes of these models are their capabil-ities to represent ‘‘details” of the phase diagrams and also to pre-dict (often very successfully) multicomponent multiphase (e.g.VLLE) equilibria over extensive temperature/pressure and concen-tration ranges using binary parameters obtaining exclusively frombinary data. For example, with CPA over 100 multicomponent sys-tems have been studied since 1999 e.g hydrocarbons-water-MEG,methanol (LLE, VLLE, hydrate curves), oil-water-MEG, acid gases(CO2, H2S)- water –hydrocarbons– glycols, alcohols, mixtures with

Fig. 3. DEG (Diethylene glycol)-CH4-Water system: Experimental data and CPApredictions.CH4 solubility in the solution (liquid phase) and water solubility in CH4

(gas phase) vs DEG mole% in the binary solution of DEG and Water at 298 K and 30/50/80 bar. Lines (solid lines for CH4 solubility and dashes lines for water solubility)represent the prediction of CPA. Symbols represent the experimental data. Pure andbinary parameters are from Tsivintzelis et al. (2012) while the DEG kij’s with methaneand water are 0.1638 and �0.115, respectively.

Fig. 4. MEG (ethylene glycol) – Water – n-Octane LLE. Experimental data (points) andCPA predictions (lines). Pure fluid parameters were adopted from Tsivintzelis et al.2018, while the kij used for CPA are 0.0251 for MEG-octane, �0.0165 for water-octaneand �0.028 for water-MEG; the first two obtained from fitting binary LLE and the latterfrom binary VLE data. All are temperature-independent and have been used for varioustemperatures (both for the binary and ternary systems).

Fig. 5. P-xy plot for methanol-propane at 20 �C using the simplified PC-SAFT and CPAequations of state. Both models accurately represent the azeotrope at very low methanolconcentrations which is important in the design of relevant separation processes(Propene/ Propane Splitters). Conventional cubic EoS fail to predict the azeotropeformation at these extreme conditions. There are no experimental data at the area ofazeotrope but there is industrial evidence for its existence. The CPA parameters are takenfrom Tsivintzelis et al., 2012, the PC-SAFT parameters of propane are from Gross andSadowski (2001), the PC-SAFT parameters of methanol are from Gross and Sadowski(2002) and Liang et al. (2012).


mercaptans and other sulfur compounds as well as for a widerange of chemicals (water-DME-ethanol, biodiesel and esters-water-alcohols, acetic acid-water-polar chemicals, CO2-dimethylcarbonate-water-methanol, octanol, benzaldehyde, benzyl alco-hol,..). Equally good results are obtained with other associationequations of state like PC-SAFT, SAFT-VR, soft-SAFT, SAFT-VR Mie,SAFT c - Mie and NRHB, which is an advanced lattice theory(Kontogeorgis and Folas, 2010, McCabe and Galindo, 2010).

Some typical examples of these capabilities of the associationmodels are shown in Figs. 2–4, 6–7 for CPA, Figs. 5 and 8 forCPA, NRHB and/or PC-SAFT, Fig. 9 for CPA and NRHB and Fig. 10for NRHB and PC-SAFT. In reality, we expect that both CPA/SAFT

and other association EoS with explicit hydrogen bonding terms,such as the quasi-chemical NRHB theory, will behave similarly atleast for phase equilibrium (see also Figs. 5, 8, 9 and 10), and wediscuss this issue later.

More than 90% of the applications with advanced equations ofstate refer to phase equilibria (possibly due to its importance fornumerous practical applications) and in few cases other propertiese.g. thermal, derivative ones, etc. have been considered.

As mentioned, what is particularly encouraging with the suc-cessful multiphase multicomponent calculations is that these arepredictions using solely binary interaction parameters obtainedfrom binary data. In many cases, just one binary interaction param-eter is needed per system. On the other hand, for models like cubicEoS with advanced mixing rules (e.g. SRK/Huron-Vidal), mentionedearlier, 2–3 binary adjustable parameters are required.

There are cases, where two parameters are sometimes used forthe so-called ‘‘solvating” systems i.e. those where there is crossinteraction (acid-base type), but only one or none of the compo-nents are self-associating e.g. alkanolamines, glycols or water witharomatic hydrocarbons. In this case, a cross association-volumeparameter is needed in addition to the interaction parameter (kij)in the physical term of the model. The physics of the associationterm can be used to our advantage here and it has been shown(Kontogeorgis and Folas, 2010) that, in many cases, the use of asecond interaction parameter can be avoided using the so-calledhomomorph concept (Breil et al., 2011). This means that e.g. forMEA-benzene we can adopt the kij from MEA-heptane and thenonly fit the cross-association volume. The same can be done forwater-BTEX compounds (Benzene-Toluene-Ethylbenzene-Xylene),glycol-BTEX compounds and in other cases. This is a useful engi-neering utilization of the association term of these models. Theresults are often as successful as when two interaction parametersare used. This was important to investigate in order to maintainthe (semi)theoretical aspect of these models. If the good resultsare obtained at a price of two adjustable binary parameters persystem, this theoretical value of the models would be questioned(recall that cubic EoS with two adjustable parameters can also per-form very well).

Fig. 6. Water – Ester LLE with CPA. Aqueous mixtures with propyl esters (a) and butyl esters (b). Experimental data (points) and CPA correlations (lines) using a single fitted parameter(kij) and with constant cross-association volume (Pure and binary parameters from Tsivintzelis et al., 2016).

Fig. 7. Water – octanoic acid LLE with the CPA equation of state. Experimental data(points) and CPA correlations (lines), using one temperature independent binaryinteraction parameter (Pure and binary parameters from Tsivintzelis et al., 2016).

Fig. 8. Liquid-liquid equilibria of water-hexane with three thermodynamic models(CPA, PC-SAFT and NRHB). From Liang et al. (2016). All models use a single adjustableparameter. The minimum of the hydrocarbon solubility in the aqueous phase cannot becaptured with a single parameter (but it can with a temperature-dependent adjustableparameter). Reprinted with permission.

