Implementation of SAFT + Cubic, PC-SAFT, and SoaveâBenedict

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Implementation of SAFT + Cubic, PC-SAFT, and SoaveBenedictWebbRubin Equations of State for Comprehensive Description of Thermodynamic Properties in Binary and Ternary Mixtures of CH4, CO2, and n-C16H34 Ilya Polishuk* Department of Chemical Engineering and Biotechnology, Ariel University Center of Samaria, 40700, Ariel, Israel ABSTRACT: The binary and ternary mixtures considered in the present study are the most asymmetric ones for which the sound velocity and compressibility data are currently available, and their complete description, including auxiliary and phase equilibria properties, puts a challenging test for Equation of State (EoS) models. The recently proposed SAFT + Cubic EoS passes this test relatively successfully (AAD% for the single phase properties less than 6%), proving its robustness as a predictive tool. PC-SAFT appears to be the less reliable estimator of the data, whose AAD% might exceed 22%. Although the overall precision of the SoaveBenedictWebbRubin (SBWR) model in predicting thermodynamic properties is better than of many popular EoS, it is not as advantageous as SAFT + Cubic (AAD% for the single phase properties less than 12%). The major difficulty of SBWR is modeling phase equilibria in asymmetric systems due to the prediction of the unrealistic U-type LLE critical loci. Nevertheless, the significant practical potential of SBWR for industrial applications should not be neglected, and this model deserves therefore further evaluation and development.

’ INTRODUCTION A complete description of mixtures, including volumetric, auxiliary and phase equilibria properties by analytical Equation of State (EoS) models is receiving a growing attention in literature113 due to its practical importance in science and industry. The latter task becomes particularly challenging in the case of asymmetric mixtures at high pressures. Unfortunately, the pertinent information on the properties such as heat capacities,14 sound velocities,15,16 isothermal and isentropic compressibilities are scarce. One of the most valuable contributions to this currently small data collection is the comprehensive experimental investigation of the high pressure thermodynamic properties in the asymmetric ternary system CH4(1)CO2(2)n-C16H34(3)17,18 and sound velocities in two constituent binary systems CH4(1)n-C16H34(2) and CO2(1)n-C16H34(2).19 These data present a severe test for different EoS models.19 The current study applies this test to the recently proposed SAFT + Cubic EoS,20,21 a model that intends addressing an issue of simultaneous prediction of various thermodynamic properties with a minimal number of pure compound adjustable parameters. Previously,22 SAFT + Cubic has been compared with the successful modification of the BenedictWebbRubin Equation of State proposed by Soave23 (SBWR) while considering several relatively symmetric mixtures. An accuracy of SBWR in modeling asymmetric systems involving VLE, LLE, and VLLE presents doubtless interest as well. Unlike the entirely empirical SBWR and the semiempirical SAFT + Cubic, the structures of the theoretically advanced SAFT models and the related approaches2428 are focused on representing the molecular simulation rather than the experimental results. As a consequence, they are typically unable to fit the pure compound critical and subcritical data simultaneously.2933 r 2011 American Chemical Society

Figure 1. Sound velocities in n-hexadecane predicted by PC-SAFT41 and Trebble-Bishnoi-Salim45 EoS (TBS). One-phase experimental data:46 (black left-half circle) 313.15 K, (blue right-half circle) 373.15 K, (red top-half circle) 433.15 K; (open circle) saturated data.47 Predictions: (solid lines) PC-SAFT; (dashed lines) TBS.

In addition, they might be affected by the undesired numerical pitfalls, such as predicting the unrealistic phase equilibria and the nonphysical negative values of heat capacities.3440 Nevertheless, these models are particularly important for both science and industry. PC-SAFT41 is one of the most successful and widely used theoretically based fluid phase equations. It is affected by the undesired numerical phenomena at extreme conditions, typically outside the range of most practical implementations. Therefore this model is included in the current study. Received: August 30, 2011 Accepted: November 9, 2011 Revised: November 3, 2011 Published: November 09, 2011 14175

