It

ID

a

ARRA

KESHPSC

1

da[dt[metmy

odrfbArnoo

0d

J. of Supercritical Fluids 62 (2012) 47– 54

Contents lists available at SciVerse ScienceDirect

The Journal of Supercritical Fluids

jou rn al h om epage: www.elsev ier .com/ locate /supf lu

mplementation of SAFT + Cubic and PC-SAFT for comprehensive description ofhermodynamic properties of n-octane and its mixtures

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 9 September 2011eceived in revised form 7 November 2011ccepted 8 November 2011

a b s t r a c t

The paucity of the available sound velocity and other auxiliary property data on supercritical mixturesrestricts the comprehensive examination of the EoS models to just a few systems. In the current studypure n-octane(2) and its several mixtures such as methane(1)–n-octane(2) and nitrogen(1)–n-octaneare considered. It is demonstrated that the recently proposed SAFT + Cubic Equation of State (EoS) hasa doubtless advantage over the popular PC-SAFT model in predicting the elevated pressure densities.

eywords:quation of statetatistical association fluid theoryigh pressurehase equilibriaound velocity

In addition, SAFT + Cubic appears as a robust estimator of sound velocities and compressibilities bothat low and high pressures, which is not a case of PC-SAFT. Although SAFT + Cubic is capable of onlyrough estimation of heat capacities of pure n-octane, its results are closer to the experimental data thanthe predictions of PC-SAFT. Additional research should be performed for drawing grounded conclusionsconcerning the accuracy of both equations in modeling phase equilibria.

ompressibility

. Introduction

During the recent decades the supercritical phase equilibriaata collection has been enriched by nearly 3000 references [1–5]nd significant progress in modeling these data has been achieved6–8]. However, in addition to phase equilibria, modern processesign requires information on auxiliary thermodynamic proper-ies, such as compressibilities, heat capacities and sound velocities9–12]. Thus, development of analytical Equation of State (EoS)

odels capable of simultaneously accurate estimation of large vari-ty of properties of pure compounds and their mixtures in the entirehermodynamic phase space is one of the challenging problems of

odern thermodynamics which has not been satisfactorily solvedet.

Unfortunately, the database on supercritical auxiliary propertiesf mixtures is currently scarce in comparison with the pertinentatabase on phase equilibria. This paucity of experimental dataestricts the comprehensive evaluation of EoS models to just aew mixtures for which the high pressure phase equilibria haveeen published synchronously with densities and sound velocities.mong the most interesting cases of the kind are the two asymmet-ic n-octane mixtures, namely methane(1)–n-octane(2) [13] and

itrogen(1)–n-octane(2) [14]. In addition, representative data setsn thermodynamic properties of pure n-octane [15–21] and somef its symmetric mixtures [22–28] are available as well.

∗ Tel.: +972 3 9066346; fax: +972 3 9066323.E-mail addresses: polishuk@ariel.ac.il, polishyk@bgumail.bgu.ac.il

896-8446/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.supflu.2011.11.009

© 2011 Elsevier B.V. All rights reserved.

In the current study the experimental data above have beenused for examining two EoS models, namely the recently proposedSAFT + Cubic [29] and PC-SAFT [30], which is yet probably the mostsuccessful and widely used theoretically based fluid phase equation[7]. Having a more solid theoretical background than SAFT + Cubic,PC-SAFT is however affected by certain numerical pitfalls [31–33].At the same time, it should be realized that these undesired phe-nomena take place at extreme conditions, typically outside therange of most practical implementations (which cannot be con-cluded concerning some other versions of SAFT, such as SAFT-CP[34] or SAFT-VR-Mie [35], for more details see [29,36]). A majoradvantage of SAFT + Cubic over PC-SAFT is its capability of simulta-neous modeling of critical and sub-critical data in pure compounds.However, again, it should be pointed out that PC-SAFT tends tooverestimate the critical temperatures and pressures in a smallerextent than the variations of SAFT-VR [35].

