Hybridizing SAFT and Cubic EOS: What Can Be Achieved? - Industrial

ARTICLE pubs.acs.org/IECR

Hybridizing SAFT and Cubic EOS: What Can Be Achieved? Ilya Polishuk* Department of Chemical Engineering & Biotechnology, Ariel University Center of Samaria, 40700 Ariel, Israel ABSTRACT: This study deals with creating a concept of the hybrid model gathering the advantages of both cubic EOS and SAFT approaches. The proposed idea is a revision of the Chapman’s et al. SAFT by addressing the problem of its numerical pitfalls and the issue of the space available for dispersive interactions with further attaching the SAFT part by the cubic EOS’s cohesive term. It is demonstrated that the resulting model on one hand preserves the characteristic for SAFT accuracy in estimating the liquid compressibility, and on the other one - the characteristic for cubic equations capability of simultaneous modeling of critical and subcritical data. Moreover, on the basis of the comprehensive set of thermodynamic properties of 8 challenging for modeling compounds (including n-hexatriacontane, water, and methanol) it has been demonstrated that the proposed EOS has a superiority comparing even to one of the most successful versions of SAFT such as the SAFT-VR-Mie.

’ INTRODUCTION Development of robust analytical Equation of State (EOS) models, suitable for engineering practice, reliable in the entire thermodynamic phase space, and capable of accurate prediction of the entire set of thermodynamic properties is a challenging problem that has not been satisfactorily solved yet. The current progress in developing EOS models has been surveyed in several comprehensive reviews and monographs (see for example refs 1-7). It comes into view that the most widely used and the comprehensively investigated EOS models are the cubic ones. Cubic EOS models are empirical in nature rather theoretically based. Nevertheless they have certain doubtless advantages which explain their success. In particular, the simplicity of cubic EOSs allows generalizations, which might remove a need for evaluating the particular compounds parameters. It means that these models might be implemented in the entirely predictive manner. In addition, cubic equations might be entirely free of numerical pitfalls and nonphysical predictions. Another significant advantage of these models is their capability of yielding a mostly reasonable description the subcritical PVT with the parameters evaluated at the critical pressures and temperatures. This feature typically allows a simultaneous consideration of the critical and the subcritical data both in pure compounds and their mixtures. From the author’s viewpoint the major disadvantage of cubic EOS models is their tendency to generate a relatively close proximity between their covolumes and the predicted saturated liquid molar volumes at low temperatures, which could be explained by the curvature of the van der Waals potential. As a result, cubic equations underestimate the liquid compressibility -(∂V/∂P)T and, consequently, overestimate-(∂P/∂V)T. The inaccurate pressure-volume interrelation established by cubic EOSs eventually affects the predictions of other properties as well.8 These facts outline the importance of the more realistic description of the intermolecular potential provided by another family of thermodynamic models, namely the equations of state based on the Statistical Association Fluid Theory (SAFT). Indeed, SAFT models might describe the liquid phase compressibility and the related thermodynamic properties more accurately that cubic equations.8 r 2011 American Chemical Society

However, often there is a price to pay for the increased model’s complexity. In particular, most versions of SAFT are not free of the undesirable numerical pitfalls responsible for inaccurate and sometimes even nonphysical predictions. One of these numerical pitfalls generated by the excessively complex dispersion terms is the multiple phase equilibria for pure compounds. This problem affects several popular versions of SAFT, such as the ChenKreglewski’s9 SAFT of Huang and Radosz (CK-SAFT),10 the Perturbed Chain (PC-SAFT),11 and the Soft-SAFT.12 At the same time, other dispersion terms, such as those implemented by the SAFT of Chapman et al.13 and by different versions of Variable Range (SAFT-VR)14-16 result in prediction of the classical van der Waals shapes of isotherms and single phase envelopes for pure compounds.8,17-22 Thus, the problem of fictional phase splits can be avoided by the appropriate selection of SAFT’s dispersion term. The additional numerical pitfalls namely the negative heat capacities at the extremely high pressures and the intersections of isotherms generated by the temperature dependencies attached to the reduced densities might be addressed as well.8 However there is another serious drawback characteristic for SAFT models, namely the wrong estimation of the pure compounds critical pressures and temperatures. It comes into view that SAFT models typically have a limited capability of simultaneous description of the critical and subcritical PVT, comparing even with simple cubic EOSs. Indeed, as demonstrated elsewhere (see for example ref 23), the SAFT parameters being fitted to the experimental critical data might result in the particularly inaccurate estimation of equilibrium phase densities. The weaknesses of the mean-field theory in the near-critical region can be addressed by the crossover approaches. 6 Unfortunately, most of these approaches increase the complexity of the models, which might hinder their implementation for industrial purposes. In this respect a relatively simple EOS (entitled as SAFT-CP or SAFT-BACK) proposed by Chen and Mi24 presents a particular Received: December 1, 2010 Accepted: February 8, 2011 Revised: February 1, 2011 Published: March 04, 2011 4183

dx.doi.org/10.1021/ie102420n | Ind. Eng. Chem. Res. 2011, 50, 4183–4198

Industrial & Engineering Chemistry Research

ARTICLE

fact provide important practical advantages (see also refs 33-39). This observation leads to the idea of gathering the strong sides of both SAFT and cubic EOSs by hybridizing both approaches. The details are discussed below.