dx.doi.org/10.1021/ie201952n | Ind. Eng. Chem. Res. 2011, 50, 14175–14185

Industrial & Engineering Chemistry Research

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Another group of models having significant practical value is the family of Cubic Equations of State.24,25,42,43 Previously20,21 it has been demonstrated that the popular equation of Peng and Robinson44 can be characterized by a poor accuracy in predicting volumetric and auxiliary properties at high pressures comparing to various SAFT models. The more sophisticated four-parameter Cubic EoS of Trebble, Bishnoi and Salim45 (TBS) provides better volumetric predictions in the compressed liquid states and critical isotherms than the EoS of Peng and Robinson. Figure 1 depicts the prediction of sound velocities of pure n-C16H34 yielded by TBS and PC-SAFT (the results of SAFT + Cubic and SBWR have been presented previously22). As seen, TBS is a particularly inaccurate estimator of the data. This result should be explained by the inappropriate estimation of the infinity pressure densities and, consequently, the high pressure liquid compressibilities characteristic for the Cubic equations.21 Thus, despite that these equations might yield accurate modeling of particular properties, such as various phase equilibria, they are unlikely to be the harsh challengers of SAFT + Cubic in comprehensive description of thermodynamic properties in the entire thermodynamic phase space. Therefore Cubic equations are not considered in the current study.

gij ðdij Þhs ¼

and Adisp where disp ao1

disp

ao2

a v þ c

ð2Þ

∑i

where: s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε ε ε ¼ ð1 kij Þ k ij k ii k jj

σ ij ¼ ð1 lij Þ

∑i

ð3Þ

xi mii diik

ð4Þ where ε is the intersegment interaction’s dispersion energy and k is the Boltzmann’s constant. A

¼ RT

∑i, j xi xjð1 mij Þ ln½gijðdijÞ

hs

∑i xi mii

ð10Þ

ð11Þ

ð12Þ

and:

NAv is the Avogadro’s number, m is the effective number of segments, σ is the Lennard-Jones’s segment diameter and 0 1 k 1 þ 0:2977 T B C ε ii C dii ¼ σ ii B 2 C B @ A k k 1 þ 0:33163 T þ 0:0010477 T2 ε ii ε ii

chain

σ3

∑∑

where πNav ζk ¼ 6V

∑i ∑j

ε pffiffiffiffiffiffiffiffiffiffiffi xi xj mii mjj σij 3 k ij

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u pffiffiffiffiffiffiffiffiffiffiffi u xi xj mii mjj σ ij 3 u 3 i j σ ¼u t xi mii

where A is the Helmholtz free energy, v is the molar volume, a and c are parameters of the cohesive correction term. m 3ζ1 ζ2 ζ2 3 þ ζ0 1 ζ3 ζ3 ð1 ζ3 Þ2 !sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ζ2 3 d3 ðζ3 1Þ þ 2 ζ0 ln½1 ζ3 ζ3 σ 3 d 3 ζ3

ð7Þ

pffiffiffi 3 2 ½ 8:5959ζ3 6:1344ζ3 2 3:87882ζ3 3 þ 25:3316ζ3 4 ¼ π ð8Þ pffiffiffi 3 2 ½ 1:9075ζ3 þ 13:4675ζ3 2 40:5171ζ3 3 þ 39:1711ζ3 4 ¼ π ð9Þ

ε ¼ k

ð1Þ

AHS ¼ RT

0 1 disp ε ! B ao2 ε B disp 2Achain C k C C 1 þ HS Ba þ ¼ mR k @ o1 T A A

ð6Þ

The mixing rules are:

’ THEORY The detailed explanation of SAFT + Cubic and its mixing rules have been provided in the previous studies.20,22 For nonassociating compounds this EoS is given as follows: Ares ¼ Ahs þ Adisp þ Achain

3dii djj ζ2 1 þ 1 ζ3 ðdii þ djj Þð1 ζ3 Þ2 !2 dii djj ζ2 2 þ 2 dii þ djj ð1 ζ3 Þ3

a¼

σii þ σjj 2

∑i ∑j ∑k xi xjxk aijk

where aijk ¼

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 a a a ii jj kk

ð13Þ ð14Þ

ð15Þ

In the particular case of binary mixtures eq 14 is given as pffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ ðx3 a11 þ 3x2 ð1 xÞ 3 a11 2 a22 pffiffiffiffiffiffiffiffiffiffiffiffiffi þ 3xð1 xÞ2 3 a11 a22 2 þ ð1 xÞ3 a22 Þ ð16Þ In addition m¼

