Standardized Critical Point-Based Numerical Solution of Statistical

Article pubs.acs.org/IECR

Standardized Critical Point-Based Numerical Solution of Statistical Association Fluid Theory Parameters: The Perturbed Chain-Statistical Association Fluid Theory Equation of State Revisited Ilya Polishuk* Department of Chemical Engineering & Biotechnology, Ariel University, 40700, Ariel, Israel S Supporting Information *

ABSTRACT: The current study has aimed at developing an approach replacing the lists of the compound-specific molecular parameters attached to perturbed chain-statistical association fluid theory (PC-SAFT) by the entirely transparent and universal method for their derivation. The proposed approach requires limited data for the numerical solution of the PC-SAFT parameters, namely, the critical constants and the triple point liquid density. Its implementation has necessitated a careful re-evaluation of part of the PC-SAFT universal parameters matrix and some additional revisions. The resulting model appears to be virtually free of several undesired numerical pitfalls characteristic for PC-SAFT. The proposed equation of state (EoS) has been implemented for modeling data of nonpolar substances such as the light compounds, n-alkanes, and 1-alkenes and their mixtures. Using a large experimental database (more than 6000 points) it has been demonstrated that it exhibits remarkable precision in predicting the high-pressure liquid phase densities and sound velocities, with AAD hardly exceeding 3% even in the cases of complex asymmetric mixtures. However, a major drawback of the proposed model is a poor accuracy of predicting the vacuum vapor pressures of heavy compounds away from their critical points.

I. INTRODUCTION The experience shows that various thermodynamic properties of real substances in broad PVT range currently cannot be precisely described even by the most advanced and sophisticated molecular theories. As a result, nearly all theoretically based approaches pretending to have a quantitatively accurate character unavoidable including certain empirical elements since their substance-dependent microlevel molecular parameters are typically obtained by fitting the macro-level experimental data, such as the vapor pressures and the liquid densities. Several versions of the statistical association fluid theory (SAFT) equation of state (EoS) should be recognized as better molecularly grounded because their expressions and the matrixes of universal parameters have been derived on the basis of different theories, such as the Percus−Yevick molecular approximation1 in the case of PC-SAFT.2 Other versions, such as the later and widely implemented form of this model3 increase the empirical constituent by fitting also the universal parameters matrix to the macro-level real compounds data, instead of approximating the molecular theory. Recently Liang et al.4,5 have proposed refitting PC-SAFT’s universal parameters matrix while considering the additional macro-level data, specifically the sound velocities. Remarkably, this novel approach has virtually removed6 a well-known numerical pitfall, namely, the appearance of additional fictitious pure compound critical points.7−12 Another important issue is related to the contribution of the theoretical basis to the predictive capacity of the industrialoriented fluid EoS models. Indeed, the molecular parameters of the advanced SAFT-based equations are usually evaluated considering relatively large and vague experimental databases. However, as indicated by the recent survey of the industrial requirements for thermodynamics and transport properties,13 © XXXX American Chemical Society

the latter practice often leads to lack of standardization and transparent implementations of these models, which currently hinder their wide implementation in industrial simulators. Unlike that, the popular cubic equations typically require just the experimental values of the pure compound critical temperatures, pressures, and the acentric factors. This significant cut of the data input is achieved due to the standardized parametrization of cubic equations at the pure compound critical points.14,15 Unfortunately, rescaling the SAFT’s molecular parameters in that way typically leads to significant deterioration of their accuracy.16−21 Although an attempt to solve this problem by modifying SAFT22 empirically has recently been made,23−30 the pertinent standardized parametrization procedure has been proposed just for the heavy compounds (carbon number ≥18) and the ionic liquids.31,32 The current study is a step toward developing an entirely transparent standardized critical point-based procedure for obtaining the substance-specific molecular parameters of SAFT. At this preliminary stage of research the PC-SAFT3 EoS has been considered. Obviously, in the case of this complicated model the simple analytical expressions for the model’s parameters could hardly be derived. Hence the proposed concept deals with replacing of the tedious and nontransparent fitting procedures by a numerical solution of the clearly defined equations, requiring the minimal amount of the input data similarly to the popular cubic equations. The details of the proposed approach and the pertinent modifications of the PC-SAFT EoS are listed below. Received: July 2, 2014 Revised: August 11, 2014 Accepted: August 21, 2014

