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Fluid Phase Equilibria 301 (2011) 123–129

Contents lists available at ScienceDirect

Fluid Phase Equilibria

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

ddressing the issue of numerical pitfalls characteristic for SAFT EOS models

lya Polishuk ∗

epartment of Chemical Engineering & Biotechnology, Ariel University Center of Samaria, 40700 Ariel, Israel

r t i c l e i n f o

rticle history:eceived 5 August 2010eceived in revised form8 November 2010ccepted 24 November 2010

a b s t r a c t

This study demonstrates the SAFT models can have doubtless advantages in predicting thermodynamicproperties comparing to cubic equations of state. However they might be affected by two kinds of numer-ical pitfalls, namely the erroneous shapes of their isotherms resulting in the multiple phase equilibriapredicted for pure compounds, and the undesired consequences of attaching their reduced densities bytemperature dependencies. The latter are the negative heat capacities at very high pressures and the

vailable online 30 November 2010

eywords:quation of statetatistical association fluid theoryigh pressure

intersections of isotherms. Some versions, such as the SAFT of Chapman et al. [17] and SAFT-VR-SW ofGil-Villegas et al. [19] are found to be free of the pitfalls of the 1st kind. However the unsafe tempera-ture dependencies are still essential for their accuracy. The present study proposes a modification of theCarnahan–Starling repulsive term maintaining the reduced densities temperature-dependent while mak-ing the covolumes temperature-independent. It is demonstrated that this manipulation might address

l pitfa

eat capacityound velocity

the problem of numerica

. Introduction

The EOS models based on the statistical association fluid the-ry (SAFT) are among the most important approaches for modelinghermodynamic properties of pure substances and their mixtures.lthough major attempts have been devoted so far for develop-

ng different versions of SAFT [1–5], some issues have not beenatisfactorily solved yet. In particular, SAFT EOSs often have a lim-ted capability for the simultaneous and accurate description of theritical and sub-critical data of pure compounds, comparing evenith simple cubic EOS models. In addition, being attached by flex-

ble temperature dependencies, cubic EOSs might be superior inorrelating vapour pressures.

The latter issues will be treated in the subsequent studies, afterddressing the most fundamental problem characteristic for manyAFT models, namely lacking of robustness in the entire thermo-ynamic phase space. Indeed, thus far it has been demonstrated6–11] that many versions of SAFT are not free of the undesir-ble numerical pitfalls responsible for inaccurate and sometimesven non-physical predictions. One of those numerical pitfalls is

he generating of multiple extrema for isotherms and, as a con-equence, multiple phase equilibria for pure compounds. Whilehis problem affects several popular versions of SAFT, such as thehen–Kreglewski’s [12] SAFT of Huang and Radosz [13], the PC-

Abbreviations: EOS, equation of state; HS, hard sphere; SAFT, statistical associ-tion fluid theory.∗ Tel.: +972 3 9066346; fax: +972 3 9066323.

E-mail addresses: polishuk@ariel.ac.il, polishyk@bgumail.bgu.ac.il

378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2010.11.021

lls and preserve the outstanding performance of the models.© 2010 Elsevier B.V. All rights reserved.

SAFT [14] and the Soft-SAFT [15], the other ones, such as thesimplified SAFT [16] and the SAFT of Chapman et al. [17] pre-dict the classical van der Waal’s shapes of isotherms and singlephase envelopes for pure compounds [11]. These facts indicatewhich versions of SAFT should be selected for further considera-tion.

Additional numerical pitfalls, namely the negative heat capac-ities at the extremely high pressures [18] and the intersectionsof isotherms [11] are generated by the temperature dependenciesattached to the reduced densities of all the models above. The ver-sion of SAFT being apparently free of all kinds of the numericalpitfalls currently known to the author is the SAFT with square-wellpotentials of variable range (SAFT-VR-SW) [19,20].

Evidently, this result is achieved not only by the appropriateformulation of the dispersion term, but also by implementationof the temperature-independent reduced densities. In what fol-lows let us consider a contribution of the temperature dependenceabove.