ð5Þ c¼

where the segment radial distribution function given as 14176

∑i xi mi ∑i xi ci

ð17Þ ð18Þ

dx.doi.org/10.1021/ie201952n |Ind. Eng. Chem. Res. 2011, 50, 14175–14185


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Table 1. Pure Compound Parameters of SAFT + Cubic adjustable parameter

parameters calculated using eqs 1921

m

compound n-C16H34

3.3830

c [L/mol]

a [bar-mol/L]

σ [Å]

ε/k [K]

0.58789

147.20

5.2040

563.31

adjustable parameters m

compound

parameters calculated using eqs 20 and 21

c [L/mol]

a [bar-mol/L]

σ [Å]

ε/k [K]

CH4

1

0.117981

1.40838

3.68166

132.054

CO2

1.77

0.058217

2.8997

2.93263

216.743

Table 2. Pure Compound Parameters of SBWR compound

CH4

CO2

n-C16H34

Zc

0.290055

0.27208

0.2337

ω

0.0115

0.215

0.7174

Figure 3. Critical loci in the system CH4(1)CO2(2). Experimental data:56,57 (b). Predictions: (black solid lines) SAFT + Cubic; (red dashed lines) SBWR; (blue dotted-dashed lines) PC-SAFT.

densities. The remaining four parameters (c, a, σ, and ε/k) are not adjustable but are obtained by solving a system of four equations: c ¼ 1:6049 þ 1:3440ðσ 109 Þ 0:3943ðσ 109 Þ2 þ 0:0417ðσ 109 Þ3 ∂P ¼ ∂v Tc

∂P2 ∂2 v

! Tc

¼ 0

ð20Þ vc, EOS ¼ 1:1vc, experimental

Pc, EOS ¼ Pc, experimental

Figure 2. VLE in the system CH4(1)CO2(2). Experimental data:53,54 (right black-half circle) 230 K, (left blue-half circle) 250 K, (top red-half circle) 270 K. Predictions: (solid lines) SAFT + Cubic; (dashed lines) SBWR, PC-(dotted-dashed lines) SAFT.

ð19Þ

ð21Þ

In the cases of other compounds, the fitting of both m and c should be performed, while the nonadjustable a, σ, and ε/k are obtained by solving a system of eqs 2021. For the light gases such as CH4 and CO2 the critical volume displacement of 1.065 instead of 1.1 (see eq 20) was selected. All pure compound parameters need to be determined together at once. Table 1 lists the SAFT + Cubic parameters evaluated for the compounds considered in the present study. The detailed description of SBWR was provided in the original reference.23 Its brief overview is given by eqs 2225: Z¼

In this study eqs 1418 are not attached by the binary adjustable parameters. Adjusting SAFT + Cubic to the data of most pure alkanes requires fitting of only one adjustable parameter m to the liquid

Pv RT

B D E F F ¼ 1 þ þ 4 þ 2 1 þ 2 exp 2 v v v v v 14177

ð22Þ



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Table 3. Binary Adjustable Parameters CH4(1) +

CH4(1) +

CO2(1) +

model

parameters

CO2(2)

n-C16H34(2)

n-C16H34(2)

SAFT+Cubic

k12

0

0.0220

0.2585

l12

0

0.0370

0.0050

k12

0

0.0475

l12

0

0.0350

0.0210

k12 l12

0 0

0.0260 0.0200

0.5380 0.1800

EoS

SBWR

PC-SAFT

0.010

where the parameters B, D, E, and F are the generalized functionalities of other parameters, namely S1, S2, and S3, which in turn are given in terms of the critical constants. In the original reference23 only one binary adjustable parameter kij has been provided: S1 ¼

T T

ci cj ð1 þ 1:25ωi Þð1 ∑i ∑j xi xj ð1 kijÞ pffiffiffiffiffiffiffiffiffiffi Pci Pcj

þ 1:25ωj Þ S2 ¼

∑i ∑j

sffiffiffiffiffiffiffiffiffiffiffi Tci Tcj xi xj ð1 kij Þ ð1:25ωi Þð1:25ωj Þ Pci Pcj