A

dx.doi.org/10.1021/ie502633e | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Industrial & Engineering Chemistry Research

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Table 1. Universal Model Constants for Equations 7 and 8

■

i

a0i

a1i

a2i

b0i

b1i

b2i

0 1 2 3 4 5 6

0.880823927666 1.26235042398 −2.88916037036 −0.791682734039 31.4414035626 −67.7739765931 37.6471023573

−0.349731891574 1.06133747189 −9.92662697237 55.1147516007 −158.619888888 237.469601780 −146.917589624

−0.041574194083 −0.828880456022 10.6610090572 −42.2676046130 93.3498157944 −119.982855050 69.3982688833

0.7240946941 2.2382791861 −4.0025849485 −21.003576815 26.855641363 206.55133841 −355.60235612

−0.5755498075 0.6995095521 3.8925673390 −17.215471648 192.67226447 −161.82646165 −165.20769346

0.0976883116 −0.2557574982 −9.1558561530 20.642075974 −38.804430052 93.626774077 −29.666905585

THEORY A. Revision of the PC-SAFT EoS. The residual Helmholtz energy obtained from PC-SAFT2−5 EoS for pure nonpolar compounds can be expressed as follows: Ares = AHS + Achain + Adisp

In the current study four modifications of the original PC-SAFT3 EoS have been performed: (1) Following the idea of Liang et al.,4,5 the matrix of the PC-SAFT’s universal parameters has been re-evaluated. However, this time the model’s capability of simultaneous description of the critical temperatures, pressures, and the liquid densities away from the critical points has been targeted instead of the sound velocities and the vacuum vapor pressures. At this preliminary stage of research the original3 values of b0−2i have been remained unchanged and values of a0−2i have been refitted (see Table 1). Unsurprisingly, all the refitted to the macro-level data versions of PC-SAFT are less accurate in describing the Monte Carlo simulated square well chain data in comparison to the molecularly grounded approach,2 especially in the case of VLE (see Figure 1). At the same time,

(1)

where the hard-sphere, chain, and dispersion contributions are AHS = mRT

4η − 3η2 (1 − η)2

Achain = RT (1 − m) ln

disp

A

(2)

1 − η /2 (1 − η)3

(3)

⎛ ⎜ 2π (ε /k)m2σ 3 I1 = −RNAv ⎜ v ⎜ ⎝ π (ε /k)2 m3σ 3

+

(

8η − 2η 2

vT 1 + m (1 − η)4 + (1 − m)

20η − 27η2 + 12η3 − 2η 4 ((1 − η)(2 − η))2

)

⎞ ⎟ I2 ⎟ ⎟ ⎠ (4)

In eqs 2−4, m is the number of segments and σ is the segment diameter (Å), ε/k segment energy parameter divided by Boltzmann’s constant. As indicated previously, these substance-dependent molecular parameters are not necessarily attached by their genuine microlevel values but are typically fitted to the macro-level fluid data. η is the reduced density given as η=

πNAv 3 md 6v

(5)

where d = σθ and ⎛ 3(ε /k) ⎞ ⎟ θ = 1 − 0.12 exp⎜ − ⎝ T ⎠

Figure 1. Square well chains data. Solid lines, the Monte Carlo simulation based form of PC-SAFT2; dashed lines, PC-SAFT3; dotted lines, approach of Liang et al.;4,5 dashed-dotted-dotted lines, the proposed approach. Points, Monte Carlo simulations.33,34

(6)