Fig. 9. CO2 – ethylene glycol (MEG) vapor – liquid equilibrium at 323.15 K. Compositionof the liquid (a) and the vapor phase (b). CPA and NRHB parameters from Tsivintzelisand Kontogeorgis, 2016 and Kontogeorgis et al., 2011.


Another way to apply an association model to solvating systemswith fewer adjustable parameters is by using a constant value forthe cross-association volume for a family of organic compounds,

Fig. 10. Methanol – Hexane Liquid-Liquid and Vapor – Liquid equilibrium. Descriptionof the NRHB and sPC-SAFT models using one temperature independent binaryinteraction parameter (Parameters from Tsivintzelis et al., 2008, Grenner et al.,2008).


as shown in Fig. 6 for aqueous solutions of two families of esters.Solvation is accounted for assuming that one negative site on everyester molecule is able to cross associate with the positive sites ofwater. Using the combining rule adopted, in total two binaryparameters were adjusted to the experimental data, the binaryinteraction parameter (kij) and the cross-association volume(bcross), with the latter having the same value for all esters (alsomethyl and ethyl esters, results not shown here). Moreover, in thisapproach, by keeping the cross-association volume constant, asmooth correlation of the other interaction parameter (kij) withthe number of ester carbon atoms was obtained, slightly increasingwith an increase of the ester chain length, which seems reasonable(less strong cross-associating interactions as the ester molecularweight increases) (Tsivintzelis et al., 2016). These are difficult sys-tems as they are reactive mixtures and under proper conditions,substantial hydrolysis of the ester molecules may occur. For thisreason, many contradictory experimental data can be found inthe literature (Tsivintzelis et al., 2016). The performance of themodel (in this case CPA) is satisfactory under these conditions.

A very important successful characteristic of the associationmodels is the transferability of parameters so that they can be usedunchanged for a wide range of systems. For example, the samewater parameters for CPA used in Figs. 3, 4 and 6 are also usedin Figs. 7 and 8 for aqueous mixtures with acids and hydrocarbons.These are the same water parameters for CPA published 20 yearsago (Yakoumis et al., 1998). The performance is very satisfactory.In all of these cases, only one temperature independent adjustableparameter is used (a combining rule is used for the water-acidcross-association).

Another positive characteristic of the association models is thatwith most of these models we know by now that using suitablepure component parameters for water, they can successfully corre-late water-hydrocarbon LLE over extensive temperature ranges (deHemptinne et al., 2012; McCabe and Galindo, 2010; Kontogeorgisand Folas, 2010; Tsivintzelis et al., 2018). Until relatively recently,successful SAFT results were not available for water-alkanes due tolack of appropriate SAFT parameters for water. This problem hasbeen now overcome and several research groups have presentedsuccessful SAFT applications for the phase equilibria of aqueousmixtures of hydrocarbons (de Hemptinne et al., 2012; McCabeand Galindo, 2010; Kontogeorgis and Folas, 2010; Tsivintzeliset al., 2018). For example, Liang et al. (2016) have presented suc-cessful results for CPA, PC-SAFT and NRHB, as shown in Fig. 8.

Finally, such association models succeed in describing both theVLE and the LLE of mixtures with hydrogen bonding fluids, with aminimum number of binary parameters (one temperature inde-pendent binary interaction parameter in most cases, see Fig. 10)

The successful performance for binary aqueous systems hasopened the way for the successful application of diverse associa-tion models for multicomponent aqueous solutions. In our andother research groups, extensive comparisons have been over therecent years carried out of the performance of CPA and PC-SAFT(and other SAFT variants) for a wide range of systems e.g. water,alcohols and glycols with hydrocarbons (for a recent comparisonsee Liang et al. (2014) and (2017)). Other advanced models, e.g.PC-SAFT and NRHB, have also been compared over an extensivedatabase (Tsivintzelis et al., 2008; Grenner et al., 2008).Comparisons have been also performed (CPA vs. PC-SAFT) for‘‘ill-defined” systems such as oil mixtures and asphaltenes (Lianget al., 2015; Arya et al., 2008).

While the overall performance of these models is very satisfac-tory and even though results with the various models may differfrom system to system, the overall performance is very similar.We see no significant differences between the various associationmodels as long as we do not consider polymers, we focus on phasebehavior and we perform the comparisons ‘‘on equal terms” i.e. themodels are developed in the same way (using the same data foradjusting pure fluid parameters, the same number of binary adjus-table parameters, the same association schemes, input data etc).Even when we have compared different models like CPA and PC-SAFT for oil mixtures and asphaltenes, very similar results areobtained if the oils are characterized in the same way and also ifthe association term is used in similar ways in both models whenmixtures of asphaltenes are considered.

While these results are overall positive, the excellent but rathersimilar performance, at least for phase equilibrium calculations,with ‘‘fundamentally different theories” has puzzled us and others.What do these results tell us? Which theory is correct (if any) orhow much of the results represent a fundamental understandingof physical behavior and howmuch constitute a successful correla-tion exercise? Can it be that none of the advanced theories is veryrealistic, can it be that all are very approximate and the limitationsare (sometimes/often) masked behind the adjustable parameters?Maybe, expressed in another way, we could ask how much wehave advanced science and engineering with the association theo-ries. In this context advancement in science implies obtaining afundamental understanding of phase behavior and a wide rangeof properties, unique for a specific theory, and beyond the use ofarbitrary adjustable parameters and advancement in engineeringmeans, for example, useful predictions of complex phase behaviorand other properties beyond the capabilities of the ”semi-empirical” classical models (like the benchmark model we haveintroduced in the previous section). We will come back to thisquestion later.

Despite the aforementioned concerns, association models havealso been extensively and successfully used in applications whereextrapolations are needed with respect to parameter estimatione.g. polymers and pharmaceuticals. For such compounds, extensivevapor pressure data are not available and diverse ‘‘indirect” meth-ods are required for estimating the pure compound parameters.Still, many successful low and high pressure VLE and LLE as wellas gas solubility results have been presented for mixtures withhomo- and co-polymers and solvents with SAFT, especially withPC-SAFT, indicating a very satisfactory use of the chain term (seePrausnitz et al., 1999; McCabe and Galindo, 2010, Kontogeorgisand Folas, 2010 for several results). Diverse association models(CPA, NRHB, PC-SAFT) have been applied also for pharmaceuticalsand good results have been obtained for relatively simple pharma-ceuticals in mixtures with single or mixed solvents (for a review


see Kontogeorgis and Folas (2010) and more recently also Motaet al. (2011) and Spyriouni et al. (2011)).

