Equations of State From the Ideas of Van Der Waals to Association Theories (1)

J. of Supercritical Fluids 55 (2010) 421437

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journa l homepage: www.e lsev ier .com/ locate /supf lu

Equations of state: From the ideas of van der Wa

Georgiosa Center for Ene ng, Teb The Petroleum Emira

a r t i c l

Article history:Received 21 JuReceived in reAccepted 16 O

Keywords:Phase equilibrvan der WaalsCubic equationExcess Gibbs eMixing rulesAsymmetric mixturesAssociating uidsSAFTCPANRHB

ltedidelof va

n thisxcessWe illed. Mlatedon of

developed using the so-called EoS/G mixing rules and we illustrate with the same methodology howthesemixing rules should best be used for size-asymmetric systems. Despite the signicant capabilities ofcubic equations of state, their limitations lie especially in the description of complex phase behavior, e.g.liquidliquid equilibria for highly polar and/or hydrogen bonding containing molecules. In these cases,advanced equations of state based on statistical mechanics that incorporate ideas from perturbation (e.g.SAFT and CPA), chemical (e.g. APACT) and lattice (e.g. NRHB) theories are preferred. Some of the mostpromising approaches are briey outlined here. Their capabilities and limitations/challenges, pointing

1. Introdu

Equationdynamic mover extensame timebe calculatof state, liteclaiming toideal liquidvan der Wation, speedwhy suchindustrial aallel with tfor the liquoped and m

CorresponE-mail add

ieconomou@p

0896-8446/$ doi:10.1016/j.out to future research needs are also discussed. 2010 Elsevier B.V. All rights reserved.

ction

s of state (EoS) represent the cornerstone of thermo-odels. They can be used to represent phase equilibriasive temperature and pressure ranges, while at theother properties like thermal and volumetric ones caned. Since the advent of the van der Waals equationrally hundreds such models have been published, allbe used with different degree of success for non-

s and gases. The elegance of cubic EoS, following theals developments, combined with ease of implementa-and accurate results in many practical cases explainmodels dominated and to some extent still do pplications, especially in the oil & gas sector. In par-he development of EoS, theories explicitly designedid phase (excess Gibbs energy models) were devel-uch knowledge was obtained. These activity coefcient

ding author. Tel.: +45 45252859.resses: [email protected] (G.M. Kontogeorgis),i.ac.ae (I.G. Economou).

models are limited to low pressure applications and an equa-tion of state should be used for describing deviations from theideal gas law in the vapor phase. As more demanding applicationsappeared combinedwith need for accurate design of industrial pro-cesses, it became apparent that cubic EoS did have limitations,especially for describing highly polar and/or hydrogen bondingcompounds. Consequently, higher order EoS rooted to statisticalmechanics became increasingly important and widely used bychemical engineers over the last 20 years. The increased com-plexity of these models, due to a stronger physical basis, resultin a higher accuracy over cubic EoS for highly non-ideal mix-tures.

This review is divided into two parts. In the rst part we illus-trate how far we can go with cubic equations of state and howmuch we can learn on their capabilities and limitations by taking acloser look at the excess Gibbs energy and the activity coefcientexpressions which can be derived from them. The very importantrole of mixing and combining rules will be illustrated. In the sec-ond part, we turn our interest on some of the recently developedadvanced EoS, especially those based on statistical mechanics forassociating uids. We will show some examples illustrating theirsignicant capabilities, but we will also outline some limitationswhich indicate areas where future research is required.

see front matter 2010 Elsevier B.V. All rights reserved.supu.2010.10.023M. Kontogeorgisa,, Ioannis G. Economoub

rgy Resources Engineering (CERE), Department of Chemical and Biochemical EngineeriInstitute, Department of Chemical Engineering, PO Box 2533, Abu Dhabi, United Arab

e i n f o

ne 2010vised form 16 October 2010ctober 2010

ia

s of statenergy equations

a b s t r a c t

The ideas of van der Waals have resuand PengRobinson (PR) which are wthought that the range of applicabilityrelatively similar in size. We employ iequations of state by looking at the ederived fromthese equationsof state.to the mixing and combining rules ussize-asymmetric systems are more rethe functionality of the cubic equati

Eals to association theories

chnical University of Denmark, Building 229, DK-2800 Lyngby, Denmarktes

to cubic equations of state like SoaveRedlichKwong (SRK)y used in the petroleum and chemical industries. It is oftenn der Waals-type models is limited to mixtures of compoundswork an approach for investigating the various terms of cubicGibbs energy and activity coefcient expressions which areustrate that the results of cubic equationsof state are sensitiveoreover, it is shown that previously reported deciencies forto the van der Waals one uid mixing rules used rather thanstate itself. Improved models for polar systems have been

422 G.M. Kontogeorgis, I.G. Economou / J. of Supercritical Fluids 55 (2010) 421437

2. Cubic equations of state

2.1. Physical concepts in cubic equations of state

The vanadvent of a

P = RTV b

In somewyears sinceare still muical industrreviewed inpractical puabbreviated

P = RTV b

and the Pen

P = RTV b

SRK anderature retmolecular dattractive tand PR conforces and calso combinchoice usedder Waals othe classic

a =i=1

j=1

b =i=1

j=1

aij =

aiaj

bij =bi + b

2

There arthem is thacess is largin detail bywhy these sical. First othe basic varelated to thetc.) of thepound paraacentric facthree-paramtication ofThe vdW1ffor the secofrom statistthe cross-etheory, as itric mean copotential (

It is alsofor mixturewill be call

they cannot represent polymer solutions. This is another miscon-ception, as will be illustrated in Section 2.4. We will see that theperformance of modern cubic EoS like SRK and PR may not be asmuch dependent on their precise functional form but relies much

he cmpotandons oande usped e

cesse equ

ougng cu1978fciedyn

n

lnii

(6) pom asingressthe te exc:(

i

+ PRT

(i

+(

R

+ PRT

(i

+(

R

ablder Waals (vdW) equation of state [1] marked thenew family of models which are since called cubic EoS:

a

V2(1)

hat different forms than the vdW EoS and almost 130van der Waals proposed his equation of state, cubic EoSch used today especially in the petroleum and chem-ies. There are numerous such cubic EoS and many areliterature [26], but the most widely used ones for

rposes are the SoaveRedlichKwong from 1972, oftenas SRK [7]:

a(T)V(V + b) (2)

gRobinson (PR) one from 1976 [8]:

a(T)V(V + b) + b(V b) (3)

PR as well as many other cubic EoS presented in lit-ain vdWs repulsive term, despite its mismatch withynamics data for hard-sphere uids, while differenterms are used. Most cubic EoS including vdW, SRKtain two parameters (a, b) reecting intermolecularo-volume effects. These EoS require mixing (and oftening) rules for application to mixtures and one popularalso by van der Waals and van Laar is the so-called vanne-uid (vdW1f) mixing rules, Eq. (4), together with

al combining rules shown in Eq. (5):

xixjaij

xixjbij(4)

(1 kij)j (1 lij)

(5)

e several misconceptions regarding cubic EoS. One oft they are completely empirical models whose suc-ely attributed to cancellation of errors. As discussedKontogeorgis and Folas [3] there are many reasons

imple models should not be considered entirely empir-f all, they account for free-volumes (Vf =Vb) and forn der Waals forces and the energy parameter can beephysical properties (polarizabilities, dipolemoments,intermolecular potential. Moreover, if the pure com-meters are estimated from the critical properties (andtor), cubic EoS are representations of the two- or theeter corresponding states principle. There is also jus-the classical mixing and combining rules of cubic EoS.mixing rules (Eq. (4)) satisfy the quadratic mixing rulend virial coefcient (B=

i

jxixjBij), which is derivedical mechanics. The geometric mean combining rule fornergy parameter, Eq. (5), has its origin on the Londoncan be shown that for the dispersion forces a geomet-mbining rule is derived for the cross intermolecularij =

ij).

often stated that cubic EoS cannot be satisfactorily useds with compounds differing signicantly in size, whiched size-asymmetric mixtures in this discussion. Thus,

upon tpure coundersequatienergyto makdevelonext.

2.2. Exanalyz

Althstudyilate asity coethermo

gE

RT= l

lni =

Eq.sion frrules. Uthe exp

For2.1, th

vdW

gE

RT=

SRK:

gE

RT=

PR:

gE

RT=hoice of mixing and combining rules and the way theund parameters are estimated. One way to analyze andthe capabilities and limitations of (cubic and other)f state is to look at the expressions for the excess Gibbsactivity coefcient which are derived from them ande of the knowledge we have for models and theoriesxplicitly for the liquidphase. This approach is discussed

Gibbs energy and activity coefcients: a method toations of state

h the use of excess Gibbs energy/activity coefcients forbic EoS started with the HuronVidal mixing rules as[9], calculating the excess Gibbs energy, gE, and activ-

nts, , from cubic EoS is straightforward using classicalamics:

i

xi lni =

i

xi lni (6)

(7)

ermits the derivation of the excess Gibbs energy expres-n EoS without any knowledge of mixing and combiningEq. (7) and a choice for themixing and combining rules,ion for the activity coefcient is derived.hree cubic EoS (vdW, SRK and PR) presented in Sectioness Gibbs energy equations are:

xi ln(

Vi biV b

))+(

1RT

(i

xiaiVi

aV

))(

V

i

xiVi

)(8a)

xi ln(

Vi biV b

))

1T

[i

xiaibi

ln(

Vi + biVi

) a

bln(

V + bV

)])(

V

i

xiVi

)(8b)

xi ln(

Vi biV b

))

1

T2

2

[i

xiaibi

ln

(Vi + (1 +

2)bi

Vi + (1

2)bi

)

n

(V + (1 +

2)b

V + (1

2)b

)])+ P

RT

(V

i

xiVi

)(8c)

G.M. Kontogeorgis, I.G. Economou / J. of Supercritical Fluids 55 (2010) 421437 423

With Eqs. (8a)(8c) as starting point, it is tempting to divide,somewhat arbitrarily, gE into three parts. From left to right, we canidentify a so-called combinatorial-free volume term (accountingfor size or entropic effects), a residual term (accounting for ener-getic differeterminologcientmodelLaar, Florypolymers [2widely used

van Laar

gE

RT= 1

RT

(

FloryHuggzero interac

gE

RT=

i

xi

with the vo

i =xiVijxjV

and is the[10].