I1 and I2 are the analytical functions representing the integrals of the radial distribution function in first and second order perturbation terms:

the departure from the molecular theory does not necessarily results in deterioration in predicting of the second and the third virial coefficients of real substances. In particular, although the approach of Liang et al.4,5 on one hand tends to overestimate the second virial coefficient and on the other one yields the unrealistic negative values for the third one at the high temperature range, the approach of Gross and Sadowski3 estimates these data sufficiently accurately (see Figures 2 and 3). In order to preserve the general robustness of the model, the

i=6

⎛ m−1 m − 1 m − 2 ⎞⎟ i I1 = ∑ ⎜a0i + a1i + a 2i η ⎝ ⎠ m m m i=0 i=6

I2 =

⎛

∑ ⎜⎝b0i + i=0

m−1 m − 1 m − 2 ⎞⎟ i b1i + b 2i η ⎠ m m m

(7)

(8)

where a0−2i and b0−2i are the universal model parameters. B



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Figure 2. 2nd virial coefficients of n-alkanes predicted by PC-SAFT3 (solid black lines) the approach of Liang et al.4,5 (dotted-dashed red lines) and by the proposed approach (dashed blue lines). Points, experimental data.35

Figure 3. 3rd virial coefficients of n-alkanes predicted by PC-SAFT3 (solid black lines) the approach of Liang et al.4,5 (dotted-dashed red lines) and by the proposed approach (dashed blue lines). Points, experimental data.35

universal parameters matrix take place near the absolute zero6 and, hence, practically cannot affect modeling of fluids. (2) The arguments of the general robustness of the model have led to its two additional modifications. In particular, it can be seen that eq 6 does not approach zero at the infinity

virial coefficients (and especially the second one) have served as an additional benchmark in the current re-evaluation of the universal parameters matrix. Besides that, Figure 4 demonstrates that similarly to the approach of Liang et al.,4,5 the additional unrealistic pure compound critical points generated by the current C



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Table 2. Values of Parameters Obtained by Solving a System of Equations 21,22−24

Figure 4. Global map of temperatures of the additional unrealistic pure compound critical points of nonpolar compounds generated by the proposed approach.

temperature limit. In addition, PC-SAFT attached by eq 6 fails to produce an acceptable Joule inversion curve,36 whose significance has been summarized by Deiters and De Reuck.37 However, it can be demonstrated that replacing eq 6 by the expression of Cotterman et al.:38 θ=

1 + 0.2977(k /ε)T 1 + 0.33163(k /ε)T + 0.0010477(k /ε)2 T 2

(9)

allows correct prediction of this curve. In addition, this expression is completely consistent with the theoretical structure of the SAFT-Mie model. Consequently, implementation of eq 9 instead of eq 6 was the second modification of PC-SAFT introduced in this study. It should however be pointed out that this modification has a minor effect on the results. (3) Another robustness issue related to the extreme conditions is the nonphysical prediction of the negative heat capacities and the isotherms intersections at very high pressures. Remarkably, in the case of PC-SAFT modified by Liang et al.,4,5 the latter phenomena takes place already in the range of

compound

m

ε/k [K]

σ [Å]

δ

Ne Ar Kr Xe N2 CO CO2 CH4 C2H6 C3H8 n-C4H10 n-C5H12 n-C6H14 n-C7H16 n-C8H18 n-C9H20 n-C10H22 n-C11H24 n-C12H26 n-C13H28 n-C14H30 n-C15H32 n-C16H34 n-C19H40 1-C6H12 1-C7H14 1-C8H16 1-C9H18 1-C10H20 1-C12H24 1-C14H28 1-C16H32 C8H18 (isooctane)

0.850615 0.940596 0.973635 1.01105 0.998798 0.998329 2.03351 1.00082 1.56358 2.41440 2.48262 3.06424 3.51081 4.07032 4.45475 4.85100 5.27013 5.71457 6.01209 6.64356 7.04293 7.31716 7.58776 8.11235 3.50560 3.61314 3.74981 5.11730 4.90374 5.42390 6.40155 6.90376 3.10606

35.2369 115.643 158.300 215.678 94.3513 99.4738 163.491 142.508 185.392 184.368 209.446 212.528 218.238 220.494 225.287 229.271 232.262 234.681 238.240 237.896 240.579 243.431 246.366 254.171 216.808 228.621 238.091 225.327 237.114 245.327 246.301 251.931 244.773