’ THEORY The current study presents a concept rather than final model, and the particular details could be reconsidered while gaining more experience with the proposed approach. Its mean idea is attaching SAFT by the attractive term of cubic EOS as follows a ð1Þ Ares ¼ Ares, SAFT V þc where a and c are the cubic EOS’s parameters and Ares, SAFT ¼ AHS þ Achain þ Adisp þ Aassoc Figure 1. Liquid densities of n-octadecane and its predictions by SAFT-CP 24 and CK-SAFT. 10 Experimental data: 32 b (blue) 383.15 K, b (purple) - 373.15 K, b (black) - 363.15 K, b (green) - 353.15 K. Solid lines - SAFT-CP, dashed lines - CKSAFT.

interest. The latter model implements the Boublik’s hard-convexbody repulsive term25 assuming that molecules consist of nonspherical segments. In addition, it modifies the CK-SAFT’s10 dispersive term proportionally to the ratio of the chain term to the hard-convex-body term while assuming the fact that the chain formation reduces the space available for dispersive interactions. SAFT-CP has been lately modified by taking into account the dipole-dipole interaction26,27 for modeling the polar compounds data. In addition, it has been implemented for modeling different thermodynamic properties of pure compounds and mixtures.28-31 Unfortunately, the modifications proposed by Chen and Mi24 do not solve the problem of the CK-SAFT’s10 multiple isotherms maxima phenomenon but extend its undesired impact. In particular, as demonstrated by Figure 1, which depicts the densities of liquid n-octadecane, the CK-SAFT10 yields physically meaningful (although inaccurate) predictions of the data in the selected PVT range. In contrast, the SAFT-CP24 generates the fictional isotherms pressure maxima (γ22) and their further plunge to negative pressures. It can be seen that as the temperature decreases, γ decreases as well. As a result the realistic liquid phase disappears below ∼350 K, and the model continues predicting the existence of only the nonrealistic phase in the vicinity of the covolume. Implementation of the Chen and Mi’s24 approach to the SAFT models free of the numerical pitfall above (SAFT of Chapman et al.13 and SAFTs-VR14-16) performed by the author has revealed certain problems, which could probably be explained by the fact that these robust versions of SAFT tend to overestimate critical points more than the CK-SAFT. In particular, it has been found that moving away from the spherical segments shape might affect the overall reliability of predictions. At the same time, modification of the dispersive term proportionally to the ratio of the chain term to the hard-sphere (HS) term might yield certain improvement in simultaneous modeling of saturated and critical data. However this improvement still does not achieve the accuracy typically yielded by the popular cubic EOS models. Paradoxically, it comes therefore into view that in spite of their weak theoretical basis, van der Waals-type equations might in

Since

res RT DA P¼ V DV T

ð2Þ

ð3Þ

the pressure is obtained as P ¼ PSAFT -

a ðV þ cÞ2

ð4Þ

In view of the fact that the proposed EOS is based on practical observations rather than molecular theory, the relatively simple version of SAFT free of the multiple phase equilibria’s numerical pitfall,22 namely the SAFT of Chapman et al.13 has been selected while the two following modifications have been performed: 1) The hard-sphere term has been revised in order to address the issue of the numerical pitfalls generated by the temperaturedependent reduced density η by making the EOS’s covolume temperature independent (for more details see8) as follows AHS ¼ mRT

4η - 3η2 1=2 η ð1 - ηÞ3=2 1 θðTÞ

ð5Þ

where η¼

πNAv mσ 3 θðTÞ 6V

ð6Þ

NAv is the Avogadro’s number, m is the effective number of segments, σ is the Lennard-Jones’s segment diameter and 13 0 k 1 þ 0:2977 T C B ε C B θðTÞ ¼ B 2 C ð7Þ A @ k k T2 1 þ 0:33163 T þ 0:0010477 ε ε 2) Assuming the fact that the chain formation reduces the space available for dispersive interactions and following the Chen and Mi’s24 approach, the dispersive term has been multiplied by the ratio of the chain term to the hard-sphere term as follows 0 1 disp ε ! B ao1 chain C ε 2A B C k disp Ba þ C 1 þ HS ð8Þ Adisp ¼ mR k @ o1 T A A

4184

dx.doi.org/10.1021/ie102420n |Ind. Eng. Chem. Res. 2011, 50, 4183–4198


ARTICLE

Table 1. Parameters for the Compounds with the Experimentally Available Critical Points independent adjustable parameters compound

m

c (L/mol)

dependent calculated parameters εAB/k (K)