Wilsonin a form scontributio

gE

RT= x1 ln

x1 l

x2 l

UNIQUA

gE

RT= x1 ln

x1q

x2q

where si

tions calculusing van d[16] or UNI

NRTL [13

gE

RT= x1x2

[Gij = exp(

In all theters in thdifferencesdata.

EntropicUNIQUAC) [

gE

RT=

i

xi

Vw is thevanderWaals volume, taken fromBondi [16]orUNIFAC[17] tables.

As can be seen in Eqs. (9a)(9h), the van Laar and NRTL mod-els have no combinatorial-free volume term; they are essentially

etics shoatoral towithtingaboquatombiy coeare Ee) terf weic-FV/matthaned las mixented

e vann the

an bing th

comcan

i

xi

xiVjx

lumexp

the sij = lij

ln

(

Laarer We (E=bi aivedGibb

1RT

(

(biRT

solt of ve mo

xibi

jxjb

r, Hptionnces) and an excess volume term. This division and thisy is inspired by the one used in explicit activity coef-s, derived for the liquid (condensed) phases like the vanHuggins, local composition models and Entropic-FV for,3]. The expressions for some of the most known andexplicit activity coefcient models are given below:

:i

xiaibi

ab

)(9a)

ins and regular solution theory (for binary mixtures,tion parameters):

lnixi

+ VRT

ij(i j)2 (9b)

lume fraction dened as:

j(9c)

solubility parameter as dened byHildebrand and Scott

[3,11] equation (shown here for binary mixtures anduitable for identifying the combinatorial and residualns):

1x1

+ x2 ln2x2

n(1 + 2 exp

(21

RT

))n(2 + 1 exp

(12

RT

)) (9d)

C [12] (for binary mixtures):

s1x1

+ x2 lns2x2

+ Z2

(x1q1 ln

1s1

+ x2q2 ln2s2

)

1 ln(1 + 2 exp

(U21

RT

))

2 ln(2 + 1 exp

(U12

RT

))(9e)

i are, respectively, the segment and surface area frac-ated in a similar way to volume fractions in Eq. (9c) buter Waals volume and surface areas (taken from BondiFAC [17] tables) instead of volumes.] (for binary mixtures):

21G21x1 + x2G21

+ 12G12x2 + x1G12

](9f)

aijij) ij =gij gjj

RT(9g)

ree models (Wilson, NRTL and UNIQUAC), the param-e exponential terms denote the energy parameterand they are typically estimated from experimental

-FV (the residual term could be taken from UNIFAC or14,15]:

lnf v

i

xi+ g

E,res

RT=

i

xi ln

(Vi Vw,iV Vw

)+ g

E,res

RT(9h)

energmodelcombinidenticis usedinterac

Thecubic ebasic cactivit(compvolumterm oEntroplogicalratherobtainvariouis pres

2.3. Thsolutio

It cassuming andenergy

gE

RT=

f vi

=

(The voThe

underwith k

lni =

Van(van dthe timuidsVihe derexcess

gE

RT=

lni =

Thecontexthan th

i = Late

assummodels, as is the regular solution theory. The otherwn in Eqs. (9a)(9h) as well as UNIFAC [17] have both aial and a residual term. UNIFACs combinatorial term isthat of UNIQUAC, whereas a group-based residual termtwo (ormore) group-based interaction parameters pergroups.ve analysis sheds some light on the capabilities ofions of state. Even the simplest cubic EoS contain thenatorial-free volumeand residual contributions like thefcientmodels explicitly developed for the liquid phaseqs. (8) and (9)). Moreover, the size (combinatorial-freem of cubic EoS is functionally similar to the equivalentll-established polymer models, like FloryHuggins and. The above analysis is based, however, on phenomeno-hematical similarities and may be more of indicativeof conclusive character. Additional evidence will be

ter by looking at the activity coefcient values usinging rules. First the special case of the van der Waals EoS.

der Waals EoS against the van Laar and the regularories

e shown for the vdW EoS that starting from Eq. (8a),at the excess volume is zero and that the vdW1f mix-bining rules apply (Eqs. (4) and (5)), the excess Gibbs

be equivalently written as:

lnf v

i

xi+ V

RTij(i j)2 (10a)

fi

jVfjVfi = Vi bi i =

ai

Vi(10b)

e fraction is dened in Eq. (9c))ression for the activity coefcient from the vdW EoSame assumptions (vdW1f mixing and combining rules=0) is:

f vi

xi

)+ 1

(f v

i

xi

)+ Vi

RT(i j)22j (11)

s starting point was the equation of state of his teacheraals) and the same mixing and combining rules used atqs. (4) and (5)). Van Laar further assumed that for liq-nd the excess volume is zero. Under these assumptions,a liquid phase activity coefcient model, which for thes energy and the activity coefcient is expressed as:

i

xiaibi

ab

)= b

RTij(i j)2 (12a)

(i j)22j)

(12b)

ubility parameters and volume fractions are, in thean Laars model, dened using the co-volumes ratherlar volumes:

ji =

ai

bi(12c)

ildebrand and Scott [10] maintained some of the basics of regular solutions (gE =hE, VE = SE =0). Their regu-


lar solution theory is shown in Eqs. (13a)(13c) for binarymixtures.In absence of any interaction parameters, the regular solution the-ory is identical to the van Laar model with the co-volume beingreplaced by the molar volumes and using the experimental solu-bility param

gE

RT= V

RT

lni =(

ViRT

Now thethe usual w

=

Hv

It is, thution theorievdW, theybe used fotions. In cocombinatorbe used forwith the regdened in E

gE

RT=

i

xi

lni = ln(

All the etion are bas(Eqs. (4) anvan der Waand (14b).an activitytures seenas FloryHuimportant fcommon, arcombinatora free-volumand an enerregular solucubic EoSbeand, if not,EoS (or both

2.4. Mixing

A moreexaminingEoS using aconsider ththe activity(15a)(15c)mixing rule

SRK:

lni = ln[

i

PR:

lni = ln[Vi biV b

]+ 1

[Vi biV b

]+ i

2

2ln

[Vi + (1 +

2)bi

Vi + (1

2)bi

]

2

a

Tb + 2

a

RT

[willite dixing=

etering e:

= ln

+

= ln

+

+

wh

nctio)turecanbof thmpaEoS trentutioexp

natorr ananatorcom1

ue toe re

residEoS aetric

bic Ee deanuneters rather than those obtained from the vdW EoS:

ij(i j)2 (13a)

(i j)22j)

(13b)

volume fractions are dened not via Eq. (12c) but inay via Eq. (9c) and the solubility parameter is given as:

ap RTV

(13c)

s, clear that both the van Laar and the regular solu-s follow the spirit of the vdW EoS. However, unlikeare purely energetic models and alone they cannotr size-asymmetric mixtures including polymer solu-mbination with the FloryHuggins or the Entropic-FVial/free-volume terms, see Eqs. (9b) and (14), they canpolymers [2,3,18,19]. The Entropic-FV model combinedular solution theory canbewritten as (volume fractionsq. (9c)):

lnf v

i

xi+ V

RTij(i j)2 (14a)

f vi

xi

)+ 1

(f v

i

xi

)+ Vi

RT(i j)22j (14b)

quations presented for activity coefcients in this sec-ed on the vdW1fmixing and classical combining rulesd (5)). Under these assumptions, we can compare theals EoS in the form of Eqs. (10) and (11) with Eqs. (14a)We conclude that the vdW EoS also when written ascoefcient model contains many of the essential fea-in successful and well-known polymer models suchggins, regular solution theory and Entropic-FV. Theseeatures, which vdW EoS and polymer models have ine an explicit representation of the size-asymmetry via aial-free volume term which has the same functionality,e denition similar to the one used in polymer models

gy term which is mathematically similar to that of thetion theory. Thus, the following question is raised: canused for size-asymmetricmixtures includingpolymersis this attributed to the size or the energy term of the)?

and combining rules in the van der Waals spirit

detailed analysis of the cubic EoS can be obtained bythe expression for the activity coefcient derived fromspecic set of mixing and combining rules. At rst wee classical vdW1f mixing rules, Eq. (4), and using themcoefcient equationswith SRK and PR are shown in Eqs.. Eqs. (15a) and (15b) can be further simplied if a linearis used for the co-volume parameter, b=

ixibi

Vi biV b

]+ 1

[Vi biV b

]+ i ln

[Vi + bi

Vi

]

ln[V + b

V

]+ (bVi Vbi)

V(V + b) (15a)

where

=bR

bi =

i = b

Weat innuid meter, bparamfollow

SRK

ln1

PR:

ln1

(The fuscripts

Mixwhichbutionand cocubicof diffecontribenergycombiIn theicombi(16) (lnb2))) done. Thgetic (the PRasymm

(i) Cutivth i2ln

[V + (1 +

2)b

V + (1

2)b

]+ (bVi Vbi)

V(V + b) + b(V b) (15b)

j

xjbij ai = a + 2

j

xjaij

aia

+ 1 bib

] (15c)

simplify these expressions by considering the equationsilution for binary mixtures using the van der Waals onerules, the linear mixing rule for the co-volume param-

ixibi, and the geometric mean rule for the cross-energy(with kij =0; aij =

aiaj). Under these assumptions, the

quations can be derived:

(V1 b1V2 b2

)+ 1

(V1 b1V2 b2

)+ a1

b1RTln

(1 + (b1/V1))(1 + (b2/V2))

a2b2RT

(V1b2 V2b1V22 + b2V2

)+ b1

RT

[a2

b2

a1

b1

]2ln(1 + b2

V2

)(16a)

(V1 b1V2 b2

)+ 1

(V1 b1V2 b2

)a1

2

2b1RTln

f (b1, V1)f (b2, V2)

+ a2b2RT

(V1b2 V2b1

V22 + 2V2b2 b22

)

b1

2

2RT

[a2

b2

a1

b1

]2ln f (b2, V2)

ere f (bi, Vi) =[

Vi + (1 +

2)biVi + (1

2)bi

](16b)

n f(b2, V2) is the same as f(b1, V1), with change of sub-

s containing alkanes are nearly athermal solutionseused for testing the combinatorial-freevolumecontri-ermodynamic models. It is, thus, of interest to computere the innite dilution activity coefcient values fromo experimental data for mixtures containing alkanessize as well as examining the values of the different

ns. Unlike what was the case with the excess Gibbsressions, it may be difcult to identify which are theial/free-volume and residual parts in Eqs. (15) and (16).lysis from 1998, Kontogeorgis et al. [20] considered theial-free volume to be the rst three terms of Eqs. (15) orbFV, = ln((V1 b1)/(V2 b2)) + 1 ((V1 b1)/(V2 similarities with polymer models like the Entropic-FV

maining terms of Eqs. (15) or (16) constituted the ener-ual) term in that analysis. Kontogeorgis et al. [20] usednd the most important conclusions of their analysis formixtures of alkanes are:

oS with vdW1f mixing rules and kij =0 result to posi-viations from Raoults law (activity coefcients higherity), in contradiction to theexperimentaldata (negative


0,5

0,6

0,7

0,8

0,9

1

1,1

1,2

1,3

14 16 18 20 22 24 26 28 30 32 34 36

Carbon number of Alkane

Infi

nit

e D

ilu

tio

n A

cti

vit

y C

oe

ffic

ien

t o

f H

ep

tan

e exp. data

Modified UnifacPR EoS (vdw1f mix. rule)

PR EoS (a/b mix. rule)

Fig. 1. Activit293373K asequation of staand (5)) and ththe modied U

deviaticientssee also

(ii) The comunity (much htive vavalidwmean fco-volu(Berthe

(iii) It is iminteracto corrresiduacient foresultsto veryseriesobtaine

At a rssmooth kij-that cubicalthough ththe modelsand activitycombinatorafter.