2. Theory

The detailed description of PC-SAFT EoS and the values of thepure compound adjustable parameters are provided in the origi-nal reference [30]. Similarly to other versions of SAFT, PC-SAFT isexpressed in terms of the following contributions to the residualHelmholtz free energy:

res hs chain disp assoc

A = A + A + A + A (1)

where the superscripts hs, chain, disp and assoc correspond tothe hard sphere, chain, dispersion and association contributions,respectively. These contributions, in turn, are functions of the

48 I. Polishuk / J. of Supercritical Fluids 62 (2012) 47– 54

Table 1Pure compound parameters of SAFT + Cubic.

Compound Adjustable parameter Parameters calculated using Eqs. (19)–(21)

m c [L/mol] a [bar-mol/L] � [A] ε/k [K]

Cyclohexane 2.1300 0.16928 19.093 4.1804 403.72n-Hexane 2.3100 0.19889 23.498 4.2991 370.56n-C8H18 2.5843 0.26708 39.990 4.5259 422.09n-C12H26 3.0470 0.41848 85.672 4.8978 502.85n-Hexadecane 3.3830 0.58789 147.20 5.2040 563.31

Compound Adjustable parameters Parameters calculated using Eqs. (20) and (21)

Nitrogen 1.1300 0.07322 0.7555 3.4089 86.9591.4084 3.6817 132.05

20.221 4.1389 394.7723.200 4.1439 460.08

f�εkicttp

hp

A

wc

A

w

�

Fet

W(m/s)270018009000

.0001

.001

.01

.1

1

10

100

1000

10000

P(b

ar)

2 phases

1 phase

800 1300 1800 2300

1000

2000

3000

n-Octane

Methane 1.0000 0.11798

1-chlorobutane 2.1000 0.19501

Chlorobenzene 2.1400 0.17700

ollowing adjustable parameters: m, effective number of segments;, Lennard–Jones temperature-independent segment diameter;/k, segment energy parameter divided by Boltzmann’s constant;AB, volume of interaction between the association sites and εAB/k,nteraction of association energy. Thus, for the associating polarompounds the model requires five adjustable parameters. Inhe case of the non-associating compounds the association con-ribution is deleted and the model stays with three adjustablearameters.

The detailed explanation of SAFT + Cubic and its mixing rulesave been provided in the previous studies [29,37,38]. For the com-ounds considered as non-associating this EoS is given as follows:

res = Ahs + Adisp + Achain − a

v + c(2)

here A is the Helmholtz free energy, v is the molar volume, a and are the parameters of the cohesive correction term.

HS = RTm

�0

(3�1�2

1 − �3+ �3

2

�3(1 − �3)2+(

�32

�23

− �0

)ln[1 − �3]

)

×√

d3(�3 − 1)�3�3 − d3

(3)

here

k = �Nav

6V

∑i

ximiidkii (4)

ρ (g/L)906003000 0

.0001

.001

.01

.1

1

10

100

1000

10000

P(b

ar)

470 57 0 67 0 77 0 87 0

160 0

320 0

480 0

1 phase

2 phases

n-Oc tane

ig. 1. Densities of pure n-octane. Saturated experimental data [15]: ©; single phasexperimental data: – 298.03 K [16], – 348.14 K [16], – 521.55 K [17]; predic-ions: SAFT + Cubic – solid lines, PC-SAFT – dashed lines.

Fig. 2. Sound velocities in pure n-octane. Saturated experimental data [18]: ©;single phase experimental data [19]: – 313.15 K, – 353.15 K, – 393.15 K;Predictions: SAFT + Cubic – solid lines, PC-SAFT – dashed lines.

NAv is the Avogadro’s number and:

dii = �ii

(1 + 0.2977(k/ε)iiT

1 + 0.33163(k/ε)iiT + 0.0010477(k/ε)ii2T2

)(5)

Achain = RT∑

i,j

xixj(1 − mij) ln[gij(dij)hs] (6)

P(b

ar)

2.52.32.11.9100

1000

10000

CP(J/g-K)

Fig. 3. Isobaric heat capacities of pure n-octane at elevated pressures and 299 K.Experimental data [20]: ©. Predictions: SAFT + Cubic – black solid line, PC-SAFT –red dashed line.

I. Polishuk / J. of Supercritical Fluids 62 (2012) 47– 54 49

T (K )385360335310

250

280

310C

P(J/

mol

-K)

Fig. 4. Isobaric heat capacities of pure n-octane at moderated pressures. Experimen-tal data [21]: – 1 bar, – 100 bar; Predictions: SAFT + Cubic – solid lines, PC-SAFT– dashed lines.