2. Theory

2.1. The covolumes of SAFT: what can be learned?

In the present study the predictions of the thermodynamic prop-

erties of methane at high pressures yielded by the SAFT of Chapmanet al. [17] and SAFT-VR-SW [19] will be investigated. It should beemphasized that very similar patterns of behavior of the modelsunder consideration are observed also in the cases of other simplemolecules. Previously [19] it has been demonstrated that SAFT-VR-

124 I. Polishuk / Fluid Phase Equilibria 301 (2011) 123–129

T(K)

8006004002000

0

200

400

600

ρ(g/L)a

T(K)

8006004002000

500

1000

1500

2000

2500

3000

3500

4000

4500

W(m/s) b

T(K)

8006004002000

1.8

2.1

2.4

2.7

3.0

3.3

Cv(J/g-K)

c

T(K)

8006004002000

2.7

3.0

3.3

3.6

3.9

4.2

4.5

Cp(J/g-K)d

2000bar

100

00b

ar

500

0b

ar

2000

bar

Fig. 1. The thermodynamic properties of methane at high pressures predicted by cubic EOS models: dotted lines – Peng–Robinson EOS [21]; solid lines – cubic EOS [22].Experimental data: � – 100 bar, © – 200 bar, � – 500 bar, � – 1000 bar, � – 2000 bar, � – 5000 bar, � – 10,000 bar [24]; ♦ – 2000 bar; � – 5000 bar; × – 10,000 bar [25].

quilibria 301 (2011) 123–129 125

Sdblc

cPfpstir

pcubeyp0hi(iisast[

tmaetia

btptSdsSe

2c

pCticetib

T(K)

8006004002000

0

3e+6

6e+6

−bar molL

( )− ∂ ∂ TP V

Fig. 2. The −( ∂ P/∂ V)T data of methane at high pressures predicted by cubic EOS

to this problem.

8006004002000

0

1e+6

2e+6

−bar molL

( )− ∂ ∂ TP V

I. Polishuk / Fluid Phase E

W is superior in modeling vapour pressures and coexisting phaseensities of n-alkanes and n-perfluoroalkanes. However it shoulde realized that both models are usually less successful in corre-

ating the saturated PVT of pure compounds comparing to simpleubic EOS models.

Generally speaking, cubics might serve as a good reference dis-ussing different aspects of SAFT. Fig. 1 depicts the prediction ofeng–Robinson EOS [21] and the recently proposed cubic EOS [22]or density, heat capacities and speed of sound of methane at highressures. It can be seen that although the more recent EOS has alight over-all advantage, the results of both models are affected byhe same problems. In particular, as the pressure rises, the alreadynitially imperfect accuracy in predicting the sound velocities dete-iorates even more.

The latter point deserves in depth consideration. As discussedreviously [23], cubic equations tend to generate a relativelylose proximity between their covolumes and the predicted sat-rated liquid molar volumes at low temperatures, which coulde explained by the curvature of van der Waals potential. Forxample, in the case of methane the EOS of Peng–Robinson [21]ields b = 0.0268 L/mol and the liquid molar volume at the tripleoint is 0.0315 L/mol. The recently proposed cubic EOS [22] yields.0285 and 0.0343 L/mol respectively. As a result, under veryigh pressures cubic equations underestimate the compressibil-

ty of fluids −( ∂ V/∂ P)T and, consequently, overestimate −( ∂ P/∂ V)T

see Fig. 2). Since sound velocity is proportional to√

−(∂P/∂V)T ,t is overestimated as well. The inaccurate pressure–volumenterrelation established by cubic EOSs at very high pressureshould eventually affect also the predictions of heat capacitiest such conditions. However the quality of these predictionseems to be hard for evaluation in the present case due tohe significant disagreement between the available data sources24,25].

Remarkably, attaching the covolumes of cubic EOSs by tempera-ure dependencies (besides creating the numerical pitfalls [26,27])

ight have a positive impact on modeling the PVT data at moder-te pressures. However such practice would not have any notableffect on the proximity between the saturated liquid volumes at lowemperatures and the covolumes and, therefore, would not resultn any major improvement in predicting the compressibility datat very high pressures.