Table 1Percentage decoefcients at(with the vdWactivity coefc

Mixture

nC5/nC6nC5/nC20nC5/nC22nC5/nC24nC5/nC28nC5/nC32nC5/nC36

Following the ideas of van Laar and HuronVidal, we rec-ognize that at the innite pressure limit (when Vb) thecombinatorial/free-volume termvanishes and the remaining termscorrespond to the residual (or energetic) part. Thus, for all threeEoS consideinnite pre

gE,EoS,P

RT=

= RT

(

= 1 (vdW = ln 2 (S = 0.623

SRKrt oatorhic

=bi):

= ln

+

= lnc1

+2

the fith Snd (1parm ofyingactlyility p

es =y coefcients at innite dilution of n-heptane in alkane solvents ata function of the alkane carbon number using the PengRobinsonte. Results are shown using vdW1f mixing rules (with kij =0, Eqs. (4)e a/bmixing rule, Eq. (19). For comparison are shown the resultswithNIFAC activity coefcient model [47].

ons from Raoults law). Moreover, these activity coef-become even higher with increasing size asymmetry;Fig. 1 and Table 1.binatorial/free-volume term has values slightly below

close to one), while the residual term reaches valuesigher than one, which explains the very large posi-

lues for the activity coefcients. These conclusions arehen the classical combining rules are used (geometricor the cross energy, arithmetic mean rule for the crossme, Eq. (5)), while the other combining rules testedlot, Lorentz, Sandler) failed (in the context of PR).portant to correct the combining rule for b12 (via antion parameter), which can result at the same timeect values for the combinatorial/free-volume and thel contributions, thus also for the total activity coef-r alkane mixtures. This approach results to good VLEaswell e.g. a single lij (=0.02) togetherwith PR can resultgood correlation of VLE for the whole ethanealkane

(up to ethaneC44). Equally good results cannot bed using a single kij value.

t look, these and other results, for example the noncarbon number trends for gas/alkanes, may indicateEoS cannot represent size-asymmetric mixtures well,

Forthe pacombin(16b) wand Vi

SRK

ln1

PR:

ln1

wherement w(18a) a

Comlast teridentifEoS exsolub

ln,r1is could be attributed more to the energetic term of. A closer look [3,21,22] at the excess Gibbs energycoefcient equations reveals the true identity of the

ial/free-volume (size-related) term, as explained here-

viations between the experimental and calculated pentane activityinnite dilution in various alkane solvents at 373K using the PR EoS1f and the a/b mixing rule, Eq. (19)) and with the modied UNIFACient model [47].

PR (vdW1f, kij =0) PR (/b rule) Modied UNIFAC

6 4 424 1 1028 5 1139 3 753 6 667 10 585 13 3

Consequsize-asymmity coefcie(18a) or (18the followin

a

b=

i

xia

b

b =

i

xib

It can beEqs. (15a) aple a/b mixthe simpleing the resFigs. 14 anrules (Eq. (1size asymmred (vdW, SRK and PR) here, the excess Gibbs energy atssure can be written as:

gE,res

RT=

(1RT

(i

xiaibi

ab

))

i

xii )

= ab

, vanLaar)RK)(PR)

(17)

and PR at innite dilution, it can be shown thatf Eqs. (16a) and (16b) which corresponds to theial/free-volume term (i.e. the part of Eqs. (16a) andh would disappear at innite pressure by setting V=bcan be expressed as:

combFV1 = ln(

V1 b1V2 b2

)+ 1

(V1 b1V2 b2

)a1

b1RTln

(1 + (b1/V1))(1 + (b2/V2))

+ a2b2RT

(V1b2 V2b1V22 + b2V2

)(18a)

ombFV = ln(

V1 b1V2 b2

)+ 1

(V1 b1V2 b2

)a12b1RT

lnf (b1, V1)f (b2, V2)

+ a2b2RT

(V1b2 V2b1

V22 + 2b2V2 b22

)(18b)

(bi,Vi) function is given in Eq. (16b). This is also in agree-akomani and Brignole [21] who called these terms Eqs.8b) as non-residual.

ingwith Eqs. (16a) and (16b), it can be seen that only theEqs. (16a) and (16b) remains even at innite pressure,as the residual term (true energetic tem) of the cubicthis last contributionwhich contains the EoS-generatedarameters (i = (

ai/bi)) e.g. for SRK:

b1RT

[a2

b2

a1

b1

]2ln(1 + b2

V2

)ently, an assessment of the true value of cubic EoS foretric mixtures can be obtained by computing the activ-nts from Eqs. (18a) and (18b). It can be shown that Eqs.b) are the total activity coefcients fromcubic EoSusingg simple mixing rules:

i

i

i

(19)

proved that Eqs. (18a) and (18b) can be derived fromnd (15b) at innite dilution conditions using the sim-ing rule shown in Eq. (19). Thus, based on the above,mixing rules of Eq. (19) have the property of eliminat-idual term of cubic EoS. Typical results are shown ind Table 1 illustrating that cubic EoS with these mixing9)) predict reliable values of the activity coefcients foretric alkane mixtures, even with kij =0. The predictions


10

15

20

25

30

35

40

0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8mole fraction of Hexane

Pre

ss

ure

(b

ar)

exp (liquid phase)exp (vapor phase)PR eos (a/b rule)PR eos (vdw1f rule)

Fig. 2. Vaporliquid equilibria for hexane/nC16 at 623K using the PengRobinsonequation of state. Results are shown using the vdW1f mixing rules (with kij =0, Eqs.(4) and (5)) and the a/b mixing rule, Eq. (19).

5

15

25

35

45

55

0

Pre

ssu

re (

bar

)

Fig. 3. Pressu373.15K usingvdW1f mixing

are in qualimental valubetter commixing ruleequilibriumhave a perc

5

15

25

35

45

0

Pre

ssu

re (

bar

)

Fig. 4. PressurPengRobinso(with kij =0, Eq

with the vdW1f rules. Similarly the errors (respectively with Eq.(19)/vdW1f) are 12% and 20% for C2/C28 and 13% and 33% forC2/C36 (373K). These are typical results and similar deviations areobtained for other systems as well. All results with both mixingrules are prto zero.

This invfor cubic Eoattributedmcially that ofunctionalitmixing rulenext.

2.5. Excessclassical to a

As illustof the vdWtems like aprovided thOne suitabl

vdWexpractHur] andhichparaexp. data

PR eos (a/b mix. rule)

PR eos (vdw1f mix. rule)

nor thegin areno interst by[2426ruleswenergy0,30,20,1

mole fraction of methane

reliquid phase mole fraction plot for methane/nC36 mixture atthe PengRobinson equation of state. Results are shown using therules (with kij =0, Eqs. (4) and (5)) and the a/b mixing rule, Eq. (19).

tative and often quantitative agreement to the experi-es. Moreover, these predictions are shown to be muchpared to those of cubic EoS using the classical vdW1fs (i.e. using Eqs. (16a) and (16b)), even for vaporliquidcalculations. For example, for the system C1/C36 we

entage deviation in pressure 3% with Eq. (19) and 14%

0,50,40,30,20,1

mole fraction of Ethane

exp (liquid phase)

PR eos (a/b rule)

PR eos (vdw1f rule)

eliquid phase mole fraction plot for ethane/nC28 at 573K using then equation of state. Results are shown using the vdW1f mixing ruless. (4) and (5)) and the a/b mixing rule, Eq. (19).

excess Gibband from anreference p(

gE

RT

)EoSP

=

The Hurand in this cthe size terthe EoS van(

gE

RT

)EoS

=

which in th

a

b=

i

xia

b

vdW = 1SRK = lnPR = 1

2

It is thusapproach (model whic(21) e.g. NRrule is empshown thatto an excellrily phase ephase behationparameshouldbe reinnite refeavailable inpressure daedictions, i.e. the interaction parameters are set equal

estigation illustrates that problems previously reportedS in representing size-asymmetric systems should beore to the use of specicmixing/combining rules espe-f the energy parameter, rather than the van der Waalsy of the vdW repulsive term. The studies of the EoS/GE

s will shed some additional light on this, as explained

Gibbs energy at zero and innite pressures: fromdvanced mixing rules (EoS/GE)

rated from the analysis in Sections 2.22.4, cubic EoS-type can be used successfully for size-asymmetric sys-ctivity coefcient calculations for mixtures of alkanesat mixing rules other than the vdW1f ones are used.e choice is the simple a/b rule, Eq. (19), but neither this1f mixing rules with their random-mixing based ori-

ected to work well for polar mixtures, especially whenion parameters are used. Another approach, pioneeredon and Vidal [9] and later by Mollerup [23], MichelsenWongSandler [27,28] is the so-called EoS/GE mixingprovide away to derivemixing rules for, especially, themeter of cubic EoS. The starting point is equating thes energy (or the excess Helmholtz energy) from the EoSexplicit activity coefcient model (*) at some specic

ressure, P:(gE

RT

)model,P

(20)

onVidal approach assumes innite reference pressurease in accordance to the discussion in Sections 2.22.4,ms (combinatorial/free-volume and excess volume) ofish and Eq. (20) reduces to:(

gE

RT

)model,

(

gE

RT

)EoS,res=(

gE

RT

)model,res(21a)

e case of vdW, SRK and PR can be written as:

i

i 1

gE,

2

2ln

(2 +

2

2

2

) (21b)

understood that in combination with the HuronVidalinnite pressure limit), an explicit activity coefcienth only has a residual-type term should be used in Eq.TL or the residual term of UNIQUAC. The linear mixingloyed for the co-volume parameter. Many studies haveSRK and PR using the HuronVidal mixing rule resultent correlation model capable of describing satisfacto-quilibria for binary complexmixtures and for predictingvior of multicomponent mixtures [3,9,29]. The interac-ters of the explicit excessGibbs energymodel, e.g.NRTL-estimated togetherwith the cubic EoS since, due to therence pressure used, existing parameters of the modelsdatabases like Dechema which are obtained from lowta cannot be used.