4. Future challenges

The results obtained with association theories are, at a firstglance, very promising, even impressive sometimes, and can bevery useful in many engineering applications, but ‘‘there is no freelunch” and much research is needed before these results areobtained and before the models are available for (routine) engi-neering applications. Development of association models includingestimation of pure component, mixture parameters and choice ofassociation schemes takes time, and requires patience as it is oftennecessary to select the final set of parameters among several suit-able ones, all representing primary properties like vapor pressuresand liquid densities equally well, which, however, may not all beequally successful for calculations of mixtures’ properties.

Appendix A illustrates this for the case of MEA (studied beforebut here revisited) and is demonstrated how diverse pure compo-nent parameters can be obtained from vapor pressures and liquiddensities and how LLE data can be used to ‘‘guide” the parameterselection. This is a useful but a not so easy and sometimes a bitrisky procedure. Anyhow, ‘‘we cannot get something from nothing”and first of all extensive experimental data are needed at the startof the model development, often in terms of vapor pressures andliquid densities over extensive temperature ranges. For many ‘‘or-dinary” compounds such data have been measured, but this is notthe case for several complex and/or heavy compounds e.g poly-mers, pharmaceuticals, biomolecules and electrolytes includingionic liquids. For such compounds, vapor pressure data are oftennon-existent. In some cases, indirect estimation methods havebeen developed e.g. based on group-contributions especially forpolymers, but in most cases the problem is not solved and the finalchoice of parameters is –at least to some extent- based on use ofmixture data as well. For some years, we had hopes that monomerfraction data obtained from spectroscopy could provide an addi-tional or even alternative way to estimate pure compound param-eters for associating compounds. Unfortunately, it turned out thatsuch data are hard to find, are available for only few compounds(mostly alcohols, glycolethers and water), the quality of the data

Fig. 11. Free site fraction of saturated water using thermodynamic models (CPA, PC-SAFT, NRHB), experimental data, a new theory and recent molecular simulation data.Water is assumed to have a tetrahedral structure with all models. Re-arranged fromLiang et al. (2016) where references to all models and molecular simulation data areprovided. Reprinted with permission.

is not clear either and they are not always easy to interpret inthe way needed in association theories. The agreement with theassociation theories (see e.g. Fig. 11 for water) is often qualitatively(but not quantitatively) correct. In the case of water all theoriespredict much ”more” hydrogen bonding or less monomer fractioncompared to the experimental data of Luck. It is possible to ”force”the theories to fit the monomer fraction data but this can only bedone at the cost of getting unusual parameters, which moreoveryield poor phase behavior results e.g. LLE for water-alkanes. Thisproblem may reflect the quality of the theories or the monomerfraction data themselves and their interpretation; we do not knowas yet what is the case (Liang et al., 2016; Tsivintzelis et al., 2014b).

The development of association models is, thus, a time consum-ing task, starting with the estimation of pure component parame-ters. Some of the challenges due to lack of especially vaporpressure data and difficulty in using the monomer fraction datahave been mentioned.

But there are several more issues on pure compound parameterestimation. With the current approaches, the critical point is ‘‘sac-rificed” and it is actually overestimated, something that was not aproblem with the cubic EoS. Many researchers have used the so-called crossover approaches, a scientific elegant method, whichsomehow solves this problem, but introduces extra parametersand significant (computational) complexity in the final models.We do not believe that such approaches can easily find acceptancefor industrial applications, as they stand today. Coutinho, Quei-mada and co-workers (Palma et al. 2017, 2018a, 2018b) haverecently proposed a more elegant approach (with CPA) whichincludes all properties (vapor pressure, densities, heat capacitiesand critical point) in the parameter estimation. They used heatcapacities in the parameter estimation and a more complextemperature-dependency for the energy parameter of the model.The results seem promising, also for the critical point of mixturesand deserve further study. Polishuk and co-workers (Polishuk,2014, 2015, Polishuk et al. 2017) have revised the PC-SAFT EoSincluding part of the so-called universal constant matrix, in whichthe critical point is used in parameterization, and the model hasbeen used for different properties of various systems. The resultsare promising but not entirely conclusive as not all properties arerepresented equally satisfactorily.

This brings us to a related topic – exactly how should we esti-mate the parameters of the association theories. As these modelshave, as mentioned, 5–6 parameters for associating fluids, a largenumber of data – several properties are needed. Is the use (alone)of vapor pressures and liquid densities the optimumway? As men-tioned, Coutinho and co-workers (Palma et al. 2017, 2018a, 2018b)have included heat capacities in the parameter estimation. Fewsystematic studies exist where various objective functions havebeen tried and we have in our group recently made a parametricanalysis for acetic acid (Ribeiro et al. 2018) and for hydrofluo-roolefins (Kang et al. 2018). Several objective functions have beentried in both cases. Even when focusing only on these specific fam-ilies of compounds considered, it is difficult to identify whichobjective function is the best for obtaining reliable parameters.This is particularly difficult as even the advanced association theo-ries cannot describe all properties, especially thermal and deriva-tive ones, particularly well. It appears that keeping the vaporpressure in the parameter estimation is crucial. SAFT approachesare also expected to perform better than CPA ones for certainderivative properties such as speed of sound, especially at highpressures (but there are exceptions as the studies above have indi-cated). Much more work is needed in the area of derivative/ther-mal properties and whether and how they can be used in theparameter estimation for association models.

Particularly in the area of derivative properties much attentionhas been given over the recent years. SAFT-VR Mie has indeed


showed nice results for derivative properties, including heat capac-ity, heat of vaporization and speed of sound, etc., for chosen sys-tems (Lafitte et al. (2013) and Dufal et al. 2015), despite somelimitations shown by Polishuk and Garrido (2018, 2019).