3.02407 3.50130 3.69107 3.95851 3.61590 3.64373 2.81786 3.74760 3.57406 3.39176 3.65040 3.62421 3.65575 3.63515 3.67868 3.70469 3.72188 3.73086 3.76983 3.73387 3.75151 3.78620 3.81901 3.94690 3.57773 3.72709 3.84710 3.59937 3.77234 3.86745 3.84386 3.91434 4.14356

1.06918 1.10273 1.11662 1.12802 1.10537 1.07880 1.17358 1.12673 1.16657 1.15188 1.15976 1.16385 1.16091 1.16631 1.17934 1.18722 1.20260 1.21409 1.22522 1.24179 1.26367 1.27171 1.28604 1.30163 1.13535 1.15178 1.15197 1.19855 1.18238 1.22450 1.22819 1.24588 1.17282

Figure 5. Vapor pressures. Points, data obtained from the DIPPR database. Lines, predictions of the proposed approach. D



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Figure 6. Densities of the light compounds. Points, literature data.50−55 Lines, predictions of the proposed approach.

Achain = RT ∑ xixj(1 − mij)ln[gij(dij)hs ]

10−20 kbar, which can affect estimation of densities at much lower pressures as well. In order to address this problem, in the current approach eq 2 has been replaced by the previously proposed expression:39 AHS = mRT

i,j

where the segment radial distribution function given as ⎛ diidjj ⎞2 3diidjjζ2 1 ⎟⎟ gij(dij) = + + 2⎜⎜ 1 − ζ3 (dii + djj)(1 − ζ3)2 ⎝ dii + djj ⎠

4η − 3η2 (1 − η)3/2 (1 − η /θ 3)1/2

hs

(10)

(4) Since the flexibility introduced by the unlike size parameter in original PC-SAFT3 mixing rules seems to be insufficient,25,29 various options proposed by Schnabel et al.40 have been considered. At the current preliminary stage of research, the following mixing rules25 have been implemented: AHS = RT

ζ2 2 (1 − ζ3)3

σ=

σ 3(∑i ximii)2

3

(∑i ximii)2

∑ ximi i

∑ ximiidiik i

(15)

∑i ∑j xixjmiimjjσij 3

(11)

m= πNav 6v

∑i ∑j xixjmiimjjσij 3(ε /k)ij

ε/k =

where ζk =

(14)

In addition,

ζ2 3 m ⎛ 3ζ1ζ2 ⎜⎜ + ζ0 ⎝ 1 − ζ3 ζ3(1 − ζ3)2

⎞ d3(ζ − 1) ⎛ζ 3 ⎞ 3 + ⎜ 22 − ζ0⎟ln[1 − ζ3]⎟⎟ 3 3 ⎝ ζ3 ⎠ ⎠ ζ3σ − d

(13)

(16)

(17)

where

(12)

(ε /k)ij = (1 − kij) (ε /k)ii (ε /k)jj

The chain contribution for mixtures is E

(18)



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Figure 7. Sound velocities in the light compounds. Points, literature data.50−55 Lines, predictions of the proposed approach.

mij = (1 − lij)

σij =

identified also by Segura et al.42 The pertinent system of four equations is

mii + mjj 2

(19)

⎛ ∂P 2 ⎞ ⎛ ∂P ⎞ ⎜ ⎟ =⎜ 2 ⎟ =0 ⎝ ∂v ⎠Tc ⎝ ∂ v ⎠ Tc

σii + σjj 2

(20)