About 10 years after Vidal, Michelsen [24,25] showed how anEoS/GE mixing rule can be derived at zero reference pressure. Herethe mathematical analysis is a bit more complicated comparedto the innite pressure limit and neither the combinatorial/free-volume noran implicitthe energy(

gE

RT

)mode0

where the rthe qe() fuvalues >function is0.9. Thus, eexplicit excoften simplMHV1andMimation ofequations [

MHV1:

= 1AM

(g

MHV2:

q1

(

i

=(

gE

RT

)

The origmodels [2,3

The MHexpressionsbecause ofderived from

gE,EoS,0P

RT

where the ction of state

Notice tonly by thcombinatorzero referention can befractions ofet al. [31] anMHV1/MHVVLE for comup to 200bmodel andwsure data. Nbeen a breaFor mixtureas noUNIFAeters, e.g. Ccubic EoS w

As Kalospressure mduce the ex

model they are combined with. Moreover, the difference from thereference activity coefcient model increases with increasing size-asymmetry of the mixture components [33]. Another issue, moreof theoretical nature, is the inability of EoS/GE models to repro-

e thoefhisorule

ults wfact. Actuuce ts, inhat tn evcubis, e.g

ws nohis lg rulis prry exs in.

S/GE

es

sawre limal (enre Eore M

ixiequat actice patorthe. This, asmaterials: Thcel o

nd Aferene-asyan eymmf MH

2 inapo

to revani

s, likode

q. (20our EM [3

(gE

RTthe residual term of the EoS vanish. In the general case,mixing rule is obtained at zero reference pressure forparameter:

l,+

i

xi ln(

b

bi

)= qe()

xiq

ei (i) (22)

educed energy parameter is dened in Eq. (15c), whilenction depends on the EoS used and it is dened only forlim. This excludes mixtures with gases, since the qe()dened for reduced temperature Tr-values up to aboutven though Eq. (22) ensures full reproduction of theess Gibbs energy model the cubic EoS is combined with,er or even explicit mixing rules are used in practice. TheHV2rulesarebasedona linearandaquadratic approx-

the qe() function and they are given by the following26,31]:

E,

RT+

i

xi ln(

b

bi

))+

i

xii (23a)

xii

)+ q2

(2

i

xi2i

)

+

i

xi ln(

b

bi

)(23b)

inal publications [26,30] and books on thermodynamic] contain values for q1, q2 and AM for both models.V1/PSRK mixing rules correspond to the followingof the excess Gibbs energy at zero pressure. Exactly

the assumptions made, this is the excess Gibbs energya cubic EoS at approximately zero pressure:

i

xi lnbib

+(

q

RT

(i

xiaibi

ab

))(23c)

onstant q depends on the functional form of the equa-and the approximation used.

hat MHV1 differs from the HuronVidal mixing rulee (

ixi ln(b/bi)) term which is a FloryHuggins typeial term, originating from the cubic EoS at approximatecepressure, as canbe seen inEq. (23c). TheMHV1equa-also derived [23] assuming equal values for the packingpure compounds and of the mixture. As shown by Dahld Dahl andMichelsen [26], using either the exact or the2 mixing rules, it is possible to predict high pressureplex mixtures like acetonewater and ethanolwater

ar employing UNIFAC as the explicit activity coefcientith UNIFAC parameters solely obtained from lowpres-o additional adjustable parameters are used. This has

kthrough development in the eld of predictivemodels.s with gases, e.g. CO2/alkanes or N2/alkanes, however,C parameters are available for gases, newgroupparam-O2/CH2 and N2/CH2 should be determined from theith the incorporated EoS/GE mixing rule.piros et al. [32] showed, the approximate zero referenceodels (MHV1, MHV2, PSRK, etc.) do not fully repro-cess Gibbs energy and especially the activity coefcient

duce thvirial crules. Tmixingful resto thecientreprodmid 90cated tfrom acess ofmodelperformalkanebons. T(mixinthat thisfactomodelsection

2.6. Eomixtur

Wepressuresidupressupressution (

Whenexplicireferencombine.g. incel outsystembe estibinatowritesple can[39] athe difing sizofferssize-ascase oCO2/CHtorily vordershouldmodelthese mfrom Ethese f

LCV

= C1eoretically correct quadratic mixing rule for the secondcient, which is satised by the simpler vdW1f mixingbservation led to thedevelopmentof theWongSandlers [27,28], but it has not been veried that the success-ith the WongSandler mixing rules can be attributed

they satisfy the mixing rule for the second virial coef-ally, the WongSandler mixing rules also fail to fullyhe activity coefcient model incorporated [34]. In thea number of publications [3538] it has been indi-

he approximate zero reference pressure models sufferen more serious problem, which is, in light of the suc-c EoS discussed previously, somewhat surprising. These. MHV1, MHV2, PSRK but also WongSandler do notell for size-asymmetric mixtures, neither mixtures ofr mixtures of gases (ethane, CO2, N2) with hydrocar-ed to the development of various EoS/GE-type modelses) for correcting this deciency. We believe, however,oblem must be addressed generally for obtaining a sat-planation of the limitation which can result to bettera systematic way. This issue is discussed in the next

mixing rules: the special case of size-asymmetric

in the previous section, Eq. (21), that in the inniteit for EoS/GE models, we have in reality an equation ofergetic) terms. This is not the case in the zero referenceS/GE models, where, for the approximate zero referenceHV1 and PSRK models, a FloryHuggins type contribu-ln(b/bi)) results from the EoS, see Eqs. (23a) and (23c).ting the excess Gibbs energy from an EoS to that of anvity coefcient model for deriving a mixing rule at zeroressure, it would be useful if this FloryHuggins typeial and that of the explicit activity coefcient models,case of UNIQUAC/UNIFAC (

ixi ln(r/ri)), would can-

s is especially important in the case of gas-containingthe new group interaction parameters that need tod should not carry this difference of the two com-in the parameter estimation. Mollerup [23] in 1986

e two FloryHuggins combinatorials should in princi-ut. However, as shown by Kontogeorgis and Vlamoshlers and Gmehling [40], they dont! And moreoverce of the two combinatorials increases with increas-mmetry, e.g. for mixtures of alkanes. This observation

xplanation why models likes MHV1 and PSRK fail foretric systems. As Coniglio et al. [41] showed, for theV1, unique group interaction parameters, e.g. for theteraction cannot be obtained for representing satisfac-rliquid equilibria for the whole CO2/alkane series. Insolve this problem, the double combinatorial effectsh and this is what in essence is done in various EoS/GE

e LCVM, k-MHV1, GCVM and CHV (although some ofls have been derived semi-empirically, i.e. not starting)). The equations for the reduced energy parameter foroS/GE models are shown below:537]:

, C2

C1

gE,FH

RT

)+

i

xii (24a)


where

C1 =

AV+ 1

AM

C2 =1 AM

AV = 0.62LCVM usk-MHV1

= 1AM

(g

R

Using thof the param

= C1(

gE

RT

where

C1 =1

AMAM = 0.53

For PR aCHV (Or

= 1C

(gE

R

where C* =Based on

parameterent dependcoefcientparametersFAC they shConiglio etbinatorials(0.640.72port the hysuccessful ris the elimi

This coninnite diluing rules anand a/b mixmodel usedtorial stemmactivity coe

As discEoS/vdW1ftures, whilvanishes. Tcombinatorapplied onlWe saw thacubic EoS sTheHuronfrom EoS, ifactivity coemodels likeis eliminate

Recentlyon the prinhave been

11

bution

toth

e

activi

tyco

efci

ent.

Mden

otes

the

explici

tex

tern

alac

tivi

tyco

efci

entm

odel

use

din

the

EoS/

GE

mix

ing

rule

.FH

den

otes

the

FHco

mbi

nat

oria

lst

emm

ing

from

the

equat

ion

ofst

ate.

ln

,c

ombF

V1

(ext

ra)is

the

additio

nal

co

mbi

nat

oria

l/free

-vol

um

eco

ntr

ibution

stem

min

gfrom

the

explici

tac

tivi

tyco

efci

entm

odel

inse

rted

via

the

EoS/

GEm

ixin

gru

le.

Mod

elln

,com

bF

V1

ln

,c

ombF

V1

(ext

ra)

ln

,r

es1

vdW

1fm

ixin

gru

les,

Eqs.

(4)an

d(5

)ln

( V 1

b1

V2

b2

) +1( V 1

b1

V2

b2

) +a 1

b1RT

ln(1

+(b

1/V1))

(1+

(b2/V2))

+a 2

b2RT

( V 1b2

V2b1

V2 2

+b2V2

)b1

RT

[ a 2 b 2

a1

b1

] 2 ln(1

+b2

V2

)a/

bm

ixin

gru

les,

Eq.(

19)

ln

( V 1

b1

V2

b2

) +1( V 1

b1

V2

b2

) +a 1

b1RT

ln(1

+(b

1/V1))

(1+

(b2/V2))

+a 2

b2RT

( V 1b2

V2b1

V2 2

+b2V2

)Not

hin

g

Huro

nV

idal

,Eq.