T (K )16001200800400

10000

100000

1000000

10000000

100000000

P(b

ar)

CV<0

PC-SAFT

SAFT + Cubic

Fig. 5. Lines of zero CV. SAFT + Cubic – black solid line, PC-SAFT – red dashed line.

ρ (g/L)880800720640

0

1000

2000

3000

4000

5000

P(b

ar)

Fig. 6. Densities of the equimolar n-octane(1)–n-dodecane(2) mixture at elevatedpressures. Experimental data [16]: – 298.19 K, – 323.15 K, – 348.18 K, –373.06 K; Predictions: SAFT + Cubic – solid lines, PC-SAFT – dashed lines.

W(m

/s)

1.0.8.6.4.20.0900

1150

1400

1650

1900

x1

298.15 K

Fig. 7. Sound velocities in the system n-octane(1)–n-dodecane(2). Experimentaldata [22]: – 1 bar, – 500 bar, – 1000 bar. Predictions: SAFT + Cubic – solidlines, PC-SAFT – dashed lines.

ρ(g

/L)

1.0.8.6.4.20.0690

750

810

870

930

x1

298.15 K

ρ(g

/L)

1.0.8.6.4.20.0660

720

780

840

900

x1

328.15 K

Fig. 8. Densities of the system 1-chlorobutane(1)–n-octane(2). Experimental data[23]: – 21.3 bar, – 203.7 bar, – 386 bar. Predictions: SAFT + Cubic – solid lines,

PC-SAFT – dashed lines.

where the segment radial distribution function given as:

gij(dij)hs = 1

1 − �3+ 3diidjj�2

(dii + djj)(1 − �3)2+ 2

(diidjj

dii + djj

)2�2

2

(1 − �3)3

(7)

50 I. Polishuk / J. of Supercritical Fluids 62 (2012) 47– 54

ρ (g/L)810760710660

0

300

600

900

1200

P (b

ar)

W (m/s)169013601030700

0

400

800

1200

P (b

ar)

CV (J/kg- K)2400220020001800

0

400

800

1200

P (b

ar)

κT (GP a-1)2.82.01.2.4

0

400

800

1200

P (b

ar)

F ixtur– : SAFT

a

A

w

a

a

�

w(a

�

a

ig. 9. Thermodynamic properties of the equimolar n-octane(1)–n-hexadecane(2) m 353.15 K [24], – 373.15 K [25], – 393.15 K [24], – 433.15 K [25]. Predictions

nd

disp = mR(

ε

k

)(adisp

o1 + adispo2 (ε/k)

T

)(1 + 2Achain

AHS

)(8)

here

dispo1 = 3

√2

�[−8.5959�3 − 6.1344�2

3 − 3.87882�33 + 25.3316�4

3]

(9)

dispo2 = 3

√2

�[−1.9075�3 + 13.4675�2

3 − 40.5171�33 + 39.1711�4

3]

(10)

The mixing rules are:

ε

k=∑

i

∑jxixj

√miimjj�ij

3(ε/k)ij

�3∑

iximii(11)

= 3

√∑i

∑jxixj

√miimjj�ij

3∑iximii

(12)

here

ε

k

)ij

= (1 − kij)

√(ε

k

)ii

(ε

k

)jj

(13)

nd�ii + �jj

ij = (1 − lij) 2(14)

=∑

i

∑j

∑k

xixjxkaijk (15)

e at elevated pressures. Experimental data: – 313.15 K [24], – 333.15 K [25], + Cubic – solid lines, PC-SAFT – dashed lines.

In the particular case of binary mixtures Eq. (15) is given as:

a =(

x3a11 + 3x2(1 − x) 3√

a112a22 + 3x(1 − x)2 3

√a11a22

2

+(1 − x)3a22

)(16)

In addition:

m =∑

i

ximi (17)

c =∑

i

xici (18)

In this study Eqs. (15)–(18) are not attached by the binaryadjustable parameters.