κAB

a (bar-mol/L)

σ (Å)

ε/k (K)

CO2

1.87

0.049

-

-

2.87365

2.89986

213.210

n-C6H14

2.35

0.197

-

-

23.8100

4.27620

371.134

n-C15H32

4.00

0.297

-

-

134.192

4.79401

473.630

CH3OH

1.25

0.145

0.02

2100

8.95664

3.52470

223.559

H2O

1

0.07

0.08

1900

3.52755

2.96992

102.700

ε is the intersegment interaction’s dispersion energy, and k is the Boltzmann’s constant. The original Chapman’s et al.13 disp expressions for adisp o1 and ao1 might be reduced to pffiffiffi 3 2 disp ao1 ¼ ½-8:5959η - 6:1344η2 - 3:87882η3 þ 25:3316η4 π ð9Þ disp

ao2

pffiffiffi 3 2 ¼ ½-1:9075η þ 13:4675η2 - 40:5171η3 þ 39:1711η4 π ð10Þ

The original SAFT’s chain and the association terms have been remained unchanged Achain ¼ RTð1 - mÞln

A

assoc

¼ RT

∑A

1 - η=2 ð1 - ηÞ3

XA ln X 2

!

A

þ

M 2

ð12Þ

A 3 A 2 A Aassoc 3B ¼ RTðln½2ðX3B Þ - ðX3B Þ - 2X3B þ 2Þ

A X3B ¼

! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ2 þ 6ΔV þ V 2 -1 ΔþV V2 4Δ

The association scheme 4C is typically referred to water ! A X 1 A 4C Aassoc þ 4C ¼ 4RT ln X4C 2 2

ð13Þ

A X4C

dependent calculated parameters

compound

m

σ (Å)

a (bar-mol/L)

ε/k (K)

n-C18H38

5.00

4.748

136.605

365.289

n-C28H58

6.37

5.080

291.469

291.469

n-C36H74

7.27

5.295

445.289

451.902

1 !3 εAB C 7 B C B 6 7 6 1 - η=2 AB B 6 k 7 C 3 C ð17Þ B 7 6 1 k exp Δ ¼ NAv σ θðTÞ 3 C B 7 6 ð1 - ηÞ A @ 4 T 5

κAB is the volume of interaction between the sites A and B, and εAB is their interaction’s association energy. It can therefore be seen that the proposed model has the following substance-dependent parameters: a, c, m, σ, and ε/k. In the case of polar compounds two other parameters, namely κAB and εAB/k, are added. It is proposed to treat c, m and, if appropriate, the two additional polar interaction’s parameters as the adjustable ones. The remaining three parameters (a, σ, and ε/k) can be obtained by the procedure typically used for cubic equations, namely solving the system of three equations: the two critical point conditions (setting the first and the second derivatives of P with respect to V equal to zero at the experimental critical pressure and temperature) and Vc, EOS ¼ ξVc, experimental

ð14Þ

ð15Þ

where rffiffiffiffiffiffiffiffiffiffiffiffiffiffi Δ 1þ8 -1 V ¼ Δ 4 V

independent adjustable parameters

Δ is the “association strength” defined as 0 2

ð11Þ

where M is the number of association sites on each molecule, XA is the mole fraction of molecules not bonded at site A, and ∑A represents a sum over all associating sites on the molecule (additional details are provided elsewhere7). Different rigorous association schemes are related to different polar molecules. For example, the scheme 3B is typically assumed for alcohols, and the pertinent equations are given as

and:

Table 2. Parameters for the Compounds with the Experimentally Unavailable Critical Points (c = 0)

ð18Þ

In the present study ξ = 1.1. For the heavy compounds decomposing below their critical temperatures it seems expedient to simplify the model by taking c = 0 and removing the adjustment to the imaginary experimental critical pressure. In such case the independent adjustable parameters are m and σ, while a and ε/k - the dependent calculated ones (see Tables 1 and 2). It comes therefore into view that the proposed approach requires even less substance-dependent adjustable parameters than the SAFT of Chapman et al.13 which involves fitting of m, σ, and ε/k. Further elimination of adjustable parameters could be achieved by developing generalization schemes. For example, for many organic compounds c might be obtained as follows c ¼ - 1:6049 þ 1:3440ðσ109 Þ - 0:3943ðσ109 Þ2 þ 0:0417ðσ109 Þ3 ð19Þ

ð16Þ

4185



ARTICLE

Figure 4. Saturated phase densities of selected n-alkanes. O - Pseudoexperimental data.48 Black lines - the proposed EOS, red lines SAFT-VR-Mie, blue lines - PR EOS.

Figure 2. Contributions of the EOS’s parts to the critical isotherm (507.6 K) of n-hexane. Solid lines - the proposed EOS and PR EOS, dashed lines - SAFT-VR-Mie.

Figure 3. Vapor pressures. O - Pseudoexperimental data.40-42,48 Black lines - the proposed EOS, red lines - SAFT-VR-Mie, blue lines - PR EOS.