(21b

)ln

( V 1

b1

V2

b2

) +1( V 1

b1

V2

b2

) +a 1

b1RT

ln(1

+(b

1/V1))

(1+

(b2/V2))

+a 2

b2RT

( V 1b2

V2b1

V2 2

+b2V2

)

1 AV

ln

( V 2+

b2

V2

) lnM

,com

b1

1 AV

ln

( V 2+

b2

V2

) lnM

,res

1

MHV1,

Eq.(

23a)

ln

( V 1

b1

V2

b2

) +1( V 1

b1

V2

b2

) +a 1

b1RT

ln(1

+(b

1/V1))

(1+

(b2/V2))

+a 2

b2RT

( V 1b2

V2b1

V2 2

+b2V2

)(ln

M

,com

b1

ln

FH 1

)

1 AM

ln

( V 2+

b2

V2

)

1 AM

ln

( V 2+

b2

V2

) lnM

,res

1

k-M

HV1,

Eq.(

24b)

ln

( V 1

b1

V2

b2

) +1( V 1

b1

V2

b2

) +a 1

b1RT

ln(1

+(b

1/V1))

(1+

(b2/V2))

+a 2

b2RT

( V 1b2

V2b1

V2 2

+b2V2

)(ln

M

,com

b1

kln

FH 1

)

1 AM

ln

( V 2+

b2

V2

)

1 AM

ln

( V 2+

b2

V2

) lnM

,res

1

LCVM

,Eq.

(24a

)ln

( V 1

b1

V2

b2

) +1( V 1

b1

V2

b2

) +a 1

b1RT

ln(1

+(b

1/V1))

(1+

(b2/V2))

+a 2

b2RT

( V 1b2

V2b1

V2 2

+b2V2

)([ A

V+

1

AM

] lnM

,com

b1

[ 1

AM

] lnFH 1

) ln

( V 2+

b2

V2

)[ A V

+1

AM

] lnM

,res

1

ln

( V 2+

b2

V2

)CHV,E

q.(2

4d)

ln

( V 1

b1

V2

b2

) +1( V 1

b1

V2

b2

) +a 1

b1RT

ln(1

+(b

1/V1))

(1+

(b2/V2))

+a 2

b2RT

( V 1b2

V2b1

V2 2

+b2V2

)(ln

M

,com

b1

(1

)

ln

FH 1)

1 Cl

n

( V 2+

b2

V2

)

1 Cl

n

( V 2+

b2

V2

) lnM

,res

13, AM = 0.52es =0.36 (C2/C1) = 0.68[42]:

E,

T k g

E,FH

RT

)+

i

xii (24b)

e t-modied PR+original UNIFAC, the optimum valueeter k=0.65 GCVM (Coniglio et al. [41]):

, C2

C1

gE,FH

RT

)+

i

xii (24c)

C2 =1 AM

, = 0.285nd UNIFAC VLE the ratio proposed is: C2/C1 =0.715bey and Sandler [43], Zhong and Masuoka [44]):

,

T (1 )g

E,FH

RT

)+

i

xii (24d)

0.6931 and the proposed values for 1 =0.640.7.the above explanation, it can be understood why the

C2/C1 in LCVM or the k in k-MHV1 must be differ-ing on the combinatorial term of the explicit activitymodel used. Thus, in the case of original UNIFAC theseshould be around 0.650.68, while for modied UNI-ould around 0.3. Similar values have been reported byal. [41] for both the original and modied UNIFAC com-and by Orbey and Sandler [43]/Zhong and Masuoka [44]for original UNIFAC). These similar values further sup-pothesis that the key concept and explanation of theesults of these EoS/GE models for asymmetric systemsnation of the double combinatorial effect.clusion can be further understood by looking at thetion activity coefcients from SRK using the abovemix-d compare them to HuronVidal, MHV1/PSRK, vdW1fing rules (M is the explicit external activity coefcientin the mixing rule, while FH denotes the FH combina-ing from the equation of state). These innite dilution

fcients are shown in Table 2.ussed previously, the residual term from cubicdoes not perform satisfactorily for asymmetric mix-

e when using the a/b mixing rule the residual termhus, in the latter case the cubic EoS has onlyial/free-volume contributions and can be rigorouslyy to nearly athermal mixtures like mixtures of alkanes.t the results are satisfactory in this case, indicating thatatisfactorily capture size effects in athermal solutions.Vidalmixing rule has only the same combinatorial terman extra combinatorial term is not used in the externalfcientmodel. Theapproximate zero referencepressureMHV1 and PSRK have a double combinatorial whichd in other mixing rules, e.g. LCVM and CHV., more approaches which also fully or partially relyciple of eliminating the double combinatorial term,proposed; these approaches, e.g. PSRK/Li [45], VTPR Ta

ble

2In

nite

dilution

activi

tyco

efci

ents

calc

ula

ted

from

SRK

using

the

variou

sm

ixin

gru

les.

ln

,c

ombF

Vis

the

com

binat

oria

l/free

-vol

um

eco

ntr

ibution

toth

eac

tivi

tyco

efci

entan

dln

,res

isth

ere

sidual

contr

i


[40] and UMR-PR [46] also result to EoS/GE models suitable forsize-asymmetric systems. In some of these approaches the combi-natorial termof the activity coefcient and of the EoS (FH-term) arebothdroppedout or/anda combining rule other than the arithmeticmean comb

2.7. Capabi

The discbilities ofsize-asymmbinary andEoS have alblends. Thenitz and co-term. Mostnal vdW redeveloped fing severalMasuoka [5sion of the Sused the quan additionand co-voluwith cubic E

In combEoS represeespecially fmixtures. Cequilibria acorrelated wrections toStill, even wple performthey are comasymmetricgood perforvolume term

Thus, thtively of the[2,3,60]:

Correlatioplex mixthydrocarbusing tem

Descriptiosame set

Correlatiocible sywaterpe

PredictionLLE and Sdata.

Very combiomolec

To handneeded whplesof thesethe chemicailarities to tworkingat tnew concepgenerated w

the aforementioned complex types ofmixtures andphasebehavior.In some sense, we could state that van der Waals-type equationsof state really can take us far but not further, in the sense there is aplethora of systemswhere the physical approaches (vdWand the

ouldctorysful aare dl as p

ociat

igher

deles ationequilulatsta

anceal enf theecadtatise calecurest tpartles (

dp

rN ales,tionaolec

oniacoordsum:

) =

rstraturnds. Intas ha

pley tolar i

m thdynfor thble)

) = A isties iasedrtitiowsimining rule is used for the cross co-volume parameter.

lities and limitations of cubic equations of state

ussion so far has illustrated the tremendous capa-cubic equations of state, capabilities which includeetric mixtures, even polymers, as well as polar VLE ofmulticomponent mixtures. Since the early 90s cubicso been used for modelling of polymer solutions andrst cubic EoS for polymers was proposed by Praus-workers [48], based on a modied vdW-type repulsiveother cubic EoS employed to polymers use the origi-pulsive term. The vdW, SRK and PR EoS have all beenor polymers and various mixing rules are used includ-EoS/GE ones [4959]. It is interesting that Zhong and9]haveused thea/bmixing rule (Eq. (19)) in their exten-RK and PR equations of state to polymer solutions. Theyadratic mixing rule for the second virial coefcient asal equation to obtain a mixing rule for both the energymeparameters. They obtained good results, better thanoS using the conventional vdW1f mixing rules.

ination with especially the EoS/GE mixing rules, cubicnt an excellent correlative and even predictive toolor low and high pressure VLE for a wide range ofertain complex types of phase behavior, e.g. solidgasnd VLE of complex mixtures like H2Swater can beell using two adjustable interaction parameters cor-

the classical vdW1f combining rules, Eqs. (4) and (5).ith the EoS/GE mixing rules, cubic EoS cannot in princi-muchbetter than the explicit activity coefcientmodelbinedwith. Asmentioned, however, for athermal size-mixtures satisfactory results are obtained due to the

mance of the underlying cubic EoS combinatorial/free-, see the results for the LCVM model for instance.

e most serious problems with cubic EoS, and irrespec-choice of mixing and combining rules, are expected for

n of vaporliquid and liquidliquid equilibria of com-ures containing hydrogen bonding compounds withons and similar ones, e.g. alcohol-alkanes, especiallyperature-independent interaction parameters.n of VLE and LLE of e.g. methanol/alkanes using the

of interaction parameters.n of liquidliquid equilibria of highly immis-stems, e.g. wateralkanes, glycolalkanes andruoroalkanes.of multiphase, multicomponent equilibria, e.g. VLLE,

LE using parameters solely obtained from the binary

plex mixtures such as those containing electrolytes andules.

le such complex mixtures, new advanced models areich go truly beyond the ideas of van der Waals. Exam-newconcepts stemfromtheworkofWertheimor froml and lattice theories. Many of these theories bear sim-he early chemical approach by Dolezalek [61] who washesametimeasvanderWaals andvanLaar.Using thesets, families of advanced equations of state have beenhich can describe better than the vdW-type cubic EoS

like) shsatisfasuccesyears,as wel

3. Ass

3.1. H

Themolecucorrelaphaseare simappliedappearchemicsome othree d

In stem arthe moof inteso themolecu

Q = c

wheremolecuproporable) mHamiltof theas theso that

H(rNpN

ThetempeU depeactions(suchate comthe wamolecuetc.

Frothermowhichensem

A(NVT

whereproperlated bthe pavery febe combined with the chemical ones for arriving to arepresentation of the phase behavior. Some of themostdvanced equations of state, developed over the last 30iscussed next together with characteristic applications,resentation of their capabilities and limitations.

ion theories

order EoS rooted to statistical mechanics

velopment of chemical processes using complexnd the need for accuratemodels able to provide reliableand prediction of the thermodynamic properties andibria of the mixtures involved so that these processesed and optimized, together with signicant advances intistical mechanics in the 1980s and onwards led to theof higher order EoS that gradually gained popularity ingineering community. In this section, we will reviewmost popular higher order EoS developed in the last

es based on statistical mechanics.tical mechanics, the properties of a bulk chemical sys-lculated based on the collective interactions betweenles that make up the system. Almost all of the systemso chemical engineering follow Boltzmann statistics andition function (Q) of a system of constant number ofN) in a specic volume (V) and temperature (T) is [62]:

N drN exp

[H(rNpN)

kBT

](25)

nd pN denote the coordinates and momenta of all NH(rNpN) is the Hamiltonian of the system and c is ality constant. For a systemofN identical (indistinguish-ules: c=1/(h3NN ! ) where h is Plancks constant. Then provides the total energy of the system as a functioninates and the momenta of the molecules and is givenof the kinetic energy (K) and the potential energy (U),

i

p2i

(2mi) + U(rN)(26)

term on the right hand side of Eq. (26) provides thee dependence of the Hamiltonian. The potential energystrongly on the nature (complexity) of molecular inter-ermolecular potentials range from primitive potentialsrd sphere, square well, etc.) to potentials of moder-xity (such as Lennard-Jones, Stockmayer, etc.) and allcomplex potentials that account for intra- and inter-nteractions, many body effects (polarizable potentials),

e partition function, one may calculate macroscopicamic properties using the so-called bridge function,e case of the constant NVT system (canonical statistical

is:

kBT lnQ (NVT) (27)

the Helmholtz free energy. Additional thermodynamicncluding pressure, chemical potential, etc. can be calcu-on standard thermodynamic relations. Unfortunately,n function Q can be calculated analytically only for aple systemsandsignicantapproximationsareneeded


along the way in order that Eq. (27) can lead to engineering mean-ingful results. The van der Waals and other cubic EoS can be alsoderived from statistical mechanics by assuming a simple inter-molecular potential [62].