ð23Þ ð24Þ

This binary parameter has the same effect as in SAFT and Cubic EoS models: its increase affects the cohesive interactions and, therefore, results in enlargement of the areas of both VLE and LLE phase splits on phase diagrams. S3 has been originally given23 without the binary parameter: S3 ¼

∑i xi Pcici T

ð25Þ

For the fair comparison with SAFT + Cubic in the cases of asymmetric systems, the following modification is proposed: Tc ð1 l12 Þxi xj ð26Þ S3 ¼ Pc ij i j

∑∑

where Tcj Tci þ Pci Pcj Tc ¼ Pc ij 2

ð27Þ

Since l12 is proportional to volume, it influences the results in a manner analogous to other EoS models, namely its negative values increase extend of LLE and decrease VLE. Similarly to SAFT + Cubic applied to the nonpolar compounds, SBWR requires fitting of two adjustable parameters (ω and Zc). The values of these parameters evaluated for the compounds considered in the present study are listed in Table 2. PC-SAFT is implemented without modifications and using the pure compounds parameters provided in the original reference.41

’ RESULTS First, let us consider phase equilibria in three binary systems, CH4(1)CO2(2), CH4(1)n-C16H34(2) and CO2(1) n-C16H34(2). The VLE in the first systems was previously

Figure 4. VLE in the system CH4(1)n-C16H34(2). Experimental data:63 (right black-half circle) 300 K, (blue left-half circle) 330 K, (red top-half circle) 350 K; 64 (pink bottom-half circle) 542.65 K. Modeling: solid lines.

correlated using different EoS models and mixing rules with binary adjustable parameters.4852 Adjusting binary parameters to experimental data removes the predictive character of models. Therefore the use of these parameters should be reduced to the minimal extent. Figure 2 depicts the results yielded by SAFT + Cubic and SBWR for VLE of CH4(1)CO2(2) without the binary parameters. As seen, SAFT + Cubic predicts the data satisfactorily accurate with certain underestimation of y1. The predictions of SBWR and PC-SAFT are substantially less successful: the methane’s concentration is overestimated in both phases, especially in the liquid one. 14178



Figure 5. Predictions of critical loci in the system CH4(1) n-C16H34(2). (Black solid line) SAFT + Cubic; (red dashed lines) SBWR; (blue dotted-dashed lines) PC-SAFT.

Figure 6. VLE in the system CO2(1)n-C16H34(2). Experimental data:69 (right black-half circle) 353 K, (left blue-half circle) 373 K, (top red-half circle) 393 K. Modeling: (solid lines) SAFT + Cubic; (dashed lines) SBWR; (dotted-dashed lines) PC-SAFT.

As known, the absolute liquidliquid immiscibility is characterized by the nearly vertical slopes of phase boundaries. Therefore, in the vicinity of LLE, the VLE bubble pressure lines typically tend to approach the vertical slopes as well. In other words, inaccurate description of the liquid phase compositions usually indicates establishing the inappropriate inter-relation between VLE and LLE (the latter ones might however be interrupted by solidification). In the cases of SBWR and PC-

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Figure 7. Critical loci in the system CO2(1)n-C16H34(2). (O) Experimental data.69 Modeling: (black solid line) SAFT + Cubic; (red dashed lines) SBWR; (blue dotted-dashed lines) PC-SAFT.

Figure 8. LLVE in the system CO2(1)n-C16H34(2). Experimental data:70 O,b,2. Predictions: (black solid lines) SAFT + Cubic; (red dashed lines) SBWR; (blue dotted-dashed lines) PC-SAFT.

SAFT the results for the VLE clearly signify substantial underestimation of LLE. An issue of the possible presence of LLE in the system under consideration has been investigated by Rainwater.55 It has been estimated that, if freezing were suppressed, the metastable LLE were supposed to occur around ∼180 K. As seen (Figure 3), SAFT + Cubic yields a relatively 14179


Industrial & Engineering Chemistry Research Table 4. Absolute Average Deviations

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0 B B BAAD% ¼ B @

properties

¼N ∑ii ¼ 1

1 X EoS i expt 1100C Xi C C N C A

SAFT + Cubic

SBWR

PC-SAFT

71

number of points (N)

densities of CH4(1)CO2(2)