Implementation of eq 19 along with possible ideas for generalizing m will be considered in the forthcoming studies. It would be quite unrealistic to expect from the currently proposed EOS for the precision in modeling of the complete set of thermodynamic properties in the entire thermodynamic phase space as achieved by the multiparameter expressions

Figure 5. Densities and sound velocities of liquid n-hexane. Experimental data:49 O (black) - 293.15 K, O (blue) - 333.15 K, O (red) 373.15 K. Solid lines - the proposed EOS, dashed lines - SAFT-VRMie, dot-dashed lines - PR EOS.

implemented for evaluating the pure compound pseudoexperimental data (see for example refs 40-42). Its obvious and tough competitor could rather be SAFT-VR-Mie,14 which however presents a more complex and deeper rooted in molecular theory approach. The doubtless advantages of the latter model in predicting the auxiliary thermodynamic properties have been demonstrated by Lafitte et al.15,16 The outstandingly accurate 4186



ARTICLE

Figure 6. Thermodynamic properties of liquid n-pentadecane. Experimental data: O (black) - 303.15 K,50 O (blue) - 343.15 K,50 O (red) - 383.15 K,50 b (blue) - 343.15 K,51 b (red) - 383.15 K,51 b (green) - 433.15 K.51 Solid lines - the proposed EOS, dashed lines - SAFT-VR-Mie, dot-dashed lines - PR EOS.

predictions of the very sophisticated property such as the sound velocity in polar mixtures recently achieved using this model by Khammar and Shaw43 should be noticed as well. Taking into account the fact that SAFT-VR-Mie does not exhibit the nonrealistic phase splits, it should be considered as one of the most

successful and promising versions of SAFT. The EOS of Peng and Robinson44 (PR) in its original form attached by the Soave’s temperature dependency has also been selected for the comparison as a reference model. The latter temperature dependence in spite of its inconsistency at high temperatures18 might be 4187


Figure 7. Densities and sound velocities of liquid n-octadecane. Experimental data:32 O (black) - 323.15 K, O (blue) 353.15 K, O (red) - 383.15 K. Solid lines - the proposed EOS, dashed lines - SAFT-VR-Mie, dot-dashed lines - PR EOS.

Figure 8. Densities and sound velocities of liquid n-octacosane. Experimental data:52 O (black) - 363.15 K, O (blue) - 383.15 K, O (red) - 403.15 K. Solid lines - the proposed EOS, dashed lines - SAFT-VR-Mie, dot-dashed lines - PR EOS.

Figure 9. Densities and sound velocities of liquid n-hexatriacontane. Experimental data:52 O (black) - 373.15 K, O (blue) 383.15 K, O (red) - 393.15 K, O (green) - 403.15 K. Solid lines the proposed EOS, dashed lines - SAFT-VR-Mie, dot-dashed lines - PR EOS.

Industrial & Engineering Chemistry Research ARTICLE

4188


Figure 10. Saturated phase properties of water. O,b - Pseudoexperimental data.41 Black lines - the Figure 11. Saturated phase properties of methanol. O,b - Pseudoexperimental data.40 Black lines proposed EOS, red lines - SAFT-VR-Mie, blue lines - PR EOS. the proposed EOS, red lines - SAFT-VR-Mie, blue lines - PR EOS.


4189


Figure 12. Saturated phase properties of carbon dioxide. O,b - Pseudoexperimental data.42 Black lines - the proposed EOS, red lines - SAFT-VR-Mie, blue lines - PR EOS.

Figure 13. Joule-Thomson inversion curve of methanol. O Pseudoexperimental data.40 Black lines - the proposed EOS, red lines - SAFT-VR-Mie, blue lines - PR EOS.


4190


Figure 14. Thermodynamic properties of water in one-phase region as predicted by the proposed EOS. Pseudoexperimental data:41 O (black) - 300 bar, O (blue) - 500 bar, O (dark green) - 750 bar, O (purple) - 1000 bar, O (light green) - 2000 bar, O (red) - 4000 bar, O (brown) - 10,000 bar. Calculated data - solid lines.

Figure 15. Thermodynamic properties of water in one-phase region as predicted by SAFT-VR-Mie. For legend see Figure 14.


4191


Figure 16. Thermodynamic properties of water in one-phase region as predicted by PR EOS. For legend see Figure 14.

Figure 17. Thermodynamic properties of methanol in one-phase region as predicted by the proposed EOS. Pseudoexperimental data:40 O (black) - 100 bar, O (blue) - 400 bar, O (red) 700 bar. Calculated data - solid lines.


4192


Figure 18. Thermodynamic properties of methanol in one-phase region as predicted by SAFT-VR- Figure 19. Thermodynamic properties of methanol in one-phase region as predicted by PR EOS. For Mie. For legend see Figure 17. legend see Figure 17.