A powerthe intermoperturbatiosummation

U(rN) = U(0

where U(0)(but yet relabation termchoice of thcations neand, to a latem underwas proposMcQuarriemodels refements resutheories forby Barker adler,Weekstheories is g

The formEoS was prPerturbed-Henergy is wrepulsive (r

A = Aig + Ar

The repbased on tAlders Tayby Donohuon Stells p[66]. Finallyory (APACTmodelled bcal theory,reactions thmaydistingrelevant co

A differeintroducednamic pertu(SAFT) usesand hydrogSAFT, hydroand the assrial balancewell potenttremendounity and wgases and htures. In adexplicitly fChain-SAFTvariable ranpolar intera(electrolyte(29) and sonumber of[3,76,77].

A different approach to model thermodynamic properties ofuids is based on lattice theory. Early attempts in this directioninclude the models proposed by Flory [78], and Fowler andGuggenheim [79]. Sanchez and Lacombe formulated a lattice

in ttingdel iids a, knondomtiong beon Ve

ram

ll ofaramulatacterlarle).

etersg. Into exasesticaled fretersily exle, thhe paequixtu

curaclly ij is

ratur

pres

herorangessurial inCT aands syue trongutuafor v5(a)n fr

nt agodelte msupraoligoE ofhere

to 2ive ren wis ne[68]ful and elegant approach to simplify the complexity oflecular potential is based on the perturbation theory. Inn theory, the intermolecular potential is written as theof two terms:

)(rN) + U(1)(rN) (28)

rN) refers to the reference system and is the dominant,tively easy to calculate term, and U(1)(rN) is the pertur-,which ismore complex and less accurately known. Thee reference system and the approximations and simpli-eded for the perturbation term are practically endlessrge extent, depend on the nature of the chemical sys-investigation. One of the earliest perturbation theoriesed by Zwanzig in 1954 [63] followed by the work ofand Katz in 1966 for high-temperature gases [64]. Bothr to relatively simple systems. Subsequent advance-lted in much more useful and accurate perturbationreal uids (gases and liquids) such as those proposednd Henderson in the period 19681972 and by Chan-andAndersen in1971. Anexcellent descriptionof theseiven by McQuarrie [62].ulation of the perturbation theory into an engineeringoposed by Donohue and Prausnitz [65] based on theard-Chain Theory (PHCT). In PHCT, the Helmholtz free

ritten as the sum of contributions due to ideal gas (ig),ep) and attractive (att) interactions, so that:

ep + Aatt (29)

ulsive and attractive contributions were calculatedhe Carnahan-Starling EoS for hard spheres and thelor series expansion, respectively. PHCT was extendede and co-workers to polar and polarizable uids basederturbation theory for dipolar and quadrupolar uids, in the Associated-Perturbed-Anisotropic-Chain The-), hydrogen bonding between molecules was explicitlyased on chemical theory [67,68]. According to chemi-hydrogen bonding is modelled as a series of chemicalat result in the formation of oligomers. In this way, oneuishbetweenphysical and chemical forces and theirntribution to the overall model.nt family of EoS based on perturbation theory wasin 1990. Rooted to Wertheims rst order thermody-rbation theory, the Statistical Associating Fluid Theorythe hard sphere as the reference uid while dispersionen bonding interactions are perturbations [69,70]. Ingen bonding is modelled using a square well potentialociation term is calculated based on a simple mate-. For the case of an innitely deep and narrow squareial, a molecular chain model is obtained. SAFT receiveds attention by the academic and industrial commu-as used for a very wide range of uids, from simpleydrocarbons to ionic liquids and pharmaceutical mix-dition, it has been extended and generalized to accountor chain formation in the reference uid (Perturbed) [71], Lennard-Jones reference term (soft-SAFT) [72],ge square well attractive interactions (SAFT-VR) [73],ctions (PC-Polar SAFT) [74], chargecharge interactions-SAFT) [75,76], etc. Additional terms are added in Eq.

the model becomes signicantly more complex. Areviews of the SAFT equation of state is now available

theoryaccounthe moto liqutheorynon-rainteracbondinbased

3.2. Pa

In anent pbe calca charmolecumolecuparambondinmodelsome cthe criobtainparambe easexampfrom t

Thenent mthe acis usuacases ktempe

3.3. Re

Higawideand prindust

APAforcesaqueouideal, dand stThe mknownIn Fig.is showexcellesite mtwo sisionalchainthe LLbilities(by 1that gbetwevalueAPACThe isothermalisobaric (NPT) statistical ensemble byfor empty sites (holes) in the lattice [80]. In this way,s able to account for pressure effects and is applicablend gases. In the most recent development of the latticewn as non-random hydrogen bonding (NRHB) theory,

distribution of molecular sites due to intermoleculars is calculated explicitly [81]. In addition, hydrogentween rst neighbouring molecular sites is calculatedytsmans statistics.

eter estimation in higher order EoS

the higher order EoS presented above, pure compo-eters have a clear physical basis and, in principle, caned from molecular considerations. Parameters includeistic dispersion energy, molecular segment size, andsize (expressed in terms of spherical segments perFor the case of associating components, additionalinclude energy and entropy (or volume) of hydrogenpractice, these parameters are calculated by tting theperimental vapor pressure and saturated liquid (and invapor) densities from low temperature up to close topoint. The hydrogen bonding parameters can be alsoom spectroscopic data. For a homologous series, thevary smoothly with molecular weight and so they cantrapolated or interpolated to missing components. Fore EoS parameters for polyethylene may be estimatedrameters of n-alkanes.ations of state are extended to binary and multicompo-res using appropriate mixing rules. In order to improvey of calculations, a binary interaction parameter (kij)tted to experimental phase equilibrium data. In mosta constant, although for highly non-ideal mixtures ae-dependent kij is used.

entative applications of higher order EoS

rderequationsof statehavebeenappliedsuccessfully toeof chemical systemsover abroad rangeof temperaturere conditions. Some representative examples of directterest are presented here.ccounts explicitly for weak dispersion forces, polarhydrogen bonding and is an appropriate model forstems. Waterhydrocarbon mixtures are highly non-o strong hydrogen bonding between water moleculeshydrophobic nature of the hydrocarbon molecules.

l solubility is very low but it should be accuratelyarious petrochemical and environmental applications.

, the liquidliquid equilibria (LLE) of watern-hexaneom 315 to 480K. APACT predictions (kij =0.0) are inreementwithexperimentaldata. The threeassociating-for water provides more accurate predictions than theodel [68]. The former model results in three dimen-molecular structureswhile the two sitemodel providesmers and is more suitable for 1-alcohols. In Fig. 5(b),waterm-xylene mixture is shown. The mutual solu-are signicantly higher than in the previous mixture

orders of magnitude) due to electrons in m-xyleneise to strong polar interactions (dipolequadrupole)ater and m-xylene molecules. A relatively small kijeded here for an accurate correlation of the data by. Despite its accuracy, APACT never gained signicant


Fig. 5. Mutual solubilities of (a) watern-hexane and (b) waterm-xylene. Experimental data (points) and APACT calculations using a 2-site and 3-site model for water (takenfrom [68]).

popularity, probably due to its relative complexity over otherEoS.

SAFT has been extensively applied to polymer mixtures. InFig. 6, SAFT correlation of hydrocarbon Henrys law constant inlow density polyethylene (LDPE) at low pressure is shown. Usinga temperature independent kij value, the model provides accuratecorrelation over a wide temperature range [82].

The polymer phase behavior becomes signicantly more com-plex at high pressure. In polyethylene technology, polymerizationtakes place at very high pressures, up to a few kbar, followedby a low pmost cases,is known tolibria signipolyethylenequilibriummixture. Poas cloud poa monodispcurve) is obdata. SAFT

1

Hen

ry's

con

stan

t (ba

r)

Fig. 6. Weightimental data (n-butane, n-he

othermal pressurecomposition curves for polydisperse PEethylene mix-23.15K. Experimental data (points) and SAFT/PC-SAFT calculations (lines).odelled as a 36-component mixture (taken from [82]).

mixture [81]. Although thismay be a surprising result givenC-SAFT uses the hard chain reference uid over the harduid used by SAFT, it is clear that an appropriate param-tion of a model can result in very accurate results.ressure separation and product recovery process. Inthe polymer produced is polydisperse. Polydispersityaffect the high pressure polymersolvent phase equi-cantly [81]. In Fig. 7, the high pressure polydispersee (PE; Mn =56,000 and Mw =99,000)ethylene phaseat 423K is shown. PE is modelled as a 36-componentlydispersity results in increasing pressure (referred toint curve) at low polymer compositions. If one assumeserse polymer, then the shadow point curve (dashedtained, that deviates signicantly from experimental

correlation is more accurate than PC-SAFT correlation

1000

0000

Fig. 7. Isture at 4PE was m

for thisthan Psphereeteriza1

10

100

300250200150100

Temperature (oC)

fraction Henrys constant of hydrocarbons in LDPE at 1 atm. Exper-points) and SAFT correlation (lines). From top to bottom: ethylene,xane, n-octane, benzene and toluene (taken from [76]).

Fig. 8. Experimpoint pressureental data (points) and tPC-PSAFT correlation (lines) for the bubble-of CHF3[bmim+][PF6] at various temperatures (taken from [83]).


