about the numerical pitfalls characteristic for saft eos models

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Fluid Phase Equilibria 298 (2010) 67–74

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About the numerical pitfalls characteristic for SAFT EOS models

Ilya Polishuk ∗

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

a r t i c l e i n f o

Article history:Received 30 May 2010Received in revised form 3 July 2010Accepted 5 July 2010Available online 13 July 2010

Keywords:Equation of stateStatistical association fluid theoryPhase equilibriaGlobal stability

a b s t r a c t

This study demonstrates that SAFT EOS models might exhibit the practically unrealistic and even non-physical predictions due to the two factors, namely the temperature dependencies of a segment packingfraction and the very high-polynomial orders by volume. The first factor is responsible for predictingthe negative values of the heat capacities at very high pressures and the intersections of isotherms athigh densities. The very high-polynomial orders of several SAFT EOS models result in prediction of theadditional stable unrealistic critical points and the pertinent fictive phase equilibria. It is demonstratedthat the unrealistic phase splits might present the globally stable states established by the models, whilethe VLE matching the experimental data might be in fact metastable. In addition, the excessive complexityof certain SAFT models might result in wrong prediction of auxiliary thermodynamic properties of theexperimentally available fluid phases. The undesired predictions discussed in the present study arisequeries regarding the robustness and the over-all physical validity of the models under consideration intheir present forms. This study discusses the ways of removing the numerical pitfalls.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

The EOS models based on the Statistical Association Fluid The-ory (SAFT) are among the most important approaches for modelingthermodynamic properties of pure substances and their mix-tures. The comprehensive reviews of different SAFT EOS modelsare available elsewhere [1–5]. Based on the advanced molecularapproaches, the SAFT EOS models are expected to be robust andreliable in the entire thermodynamic phase space. However, thusfar several studies have reported some worrying results.

In particular, Koak et al. [6] have investigated the effect of theChen–Kreglewski (CK) [7] dispersion term on the pressure–volumebehavior of two non-associating pure components predicted by theCK-SAFT of Huang and Radosz [8]. It has been demonstrated the P–visotherms exhibit multiple volume roots at low temperatures andvolumes, while the smallest practically unrealistic root is more sta-ble than the realistic ones. It has been concluded that this behaviorappears to be inconsistent with the intention of the original theoriesand it needs to be resolved (see also [9]).

In the paper discussing the numerical pitfalls introduced by theSoave-type �-functions in Cubic Equations of State [10] it has beendemonstrated that the theoretically based models such as the SAFTversion of Huang and Radosz [8] and Johnson–Zollweg–Gubbinsapproach for the Lennard–Jones fluids [11] exhibit the practicallyunrealistic predictions, namely the multiple critical points for pure

∗ Tel.: +972 3 9066346; fax: +972 3 9066323.E-mail addresses: [email protected], [email protected].

compounds. However the reasons of this behavior have not beeninvestigated with great details.

The comprehensive studies of Yelash et al. [12,13] have demon-strated that the predictions of PC-SAFT [14] are affected by crossingof isotherms, attributed to the temperature dependence of hard-sphere diameter, and by appearance of practically unrealistic phaseequilibria, attributed to the polynomial expression of the dispersionterm. It has been concluded that the unrealistic phase behavior canbe a problem when the model is applied to polymer solutions andpolymer blends. The predictions of the excessive phase equilibriafor large variety of pure compounds exhibited by PC-SAFT havebeen recently discussed in great details by Privat et al. [15].

The aim of the present study was detection and investigation ofthe numerical pitfalls exhibited by several popular SAFT EOS mod-els necessary for further development of their consistent versions.

2. Theory

The SAFT EOS models treat the compressibility factor as acombination of terms representing different inter-molecular inter-actions:

Z = Zrepulsive + Zattractive + Zassociative (1)

In most of the cases the first two terms are the polynomial expres-sions by volume, which makes their general analytical investigationrelatively easy. The associative term includes logarithmical expres-sions for the volume, which substantially hinders its analyticalinvestigation. Hence the question of an over-all consistence of thisterm has been left outside the scope of the present study and it

0378-3812/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.fluid.2010.07.003


68 I. Polishuk / Fluid Phase Equilibria 298 (2010) 67–74

Table 1Phenomena of the multiple critical points generated by the non-associative SAFTEOS models considered in the present study.