In addition to the determination of pure component parameters(from vapor pressure, density or other data), often combined withmixture data esp. LLE, a careful selection of mixing and combiningrules is needed for both the physical and association parts of themodels. These mixing and combining rules will ensure an appro-priate ‘‘balance” of the physical and association effects, but whileseveral choices are available, it is not always easy to select a correct‘‘default” a priori without some checks for relevant systems. This isparticularly the case for cross-associating systems as there is awide range of them, some with both components self-associatinge.g. water-glycols and some with only one self-associating com-pound e.g. water-aromatic hydrocarbons and even some mixturescapable of cross-association with no self-associating componentse.g. chloroform-acetone (in our terminology we have called the lat-ter two cases ‘‘solvation”).

Table 1 summarizes some challenging issues with advancedequations of state for phase equilibria and other properties (fornon-electrolyte systems) and an earlier detailed discussion canbe found in a previous study (Kontogeorgis, 2013)

It is also very important to consider the predictive character ofthe association models. Often the models are used for multicompo-nent predictions using parameters from binary data alone, but thisis not always sufficient as there are systems where no binary dataare available. Several SAFT models have been extended in group-contribution schemes for the parameters of the physical term (deHemptinne et al. 2012, McCabe et al. 2010, Kontogeorgis andFolas, 2010) and in one case (Thi et al. 2006) also for the interactionparameter. Very interesting is also the work by Haslam et al.(2008) on predicting kij using the Hudson-McCoubrey theory (butthere has not been any follow up of this study).

There have been very few and often non-systematic efforts togeneralize the hydrogen bonding parameters of the associationtheories to other -externally defined- measures of the hydrogenbonding such as the Kamlet-Taft (Kamlet et al. 1983) acid-base sol-vatochromic parameters or the Hansen’s solubility parameters andthis might be of interest in the future in order to obtain parameterswhen data are lacking. Also for mixtures of associating/solvatingcompounds, it may be of interest to link the cross association ener-gies and volumes to external, independent information e.g. theDrago-Wayland approach for the hydrogen bonding enthalpiesand entropies (Drago and Wayland, 1965). To our knowledge, thishas not been attempted either in any systematic way. Again theacid-base solvatochromic parameters appear promising, as theyhave been previously used successfully in other areas e.g. forselecting co-solvents for supercritical fluid extraction (Walshet al., 1987). According to Walsh et al. (1987) significant increaseof solubility can be obtained in some cases where a solid-cosolvent is of complementary nature (acid-base effect as quanti-fied with the Kamlet-Taft parameters). Many data and modelingwith association theories can be interpreted with these parameterse.g. the strong chloroform (acid parameter = 0.44)-acetone (baseparameter = 0.48) association or the induced cross association ofMEG (acid parameter = 0.9, base parameter = 0.52) with the basictoluene (base parameter = 0.11). But, as indicated, there have notbeen systematic studies and generalizations.

The concept of acid-base interactions is extensively used inother fields as well, notably colloid and surface science (e.g. inter-facial theories, adhesion and adsorption studies). We believe thatthe further development and understanding of association theoriescan be profited by using or even by incorporating concepts from

colloid and interfacial science either in the development or the val-idation stage. Prediction of interfacial and adsorption properties isdiscussed in Table 1. In addition, in a few studies, association mod-els have been used to predict solubility parameters and even Han-sen’s solubility parameters with separate non-polar, polar andhydrogen bonding contributions (Stefanis et al., 2006). To ourknowledge, the association theories have not been as yet used topredict the diverse contributions to surface and interfacial tensionse.g. from dispersion or acid-base effects. This may be useful as suchacid-base theories (see Kontogeorgis and Kill (2016) for a review),e.g. in the form of the van Oss-Good and recently also the Panayio-tou approach, are popular and validated models in surface scienceand there is much knowledge and expertise with the associatedcontributions to surface tension.

One of the topics mentioned in Table 1 which deserves someextra attention is the highly polar systems, as many new develop-ments in the field of association models include applications topolar mixtures. If the involved polar compounds are also self-associating, the hydrogen bonding nature will almost entirelydominate the phase behavior (and most other properties). Forexample, methanol and formaldehyde have almost the samemolecular weight and the former has a boiling temperature of64 �C, while the latter of �21 �C (despite formaldehyde being morepolar with 2.3 D compared to 1.7 D for methanol). This is the effectof hydrogen bonding which dominates. However, the situation ismore complicated when polar non-self associating compoundse.g. ketones or esters are present (in mixtures with either inert,polar or hydrogen bonding compounds). In some of these cases,complex phase behavior may exist which calls for attention e.g.acetone (3 D dipolar moment) with hexane exhibits VLE at highertemperatures, but are immiscible at low temperatures, below230 K. Sulfolane with dipolar moment equal to 4.8 D exhibits LLEwith hydrocarbons and has infinite dilution activity coefficientswith hydrocarbons of the order of 50–100. Studies in literature(Kontogeorgis and Folas, 2010) have showed that polar versionsof association models (i.e. by adding a polar contribution) yieldgood results for polar compounds in presence with inert com-pounds, but there are problems when polar/associating com-pounds like water are also present. On the other hand, assumingthat the polar compounds like acetone are ‘‘pseudo-associating”works well in some cases e.g. again in mixtures with inert com-pounds, but this approach is less satisfactory for LLE or VLLE oreven for water–acetone VLE (Kontogeorgis and Folas, 2010,Tsivintzelis et al., 2018, Tsivintzelis and Kontogeorgis, 2012,2014). Studies for multicomponent systems e.g. water-methylmethacrylate-acetone and water–acetone-methanol show thatthe approach assuming that acetone only cross-associate withester and water (but does not self-associate) performs best(Tsivintzelis and Kontogeorgis, 2012, 2014).