A significant advantage of SAFT models over cubic EOSs is theigger difference between the saturated liquid volumes at lowemperatures and the covolumes (for liquid methane at the tripleoint the SAFT of Chapman et al. [17] predicts 0.0376 L/mol, whilehe covolume at this temperature is only 0.0156 L/mol; SAFT-VR-W [19] predicts 0.0355 and 0.0156 L/mol correspondingly). Fig. 3epicts −( ∂ V/∂ P)T predicted by both models for methane. It can beeen that this time the results are substantially more accurate andAFT-VR-SW [19] has certain superiority over the SAFT of Chapmant al. [17].

.2. The temperature dependent reduced densities: pro andontra

Fig. 4 demonstrates that in spite of the appropriateressure–volume interrelation SAFT-VR-SW [19] and the SAFT ofhapman et al. [17] have quite different exactness in predictinghe densities, heat capacities and sound velocities. In particular,t can be seen that in spite of its advantage in predicting the

ompressibility, SAFT-VR-SW [19] is in fact pretty inaccuratestimator of the thermodynamic properties, comparing even withhe simple cubics. The worst predictions are obtained the forsochoric heat capacities (CR

V ≈ 0 regardless of pressure. It shoulde kept in mind that the temperature independent repulsive term

models: dotted lines – Peng–Robinson EOS [21]; solid lines – cubic EOS [22]. Exper-imental data: � – 1000 bar, � – 2000 bar, � – 5000 bar, � – 10,000 bar [24].

has a zero contribution to CRV ; in the present case the contribu-

tion of the dispersion term appears to be very small as well). Incontrast, the outstanding accuracy of the SAFT of Chapman et al.[17] in predicting the thermodynamic should be emphasized (seeFig. 4).

As noticed, the main difference between both models under con-sideration is the temperature dependence attached to the reduceddensity of the SAFT of Chapman et al. [17]. Thus, it comes into viewthat this temperature dependence plays a key role in the outstand-ing performance of the model in predicting the thermodynamicproperties. Remarkable, the Mie version of SAFT-VR [19] (whosereduced densities are also temperature dependent) is advanta-geous over SAFT-VR-SW as well [[28], see also [29,30]].

Thus, on one hand it can be seen that the contribution of thetemperature dependent reduced densities to the over-all accu-racy of SAFT models cannot be neglected. However on other handthe corresponding numerical pitfalls resulting in the negative heatcapacities at extremely high pressures and the intersection ofisotherms [11] are inacceptable if they take place in the fluid PVTrange. In what follows let us consider one of the possible solutions

T(K)

Fig. 3. The −( ∂ P/∂ V)T data of methane at high pressures predicted by SAFT EOSmodels: dotted lines – SAFT-VR-SW [19]; solid lines – SAFT of Chapman et al. [17].Experimental data: see Fig. 2.

126 I. Polishuk / Fluid Phase Equilibria 301 (2011) 123–129

T(K)

8006004002000

0

200

400

600

ρ(g/L)a

T(K)

8006004002000

500

1000

1500

2000

2500

3000

3500

4000

4500

W(m/s) b

T(K)

8006004002000

1.8

2.1

2.4

2.7

3.0

3.3

Cv(J/g-K)

c

T(K)

8006004002000

2.7

3.0

3.3

3.6

3.9

4.2

4.5

Cp(J/g-K)

d

1000

0b

ar50

00bar

Fig. 4. The thermodynamic properties of methane at high pressures predicted by the original SAFT EOS models: dotted lines – SAFT-VR-SW [19]; solid lines – SAFT of Chapmanet al. [17]. Experimental data: see Fig. 1.