It should be pointed out that introduction of the unlike size adjustable parameter l12 has been found more sensible in eq 19 instead of eq 20. This is because the influence of this parameter in eq 19 is similar to its effect in the classical mixing rules of the cubic equations (moving the phase envelopes sideways),41 while in eq 20 l12 performs analogously to k12 (changing sizes of the phase envelopes). At the same time, since the nonzero l12 are typically required for correcting the balance between VLE and LLE in complex systems, this parameter has not been adjusted in the current study. B. Proposed Method for Calculating the Model’s Parameters. According the proposed critical point-based approach, four parameters, namely, m, σ, ε/k, and δ (the critical volume displacement, a ratio between the EoS’s and the experimental values) are solved numerically. It should be pointed out that instead of δ, the EoS’s critical volume can be defined as a stand-alone parameter. However, since the δ values vary less than Vc, introduction of δ increases the range of applicability of the initial estimations. Remarkably, the substance-dependent character of δ has been recently

vc,EoS = δvc

Pc,EoS = Pc

(21,22) (23)

ρL,EoS = ρL,experimental

at the triple point

(24)

As seen, similarly to the popular cubic equations, the proposed approach requires limited data, namely, the critical constants and the triple point liquid densities, which can be obtained from the databanks such as DIPPR. In some cases the model’s accuracy can be improved by taking other liquid density points. However, this practice involves elements of fitting, which has not been implemented in the current study. It should also be pointed out that the system of eqs 21,22−24 has no solution for hydrogen and helium, and an ordinary data fitting procedure should be performed in the particular cases of these two compounds. The major difficulty of the proposed approach in comparison even to the advanced cubic equations such as Trebble-BushnoiSalim43,44 or GEOS3C45−48 is a need to provide the divergent initial estimations for calculating the parameters of various compounds. F



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Figure 8. Densities of n-alkanes. Points−experimental data56−64 (the saturated phase densities have been obtained from DIPPR). Lines, predictions of proposed approach.

However, the same initial approximations can be used for calculating parameters of related substances, such as n-alkanes or the light compounds. In addition, computation times of the proposed approach are similar to the original PC-SAFT and are longer in comparison to cubic equations. However, this difference typically becomes significant in the complex cases, such as the multiphase equilibria in multicompound systems. In attempt to make the proposed approach entirely transparent,13 a detailed description of the numerical solution of eqs 21,22−24 written in Mathematica 7 is provided in the Supporting Information. The values of parameters obtained by solving a system of these equations for compounds considered in this study are listed in Table 2. Additional Mathematica 7 codes can be obtained from the author by request.

Generally speaking, there is a price to pay for the convenience of the critical-point based methods of evaluating the EoS parameters. This price is typically expressed in a poor accuracy of estimating the vacuum vapor pressures away from the critical points. In the case of the cubic equations, this problem is addressed by attaching these models with the α-functions.14,15 However, at the current preliminary stage of research, this alternative requiring the additional adjustable parameters49 has not been considered. As seen (Figure 5a), the proposed approach generates adequate predictions of the relatively short vapor pressures loci of light compounds. However, as the vapor pressure curves get longer, the accuracy in the vacuum range deteriorates (Figure 5b,c). Remarkably, the vapor pressures of heavy 1-alkenes are predicted more accurately than of heavy n-alkanes (Figure 5d). It should also be pointed out that in spite of the fact that the vacuum range the percentage deviations can reach orders of magnitude, their actual values are in fact not that big. The major consequence of this drawback is an inaccurate estimation of the heavy compounds boiling temperatures, while its influence on modeling the high pressure data is negligible. Figure 6 indicates that the proposed approach is particularly successful in predicting the light compounds densities in a wide P−T range. However, it tends to overestimate their sound velocities above 3000 bar (Figure 7), a result characteristic also for the previously developed SAFT + Cubic EoS.24

■

RESULTS The current approach differs from most EoS models proposed at this time. As indicated above, on one hand, it requires initial estimations for the numerical solution of its parameters, which makes it more complicated in comparison to cubic equations. On the other hand, unlike other SAFT-based approaches, it does not involve sophisticated fitting procedures to large experimental databases of particular compounds. Consequently, a direct comparison of their accuracies could be considered as inequitable, and it has not been performed in this study. G



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Figure 9. Sound velocities in n-alkanes. Points, experimental data.56,57,65−67 Lines, predictions of the proposed approach.

Figure 10. Sound velocities in 1-alkenes. Points, experimental data.65,66,68,69 Lines, predictions of the proposed approach.