A challenging issue, which deserves own discussion (and is notmentioned in Table 1), is electrolytes. Electrolyte thermodynamicsalso in connection to electrolyte equations of state (there are over50 of them developed during the last 20 years or so!) is a wholeseparate topic, a very challenging one. As discussed in a recentreview (Kontogeorgis et al., 2018), there is currently little consen-sus in the thermodynamic community how such electrolyte EoSshould be developed, which contributions for the electrostaticinteractions should be used (for ion-ion, ion-solvent interactions,etc), how parameters should be estimated and how the final modelshould be validated. Some of the challenges, highly interconnected,are graphically presented in Fig. 12. Moreover, the field of elec-trolyte thermodynamics includes several controversies, some ofwhich are of rather fundamental character (e.g. extent of full andpartial dissociation of salts and role of ion pairs in the modeling,

Table 1Some challenging issues with advanced equations of state for phase equilibria and other properties. Some of these problems have been studied with CPA, PC-SAFT and some with both models and other advanced equations of statee.g. NRHB and other versions of SAFT.

Challenge (Relevant references) Status Comments

Organic acids-water(Kontogeorgis and Folas, 2010, Kontogeorgis andEconomou, 2010, Ribeiro et al., 2018, Tsivintzelisand Kontogeorgis, 2014, Vahid and Elliott, 2010,Janecek and Paricaud, 2012, 2013)

Description of VLE over extensive temperature ranges including accuraterepresentation of relative volatilities is difficult and may need temperaturedependent parameters.Systems with organic acids, especially their mixtures with water, arenotoriously difficult to model, despite improvements of associationtheories, which were extended to account both for acid dimerization andthe formation of oligomers (Vahid and Elliott, 2010, Janecek and Paricaud,2012, 2013).

The 1A scheme for organic acids in SAFT framework may not be correct.It is difficult to represent with same theory both acid-alkanes and acid–watermixtures.Unclear role of small (acetic) acid dissociation in water, which is typicallyignored in modelling.

Aromatic acids-water(Kontogeorgis and Folas, 2010, Kontogeorgis andEconomou, 2010, Tsivintzelis and Kontogeorgis,2012)

Very difficult to represent simultaneously VLE, LLE and SLE with sameparameters.

The LLE behavior of e.g. benzoic acid–water may indicate some structuringbehavior not captured by association models

Multifunctional chemicals e.g. glycolethers andalkanolamines(Kontogeorgis and Folas, 2010, Kontogeorgis andEconomou 2010, Kontogeorgis 2013, Avlund et al.2008)

Unclear role of the importance of accounting of the different functionalgroups e.g. studies for MEA show that it is not necessary to differentiatebetween the functional groups of hydroxyl and amine (true for twoassociation models studied)Limited validation for multicomponent systems (not containing acid gases)Very difficult to represent closed-loop behavior e.g. for surfactant-watersystems

Unclear how important is to account explicitly for intramolecular associationand to use more advanced association schemes (than those typically used in theSAFT framework).Equally unclear is how important in the inclusion of hydrogen bonding co-operativity effects which are known to be present in many hydrogen bondingmolecules.

Systems with peculiar hydrogen bonding interactions,i.e. systems with intra-molecular association orcooperative hydrogen bonds(Missopolinou and Panayiotou 1998, Missopolinouet al. 2006, Avlund et al. 2011, Ghonasgi andChapman 1995, Sear and Jackson 1994)

Statistical thermodynamic models have been extended to account for suchinteractions. However, in most applications of the models, such interactionsare not accounted for.

In many cases the explicit account of hydrogen bonding, and the subsequentincrease of models’ complexity, results in an insignificant improvement ofthermodynamic properties, such as the phase behavior but improvement forsome other properties such as heats of mixing

Highly polar and quadrupolar molecules(Kontogeorgis and Folas, 2010, Kontogeorgis andEconomou 2010)

Unclear role of the importance of accounting explicitly for polar andquadrupolar effects.Particularly challenging to include simultaneously polar and associatingeffects.Unclear which values should be used in models for the dipolar andquadrupolar moments.

Detailed tests should consider not just polar or quadrupolar molecules inmixtures with ‘‘inert” hydrocarbons but also with water and multicomponentsystems – very few such detailed studies have been carried out with andwithout accounting for polarity.

Interfacial properties Density Gradient Theory (DGT) or Density Functional theory (DFT) arecommon choices used. Systematic comparisons and extensive studies arelacking.

Usually there are different ways to functionalize the different terms of an EoS inthe DFT framework, and it is quite often that different approaches are availableto functionalize one term. It is unclear which approach is best for a specific term.More information is provided by Sauer and Gross (2017) and Camacho Vergaraet al. (2019), (2020)

Complex Multicomponent Adsorption(Kontogeorgis 2013 and references there)

Additional frameworks are needed e.g. DFT (Density Functional Theory) orMPTA (Multicomponent Potential Adsorption Theory). None has beenextensively tested for the adsorption of complex associating compounds.

Results with CPA + MPTA for the multicomponent adsorption of associatingcompounds on diverse solids are ambiguous on the accuracy of the approach.Better results are obtained for non-polar systems.The importance and role of the association term has not been established.

Thermal and derivative properties(Tsivintzelis et al., 2018, Liang et al. 2012, 2014,Palma et al. 2017, 2018 a)

Very few studies for heat capacities and excess enthalpiesEsp. the residual Cv is difficult to be captured even with advancedassociation theoriesMore studies have been presented for speed of sound – in all cases it wasconcluded that the originally proposed version of association theories hadto be modified to capture the speed of sound trends esp. at high pressures

Some studies showed that inclusion of such data in parameter estimation maybe important and results to improved parameters – far too few studies to arriveto conclusive results.

Simultaneous representation of multiple properties(beyond phase equilibria) and for both binary andmulticomponent systems(Tsivintzelis et al., 2018 and for monomer fractionreferences see Tsivintzelis et al., 2008, 2014b, Lianget al., 2016)

Limited studies show some challenges. For example, for TEG-water forwhich binary and multicomponent VLE, SLE, activity coefficients and excessenthalpies are available, recent studies (Tsivintzelis et al., 2018) with oneassociation model show that accurate representation of all propertiesrequire temperature dependent kij and additional adjustment of the cross-association which cannot be taken (reliably) from combining rules.