I. Polishuk / Fluid Phase Equilibria 301 (2011) 123–129 127

V(L/mol)

.04.03.02

1e+3

1e+4

1e+5

1e+6

1e+7

P(bar)

intersection

of isotherms

260 K

620 K

1200 K

Fig. 5. Three isotherms of methane: solid lines – original SAFT of Chapman et al.[(

2m

rprcts

i

�

�[a

Z

Iatw

Ewa�i

Z

T

Tr

4.53.52.51.5

0

5e+6

1e+7

2e+7

2e+7

P (bar)Negative C

v

j = 0

j=

0.1

j=

0.5

Fig. 6. Lines of zero CV as function of j: dashed lines – the SAFT of Chapman et al.

the sound velocities at j = 1/3, and they appear to be particularly

17] with j = 0. Dotted lines SAFT of Chapman et al. [17] attached by Eqs. (4) and (5)j = 1/3, � = 3.71001 A, ε/k = 154.188 K). Experimental data [24].

.3. Leaving the reduced densities temperature dependent whileaking the covolumes temperature independent

The temperature-dependent covolumes are the major factoresponsible for the numerical troubles [11,18,26,27]. The idea pro-osed in the present study is based on the discussed above relativeemoteness of the SAFT’s covolumes. The assumption is that in suchase careful manipulations of the covolumes while maintaininghe models basically unchanged would not affect of their accuracyignificantly.

The temperature dependent reduced density of a single segments given as:

= �NAv

6V�3� (T) (1)

(T) is the temperature dependence. The Carnahan–Starling’s31] expressions for the hard sphere repulsive contributionsre:

HS = 1 + � + �2 − �3

(1 − �)3(2)

AHS

RT=

∫ �

0

ZHS − 1�

d� = 4� − 3�2

(1 − �)2(3)

t can be seen that these contributions approach infinity at � → 1nd the model’s covolume is therefore �V. Since � (T) decreases withemperature [17,32], the covolume of Eqs. (2) and (3) decreasesith temperature as well. Let us rewrite Eq. (3) as follows:

AHS

RT= 4� − 3�2

(1 − �)(2−j)(1 − �/� (T))j(0 < j < 2) (4)

q. (4) approaches infinite two times: at � → 1 and at �/� (T) → 1,hile �/� (T) > �. Hence, �V/� (T) represents the covolume of Eq. (4)

nd it is temperature-independent. The smaller volumes (includingV) obviously have no physical relevance anymore. The compress-

bility factor is yet given as:

HS = 1 + �∂AHS/RT

∂�= 1

+ (�/� (T)){2�(� − � (T) − 2) + j�(3� − 4)(� (T) − 1) + 4� (T)}(1 − �)3−j(1 − �/� (T))1+j

(5)

aking j = 0 Eq. (2) is recovered.

[17], solid lines – SAFT-VR-SW [19].

In what follows let us investigate the influence of jon performance of the SAFT of Chapman et al. [17] andSAFT-VR-SW [19] attached by � (T) of Cotterman et al.[17,32].

3. Results

The original models under consideration have been fitted tothe experimental pure compound phase equilibria data usingthe SAFT parameters �, ε/k, and in the case of SAFT-VR-SW[19] – also �. Yet another parameter j, having its own influenceon the phase envelop, is added. In order to discern its actualcontribution to the prediction of the auxiliary thermodynamicproperties it seems expedient to maintain the phase envelopsunchanged. In the present study this task has been achieved byevaluating �, ε/k and � solving the critical point conditions atthe critical data as generated by the original models, namelyTc = 211.597 K, Pc = 68.673 bar for the SAFT of Chapman et al.[17] and Tc = 204.195 K, Pc = 67.135 bar – for SAFT-VR-SW [19].It should be emphasized that the appropriate parameterizationgoes beyond the scope of the present study and it will be consid-ered in the subsequent one. However it is evident that j shouldnot be treated as an adjustable parameter but as a model con-stant.