Model Polynomial orderby volume

Appearance of thephenomena

SSAFT [19] m = 1 5 NoSSAFT [19] m /= 1 6 NoOriginal SAFT [21] m = 1 8 NoOriginal SAFT [21] m /= 1 9 NoCK-SAFT [8] m = 1 13 YesCK-SAFT [8] m /= 1 14 YesSimplified PC-SAFT [24] m = 1 19 YesSimplified PC-SAFT [24] m /= 1 24 YesSoft-SAFT [25] m = 1 Non-applicable,

exponentialexpression

Yes

Soft-SAFT [25] m /= 1 Non-applicable,exponentialexpression

Yes

Fig. 1. Intersection of isotherms predicted by SSAFT [19] for ethane; ©,�, and �,experimental data [20].

should be considered in future studies, after formulating the appro-priate approaches for its general analysis.

The numerical pitfalls of the repulsive term are generated by thetemperature dependence of the reduced density �. For example,according to CK-SAFT of Huang and Radosz [8] it is given as:

� = �Voo(1 − C exp[−3(uo/k)/T])3

V(2)

Fig. 2. Lines of zero CV predicted by several versions of SAFT for ethane.

Fig. 3. CV of ethane predicted by SSAFT [19]: 100 bar, solid line; 9000 bar, dashedline; © and �, experimental data [20].

where uo/k and Voo are the positive compound-specific parameters,� = 0.74048 and C = 0.12. The numerator of Eq. (2) is the model’scovolume, or the volume at the infinite pressure. It can be seen thatEq. (2) results in decrease of covolume with temperature. In otherwords, as the temperature rises, the infinite pressure is approachedat the smaller values of volume. As a result, at high-pressuresvolume will decrease with temperature and the isotherms will

Fig. 4. Intersection of isotherms predicted by SAFT of Chapman et al. [21] for ethane;©,�, and �, experimental data [20].

Fig. 5. The 120 K isotherm of nitrogen predicted by CK-SAFT [8].


I. Polishuk / Fluid Phase Equilibria 298 (2010) 67–74 69

Fig. 6. The 120 K isotherm of nitrogen predicted by CK-SAFT [8] at high densities.

Fig. 7. Two vapor pressure curves of pure nitrogen predicted by CK-SAFT [8]; �,experimental data [23].

intersect in contradiction with experimental facts. The same effectis characteristic for other expressions of covolume exhibiting sim-ilar behavior.

Another serious drawback of the temperature-dependent covol-umes is the description of the caloric properties at high pressures,

Fig. 8. Two-phase envelopes of pure nitrogen predicted by CK-SAFT [8]; �, experi-mental data [21].

Table 2Critical points predicted by CK-SAFT [8] for several pure compounds.

Compound Tc1 (K) Pc1 (bar) Tc2 (K) Pc2 (bar)

Nitrogen 127.04 35.33 159.95 23870.1Argon 151.56 50.28 191.74 34125.4Methane 192.17 48.15 242.85 32643.7Carbon monoxide 136.45 39.16 146.75 26131.9Ethane 320.61 60.24 254.59 46646.7Carbon dioxide 320.71 92.50 311.20 63525.7Water 725.95 422.88 682.38 166350.8

Fig. 9. The 150 K isotherm of nitrogen predicted by CK-SAFT [8].

as recently demonstrated by Kalikhman et al. [16]. In particular, ithas been proven that any temperature dependency of the covol-ume necessarily results in infinite negative value of the isochoricheat capacity at the infinite pressure. It should be pointed outthat although CV < 0 is mathematically possible, it cannot actuallyoccur, since the mechanical stability limit will be violated beforethe thermal limit. This is because the mechanical limit representsa response of higher-order than the thermal limit [17,18]. Con-sidering a fact that the models with the temperature-dependentcovolumes exhibit the negative CV under conditions of mechanicalstability, such predictions should be considered as non-physical.

The latter phenomena might affect different models with thedifferent degree of extend. While some models, such as the CK-SAFT of Huang and Radosz [8], exhibit the negative isochoric heatcapacities at the extremely high pressures (the reason will be dis-

Fig. 10. Departure Gibbs energy – pressure projection of nitrogen generated byCK-SAFT [8] at 150 K.