Ill-defined compounds(Kontogeorgis and Folas 2010, Liang et al. 2015,Arya et al., 2008)

Characterization of heavy oil fractions is still difficult even with advancedtheories and there are many assumptions involved

14G.M

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Fig. 12. Various aspects to be considered in modeling electrolyte solutions.


the primitive vs. non-primitive modeling pathways, the role andimportance of relative permittivity and its dependency with con-centration, the standard states and the McMillan-Mayer frame-work and finally the single ion activity coefficients discussed inseveral papers by professor Vera, as summarized in a recent text-book (Vera and Wilczek-Vera, 2016).

The complexity of the electrolyte EoS field is not a new discov-ery and it can be illustrated by this statement by Loehe andDonohue (1997) more than 20 years ago:

‘‘If published reports of new EoS claiming wide applicabilitywere to provide the quality of the fits to set of standardapplications, much time and effort would be saved by non-specialists attempting to use the EoS for engineering calcula-tions. We would like to see a cessation of what has becomethe practice of promoting an equation’s strengths without a dis-cussion of its limitations”.

This is, of course, even more true today and their suggestion iseven more pressing now. We will undertake a major researchactivity hoping to lead to more fundamental understand of elec-trolyte solutions and better models during an on-going projectgranted by the European Research Council (ERC) entitled ‘‘NewParadigm in Electrolyte Thermodynamics”.

It is also of interest to compare the current status on equationsof state with the actual needs of industry, as presented by Hendrikset al. (2010) in a manuscript summarizing the conclusions from aproject carried out by the Working Party on Thermodynamics

and Transport Properties (http://www.wp-ttp.dk) of the EuropeanFederation of Chemical Engineering. The results are based onreplies by 28 companies covering a wide range of industrial sec-tors. The companies emphasized the need for a single or at leastfewer models covering a wide range of systems, properties andconditions, the need for more and better quality experimental data,and better education in thermodynamics for end users. They alsoexpect that classical e.g. UNIFAC and advanced models like theassociation theories and even the COSMO-RS (Klamt et al., 1998)or COSMO-SAC (Lin et al., 2001) approaches should continue tobe developed in parallel as both approaches are needed. Finally,far too many papers, they pointed out, are published with insuffi-cient information to reproduce the results and better standards arerequired for ensuring the use of the most successful models byindustry.

The quest or at least hope for a single thermodynamic modelwas also mentioned by Prausnitz et al. (1983), as mentioned earlierin the manuscript, but both the analysis of the results from indus-try and the discussion we offer here illustrate that we may be quitefar from such a target. Indeed, thermodynamics is a disciplinewhere we should expect that underlying principles and theoriescould be valid. Albert Einstein quote could also apply here (evenif we do not think he had the thermodynamic models in mindwhen he said that): ‘‘A theory is the more impressive the greaterthe simplicity of its premises, the more different kinds of things itrelates, and the more extended its area of applicability. Therefore thedeep impression that classical thermodynamics made upon me”. TheSAFT approach is certainly a way forward, but the many variants,unclear capabilities and lack of comparisons confuse the industrialusers who often ask us ‘‘which SAFT-variant do we recommend,which one should they use”. This is not an easy question to answer.

And there are more critical voices among some top experts e.g.from a very experienced colleague from the Technical University ofDenmark who has advised the authors over many years (personalcommunication): ‘‘I would probably characterize mixing rules as suchas a main limitation of current models. We do not have anything bet-ter than more than 100 years old concepts, based mainly on commonsense, and in most applications we cannot select a default best! Whereis the theory? For the majority of systems we end up with an adjusta-ble parameter instead. My main comment about the ’importance’ ofregression is that it can mask - in an excellent manner - the true pre-dictive ability of model variants! It seems more and more commonthat people try out combinations of mixture and pure component datawithout explicitly indicating how the mixture data were used. Well,fine for the development of a practical model, but the predictive abilityclaim becomes a fake, in my opinion. I am afraid that most of our ’ac-complishments’ can be ascribed to much faster computers that allowus to routinely regress a large number of our parameters simultane-ously and from a variety of data, rather than fundamental improve-ments in our model. The SAFT family, on the other hand, is themuch-hailed new theoretical development, expected to replace allother models! (Although many of the extra terms included later seemsto be based on a dust-off of 30 to 40 years old theory). I do not thinkwe have any hope of progressing before we get rid of the myriad ofadjustable parameters!”. We agree with these statements.

Finally, when assessing the capabilities and range of applicabil-ity of even the most advanced thermodynamic models, two addi-tional aspects should be considered, in our view. The first is thatwith the increasing sophistication of the models, the expectationsalso increase, especially considering the wide range of applicationswe wish these models to be used for. For example, Dr. FrancoisMontel from TOTAL gave a talk during the SAFT Conference in Bar-celona (September 2010) entitled: ”A single predictive thermody-namic model for all the needs of Oil & Gas Exploration, Production,Refining and Petrochemical industries”. Even though just focusingon the needs of the broader petroleum industry, about half of thementioned applications have not been addressed as yet with theadvanced association equations of state. So, there is still lots tobe done!

The second aspect is how much the advanced thermodynamicmodels, the association theories, have actually advanced ourunderstanding of the molecular behavior of pure compounds andmixtures. We would of course like to believe that, but is it so? Thisis not entirely clear to us. Let us consider water, which is a com-pound with numerous ‘‘exceptional” properties showing anoma-lous behavior e.g. the maximum of density. While the associationtheories represent well the phase behavior of many aqueous sys-tems, including pertinent multicomponent multiphase behavior,this is not the case for the anomalous behavior. No association the-ories can predict these anomalous properties of pure water andwater mixtures. The poor representation of hydrogen bonding