4193



ARTICLE

Figure 20. Thermodynamic properties of carbon dioxide in one-phase region as predicted by the proposed EOS. Pseudoexperimental data:42 O (black) - 100 bar, O (blue) - 200 bar, O (dark green) - 500 bar, O (purple) - 1000 bar, O (light green) - 2000 bar, O (red) - 4000 bar, O (brown) - 8000 bar. Calculated data - solid lines.

characterized by certain advantages in modeling the auxiliary properties.45,46 The compounds investigated in the present study have been chosen according to their challenge for modeling (chained shape and polarity), accessibility of sufficient sets of experimental and pseudoexperimental data and availability of the SAFT-VR-Mie’s parameters in the literature.15,16,47 Although both the proposed approach and SAFT-VR-Mie are the polynomials of high order, their numerical behavior is analogous

to cubic equations. Hence, the similar procedures have been implemented for all the considered models. The calculations have been performed using the Mathematica 7 software (the pertinent routines can be obtained from the author by request).

’ RESULTS Figure 2 illustrates the numerical contributions of the parts constituting the models considered in the present study. As 4194


Figure 21. Thermodynamic properties of carbon dioxide in one-phase region as predicted by SAFT-VR-Mie. For legend see Figure 20.

Figure 22. Thermodynamic properties of carbon dioxide in one-phase region as predicted by PR EOS. For legend see Figure 20.


4195


Industrial & Engineering Chemistry Research follows from the figure, the cubic EOS’s cohesive term has a small numerical input comparing to the SAFT’s parts. It comes therefore into view that the latter parts play a dominant role in the proposed approach. Figure 3 depicts the vapor pressure lines of several compounds considered in the present study. It can be seen that all three models can be characterized by a comparable accuracy in modeling vapor pressures. The proposed EOS is the best estimator of the carbon dioxide’s vapor pressure data while it is the less successful one in the case of methanol. If necessary, developing a cubic EOS-type temperature dependency for the parameter a in eq 1 that significantly improves its precision might be considered. Figure 4 shows the saturated phase densities of three nalkanes. It comes into view that PR EOS yields satisfactorily good predictions of the n-hexane’s data; however, its accuracy deteriorates in the case of heavier homologues (this issue will be discussed henceforth). Neglecting the problem of wrong estimation of critical data, SAFT-VR-Mie is a sufficiently precise estimator of the saturated phase densities. However the proposed EOS has yet a doubtless advantage. Figure 5 presents the densities and sound velocities of liquid nhexane. It can be seen that both PR EOS and SAFT-VR-Mie do not succeed in precise description of these data. PR EOS is a particularly inaccurate estimator of the sound velocities. In contrast, the proposed EOS predicts the data in a very accurate manner. The same should be concluded considering the results for heavier homologues (see Figures 6-9). In particular, it appears that PR EOS becomes especially imprecise, which makes this model irrelevant for design of high pressure processes implementing these compounds. One could consider replacing the generalized expressions of the PR EOS’s parameters by the compound-specific ones as in the case of SAFT models. Indeed, this practice could improve the accuracy in modeling the saturated and the low-pressure liquid densities. However it could not address the problems of high pressures and auxiliary properties due to the principal difficulty in description the liquid compressibility characteristic for cubic equations.8 These facts outline the fundamental advantage of SAFT models. It particular, it can be seen that SAFT-VR-Mie exhibits stably satisfactorily, although imprecise predictions for both densities and sound velocities regardless of the chain length. It comes into view that SAFT-VR-Mie establishes not entirely exact slopes of both isobars and isotherms. These facts could raise queries regarding the obviousness of the superior consideration of advanced molecular approaches and simulated data instead of tracking the trends established by the real compound’s experimental results. Indeed, it appears that in spite of its weaker theoretical basis, the proposed EOS is capable of the almost precise modeling of both densities and sound velocities in the entire experimentally available range for all the heavy n-alkanes considered. Thus, one could assume that this model could be a reliable estimator of the currently unavailable data, for example at higher pressures and temperatures. The proposed EOS has also an apparent advantage over PR EOS and SAFT-VR-Mie in predicting the heat capacities of n-pentadecane (see Figure 6). The next compounds to be considered are water, methanol, and carbon dioxide, and they present a bigger challenge for modeling than normal paraffins. Figure 10 depicts the entire set of the saturated phase properties of water. It can be seen that the proposed model has the doubtless superiority over PR EOS and SAFT-VR-Mie; however, yet its results are not always precise. Similar conclusions could be reached also in the cases of methanol