Figs. 5 and 6 depict the influence of j on the model’s perfor-mance. In particular, it can be seen (Fig. 5) that already a smallvalue of j = 1/3 addresses the problem of isotherms intersections.In addition, j has a remarkable effect on the non-physical predict-ing on the negative isochoric heat capacities, which are restrictednow to astronomic pressures and low temperatures irrelevant forfluid phases (see Fig. 6). At j = 1 and the infinity pressure CR

V = +∞.Thus the negative isochoric heat capacities seem to be completelyremoved. However this value has already a remarkable influenceon the accuracy of the model.

Fig. 7 depicts predictions of the densities, heat capacities and

accurate. No significant difference comparing with the perfor-mance of the original SAFT of Chapman et al. [17] can be observed(see Fig. 4). These results indicate that the issue of numerical pit-falls can be treated with no noteworthy affection of the model’saccuracy.

128 I. Polishuk / Fluid Phase Equilibria 301 (2011) 123–129

T(K)

8006004002000

0

200

400

600

ρ(g/L)a

T(K)

8006004002000

500

1000

1500

2000

2500

3000

3500

4000

4500

W(m/s) b

8006004002000

1.8

2.1

2.4

2.7

3.0

3.3

Cv(J/g-K)

c

8006004002000

2.7

3.0

3.3

3.6

3.9

4.2

4.5

Cp(J/g-K)d

F by SA� , � = 3.

4

a

T(K)

ig. 7. The thermodynamic properties of methane at high pressures as predicted= 3.69611 A, ε/k = 162.218 K, � = 1.475) solid lines – SAFT of Chapman et al. (j = 1/3

. Conclusions

This study demonstrates the SAFT models can have doubtlessdvantages in predicting thermodynamic properties comparing

T(K)

FT EOS models attached by Eqs. (4) and (5): dotted lines – SAFT-VR-SW (j = 1/3,71001 A, ε/k = 154.188 K) Experimental data: see Fig. 1.

to cubic equations of state. In particular, SAFT models establishsignificantly smaller values of the covolumes. This fact allowsaccurate predictions of compressibility at high pressures. How-ever SAFT might be affected by two kinds of numerical pitfalls,

quilib

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LACCjNPRTVWZ

G�ε

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Scr

SR

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[

[[[[[

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[[[[[[[[

024509-1–024509-16.[29] T. Lafitte, M.M. Pineiro, J.-L. Daridon, D. Bessières, J. Phys. Chem. 111 (2007)

I. Polishuk / Fluid Phase E

amely the erroneous shapes of the isotherms resulting in theultiple phase equilibria predicted for pure compounds, and the

ndesired consequences of attaching their reduced densities byemperature dependencies. The latter are the negative heat capaci-ies at very high pressures and the intersections of isotherms. Someersions, such as the SAFT of Chapman et al. [17] and SAFT-VR-W of Gil-Villegas et al. [19] are found to be free of the pitfallsf the 1st kind. However the unsafe temperature dependenciesre still essential for their accuracy. The present study proposes aodification of the Carnahan–Starling repulsive term maintaining

he reduced densities temperature-dependent while making theovolumes temperature-independent. It is demonstrated that thisanipulation might address the problem of numerical pitfalls and

reserve the outstanding performance of the models. In particu-ar, the intersections of isotherms might be removed completelynd appearance of negative heat capacities restricted to condi-ions irrelevant for fluid states. Further verification of the proposedpproach will be performed in the subsequent study devoted to thever-all parameterization of SAFT.

ist of symbolsHelmholtz free energy

V isochoric heat capacityP isobaric heat capacity

scaling parameter in Eq. (4)Av Avogadro’s number

pressureuniversal gas constanttemperaturemolar volumespeed of soundcompressibility factor

reek lettersreduced density

/k segment energy parameter divided by Boltzmann’s con-stantrange of attractive forces parameter of SAFT-VR

(T) temperature dependence of reduced densityLennard–Jones temperature-independent segment diam-eter (Å)

ubscripts

critical statereduced property

uperscriptresidual property

[[[

ria 301 (2011) 123–129 129

Acknowledgment

Acknowledgment is made to the Donors of the American Chem-ical Society Petroleum Research Fund for support of this research,grant no. PRF#47338-B6.

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