0.08

Naproxen - Acetone

100806040200

0.01

Weight fraction of ethanol in solvent

293.15 K

Fig. 9. Solubil 93.15K (right). Experimental data (points) and NRHB calculations (lines).Dashed lines c the right plot) and correlations (temperature independent kij value; solidlines; taken fr

The expresulted inof the phasetative examAn IL consiwide tempeical reactiostrong dipouiddensityexperimentfor the bubinteractionment betweexception o

Thermodceutical indsolvents areon for proceceutical moencounterepolar and hof thermodRecently, Tseterizationpharmaceuthe solubilicritical uinaproxen,wmarketed unaproxen toare in goodroform andfor methanoexcellent co

A mixedorder to conaproxen inmore than3accurately pternary mixprocess sim

Evaluatidynamic mstraightforwracy of thepredictionpendent kij

15 binary mixtures was performed [86]. Although differ-were identied for specic mixtures, the overall deviationsenexperimental data andmodel predictions and correlationsimilar for the two models.ides phase equilibria, thermodynamic models that accounttly for hydrogen bonding can provide reliable prediction ofent of hydrogen bonding as a function of temperature, pres-d composition. In Fig. 10, experimental data and predictionsreefor ps arespecfurt

mpli

T inessfhydroweval in

ial chay b

8

03253203153103053002952900.00

0.02

0.04

0.06

Naproxen - Methanol

Naproxen - Ethanol

Naproxen - Chloroform

Sol

ute

mol

e fr

actio

n

Temperature / K

1E-6

1E-5

1E-4

1E-3

Sol

ute

mo

le fr

act

ion

ity of naproxen in various pure solvents (left) and in ethanolwater mixture at 2orrespond to predictions (kij =0 in the left plot and tted to single solvent data inom [84]).

licit inclusion of polar interactions into PC-SAFT hasa model (PC-PSAFT) that provides accurate descriptionbehavior of strongly polar uid mixtures. A represen-ple refers to an ionic liquid (IL)solvent mixture [83].sts of a cation and an anion and is liquid over a veryrature range. ILs are increasingly used today for chem-ns and separations. In PC-PSAFT, ILs are modelled aslar molecular uids. Model parameters are tted to liq-datawhile avery lowvaporpressure is ensured. In Fig. 8,al data and truncated PC-PSAFT correlation are shownble pressure of CHF3[bmim+][PF6]. A single binaryparameter (kij =0.031) is tted to the data. The agree-en experiments and calculations is very good, with thef high CHF3 concentration.ynamic models are increasingly used by the pharma-ustry at the product development stage when variousscreened for new pharmaceutical molecules and laterss development and optimization.Most of the pharma-lecules are signicantly more complex than moleculesd in oil and chemical industry, with multiple functionalydrogen bonding groups. Accurate parameterizationynamic models for such molecules is far from trivial.ivintzelis et al. proposed a methodology for the param-of the NRHB equation of state for several widely usedticals [84]. NRHB was subsequently used to calculatety of these pharmaceuticals in various liquid and super-d solvents and mixed solvents. An example refers tohich is a commonnonsteroidal anti-inammatorydrug

nder various trade names. In Fig. 9(left), the solubility ofvarious polar solvents is presented. NRHB predictionsagreement with experimental data for ethanol, chlo-acetone and in fair agreement with experimental data

LLE ofencesbetwewere s

Besexplicithe extsure anfrom thsentedmodelrole ofcussed

3.4. Si

SAFbe succuids,cals. HchemicessentEoS m

0.

1.

onl. Using a temperature independent binary interaction,rrelation is obtained.solvent is often used in pharmaceutical industry in

ntrol drug solubility. In Fig. 9(right), the solubility ofthe ethanolwater solvent at 293.15K that varies byorders ofmagnitude frompurewater to pure ethanol isredicted. No parameter adjustment was made for theture and so NRHB can be safely used for the relevantulation [84].on of the relative accuracy of the various thermo-odels and selection of one over another is not aard task. Recently, an extensive evaluation of the accu-simplied PC-SAFT (sPC-SAFT) and of NRHB for the

(kij =0.0) and correlation (using a temperature inde-) of the VLE of 104 binary mixtures [85] and of the

0.00.0

0.2

0.4

0.6

mo

nom

er fr

act

i

Fig. 10. Propaperatures (takpredictions (liassociating theories (sPC-SAFT, NRHB and CPA) are pre-ropanoln-heptane in the range 1555 C [87]. All threein very good agreement with spectroscopic data. Thetroscopic data in parameter estimation in EoS is dis-

her in the next section devoted to the CPA EoS.

ed association theories: the CPA equation of state

its numerous variations and NRHB have been proven toulmodels formany applications ranging fromnon-polarogen bonding ones up to polymers and pharmaceuti-er, for many practical applications in the oil & gas anddustries, a simpler approach which retains some of thearacteristics of association models but yet rely on cubice useful. In this direction the Cubic-Plus-Association

Exp. data 15 oC

Exp. data 35 oC

Exp. data 55 oC CPA NRHB sPC-SAFT0.2 0.4 0.6 0.8 1.0

mole fraction of propanol

nol monomer fraction in propanoln-heptane mixture at three tem-en from [87]). Experimental data (points), NRHB, CPA and sPC-SAFTnes).


(CPA) EoS has been developed. CPA is a simplied version of SAFT,which has been shown to be particularly useful for applicationsin the petroleum and chemical industries. In CPA, the physicalterm of SAFT is replaced by SRK, PR or another cubic EoS. TheCPA modelrecent reviepublicationchemicals [from 1996used the PRtions, the vthe associaneeded forassociatingmatic hydrCO2water.

While mglycolshydorganic acigases (CO2,alkanolamicompoundshad additio(waterNaChave beenasphaltenestions mentiof a characculations al[90,11211WaalsPlat

3.4.1. The pWe pres

tioned invethe limitatiit can be stcapable of ppredictionshydrocarbodata. Excephighly immcarbons, asThe CPA Eoequallywelimum of thaccording tfrom 1983zouk et al.needed in tparameterwater or gparametervation (ind[116], howeing similarkij from wwaterbenzinteraction

Associat(number ofbeen establperforms bemum schemTheefforts [monomer f

280 290 300 310 320 330-5

-4

-3

0.01

Exp. data Derawi et al. (2002) Exp. data Razzouk et al. (2010)

CPA, kij=0.059

Hexane in MEG rich phase

Temperature / K

MEG in HC rich phase

MEGhexane LLE. Experimental data (points) and CPA correlation (lines,9). The experimental data are from Derawi et al. [128] and the recent datazzouk et al. [126].

e selection led to conclusions of mostly qualitative nature,hough satisfactory results are obtained in many cases (see). Suulatemee is pto ded pic da

y wahe sle accor mbe phaspropumb

ines,harm

280 300 320 340 360 380 400 420 440 460 480-7

-6

-5

-4

-3

01

.1

Exp. data Tsonopoulos et al. (1983) Exp. data Razzouk et al. (2010)

CPA, k =0.0355

CPA, k =0.0

Water in HC rich phase

Temperature / K

Hexane in water rich phase

Waterhexane LLE. Experimental data (points) and CPA correlation (lines,55). The experimental data are from Tsonopoulos and Wilson [127] and theata from Razzouk et al. [126].is extensively described in literature including severalws [8789] and two books [2,3], while several recents present specic applications related to oil & gas and9092]. The CPA version by Kontogeorgis et al. [93]is based on SRK, while other researchers [94,95] have

or the ESD [96] EoS. In the majority of the applica-dW1f mixing rules are used (Eqs. (4) and (5)), whiletion term needs no mixing rules. Combining rules arecross-associating mixtures, e.g. waterMEG or induced(solvating)mixtures likewaterBTEX (BTEX= thearo-ocarbons; benzenetolueneethylbenzenexylene) or

ost original publications focused on wateralcohols orrocarbons VLLE, several recent investigations includeds [9799], uorocarbons [100], mixtures with acidH2S) [101,102], glycolethers [103], amines [104] and

nes [105], and complex multifunctional polyphenolic[106,107]. The PR-CPA from Wu and Prausnitz [94]

nal electrostatic terms for applications to electrolyteslmethane), while recently more electrolyte CPA EoSdeveloped [108,109]. Finally, an extension of CPA tohas been reported [110]. In addition to the applica-

oned above, oil applications include the developmentterization method for CPA [111] and gas hydrate cal-so in presence of inhibitors like methanol and glycols5]. In the latter case, CPA is combined with the van derteeuw model.

erformance of CPA in briefent now our conclusions on some of the aforemen-stigations with CPA illustrating both the successes andons/challenges of the model at its current state. In brief,ated that CPA is a successful thermodynamic model,roviding satisfactory multiphase (VLLE, LLE, SLE, etc.)for mixtures containing water, alcohols or glycols andns based solely on binary parameters tted to binarytionally good correlations are obtained for the LLE ofiscible systems like glycols and water with hydro-shown for two typical mixtures in Figs. 11 and 12.S can correlate the solubilities in both liquid phasesl over extensive temperature ranges, except for themin-e hydrocarbon solubility in water which is observedo the experimental data of Tsonopoulos and Wilson[127] but not in the recent measurements of Raz-

[126]. In the typical case, one interaction parameter ishe calculations with CPA (the kij in the cross-energyin Eq. (5)) but for solvating mixtures for examplelycols with BTEX compounds we need an additionalto account for the enhanced interactions due to sol-uced association) effects. It has been recently shownver, that the kij can be obtained from a mixture hav-physical interactions as the solvating system, e.g. theaterhexane or MEGhexane can be used to modelene and MEGbenzene, respectively, leaving only oneparameter to t (the cross-association volume).ionmodels like CPA require that the association schemesites and location) is known. Over the years it has

ished that, in the CPA model, the two-site (2B) schemetter for methanol (and other alcohols), while the opti-e for water and glycols is a four-site (4C) one [2,3].

86,87,117119] inusing spectroscopy results especiallyraction data towards better parameter estimation and

1E

1E

1E

mo

le fr

act

ion

Fig. 11.kij =0.05from Ra

schemeven tFig. 10ior calc2B schschemmatureimprovtroscop(mostlcases ttionab[120] fcannot

CPAacetic,large nnolamsmall p

1E

1E

1E

1E

1E

0.