ARTICLE

and carbon dioxide (Figures 11 and 12). It can be also seen that the overestimation of critical points might significantly affect the accuracy of SAFT-VR-Mie. In addition, the proposed EOS is not a particularly successful estimator of the carbon dioxide’s CV. Figure 13 shows the Joule-Thomson inversion curve of methanol. Concerning this figure it should be pointed out that the pertinent steam table keeps going only until 700 bar. Thus, the evaluation of the Joule-Thomson inversion data at the higher pressures might be considered as questionable. Anyways, the advantage of the proposed EOS over PR EOS and SAFT-VR-Mie and its particular accuracy in the confident data range should be pointed out. The doubtless superiority of the proposed EOS persists also in predicting the high pressure single phase data (see Figures 1422). However its tendency of underestimating the high-density heat capacity data might raise concerns. It appears that the success in predicting densities and the property meanly based on the first derivates such as sound velocities does not always guarantee the same accuracy in estimating the property based on the second derivatives, such as the isochoric heat capacity. In addition, it can be seen that the association terms attached to both SAFT models do not adequately address the complicated trends established by the auxiliary properties of water and methanol. Moreover, it comes into view that the proposed EOS does not have significant advantage over PR EOS in predicting the data carbon dioxide’s, while the trends established by SAFT-VR-Mie for this compound at very high pressures seem unacceptable.

’ CONCLUSIONS The aim of the present study was creating a concept of the hybrid model gathering the strong sides of both cubic EOS and SAFT approaches. Whereas the popular Cubic Plus Association equation7 considers a cubic EOS as a basis model and attaches it by the SAFT’s association term, the proposed idea was attaching a revised SAFT of Chapman et al.13 by the cubic EOS’s cohesive term. Such a selection is explained by the fact that the latter version of SAFT does not exhibit the nonrealistic phase splits characteristic for several others versions.17-22 In order to address the issue of the numerical pitfalls generated by the temperaturedependent reduced densities, the Carnahan-Starling’s HS term has been modified by making the covolumes temperatureindependent.8 Assuming the fact that the chain formation reduces the space available for dispersive interactions and following the proposal of Chen and Mi,24 the Chapman’s et al.13 dispersive term has been multiplied by the ratio of the chain term to the hard-sphere term. The proposed model contains 5 substance-characteristic parameters: two of them are independent and should be acquired by fitting experimental data and three others are dependent and be obtained by the procedure typically used for cubic equations, namely solving the critical point conditions. Modeling polar compounds implies the association term and its two additional adjustable parameters. It is demonstrated that the resulting model on one hand preserves the characteristic for SAFT accuracy in estimating the liquid compressibility, and on the other one - the characteristic for cubic equations capability of simultaneous modeling of critical and subcritical data. Moreover, on the basis of the comprehensive set of thermodynamic properties it has been demonstrated that the proposed concept has a superiority comparing even to one of the most successful versions of SAFT such as the SAFT-VR-Mie. 4196



’ AUTHOR INFORMATION Corresponding Author

*Phone: þ972-3-9066346. Fax: þ972-3-9066323. E-mail: [email protected]; [email protected].

’ ACKNOWLEDGMENT Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for support of this research, grant N° PRF#47338-B6. ’ LIST OF SYMBOLS A Helmholtz free energy CV isochoric heat capacity CP isobaric heat capacity m number of segments M the number of association sites on each molecule NAv Avogadro’s number P pressure R universal gas constant T temperature V molar volume W speed of sound XA the mole fraction of molecules not bonded at site A Greek letters

Δ association strength κAB the volume of interaction between the sites A and B εAB interaction of association energy η reduced density ε/k segment energy parameter divided by Boltzmann’s constant θ(T) temperature dependence of reduced density σ Lennard-Jones temperature-independent segment diameter (Å) Subscripts

c critical state Superscripts

res residual property Abbreviations

EOS equation of state HS hard sphere SAFT statistical association fluid theory

’ REFERENCES (1) Anderko, A.; Cubic and Generalized van der Waals Equations. In: Equations of State for Fluids and Fluid Mixtures. Part I; Elsevier: Amsterdam, 2000. (2) Wei, Y. S.; Sadus, R. J. Equations of State for the Calculation of Fluid-Phase Equilibria. AIChE J. 2000, 46, 169. (3) M€uller, E. A.; Gubbins, K. E. Molecular-Based Equations of State for Associating Fluids: A Review of SAFT and Related Approaches. Ind. Eng. Chem. Res. 2001, 40, 2193. (4) Economou, I. G. Statistical Associating Fluid Theory: A Successful Model for the Calculation of Thermodynamic and Phase Equilibrium Properties of Complex Fluid Mixtures. Ind. Eng. Chem. Res. 2002, 41, 953. (5) Valderrama, J. O. The State of the Cubic Equations of State. Ind. Eng. Chem. Res. 2003, 42, 1603. (6) Tan, S. P.; Adidharma, H.; Radosz, M. Recent Advances and Applications of Statistical Associating Fluid Theory. Ind. Eng. Chem. Res. 2008, 47, 8063.