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shown in Fig. 11 could be attributed to the quality of experimentalmonomer fraction data (in terms of fraction of free molecular sites)or to inadequacies of the association theories. A key problemwhicharises is which structure shall be used in the models. Over100 years we have debated on water’s hydrogen bonding-relatedstructure. It is still heavily debated whether liquid water maintainsthe tetrahedral structure as we know it from ice (the one typicallyused in association models), or whether water should be describedby a two-state model where most molecules are in the form ofrings or chains. The latter theory has been recently proposed bythe group of Nilsson and Pettersson (Wernet et al., 2004; Perakiset al., 2017), idea matured over the recent years, according towhich water is best described as a two-state model and only asmaller portion is tetrahedral. Actually, already in 1892, WilhelmRöntgen theorized that water is actually a two-state system con-sisting of a bi-phasic liquid. Nilsson and Pettersson have even‘‘proved” the transition between the two liquid states of water, aphenomenon which could be responsible for a second critical pointin water. While Nilsson and Pettersson have stated that water is inessence a mixture of two liquids in a complex relationship, show-ing all these with a wide range of advanced spectroscopic tech-niques, and the topic has been popularized (Sanderson, 2018), itis still a heavily debated topic with numerous papers in Natureand Science during the 21st century focusing on this (Ball, 2008;Woutersen et al., 2018; Smith et al., 2004). And even though PhilipBall recently (March 2018) proudly stated that ‘‘evidence mountsthat water exists in two liquids forms” (Ball, 2018), indeed, itseems that the interpretation of the same/similar spectroscopicand X-ray data by different groups on the structure of water canbe done in more than one ways, leading to different conclusions!

The above considerations have not as yet (to the best of ourknowledge) been incorporated in the advanced association theo-ries and as we saw they are not needed for ‘‘bulk” phase behaviorwhich is well represented (simply assuming a tetrahedral hydro-gen bonding structure). These modern considerations may berequired if more water properties should be accurately calculated(beyond phase equilibria) and if a more complete description ofthe behavior of water should be obtained, including representationof the water’s many anomalous properties. It remains to be seenwhether this will be possible.

5. Concluding remarks and some future expectations

From this short trip on equations of state over three centuries,we should first of all conclude that cubic equations of state endure;they are powerful models, which can be used for both size-symmetric and asymmetric systems, even for polymers. The func-tional form of the well-known cubic EoS such as SRK and PR pro-vides (qualitatively at least) the right magnitude of the size(entropic) effects and when combined with local-compositionmodels using the EoS/GE mixing rules, cubic EoS can be appliedsuccessfully also to polar systems, sometimes for both VLE andLLE (with careful parameterization of the incorporated activitycoefficient model). We believe that the so-called zero referenceor approximate zero reference pressure models have completedtheir cycle and our favorite approach is the Huron-Vidal mixingrule (in combination with NRTL or the residual term of UNIQUACor even the group-based residual term of UNIFAC in the recentVTPR model). This approach should be considered the ‘‘bench-mark” classical approach against which more advanced modelsshould be compared to.

We can also conclude that demanding applications in the petro-leum and chemical industries may sometimes require advancedmodels, better than the cubic EoS (even when combined withactivity coefficient models and advanced mixing rules). This is

especially the case for hydrogen bonding systems. Associationmodels like CPA and SAFT are promising and are being adoptedby industry and perform typically better than conventional models,especially for phase equilibria of multicomponent mixtures, andthey use fewer adjustable parameters. It should be emphasizedthat the development of such advanced models is not easy (insome sense, advanced models come at a price) and much work isneeded from researchers prior to having models readily availablefor engineers. For example, numerous parameter sets for purecompounds, performing equally well for a number of properties,should be checked against mixture data prior to the final selection.And we do not as yet have the ‘‘optimum” objective function forparameter estimation of these advanced models; much stilldepends on what we want to do or even a researcher’s taste onwhat is important and the data availability.

Another issue we have noticed is that, in many cases, the differ-ences in the performance among many advanced equations of stateare small when the model development is carried out in the sameway. Moreover, this is the case even when we consider not only thediverse SAFT variants but also quasi-chemical and chemical theo-ries as well. It often seems indeed that the parameter estimationis either as or more important as the models’ functionality, orcan mask the importance of the actual functionality, or even sepa-rate ‘‘physical background” of the models, even for the diverseassociation theories. Many studies indicate this and some charac-teristic references are cited here (Kontogeorgis and Economou,2010; Tsivintzelis et al., 2008, 2014b, 2018; Tsivintzelis andKontogeorgis, 2014; Economou and Tsonopoulos, 1997; Lianget al., 2012; Grenner et al., 2008; Tsivintzelis and Kontogeorgis,2012).

Moreover, in our view, possibly due to the importance ofparameter estimation and number and nature of the adjustableparameters, it is currently unclear how much more ‘‘physics” isneeded for improving the advanced theories and the effects of e.gexplicit inclusion of polarity, advanced/new association schemes,intramolecular association and co-operativity are all unclear, atleast from the practical point of view. We have as yet to see clearconclusions that such effects have improved the results with asso-ciation theories (at least for phase equilibria).

Finally, compared to phase equilibria, less attention has beenpaid to thermal, interfacial and derivative properties (but the situ-ation is slowly being improved) and studies so far are not entirelyconclusive on the performance of the advanced models for suchproperties. Even worse is the situation for mixtures with elec-trolytes where numerous electrolyte versions of advanced modelsare today available with little (if any) consensus even for ‘‘thebasics” such as, for example, how many and which contributionsfor the electrolyte effects should be included and how the param-eters of the electrolyte equations of state should be estimated.

As a last note, the dissemination of such advanced models toindustry has started, but is slow as only limited versions of thesemodels and for only few systems are available in commercial pro-cess simulators. Often the implementation is not optimum in thesesimulators. To our knowledge, these advanced models are not yetavailable in reservoir simulators. Nevertheless, we consider it pos-itive that, despite earlier reservations (Letcher et al., 2004), simula-tor vendors have started adopted the association models in theprocess simulators, as can be seen by the fact that both CPA, PC-SAFT and other SAFT models can now be found in several commer-cial process simulators.

We would also like to comment on the predictions by professorJohn Prausnitz and co-workers (Prausnitz et al., 1983) mentionedearlier in the manuscript. We believe that several of the predictionscame true (with the advent of EoS/GE mixing rules and SAFT),though not the last two on critical behavior and electrolytesystems.

Fig. A1. Liquid-Liquid equilibrium for binary system of MEA and n-Heptane (nC7)at 1.01325 bar. Solid, dotted and dashed lines are CPA results for set-1, set-2 andset-3 respectively.