Adjusting SAFT + Cubic to the data of most pure alkanes requiresfitting of only one adjustable parameter m to the liquid densities.The remaining four parameters (c, a, �, and ε/k) are not adjustablebut are obtained by solving a system of four equations:

c = −1.6049 + 1.3440(� × 109) − 0.3943(� × 109)2

+0.0417(� × 109)3

(19)

(∂P

∂v

)Tc

=(

∂P2

∂2v

)Tc

= 0

∣∣∣∣vc,EOS=1.1vc,experimental

(20)

Pc,EOS = Pc,experimental (21)

In the cases of other compounds fitting of both m and c should beperformed, while the non-adjustable a, �, and ε/k are obtained bysolving a system of Eqs. (20) and (21). For the light gases such as N2and CH4 the critical volume displacement of 1.065 instead of 1.1

itical Fluids 62 (2012) 47– 54 51

(tpsbopPb

3

ndteecbhdiraarp

v

F2–

T (K)385350315280

0

100

200

300

P(b

ar)

I. Polishuk / J. of Supercr

see Eq. (20)) was selected. All pure compound parameters needo be determined together at once. Table 1 lists the SAFT + Cubicarameters evaluated for the compounds considered in the presenttudy. All the calculations (including the fitting of parameters) haveeen performed in the Mathematica 7® software. Implementationf both models under consideration usually requires similar com-utation times. However the critical loci are calculated faster withC-SAFT. The pertinent routines can be obtained from the authory request.

. Results

Fig. 1 depicts the saturated and one-phase densities of pure-octane. As seen, both models under consideration accuratelyescribe the saturated liquid density, in exception of the predic-ions yielded by PC-SAFT for the critical region. However at thelevated pressures the advantage of SAFT + Cubic becomes evidentven more. In particular, it can be seen that PC-SAFT signifi-antly over-estimates the data above 1000 bar. This result shoulde explained by the too low co-volume and, consequently, tooigh infinity pressure density established by this model (for moreetails see Ref. [38]). Overestimation of densities typically results

n underestimation of sound velocities. However yet the inaccu-acy of PC-SAFT is not restricted by the elevated pressures, butffects the entire pressure range, including the saturated liquidt deep vacuum (see Fig. 2). Unlike PC-SAFT, SAFT + Cubic accu-

ately describes the sound velocity data at both low and highressures.

Unfortunately, the success in predicting densities and soundelocities does not guarantee the same degree of accuracy in the

x1,y1

1.0.8.6.4.20.00

100

200

300

P(b

ar)

v (L/mol).9.6.30.0

0

100

200

300

P(b

ar)

ig. 10. VLE in the system methane(1)–n-octane(2). Experimental data [41]: –23.15 K, – 323.15 K, – 423.15 K; Modeling: SAFT + Cubic – solid lines, PC-SAFT

dashed lines.

Fig. 11. Dew pressures of the {0.98 methane(1) + 0.02 n-octane(2)} mixture. Experi-mental data [13]: ©. Modeling: SAFT + Cubic – black solid line, PC-SAFT – red dashedline.

case of isobaric heat capacities. As seen (Fig. 3), the experimen-tal data exhibits a pressure minimum around 1500 bar. Althoughboth models under consideration predict this minimum, they failto do it quantitatively. In particular, PC-SAFT estimates it around400 bar and SAFT + Cubic – at the extremely high pressure near250,000 bar. Analysing the available isobaric heat capacity data fordifferent compounds, Czarnota [20] has concluded that these pres-sure minima are in fact related to the solidification pressures. Inparticular, heavier compounds with the lower solidification pres-sures exhibit the isobaric heat capacity minima at lower pressuresas well. Randzio [39] has explained these minima by the changein the shape of effective intermolecular potential in dense liquidscaused by pressure, resulting also in a change of sign of the temper-ature derivative of the thermal expansion coefficient. As seen, bothSAFT + Cubic and PC-SAFT are incapable of quantitative descriptionof these phenomena, yielding only rough estimation of the isobaricheat capacity data (see also Fig. 4). At the same time, it should bepointed out that both models under consideration start predict-ing the non-physical negative heat capacities at the tremendouslyhigh pressures, far above the estimated chemical decomposition of

hydrocarbons [40] (see Fig. 5).