3.364

6.161

5.209

90

densities of CH4(1)CO2(2)n-C16H34(3)18

1.641

3.966

1.077

389

sound velocities in CH4(1)n-C16H34(2)19

3.401

3.200

13.33

396

sound velocities in CO2(1)n-C16H34(2)19 sound velocities in CH4(1)CO2(2)n-C16H34(3)17

3.838 4.122

5.388 6.291

15.36 15.90

404 397

isochoric compressibilites of CH4(1)CO2(2)n-C16H34(3)17

5.094

10.06

22.79

398

isobaric compressibilites of CH4(1)CO2(2)n-C16H34(3)17

5.733

11.36

19.49

398

realistic prediction of the possible metastable LLE while using the zero binary adjustable parameters, which is not the case of SBWR and PC-SAFT. The binary system CH4(1)n-C16H34(2) is a subject of ongoing theoretical investigation.52,5862 Modeling of this asymmetric system is particularly sensitive to parametrization, which compels the use of k12 and l12 (see Table 3). Figures 4 and 5 depict the phase equilibria and the critical lines estimated by the models under consideration. As seen, PC-SAFT and SAFT + Cubic yield the satisfactorily accurate description of the experimental data, while PC-SAFT appears to be more precise. Unlike that, it was impossible to approximate the data acceptably using SBWR. This unfortunate result should be explained by Figure 5, which depicts an unusual U-type shape of the LLE critical loci established by the model. This behavior corresponds to the type transition via the mathematical double point (MDP) of the second kind65 and it was experimentally detected by Schneider66 in certain aqueous solutions exhibiting the closed loops of liquidliquid immiscibility. Such kind of behavior could hardly be expected in the nonassociating system under consideration. According to the author’s unpublished observations, similar numerical artifacts are often obtained if the classical mixing rules are implemented to other multiparameter models, such as different variations of SAFT. However this undesired phenomenon so far has not been detected while using the mixing rule of Mansoori et al.67 In addition, the zero covolume of SBWR might probably have a negative impact on the slope of the LLE critical curve as well. Modeling CO2(1)n-C16H34(2) is a challenging task because all kinds of fluid phase equilibria, namely VLE, LLE, and LLVE, are present in this system. The most successful attempt to describe all those equilibria simultaneously was recently made by Cismondi et al.68 implementing an accurate Cubic EoS and advanced mixing rules with 6 binary adjustable parameters. Figures 68 depict the results of SAFT + Cubic SBWR and PC-SAFT obtained by fitting k12 and l12 (see Table 3) to the critical data. As seen, SAFT + Cubic describes the slope of the LLE critical loci accurately; however, it is less successful in modeling the CO2-rich liquid phase (Lβ in Figure 8) of the three-phase equilibrium. As seen (Table 3), adjustment of PCSAFT to the critical data requires the exceptionally big values of the binary parameters. Nevertheless, the accuracy in modeling the slope of the LLE critical curve and the properties of Lα is still particularly poor. In the case of SBWR once again it was not

Figure 9. Densities of two CH4(1)CO2(2) mixtures. Experimental data:71 (right black-half circle) 225 K, (left blue-half circle) 250 K, (center dot circle) 260 K; (top red-half circle) 275 K; (bold outline circle) 280 K; (bottom pink-half circle) 300; (cross within circle) 350 K. Predictions: (solid lines) SAFT + Cubic; (dashed lines) SBWR; (dotteddashed lines) PC-SAFT.

found possible to correlate the data accurately, mainly due to the unusual phase behavior established by this model for asymmetric systems. The depicted phase diagram is situated shortly after passing the MDP of the second kind, that is, after connecting of the LLE and the VLE critical loci. Since the low temperature part of phase split region has just appeared, Lα and Lβ are located very close to each other (see Figure 8). 14180



Figure 10. Pressure dependence of CP in the mixture CH4(1)CO2(2), x1 = 0.492. Experimental data:72 (right black-half circle) 325.2 K, (left blue-half circle) 302.2 K. Predictions: (solid lines) SAFT + Cubic; (dashed lines) SBWR; (dotted-dashed lines) PC-SAFT.

Figure 11. Sound velocities in the mixture CH4(1) n-C16H34(2), x1 = 0.679. Experimental data:19 (right black-half circle) 313.15 K, (blue lefthalf circle) 340.15 K, (red top-half circle) 373.15 K; (pink bottom-half circle) 413.15 K. Predictions: (solid lines) SAFT + Cubic; (dashed lines) SBWR; (dotted-dashed lines) PC-SAFT.