Fig. 11. The minus departure Gibbs energy – pressure projection of nitrogen gen-erated by CK-SAFT [8] at 120 K.

cussed below), irrelevant for most practical implementations, theother ones might yield the non-physical predictions at much lowerpressures [16].

The attractive terms of SAFT EOS models typically present poly-nomials adjusted to the theoretical expressions, whose analyticalcalculation is not always possible. As the polynomial order becomeshigher, the EOS models yield more solutions for the volume. Thecomplex solutions and the solutions smaller than the covolumeshould be considered as the “numerical noise” having no impacton the performance of the model. However, if the polynomialorder and, consequently, the number of the solutions increases,the additional practically improbable real solutions bigger thanthe covolume appear. Table 1 lists several popular SAFT EOS mod-els, their polynomial order (when applicable) and appearance ofunrealistic volume solutions. In what follows let us consider theparticular models.

3. Results

3.1. Simplified SAFT (SSAFT) of Fu and Sandler [19]

This model has the lowest polynomial order among the SAFTmodels (see Table 1) and it does not exhibit the additional practi-cally unrealistic solutions for the volume. However the temperaturedependence attached to its covolume still affects its robustness.

Fig. 12. Two vapor pressure curves of pure ethane predicted by CK-SAFT [8]; ©,experimental data [20].

Table 3Critical points predicted by CK-SAFT [8] with D29 = 0.

Compound Tc1 (K) Pc1 (bar) Tc2 (K) Pc2 (bar)

Nitrogen 125.54 33.26 16.54 36477.3Argon 149.74 47.32 17.1 45041.2Methane 189.87 45.32 22.53 44798.2Carbon monoxide 135.78 38.27 16.02 43542.4Ethane 320.29 59.97 29.15 86343.5Carbon dioxide 319.87 91.31 45.87 144762.0Water 721.06 409.18 60.88 260270.1

Fig. 1 depicts the intersection of isotherms predicted by SSAFT[19] for ethane. Fig. 2 presents the zero values of CV predictedby several versions of SAFT for ethane. It can be seen that SSAFTstarts to predict the non-physical negative values at the low-est pressures, which are still very high (14,900 bars and above).However it would not be surprising that these undesired phe-nomena have an impact on modeling the experimentally availablecaloric data. In particular, Fig. 3 demonstrates that SSAFT substan-tially underestimates the experimental CV data at 9000 bar, whichshould be explained by the behavior of the model at higher pres-sures.

3.2. SAFT of Chapman et al. [21]

Although this model has a relatively high-polynomial order (seeTable 1), the additional practically unrealistic solutions for the vol-ume have not been detected. However the model is still affected bythe temperature-dependent covolume. Fig. 4 depicts the intersec-tion of isotherms predicted for ethane and Fig. 2 presents the zerovalues of CV.

3.3. CK-SAFT of Huang and Radosz [8]

The polynomial order of this model is higher than in the previ-ously considered cases (see Table 1) and analysis shows that it mayexhibit up to five real solutions for molar volume bigger than thecovolume. These multiple real solutions are not only excessive onthe thermodynamic phase scene, but might also be dangerous likethe famous Chekhov’s gun (“If in Act I you have a pistol hanging onthe wall, then it must fire in the last act” [22]).

Fig. 5 depicts the isotherm predicted by the Huang and Radosz’sversion of CK-SAFT [8] for nitrogen at 120 K. The points A, B and Crepresent the realistic phase envelope matching the experimen-tal data (A is the vapor and C is the liquid solution), and theextrema ˛ and ˇ represent the spinodal. At the volumes lower thatC the isotherm should approach the infinite pressure. However, as

Fig. 13. Two-phase envelopes of pure ethane predicted by CK-SAFT [8]; ©, experi-mental data [20].



Fig. 14. Departure Gibbs energy – pressure projection of ethane generated by CK-SAFT [8].

noticed, SAFT exhibits five real solutions bigger than its covolume.It can be seen that the two excessive solutions (E and D) deformthe thermodynamic phase space established by the model. As aresult, two practically unrealistic extrema � and ı appear (addi-tional details are shown in Fig. 6). Thus, two spinodal loci andtwo-phase equilibria are generated simultaneously: the saturationpressure 1 in the vicinity of experimental datum and the unrealisticsaturation pressure 2 located at the deep vacuum.