In the following discussion, let us proceed to the considera-tion of mixtures. The symmetric systems are predicted here withthe zero values of binary parameters. For the asymmetric ones

x1,y 1

1.0.8.6.4.20.00

200

400

600

P (b

ar)

Fig. 12. VLE in the system nitrogen(1)–n-octane(2). Experimental data [42]: –344.5 K, – 424 K, – 473.5 K; modeling: SAFT + Cubic – solid lines, PC-SAFT –dashed lines.

52 I. Polishuk / J. of Supercritical Fluids 62 (2012) 47– 54

Table 2Binary adjustable parameters.

EoS model Parameters N2(1) + n-C8H18(2) CH4(1) + n-C8H18(2)

SAFT + Cubic k12 −0.0120 −0.0530l12 0.0150 0.0200

(tdmoc(F

otStdcis

maamxofacifflpheiu

FES

ρ (g/L)775725675625

200

500

800

1100

P(b

ar)

x1 = 0.207

W (m/s)170014001100800

200

500

800

1100

P(b

ar)

x1 = 0.207

Fig. 14. Densities and sound velocities in the mixture {0.207 nitrogen(1) + 0.793 n-

PC-SAFT k12 0.1109 0.1500l12 0 −0.0500

methane(1)–n-octane(2) and nitrogen(1)–n-octane(2)) fitting ofhe binary parameters has been performed (see Table 2). Fig. 6epicts the densities of the equimolar n-octane(1)–n-dodecane(2)ixture up to 5000 bar. The doubtless advantage of SAFT + Cubic

ver PC-SAFT in predicting these can be clearly seen. The sameonclusion is valid in the case of sound velocities in this systemsee Fig. 7) and the densities of 1-chlorobutane(1)–n-octane(2) (seeig. 8).

The data for another relatively symmetric system, namely n-ctane(1)–n-hexadecane(2) (Fig. 9) include auxiliary propertieshat present a paramount interest for the current study. As seen,AFT + Cubic predicts almost all these data accurately, includinghe particularly important isothermal compressibilities. Certaineviations are observed however in the case of the isochoric heatapacities. At the same time, it should be concluded that PC-SAFTs a substantially less successful estimator of the data under con-ideration.

Before proceeding to the one-phase properties of two asym-etric mixtures selected for the present study, let us analyze the

ccuracy of the equations under consideration in modeling thevailable VLE data (Figs. 10–12). As seen, PC-SAFT is superior in esti-ating VLE at moderated pressures and the two phase boundary at

1 = 0.98 of methane(1)–n-octane(2). It should however be pointedut that the relative paucity of experimental phase equilibria dataor this particular system (see Fig. 10) hinders the appropriatedjustment of the binary parameters and, therefore, makes theomprehensive comparison difficult. Situation is more favourablen the case of the second system, namely nitrogen(1)–n-octane(2)or which data are available in the entire pressure range. It has beenound that in the case of PC-SAFT different combinations of k12 and12 have yielded almost identical results. In other words, in thisarticular case, adjusting l12 has not offered major benefits. Thus it

as been decided to adopt the fitting performed by Eliosa-Jiménezt al. [42]. In the case of SAFT + Cubic a more realistic (although stillmperfect) over-all picture of phase equilibria has been obtainedsing two binary parameters. It should be pointed out that the

W (m/s)400 60 0 80 0 100 0 120 0 14 0 0

200

450

700

950

1200

P(b

ar)

x1 = 0.9 8

ig. 13. Sound velocities in the mixture {0.98 methane(1) + 0.02 n-octane(2)}.xperimental data [13]: – 293.15 K, – 333.15 K, – 373.15 K; Predictions:AFT + Cubic – solid lines, PC-SAFT – dashed lines.

octane(2)}. Experimental data [14]: – 293.15 K, – 313.15 K, – 333.15 K,–353.15 K, – 373.15 K. Predictions: SAFT + Cubic – solid lines, PC-SAFT – dashedlines.

fitting capabilities of both models under consideration might besubstantially improved by introducing a third binary adjustableparameter for the effective number of segments. However this issuegoes beyond the scope of the current study.