Figure 12. Sound velocities in the mixture CH4(1)n-C16H34(2), T = 373.15 K. Experimental data for mixtures19 and pseudoexperimental data of pure compounds:73,74 (right black-half circle) 346 bar, (left blue-half circle) 466 bar. Predictions: (solid lines) SAFT + Cubic; (dashed lines) SBWR; (dotted-dashed lines) PC-SAFT; (dotted lines) ideal mixtures.

In what follows, let us consider the available high pressure single phase data. Table 4 lists the absolute average deviations

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Figure 13. Sound velocities in the mixture CO2(1)n-C16H34(2), x1 = 0.53. Experimental data:19 (right black-half circle) 293.15 K, (blue lefthalf circle) 305.15 K, (red top-half circle) 313.15 K; (pink bottom-half circle) 333.15 K. Predictions: (solid lines) SAFT + Cubic; (dashed lines) SBWR; (dotted-dashed lines) PC-SAFT.

Figure 14. Sound velocities in the mixture CO2(1)n-C16H34(2), T = 313.15 K. Experimental data for mixtures19 and pseudoexperimental data of pure compounds:73,75 (right black-half circle) 200 bar, (left bluehalf circle) 460 bar. Predictions: (solid lines) SAFT + Cubic; (dashed lines) SBWR; (dotted-dashed lines) PC-SAFT; (dotteddotted-dashed lines) PC-SAFT with k12 = l12 = 0; (dotted lines) ideal mixtures.

(AAD%) of the models under consideration. Unfortunately, any information on sound velocity in CH4(1)CO2(2) is unavailable to the author. Figures 9 and 10 depict the densities and the dependence of CP on pressure in this binary system. As seen, PCSAFT is advantageous over SAFT+Cubic and SBWR in predicting the latter data. For the two other binary systems, namely CH4(1)n-C16H34(2) and CO2(1)n-C16H34(2), only the sound velocity data19 are available (Figures 1114). Although the predictions of SAFT + Cubic and SBWR do not match the experimental points perfectly, it should be emphasized that the deviations are smaller than in the case of various models previously considered by Ye et al.19 At the same time, PC-SAFT yields substantially less accurate predictions for sound velocities. Remarkable, unlike many mixtures investigated thus far,16 both asymmetric systems under consideration exhibit relatively big positive excess sound velocities (see Figures 12 and 14), in exception of the irregular composition dependence of CH4(1)n-C16H34(2) at high methane’s concentration. At the same time the figures demonstrate that the accuracy in predicting the sound velocities in mixtures is strongly dependent on the results 14181



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Figure 15. Densities, sound velocities, isentropic and isothermal compressibilities in the mixture CH4(1)CO2(2)n-C16H34(3); x1 = 0.11, x2 = 0.44, x3 = 0.45. Experimental data:17,18 (black right-half circle) 313.15 K, (pink bottom-half circle) 333.15 K, (blue left-half circle) 353.15 K, (red top-half circle) 393.15 K. Predictions: (solid lines) SAFT + Cubic; (dashed lines) SBWR; (dotted-dashed lines) PC-SAFT.

(Table 4), SAFT + Cubic exhibits relatively insignificant deviations from the experimental data; the results of SBWR are less accurate; PC-SAFT exhibits the most significant deviations from the sound velocity and the compressibility data but it is superior in predicting the densities. And, finally, the two-phase boundary of another ternary mixture is considered (Figure 16). As seen, SAFT + Cubic and PC-SAFT are not entirely successful in predicting these data, yielding however their qualitatively correct description. It seems, without implementation of ternary adjustable parameters, both models tend to overestimate the extent of the possible “immiscibility holes” phenomena, which is characteristic for the asymmetric ternary CO2 mixtures.76 As one might expect, the predictions of SBWR are affected by the inappropriate estimation of LLE caused by the U-type critical loci. Figure 16. Two-phase boundary of the mixture CH4(1)CO2(2) n-C16H34(3); x1 = 0.03, x2 = 0.85, x3 = 0.12. (O) Experimental data.18 Predictions: (solid lines) SAFT + Cubic; (red dashed lines) SBWR; (blue dotted-dashed lines) PC-SAFT.