Information). The data of Houck56 for n-pentane provide an interesting insight on the performance of the proposed approach at the extreme pressure conditions. As seen, its

The proposed approach exhibits even more impressive accuracy in predicting densities and sound velocities of n-alkanes (see Figures 8 and 9 and Figures S3−S10 of the Supporting H



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Figure 11. Isochoric and isobaric heat capacities of methane and n-hexane. Points, literature data.53,59,70 Lines, predictions of the proposed approach.

Figure 12. VLE in methane−n-alkanes. Points, experimental data.71−77 Lines, modeling of the proposed approach.

n-nonadecane. In the case of 1-alkenes, the similar deterioration of accuracy is observed already at 1-hexadecene, although the sound velocity data of the lighter members of the series are predicted particularly accurately (Figure 10). These results draw another limitation of this critical-point based approach. As seems, it can hardly stay precise in the cases of heavy compounds whose critical constants become imaginary. Figure 11 indicates that the proposed approach is capable of qualitative and sometimes even quantitative description of heat capacities, in exception of the near critical isochoric heat capacity data. Generally speaking, predictions of these properties are less accurate in comparison to densities and sound

predictions of densities are surprisingly accurate in the entire fluid phase pressure range up to 25 000 bar and the notable overestimation of sound velocities starts to appear only above 15 000 bar. As seen, the current generalized approach can be successfully applied for predicting the high pressure data of very different substances such as the light noble gases on one hand and the heavy compounds such as n-hexadecane on the other one. However, further increase of the compound’s molecular weight starts affecting its reliability. As seen, (Figure S10 in the Supporting Information) the proposed EoS tends to overestimate the densities and underestimate the sound velocities of I



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Figure 13. VLE in other mixtures. Points, experimental data.78−85 Lines, modeling of the proposed approach.

Figure 14. High pressure single phase properties of asymmetric binary systems. Points, experimental data.86−88 Lines, predictions of the proposed approach.

properties of the liquid phase are currently available just for a limited number of such systems. Among these systems are nitrogen (1)−n-octane (2), methane (1)−n-octane (2), methane (1)−n-decane (2), methane (1)−n-hexadecane (2), carbon dioxide (1)−n-hexadecane (2) and the ternary system methane (1)−carbon dioxide (2)−n-hexadecane (3). In addition, the high pressure liquid phase densities and sound velocities have been reported for several symmetric systems.

velocities, a result which is typical also for the previously proposed SAFT + Cubic EoS. Although modeling of pure compounds is an important test for the SAFT-based EoS models, their main practical intention is predicting mixtures. In this respect the asymmetric mixtures of light and heavy compounds present a special interest. Unfortunately, the comprehensive experimental data including the phase equilibria and the high pressure thermodynamic J



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Figure 15. High pressure single phase properties of the ternary system CH4(1)−CO2(2)−n-C16H34(3). Points, experimental data.89,90 Lines, predictions of the proposed approach.

Table 3. AAD% of the Proposed Approach in Predicting High Pressure Data of Mixtures in Single-Phase Region systems N2(1)−CH4(2)−CO2(3) N2(1)−CH4(2) N2(1)−n-C8H18 (2) N2(1)−n-C8H18(2) CH4(1)−CO2(2) CH4(1)−n-C3H8(2) CH4(1) − n-C8H18(2) CH4(1)−n-C10H22(2) CH4(1)−n-C16H34(2) CO2(1)−n-C16H34(2) CH4(1)−CO2(2)−n-C16H34(3) CH4(1)−CO2(2)−n-C16H34(3) CH4(1)−CO2(2)−n-C16H34(3) CH4(1)−CO2(2)−n-C16H34(3) n-C6H14(1)−n-C12H26(2) n-C6H14(1)−n-C16H34(2) n-C6H14(1)−n-C16H34(2) n-C7H16(1)−n-C12H26(2) n-C8H18(1)−1-C8H16(2) n-C8H18(1)−iso-C8H18(2) 1-C8H16(1)−iso-C8H18(2) n-C8H18(1)−1-C8H16(2)−iso-C8H18(3) n-C8H18(1)−n-C12H26(2) n-C8H18(1)−n-C12H26(2) n-C8H18(1)−n-C16H34(2) n-C10H22(1)−n-C14H30(2) n-C10H22(1)−n-C16H34(2) n-C10H22(1)−n-C14H30(2)−n-C16H34(3) n-C12H26(1)−n-C16H34(2) n-C12H26(1)−n-C16H34(2)