The advantage of SAFT + Cubic over PC-SAFT becomes evidentwhile moving from VLE to the homogenous liquid phase underhigh pressures (see Figs. 13 and 14). The remarkable accuracy of

1.0.8.6.4.20.0600

800

1000

1200

ρ(g

/L)

x1

n-Hexane(1)

Cyclohexane(1)

Chlorobenzene(1)

Fig. 15. Densities at ambient conditions. Experimental data: – n-hexane(1)–n-octane(2) (298.15 K) [26], – cyclohexane(1)–n-octane(2) (303.15 K) [27], –chlorobenzene(1)–n-octane(2) (298.15 K) [28]. Predictions: SAFT + Cubic – solidlines, PC-SAFT – dashed lines.

I. Polishuk / J. of Supercritical F

0.0 .2 .4 .6 .8 1.0900

1000

1100

1200

1300W

(m/s

)

x1

n-Hexane(1)

Cyclohexane(1)

Chlorobenzene(1)

Fig. 16. Sound velocities at ambient conditions (for legend see Fig. 15).

1.0.8.6.4.20.0.5

.8

1.1

1.4

1.7

κ s(G

Pa-1

)

n-Hexane(1)

Cyclohexane(1)

Chlorobenzene(1)

F

Spyitn

tbosrw

4

sctieiewncqr

[

[

x1

ig. 17. Isothermal compressibilities at ambient conditions (for legend see Fig. 15).

AFT + Cubic in predicting densities and sound velocities should beointed out. These results indicate that this model is supposed toield robust estimation of the very important properties such as thesothermal and isentropic compressibilities as well (unfortunatelyhese data for the asymmetric systems under consideration haveot been published yet).

And, finally, Figs. 15–17 depict the ambient condition proper-ies of three additional mixtures of n-octane. As seen, althoughoth models yield relatively precise estimations of the densities,nce again, the differences in predicting the auxiliary properties areignificant. In particular, SAFT + Cubic yields satisfactorily accurateesults for the sound velocities and the isentropic compressibilities,hich, unfortunately, is not a case of PC-SAFT.

. Conclusions

The experimentally measured thermodynamic properties of realubstances are defined by various molecular phenomena whichurrently cannot be precisely described even by the most advancedheories. Hence, nearly all recent molecular approaches, includ-ng different variations of SAFT, cannot be recognized as beingntirely theoretical. This is because their molecular parameters typ-cally do not have authentic values but are obtained by fitting thexperimental vapor pressures and other macro-level data. In otherords, at the current level of our understanding of molecular phe-

omena, the EoS models aiming to have a quantitatively accurateharacter unavoidable include certain empirical elements. Conse-uently, a fundamental question that should be asked is: whichatio between the theoretical and empirical constituents of EoS

[

luids 62 (2012) 47– 54 53

models or between the molecular principles and the fitting prag-matics is appropriate?

The current study examines two equations representing two dif-ferent trends in developing EoS models. Derivation of PC-SAFT hasbeen based on analytical solution of the Percus–Yevick molecularapproximation. In other words, the structure of the latter model isfocused on representing molecular simulation rather than exper-imental facts. The theoretical basis of SAFT + Cubic is weaker andit relies on generalizing the regularities exhibited by experimentaldata. At the same time it should be pointed out that in the particularcase of the latter model, the empirical part has a relatively modestnumerical contribution [29]. In addition, SAFT + Cubic is free of thewell-known disadvantages characteristic for PC-SAFT and severalrelated approaches, such as the inability to correlate the criticaland subcritical pure compound data simultaneously and generatingartificial unrealistic phase equilibria.

The paucity of the available sound velocity and other auxiliaryproperty data on supercritical mixtures restricts the comprehen-sive examination of the EoS models to just a few systems. In thecurrent study pure n-octane and its several mixtures have been con-sidered. It has been demonstrated that SAFT + Cubic has a doubtlessadvantage over PC-SAFT in predicting the elevated pressure densi-ties. In addition, SAFT + Cubic appears as a robust estimator of soundvelocities and compressibilities both at low and high pressures,which is not a case of PC-SAFT. Although SAFT + Cubic is capableof only rough estimation of heat capacities of pure n-octane, itsresults are closer to the experimental data than the predictionsof PC-SAFT. Additional research should be performed for drawinggrounded conclusions concerning the accuracy of both equationsin modeling phase equilibria.

Acknowledgments

Acknowledgment is made to the Donors of the American Chem-ical Society Petroleum Research Fund for support of this research,grant N◦ PRF#47338-B6.

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