for the pure compounds. In particular, it appears (see also Figure 1) that PC-SAFT is a relatively imprecise estimator of sound velocities in pure n-C16H34. This fact explains the disadvantage of PC-SAFT in predicting the mixture data as well. Remarkably, the binary adjustable parameters fitted to the phase equilibria data result in further deterioration of its accuracy in modeling sound velocities (see Figure 14). Figure 15 depicts various thermodynamic properties of a ternary mixture CH4(1)CO2(2)n-C16H34(3). As seen

’ CONCLUSIONS In the current study SAFT + Cubic, PC-SAFT, and Soave BenedictWebbRubin (SBWR) EoSs have been put to the severe test of comprehensive modeling of the available data on various thermodynamic properties in challenging binary and ternary mixtures of CH4, CO2, and n-C16H34. It should be emphasized that the mixtures considered in the present study are the most asymmetric ones for which the sound velocity and compressibility data are currently available. SAFT + Cubic have estimated their single phase thermodynamic properties with the relatively small AAD% that have not exceeded 6%. In addition, SAFT + Cubic was capable of the appropriate estimation of the 14182


Industrial & Engineering Chemistry Research inter-relations between VLE and LLE in the systems under consideration. However, the limitations of SAFT + Cubic with the current mixing rules and two binary adjustable parameters have become evident as well. In particular, it was imprecise in modeling the CO2-rich liquid phase of the three-phase equilibrium in CO2(1)n-C16H34(2) and the two-phase boundary in a ternary mixture. Despite that PC-SAFT has yielded the slightly better estimations of some properties, such as the moderate pressure isobaric heat capacities in the system CH4(1)CO2(2), its overall value as a predictive and correlative tool appears to be smaller in comparison with SAFT + Cubic. In particular, PC-SAFT has significantly deviated from the sound velocity (AAD% exceeding 15%) and compressibility (AAD% exceeding 22%) data. In addition, PC-SAFT was less successful that SAFT + Cubic in modeling the global phase behavior of the systems under consideration. Although SBWR was slightly more precise than SAFT + Cubic in predicting sound velocities in the system CH4(1) n-C16H34(2), it has not exhibited an overall advantage (AAD% for the single phase properties less than 12%). Moreover, considering the previously published results for pure n-C16H34,22 one could assume that the superiority of SAFT + Cubic over SBWR might only increase at the currently uninvestigated elevated pressures above 600 bar. However, the major difficulty of SBWR was modeling phase equilibria in asymmetric systems due to prediction of the unrealistic U-type LLE critical loci. The possible reasons for this phenomenon are inappropriateness of the classical mixing rules for the multiparameter EoS models or the negative impact of the zero covolumes on the slopes of the LLE critical curves. Nevertheless, the significant practical potential of SBWR for industrial applications should not be neglected and this model deserves therefore further evaluation and development.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]; [email protected]. Tel.: +972-3-9066346. Fax: +972-3-9066323.

’ ACKNOWLEDGMENT Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for support of this research, Grant No. PRF#47338-B6. ’ LIST OF SYMBOLS A = Helmholtz free energy B,D,E,F,S1,S1,S1 = parameters of SBWR a,c = parameters of the cohesive correction term of SAFT + Cubic Cv = isochoric heat capacity d = effective Lennard-Jones segment diameter (Å) gii (dii)hs = Mansoori et al’s correction function for mixtures of hard spheres k12, l12 = binary adjustable parameters m = effective number of segments NAv = Avogadro’s number P = pressure R = universal gas constant T = temperature

ARTICLE

v = molar volume W = speed of sound x = molar fraction in liquid phase y = molar fraction in vapor phase Z = compressibility factor Greek Letters

ε/k = segment energy parameter divided by Boltzmann’s constant ζk = Mansoori et al’s correction functions for mixtures of hard spheres kT ks = isothermal and isentropic compressibilities F = density σ = Lennard-Jones temperature-independent segment diameter (Å) ω = acentric factor Subscripts

c = critical state Superscripts

res = residual property IG = ideal gas Abbreviations

EOS = equation of state HS = hard sphere SAFT = Statistical Association Fluid Theory SBWR = SoaveBenedictWebbRubin EOS

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