properties density sound velocity sound velocity density density sound velocity sound velocity density sound velocity sound velocity density sound velocity isochoric compressibility isothermal compressibility density density sound velocity sound velocity density density density density density sound velocity sound velocity density sound velocity density density sound velocity

T range (K)

P range (bar)

no. of points

k12

AAD%

refs

323.15−573.15 170−400 293.15−373.15 293.15−373.15 323.25−473.75 262.75−413.45 293.15−373.15 293.15−393.15 292.15−413.15 293.15−333.15 303.15−363.15 313.15−393.15 313.15−393.15

199.4−999.3 1.015−302.5 250−1000 250−1000 33.85−692.43 100−700 250−1000 1−1400 63−662 22−556 40−400 40−700 40−700

271 278 144 144 132 265 78 439 396 404 389 397 397

0/0/0 0 0.050 0.050 0 0 0 −0.022 −0.017 0.080 0/−0.017/0.080 0/−0.017/0.080 0/−0.017/0.080

0.8960 0.7504 0.6862 0.8153 1.521 1.831 0.9332 0.9308 2.708 3.565 1.720 2.049 4.179

91 92 87 87 93 94 94 86, 95 88 88 89 90 90

313.15−393.15

40−700

397

0/−0.017/0.080

3.038

90

298.1−373.16 298.09−373.24 298.15−433.15 292.85−318.31 298.15−373.15 298.15−373.15 298.15−373.15 298.15−373.15 298.2−373.17 298.15−433.15 298.15−433.15 298.15−358.15 298.15−433.15 298.15−358.15 298.15−358.15 298.15−433.15

1−4551 1−5028 1−1001 152−1013.2 1−4122 1−4996 1−2910 1−4146 1−5055 1−1001 1−1001 1.01−4301.04 1−1000 1.01−3172.6 1.01−3313.02 1−1001

24 159 448 419 54 50 43 43 38 91 88 104 129 96 84 93

0 0 0 0 0 0 0 0/0/0 0 0 0 0 0 0/0/0 0 0

1.342 0.8473 1.888 2.113 1.366 2.221 2.479 2.178 0.9347 2.115 1.969 0.7551 2.259 0.7090 0.6046 2.191

96 97 88, 98 99 100 100 100 100 96 101 98 104 102 104 104 103

K



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Figures 12 and 13 depict the capabilities of the proposed approach in modeling phase equilibria in mixtures and Figures 14 and 15, predictions of high pressure liquid phase thermodynamic properties. In addition, Table 3 lists their absolute average deviations. As seen, the proposed approach can yield satisfactory perditions of VLE in the systems such as methane (1)−n-propane (2), methane (1)−n-octane (2), methane (1)−carbon dioxide (2), carbon monoxide (1)−nhexane (2)/n-decane (2) in a wide range of pressures without adjusting the binary parameters. However, in the cases of more challenging systems, such as methane (1)−n-decane (2)/nhexadecane (2), carbon dioxide (1)−n-hexadecane (2), nitrogen (1)−n-alkanes (2), and argon (1)−n-alkanes (2), the fitting of the binary parameter k12 becomes desirable. It should be pointed out that being applied in the entirely predictive manner, the current approach tends to underestimate the range of the high temperature VLE and overestimate the low-temperature ones. Remarkably, the original PC-SAFT3 treats the high temperature VLE a bit more accurately due to its overestimation of the pure compound critical pressures. Moreover, similarly to its source model, the proposed approach establishes a less accurate balance between VLE and LLE in comparison with SAFT + Cubic. Generally speaking, it seems that in the case of VLE, the complexity of some SAFT-based models does not necessarily offer significant advantages over the simplicity of Cubic equations,105 which requires further development of their mixing rules. In this respect the recently proposed106 combination of the PC-SAFT with the modified UNIFAC (Dortmund)107 seems particularly promising. The advantages of SAFT in the general and of the proposed generalized approach in particular become evident in the case of the high pressure liquid phase properties. Surprisingly, it is often superior over SAFT + Cubic in predicting these data. In particular, its estimation of sound velocities in the nitrogen (1)−n-octane (2) system is nearly precise. Similarly to SAFT + Cubic, the current approach also tends to underestimate the dependence of sound velocities on pressures in the very asymmetric mixtures of n-hexadecane (Figures 14 and 15). Nevertheless, its overall AAD% for densities and sound velocities even for these complex systems are still particularly low and do not exceed 4% (see Table 3). Consequently, the predictions of the compressibilities in the ternary system ternary system methane (1)−carbon dioxide (2)−n-hexadecane (3) can be considered as satisfactorily accurate as well. Unsurprisingly, the accuracy of predicting the more symmetric systems is higher and the AAD for both the high pressure densities and the sound velocities typically do not exceed 3%.