Figs. 7 and 8 depict two vapor pressure lines and two-phaseenvelopes predicted by the CK-SAFT [8] for nitrogen. As discussedpreviously [10], the model exhibits at least two critical points for

Fig. 15. Globally stable phase diagram of methane predicted by SOFT-SAFT [25]; ©,experimental data [26].

pure compounds. Table 2 lists these critical points for certain purecompounds.

Fig. 9 depicts the isotherm at 150 K of nitrogen. It can be seenthat this time it takes a conventional form (three solutions andtwo extrema), although in total disagreement with experimentalfacts. Fig. 10 presents the excess Gibbs energy – pressure projec-tion for this isotherm. It can be seen that the stable, the metastable(A = E − � and A = E − ı) and the unstable (ı − D − �) parts exhibita robust behavior characteristic for phase equilibria in pure com-pounds. Thus, there are no doubts regarding the stability of VLE-2above the critical point 1.

However at the coexisting of two-phase equilibria the picturebecomes complicated (see Fig. 11; −�G is used for creating thelogarithmic scale). Three facts can be observed:

1. The VLE-2 take place at the conditions closer to the ideal gas stateand, therefore, at the GE closer to zero than the VLE-1.

2. There are no phase equilibria between the liquid phases.3. The liquid-2 (solution E) is more stable than the liquid-1 (solu-

tion C) and the VLE-1 at the pertinent pressure.

The interpretation of Fig. 11 is not trivial because the prac-tically unrealistic phase behavior generated by the model underconsideration cannot be explained by experimental facts. Let usconsider nitrogen at some point along VLE-1. As noticed, at suchconditions the liquid-2 (solution E) presents the most stable phase.



Fig. 16. Thermodynamic properties of methane [26] at high pressures as predicted by SOFT-SAFT [25]; �, 100 bar; ©, 200 bar;�, 500 bar;�, 1000 bar;�, 2000 bar;�, 5000 bar;�, 10,000 bar.

Thus, according to CK-SAFT [8], at the conditions of VLE-1 and inthe varying-volume vessel nitrogen is supposed to shrink into thishighly compressed liquid. However even the microscopic increaseof the vessel’s volume should result in a drastic fall of pressure(see Figs. 5 and 6) until appearance of VLE-2 at the deep vacuum.Thus, the realistic VLE-1 is actually metastable and the unrealisticVLE-2 represents the globally stable phase diagram established byCK-SAFT [8].

Table 2 shows that for some compounds CK-SAFT [8] gener-ates the realistic critical points at the temperatures higher thanthe unrealistic ones. Figs. 12 and 13 depict the globally stable partsof the ethane’s phase diagram as predicted by CK-SAFT [8]. It can

be seen that yet at least some part of the realistic phase equilibria(VLE-1 above LLVE) is globally stable. An explanation is provided inFig. 14.

In particular, it can be seen that at the T > TLLVE both VLE-1 andLLE do not face other phases with lower �G at the pertinent pres-sures, and therefore both are globally stable. However, at T < TLLVEthe situation gets different. Yet vapor has lower �G than LLE andliquid-2 has lower �G than VLE-1. Thus, only the unrealistic VLE-2 is now globally stable. Remarkably, similar behavior has beendescribed by Privat et al. [15] for PC-SAFT. However it should benoticed that in the present case the numerical pitfalls significantlyaffect the accuracy of the model.



An artificial attempt of decreasing the polynomial order of CK-SAFT [8] by considering the zero value for D29 might improve thesituation (see Table 3). Remarkably, this practice does not affect theCK dispersion term radically.

Finally, it should be pointed out that CK-SAFT [8] unavoidablyexhibits the numerical pitfalls characteristic for the previously con-sidered simpler versions of SAFT. However since approaching theinfinity pressure at low temperatures is yet hindered by the unusualforms of its isotherms, CK-SAFT [8] generates the negative isochoricheat capacities at much higher pressures (see Fig. 2).