ARTICLE

(7) Kontogeorgis, G. M.; Folas, G. K. Thermodynamic Models for Industrial Applications. From Classical and Advanced Mixing Rules to Association Theories; John Wiley & Sons: New York, 2010. (8) Polishuk, I. Addressing the Issue of Numerical Pitfalls Characteristic for SAFT EOS Models. Fluid Phase Equilib. 2011, 301, 123. (9) Chen, S. S.; Kreglewski, A. Applications of the Augmented van der Waals Theory of Fluids.: I. Pure Fluids. Ber. Bunsen-Ges. Phys. Chem. 1977, 81, 1048. (10) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (11) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (12) Pamies, J. C.; Vega, L. F. Vapor - Liquid Equilibria and Critical Behavior of Heavy n-Alkanes Using Transferable Parameters from the Soft-SAFT Equation of State. Ind. Eng. Chem. Res. 2001, 40, 2523. (13) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (14) Gil-Villegas, A.; Galindo, A.; Whitehead, P. J.; Mills, S. J.; Jackson, G.; Burgess, A. N. Statistical Associating Fluid Theory for Chain Molecules with Attractive Potentials of Variable Range. J. Chem. Phys. 1997, 106, 4168. (15) Lafitte, T.; Bessieres, D.; Pi~ neiro, M. M.; Daridon, J.-L. Simultaneous Estimation of Phase Behavior and Second-Derivative Properties Using the Statistical Associating Fluid Theory with Variable Range Approach. J. Chem. Phys. 2006, 124, 024509. (16) Lafitte, T.; Pi~ neiro, M. M.; Daridon, J.-L.; Bessieres, D. A Comprehensive Description of Chemical Association Effects on Second Derivative Properties of Alcohols through a SAFT-VR Approach. J. Phys. Chem. B 2007, 111, 3447. (17) Koak, N.; de Loos, Th. W.; Heidemann, R. A. Effect of the Power Series Dispersion Term on the Pressure-Volume Behavior of Statistical Associating Fluid Theory. Ind. Eng. Chem. Res. 2001, 38, 1718. (18) Segura, H.; Kraska, T.; Mejia, A.; Wisniak, J.; Polishuk, I. Unnoticed Pitfalls of Soave-Type Alpha Functions in Cubic Equations of State. Ind. Eng. Chem. Res. 2003, 42, 5662. (19) Yelash, L.; M€uller, M.; Paul, W.; Binder, K. A Global Investigation of Phase Equilibria Using the Perturbed-Chain Statistical-Associating-Fluid-Theory Approach. J. Chem. Phys. 2005, 123, 14908. (20) Yelash, L.; M€uller, M.; Paul, W.; Binder, K. Artificial Multiple Criticality and Phase Equilibria: an Investigation of the PC-SAFT Approach. Phys. Chem. Chem. Phys. 2005, 7, 3728. (21) Privat, R.; Gani, R.; Jaubert, J.-N. Are Safe Results Obtained When the PC-SAFT Equation of State is Applied to Ordinary Pure Chemicals? Fluid Phase Equilib. 2010, 295, 76. (22) Polishuk, I. About the Numerical Pitfalls Characteristic for SAFT EOS Models. Fluid Phase Equilib. 2010, 298, 67. (23) Blas, F. J.; Galindo, A. Study of the High Pressure Phase Behaviour of CO2þn-Alkane Mixtures Using the SAFT-VR Approach with Transferable Parameters. Fluid Phase Equilib. 2002, 194-197, 501. (24) Chen, J.; Mi, J.-G. Equation of State Extended from SAFT with Improved Results for Non-Polar Fluids Across the Critical Point. Fluid Phase Equilib. 2001, 186, 165. (25) Boublik, T. Hard-Sphere Equation of State. J. Chem. Phys. 1970, 53, 471. (26) Mi, J.-G.; Chen, J.; Gao, G.-H.; Fei, W.-Y. Equation of State Extended from SAFT with Improved Results for Polar Fluids Across the Critical Point. Fluid Phase Equilib. 2002, 201, 295. (27) Chen, J.; Mi, J.-G.; Yu, Y.-M.; Luo, G. An Analytical Equation of State for Water and Alkanols. Chem. Eng. Sci. 2004, 59, 5831. (28) Maghari, A.; Sadeghi, M. S. Prediction of Sound Velocity and Heat Capacities of n-Alkanes from the Modified SAFT-BACK Equation of State. Fluid Phase Equilib. 2007, 252, 152. (29) Maghari, A.; Safaei, Z.; Sarhangian, S. Predictions of the JouleThomson Inversion Curves for Polar and non-Polar Fluids from the SAFT-CP Equation of State. Cryogenics 2008, 48, 48. 4197