Following the concept of the Cubic equations, a substantial reduction of the data required for derivation of the model’s parameters could be achieved by implementing the standardized critical-point based approaches. The current study has aimed at developing a preliminary version of such an approach. While the Soave and Peng−Robinson equations require 3 data, namely, the critical temperature, pressure, and the acentric factor, the proposed approach basically also requires just 3 data, the critical temperature, pressure and triple point liquid density. However, currently it is not straightforward as those successful models, since its numerical modus operandi requires the changeable for different groups of compounds initial estimations. Implementation of the proposed approach to PC-SAFT has necessitated re-evaluation of the part of its universal parameters matrix. Keeping in mind the risks related to this practice, special attention has been played to the appropriate representation of the real substances virial coefficients and especially the second one. At the same time, the revised model appears to be virtually free of an undesired numerical pitfall, namely, predicting the additional unrealistic pure compound critical points. Additional modifications of PC-SAFT addressing other numerical pitfalls such as the isotherms crosses and fixing the Joule inversion curve have been performed as well. The proposed generalized approach has been implemented for modeling data of nonpolar substances such as light compounds, n-alkanes and 1-alkenes and their mixtures. Using the large experimental database (more than 6000 points) it has been demonstrated that it exhibits remarkable precision in predicting the high pressure liquid phase densities and sound velocities, with AAD hardly exceeding 3% even in the cases of complex asymmetric mixtures. At the same time, the proposed model shows a characteristic also for the original PC-SAFT accuracy in modeling the high pressure VLE and sometimes requires fitting to the experimental data of asymmetric systems. This result indicates that further development of the mixing rules should be considered. However, the major drawback of the proposed approach is a poor prediction of the vacuum vapor pressures of heavy compounds away from their critical points. Although this weakness does not affect the model’s accuracy at the high pressures, ways for its addressing should be investigated as well. Performance of the proposed model in predicting other compounds, such as the aromatics and the alkanols will be considered in the forthcoming studies.

CONCLUSIONS From the industrial viewpoint13 is seems sensible that the transformation of SAFT from a Redlich−Kwong to a Soave or a Peng and Robinson form108 should finally result in an effective model whose intricacy will not largely exceed the implementation difficulty of those popular Cubic equations. Since the current progress of computation power substantially reduces the negative impact of the sophisticated EoS expressions, the major challenge seems to be a replacement of the lists of the compound-specific molecular parameters attached to various versions of SAFT by the entirely transparent and universal methods for their derivation. Obviously, the obligation of transparence and universality presumes the minimal possible referring to the experimental data.

Detailed explanations of the Mathematica 7 procedure for the numerical solution of the revised PC-SAFT’s parameters according to the proposed approach and Figures S1−S10 depicting the predictions of the high pressure densities and sound velocities of additional compounds. This material is available free of charge via the Internet at http://pubs.acs.org.

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