3.4. PC-SAFT [14]

The case of PC-SAFT [14] has been already investigated withgreat details in the previous studies [12,13,15]. It should only benoticed that the rescaled simplified PC-SAFT [24] does not exhibitan improvement in comparison with the original version. Never-theless the fact that PC-SAFT, in spite of its very high-polynomialorder, typically generates the unrealistic phase equilibria at rela-tively low temperatures [15] should not be neglected. Hence, theundesired predictions yielded by PC-SAFT [14] are not expected toaffect most of its practical implementations. One might anticipatefor problems just in certain cases [12,13]. In addition, the unre-alistic phase equilibria could interrupt the high-pressure LLE inasymmetric mixtures at high pressures. Moreover, the probabilityof numerical problems might increase with increasing the numberof compounds in a mixture.

3.5. Soft-SAFT [25]

The Soft-SAFT [25] EOS is based on theJohnson–Zollweg–Gubbins approach for the Lennard–Jones fluids[11]. The latter approach typically exhibits the additional artificialphase equilibria at the temperatures higher than CK-SAFT [8] (seefor example [10]). The same could be concluded regarding theSoft-SAFT [25]. For example, for methane it predicts: Tc1 = 193.27 K,Pc1 = 50.97 bar, Tc2 = 320.64 K and Pc2 = 24371.8 bar. Fig. 15 depictsthe globally stable phase equilibria yielded by Soft-SAFT [25] formethane. A total disagreement with the experimental facts isevident.

Fig. 16 demonstrates that Soft-SAFT [25] could be an incredi-bly accurate model for describing the high-pressure densities, heatcapacities and sound velocities. However it can be seen that the sig-nificant part of the Soft-SAFT’s [25] fluid phase diagram is actuallymetastable because the artificial high-pressure phase split takesplace above the compound’s solidification temperatures. In addi-tion, it predicts the negative values for the heat capacities and, as aconsequence, wrong results for the sound velocities below 100 K. Itwould be tempting to find a relation these two facts [17,18]. How-ever it can be seen (Fig. 16a) that Soft-SAFT’s [25] still predicts themechanical stability at the experimentally available fluid range.Thus, the current non-physical results could rather be explainedagain by the excessive complexity of the model under considera-tion.

4. Conclusions

This study demonstrates that the SAFT EOS models might exhibitthe practically unrealistic and even non-physical predictions dueto the two factors, namely the temperature dependencies of a seg-ment packing fraction (covolume) and the very high-polynomialorders by volume. In particular, the first factor is responsible forpredicting the intersections of isotherms at high densities. In addi-tion, it results in negative values of the heat capacities at extremelyhigh pressures, which however might affect the predictions of theexperimentally available data as well.

The very high-polynomial orders of several SAFT EOS mod-els result in prediction of the additional stable unrealistic criticalpoints and the pertinent fictive phase equilibria. It is demonstratedthat the unrealistic phase splits might present the globally sta-ble states established by the models, while the VLE matching theexperimental data might be in fact metastable. In addition, theexcessive complexity of certain SAFT models might result in non-physical prediction of auxiliary thermodynamic properties of theexperimentally available fluid phases. Thus, some SAFT EOS modelsshould be re-evaluated while using fewer coefficients. Investiga-tion of the maximal equation orders free of the numerical pitfallsis essential for further progress of SAFT.

In addition, it seems highly recommendable to make the seg-ment packing fraction of SAFT models temperature-independent.It could be argued that the temperature-dependent covolumesresult in numerical pitfalls outside the range of the model’s appli-cability (extremely high pressures). However the very concept ofthe model’s applicability is characteristic for empirical correlationsrather than theoretically based approaches. The latter ones are sup-posed to be robust and reliable in the entire thermodynamic phasespace. Moreover, it does not seem possible to cancel the numeri-cal contribution of the repulsive term to the residual isochoric heatcapacity at high pressures by other model’s terms.

List of symbolsC integration constant of CK-SAFT [8]CV isochoric heat capacityk Boltzmann’s constantP pressureR universal gas constantT temperatureuo/k temperature-independent dispersion energy of interac-

tion between segmentsV molar volumeVoo temperature-independent segment volume of CK-SAFT

[8]W speed of soundZ compressibility factor

Greek letters� reduced density� segment molar volume in a closed-packed arrangement

Subscriptsc critical stater reduced property

AbbreviationsEOS equation of stateSAFT statistical association fluid theory

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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