0

mo

le fr

actio

n

Fig. 12.kij =0.03recent dch investigations provided additional (to phase behav-ions) assurance that the use of 4C scheme for water andfor most alcohols is correct, although a three-site (3B)ossibly a better choice for methanol. It is perhaps pre-etermine whether monomer fraction data will result toarameters for association models. Currently such spec-ta are available for only a few compounds andmixtures

ter, alcohols and some glycolethers). Moreover, in somepectroscopic data for monomer fractions are of ques-uracy. For example, the monomer fraction data of Luckethanol and ethanol coincide, a surprising trend whichredicted by any of the association models [87].been also applied to amines, small organic acids (formic,ionic) and recently also heavy organic acids and aer of multifunctional compounds (glycolethers, alka-polar aromatic acids, polyphenolic compounds andaceuticals and) [9799,103107]. The results are over-


all satisfactory also for these chemicals although most of theseapplications have been so far limited to binary mixtures (VLE, LLEand SLE). An interaction parameter is almost always needed par-ticularly for aqueous mixtures for which this parameter is oftennegative, tyan inabilityeffects whi1). Still thetypes of phature rangescan be reprrange of 250combiningone-site (1Aorganic acidinvestigatedthe 2B and 32B scheme

The 1A,from the noto the SAFTtional compKontogeorgof heavy glyLLE, excesswith a 6D sthe two hypler 4C scheOn theothescheme peralkanolami

Aqueouschallengingcult caseformic acidvdW1f mixcannot desctation of reAs Muro-SuHuronVidafor this pro

Some cpresented,HuronVidaand glycolcontainingwatermetclearly supresults arefewer adjus

Using arecently shpolar chemtems for wh

The sucpolar/hydrosical cubic Eabove cancombined wor limitatiowill hereaftimportant cstudies.

3.4.1.1. Parhydrogen b

established, for others especially multifunctional ones it has not.The choice may be quite difcult, partly due to lack of reliablevapor pressure and liquid density data over extensive temperatureranges. Moreover, especially when ve parameters are tted, they

t betimizpreFord an

investermd hasglycydro

alcohusedbe dless

. Theced io alkaterrs inompountomplcaseeter,ed btions(all p. Thethatonene forase,entinn extly aethanVLEmetmostions

. Thecountionractiighlytingehyapprot bd LLeternaleudor acelopms) maion sshinga m

eringpically around 0.1 to 0.2. This result may indicateto account fully satisfactorily for the cross-association

ch may be due to the combining rule used (Elliott, CR-correlation is, in most cases, satisfactory for differentse behavior (VLE and LLE) and over extensive tempera-. For instance, watermethanol and waterethanol VLEesented very well with a single kij over a temperatureC. It is not always possible to determine a prioriwhich

rule performs best. In terms of association schemes, the) (resulting to dimerization) is found to be the best fors, while the 4C is the best choice for the alkanolamines(MEA, DEA and MDEA). For amines and glycolethers,B schemes perform equally good and for simplicity the

is adopted.2B, 3B and 4C schemes are well-known and originatemenclature and recommendations made in connectionmodel byHuang andRadosz [70]. For certainmultifunc-oundsnewschemesmaybemoreappropriate. Breil andis [121] showed that many thermodynamic propertiescols like triethylene glycol can be described well (VLE,enthalpies and innite dilution activity coefcients)cheme (six equal sites for the four oxygen atoms anddrogen atoms) although the differences from the sim-me are not signicant, especially for VLE and LLE alone.r hand, as shownbyAvlundet al. [122], the four-site (4C)forms as well as more complex association schemes fornes.mixtures with other associating compounds are oftento model, as mentioned previously. A particularly dif-

is represented by the mixtures acetic acidwater andwater. In these cases, the original form of CPA with theing rules and one (or even two) interaction parametersribe satisfactorily VLE including the accurate represen-lative volatilities over extensive temperature ranges.n et al. [98] showed, a combination of CPA with thelmixing rules is a successful but empirical way to solve

blem at the cost of additional adjustable parameters.omparisons of CPA with other models have beene.g. SRK with classical [3,88], MHV2 [90] andl [3,29]mixing rules for alcohols/glycolshydrocarbonswaterhydrocarbons and with PC-SAFT for methanol-mixtures [123] (including multicomponent LLE for

hanolhydrocarbons). Compared to SRK/vdW1f, CPA iserior for associating mixtures, while overall similarobtained against SRK/HuronVidal (although CPA usestable parameters) and PC-SAFT for the systems studied.rather simple characterization method, CPA has beenown [111] to predict satisfactorily the solubilities oficals (water, methanol, glycols) in the few reservoir sys-ich such data are available.cess of association models like CPA for manygen bonding systems especially as compared to clas-oS described in Section 2 which has been summarizedbe largely attributed to Wertheims association termith careful parameterization. Some of the difculties

ns of the approach have already been presented but weer continue along these lines and outline some of thehallenging topics which need to be addressed in future

ameterization-pure compounds. While for a number ofonding compounds the association scheme has been

may nothe op(vaportainty.requireunderfor demethowater,those hheavynot beshouldmay be

3.4.1.2enhanpared twith wor estepolar cfor accrule) ctionedparamprovidpredicibilityalone)[3,91]ticompthe caslatter crepresover aexpliciand mfor thewater,be, insiderat

3.4.1.3not acinteracan intetreat hassociaacetonThesemay nVLE anparamadditioThe pstions foA devemodelextensestabliin suchconsidunique in the sense that several sets are obtained fromation which can represent pure compound propertiesssures, liquid densities) within experimental uncer-choosing the best parameter set additional data ared we nd that LLE data for the associating compoundtigation and aliphatic hydrocarbons is a successful wayining the best parameter set. This parameterizationits merits when such data are indeed available, e.g. for

ols, small alcohols, glycolethers and alkanolamines. Forgen bonding compounds like amines, small acids andols which are miscible with alkanes, this approach can-and the establishment of the optimum parameter set

etermined based on VLE data on which the parameterssensitive.

importance of solvation. There is evidence that there arenteractions resulting to high solubilities (higher com-anes) inmixtures of aromatic andolenichydrocarbons, glycols, alkanolamines and acids, mixtures of etherswater or in mixtures with acid gases (CO2, H2S) and

ounds. But is the approach we have used so far in CPAing for these enhanced interactions (the modied CR-1etely convincing? Itworks in practice for the aforemen-s, sometimes at a cost of a second adjustable interactionbut additional evidence is necessary and this could bey checking other properties or even multicomponentwhere there isnoadditional interactionparameterex-arameters should be determined from binary systemsstudy with multicomponent systems has illustratedfor the waterglycolsBTEX the solvation is for mul-t LLE as important as for binary LLE. This is not alwaysacid gaswateralcohol/glycolshydrocarbons. In the

accounting for the CO2water solvation is important forg the water solubility in CO2 for CO2watermethanetensive pressure range. However, the signicance ofccounting of the solvation for acid gases with glycolsol is, for multicomponent calculations, small. Overall,

-based multicomponent systems containing acid gases,hanol or glycols and hydrocarbons, the solvation couldcases, safely ignored (even though the solvation con-improve the representation of the individual binaries).

role of polarity (and weaker forces). The CPA EoS doest explicitly for polar (or quadrupolar) effects and suchs need to be accounted for implicitly, e.g. via includingon parameter kij. Another semi-empirical approach is topolar (non-hydrogen bonding) compounds as pseudo-ones and such approach is shown to work well fordrocarbons VLE and sulfolanehydrocarbons LLE [3].oaches (use of kij or the concept of pseudo-association)e always satisfactory. For example, acetone/hexaneE cannot be described well using a single interactionover the whole temperature range. PC-SAFT, withoutpolar terms, suffers from the same limitation [3,124].-association approach is also shown to have limita-tone-containingmixtures, e.g. acetonechloroform [3].ent towards a polar CPA (along the lines of polar SAFTy be a way to accommodate for these problems. Thishould, however, be done in a consistent way especiallywhich values for the dipole moment should be usedodel (vacuum ones or from the liquid phase) and byboth polarity and association for hydrogen bonding


compounds. A polar CPA model should be further tested for binaryand multicomponent mixtures containing polar, associating andinert compounds.

3.4.1.4. Limpounds ormat least frolimitationsgen bondinTEG and glyaccountingmixtures w

4. Conclus

We haveexcess Gibbfrom cubicing the caphave showeform similatively a correffects, domasymmetricmers. On ththe use of thwhich resuterm, whichBest resultstype term imodel, prefaccurate reuse one of tCHV, VTPR oproblem, wtic for sompressure mdo have othcoefcient,not satisfacliquidliquimixtures arsize-asymma problem.address theof multicomespecially tbasic princiapplicationbilities of thwhich indic

Acknowled

The authuseful discushown in Fthe calculatalso gratefudiscussions

References

[1] J.D. VanDisserta

[2] G.M. Kontogeorgis, R. Gani, Computer-Aided Property Estimation for Processand Product Design, Elsevier, 2004.

[3] G.M. Kontogeorgis, G.K. Folas, Thermodynamic Models for Industrial Appli-cations. From Classical and Advanced Mixing Rules to Association Theories,John Wiley & Sons, New York, 2010..S. Webria, A. Orbeate an98.

I. Saniley &. Soavate, C.-Y. Peginee.J. Hurpor-l(1979. Hild

ons In. Wilsee En713

.S. Abrew exstems. Renoons fo.S. Elbiction074

.M. Koodel fEngin. Bondiley &a. Fre

activ861Thorlelastuilibr. Lindthe

726.M. Koos, Abic eq

hemicA. Sachermeering.M. Koefcions o743

Molleodels.L. Miuatio.L. Miate, FlDahl

nifac-b.S.H. Wres ustrapoginee

.S.H. Wons of.K. Foquid-Lonentuid PhHoldsed oDahl,uatiouilibr

esearc.S. KaferenC. Vouquid-edustritations of the Wertheim theory. Finally, there are com-ixtures forwhich theWertheim theory has limitations,

m a theoretical point of view. One of the signicantis the presence of intramolecular association (hydro-g within the molecule) present in e.g. heavy glycols likecolethers. It is of interest to evaluate the signicance offor the intramolecular association in modelling of suchith SAFT-type theories. Such work is in progress [125].

ions

presented a methodological approach based on thes energy and activity coefcient expressions derivedequations of state (EoS) for analyzing and understand-abilities and limitations of these classical models. Wed that cubic

Equations of State From the Ideas of Van Der Waals to Association Theories (1)

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