Especially, based on the current status, it is difficult to concludethat we are, as yet, close to a single thermodynamic model for all oreven for many applications (which Prausnitz et al. hoped in 1983that it might be the case in the future). We also believe that theadvanced association models have enhanced somewhat our capa-bilities from an engineering point of view, while it is not entirelyclear howmuch they have enhanced our understanding of complexintermolecular interactions.

We do believe that we need possibly fewer ‘‘newer” models(some of which are actually minor variations of existingapproaches) and less emphasis on ‘‘exhausted” topics such as cor-relation of binary phase equilibria and even less emphasis only on‘‘fantastic capabilities” of any new or existing approaches withoutdiscussion of the limitations of the proposed approaches (equa-tions of state). Future studies on existing or new models shouldfocus on identifying the true frontiers of the theories beyond corre-lation exercises by considering, when appropriate, a wide range ofconditions and properties including but not limited to thermal,derivative, interfacial/adsorption, simultaneous physical andchemical equilibria and even –in combination with additionalframeworks- extension to non-equilibrium/transport properties,also a largely neglected area of research. For addressing a widerrange of difficult problems for several industrial sectors includingbiotechnology, we need significant progress in the field of elec-trolytes and even mixtures containing electrolytes, various sol-vents and sometimes also polymers and biomolecules. This willbe a formidable task but also an exciting journey for the associa-tion theories.

Two final considerations at the end. Being largely a ‘‘service”discipline, we feel, as John Prauznitz has often stated (Prausnitz,1996, 1999), that we should be concerned (mostly, only?) withtrully significant problems: ‘‘Our major responsibility is first to iden-tify truly significant problems and second to reduce our work to prac-tice. And to exhibit multidimensionality i.e. to relate thermodynamicsto other chemical engineering disciplines such as mass and heat trans-fer, nucleation and chemical kinetics. Thermodynamics comes second.First comes chemical engineering”. And we also feel that more trans-parency is often needed in model development. We do not want todiminish the excitement of model developers, but it is sometimesuseful to have in mind George E.P. Box famous quotation: ‘‘Essen-tially, all models are wrong, but some are useful”.

Declaration of Competing Interest

The authors declare that they have no known competing finan-cial interests or personal relationships that could have appearedto influence the work reported in this paper.

Acknowledgement

The authors wish to thank the European Research Council (ERC)for funding of this research under the European Union’s Horizon

Table A1CPA pure compound parameters used in this study.

Comp b a0=Rb c1 eAA=R Assoc. Volume

cm3=mol K - K -

MEA, set-1 56.56 3000 0.7012 2186 5.35MEA, set-2 54.00 1293 2.5364 1100 225MEA, set-3 55.00 2042 1.6320 1200 85

2020 research and innovation program (grant agreement No832460), ERC Advanced Grant project ‘‘New Paradigm in Elec-trolyte Thermodynamics”.

Appendix A. The MEA parameter estimation for the CPAequation of state

We present here a revision of the MEA (monoethanolamine)CPA parameters proposed by Avlund et al. (2008). This is done asthe original parameters, shown in Table A1, which were fittedbased on vapor pressures and liquid densities using the 4C schemeand having the MEA-heptane LLE data for tuning did not actuallyperform very well, see Fig. A1 and Table A2. The temperaturedependency is not captured very well and there is a crossoverbehavior at low temperatures (not shown by the data). TheAvlund et al. (2008) parameters performed well for MEA-benzeneLLE (using the solvation approach) and MEA-water VLE but theywere not tested further for multicomponent systems. The relativepoor performance of the original CPA parameters for MEA over ajust 25 �C temperature range was rather disappointing as themodel can e.g. correlate other highly immiscible systems like wateror glycol-alkane LLE very well and over more extensive tempera-ture ranges.

In this work, we have re-parameterized MEA and two new setswere obtained and compared to the literature one. Set-2 is, like set-1 (literature), based on MEA-nC7 LLE data in addition to vaporpressure and saturated liquid density data from DIPPR correlations,however, we have considered a broad range of self-association

Self-Association Scheme Percentage deviationTr ½0:4;0:9�

Reference

Psat qsat

- % %

4C 4.96 3.03 Avlund et al., 20084C 1.98 1.02 This work4C 2.65 1.00 This work

Table A2Binary parameters and deviation in CPA results with respect to experimental data.

MEA CPA Parameters Sets Binary System Equilibrium kij bij � 1000 Combining Rule AARD (%)

Set-1 MEA-nC7 LLE 0.017 NA NA 16.09(MEA in nC7)

18.09(nC7 in MEA)

MEA-Benzene LLE 0.004 7.84 MCR-1 16.23(MEA in BZ)

1.54(BZ in MEA)

MEA-Water VLE �0.161 NA CR-1 6.97


11.17(nC7 in MEA)

MEA-Benzene LLE 0.006 110 MCR-1 8.00(MEA in BZ)

3.06(BZ in MEA)



83.93(nC7 in MEA)

MEA-Benzene LLE 0.011 NA NA 5.33(MEA in BZ)

2.54(BZ in MEA)



energy and volume. While for set-3, MEA-benzene LLE data areconsidered in addition to vapor pressure and saturated liquid den-sity. Moreover, whereas Avlund et al. [94] used a reduced temper-ature range of [0.55, 0.9], we have used a broader range [0.4, 0.9]since MEA-nC7, MEA-benzene and MEA-water experimental datafall into range [0.4, 0.55]. It should be noted that for set-3, we havenot considered benzene as a cross-associating compound.

Table A1 shows the MEA parameters, Table A2 summarizes allresults and Fig. A1 shows the LLE performance for the key MEA-heptane system. We notice that the three sets are quite different,with set-2 performing better than set-1 in almost all respects. Set-3 is overall not very satisfactory, outside the MEA-benzene systemwhich was used in the parameter estimation. Other SAFT modelse.g. PC-SAFT and soft-SAFT perform similar to CPA (also showingsome crossover behavior but sometimes at high temperatures).

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equations of state in three centuries. are we closer …...inspiration for the current manuscript...

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