Industrial & Engineering Chemistry Research (30) Maghari, A.; Safaei, Z. Thermodynamic Regularities for nonPolar and Polar Fluids from the Modified SAFT-BACK Equation of State. J. Mol. Liq. 2008, 142, 95. (31) Maghari, A.; Hamzehloo, M. Second-Order Thermodynamic Derivative Properties of Binary Mixtures of n-Alkanes through the SAFT-CP Equation of State. Fluid Phase Equilib. 2010in press. (32) Dutour, S.; Daridon, J. L.; Lagourette, B. Pressure and Temperature Dependence of the Speed of Sound and Related Properties in Normal Octadecane and Nonadecane. Int. J. Thermophys. 2000, 21, 173. (33) Prausnitz, J. M. Equations of State from van der Waals Theory: the Legacy of Otto Redlich. Fluid Phase Equilib. 1985, 24, 63. (34) Kontogeorgis, G. M.; Coutsikos, P.; Harismiadis, V. I.; Fredenslund, A.; Tassios, D. P. A Novel Method for Investigating the Repulsive and Attractive Parts of Cubic Equations of State and the Combining Rules Used with the vdW-1f Theory. Chem. Eng. Sci. 1998, 53, 541. (35) Wyczalkowska, A. K.; Sengers, J. V.; Anisimov, M. A. Critical Fluctuations and the Equation of State of van der Waals. Physica A 2004, 334, 482. (36) Tian, J.; Gui, Y. Modification to the van der Waals Equation of State. J. Phase Equilib. 2003, 24, 533. (37) Roman, F. L.; Mulero, A.; Cuadros, F. Simple Modifications of the van der Waals and Dieterici Equations of State: Vapour-Liquid Equilibrium Properties. Phys. Chem. Chem. Phys. 2004, 6, 5402. (38) Kontogeorgis, G. M.; Economou, I. G. Equations of state: From the Ideas of van der Waals to Association Theories. J. Supercrit. Fluids 2010, 55, 421. (39) Mulero, A.; Cachadi~na, I. Liquid Saturation Density from Simple Equations of State. Int. J. Thermophys. 2007, 28, 279. (40) Goodwin, R. D. Methanol Thermodynamic Properties From 176 to 673 K at Pressures to 700 bar. J. Phys. Chem. Ref. D 1987, 16, 799. (41) Wagner, W.; Pruss, A. The IAPWS Formulation 1995 for the Thermodynamic Properties of Ordinary Water Substance for General and Scientific Use. J. Phys. Chem. Ref. D 2002, 31, 387. (42) Span, R.; Wagner, W. A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperature to 1100 K at Pressures up to 800 MPa. J. Phys. Chem. Ref. D 1996, 25, 1509. (43) Khammar, M.; Shaw, J. Speed of Sound Prediction in 1-nAlcohol þ n-Alkane Mixtures Using a Translated SAFT-VR-Mie Equation of State. Fluid Phase Equilib. 2010, 288, 154. (44) Peng, D. Y.; Robinson, D. B. A New Two Constant Equation of State. Ind. Eng. Chem. Fund. 1976, 15, 59–64. (45) Neau, E.; Hernandez-Garduza, O.; Escandell, J.; Nicolas, C.; Raspo, I. The Soave, Twu and Boston-Mathias Alpha Functions in Cubic Equations of State Part I. Theoretical Analysis of their Variations According to Temperature. Fluid Phase Equilib. 2009, 276, 87. (46) Neau, E.; Raspo, I.; Escandell, J.; Nicolas, C.; HernandezGarduza, O. The Soave, Twu and Boston-Mathias Alpha Functions in Cubic Equations of State. Part II. Modeling of Thermodynamic Properties of Pure Compounds. Fluid Phase Equilib. 2009, 276, 156. (47) Lafitte, T.; Mendiboure, B.; Pi~neiro, M. M.; Bessieres, D.; Miqueu, C. Interfacial Properties of Water/CO2: A Comprehensive Description through a Gradient Theory-SAFT-VR Mie Approach. J. Phys. Chem. B 2010, 114, 11110. (48) Rowley, R. L.; Wilding, W. V.; Daubert, T. E.; Danner, R. P.; Adams, M. E. Physical and Thermodynamic Properties of Pure Chemicals: DIPPR: Data Compilation; Taylor & Francis: New York, 2010. (49) Daridon, J. L.; Lagourette, B.; Grolier, J.-P. E. Experimental Measurements of the Speed of Sound in n-Hexane from 293 to 373 K and up to 150 MPa. Int. J. Thermophys. 1998, 19, 145. (50) Daridon, J. L.; Carrier, H..; Lagourette, B. Pressure Dependence of the Thermophysical Properties of n-Pentadecane and n-Heptadecane. Int. J. Thermophys. 2002, 23, 697. (51) Dovnar, D. V.; Lebedinskii, Yu. A.; Khasanshin, T. S.; Shchemelev, A. P. The Thermodynamic Properties of n-Pentadecane in the Liquid State, Determined by the Results of Measurements of Sound Velocity. High Temp. 2001, 39, 835.

ARTICLE

(52) Dutour, S.; Lagourette, B.; Daridon, J. L. High-Pressure Speed of Sound, Density and Compressibility of Heavy Normal Paraffins: C28H58 and C36H74. J. Chem. Thermodyn. 2002, 34, 475.

4198


Hybridizing SAFT and Cubic EOS: What Can Be Achieved? - Industrial

Recommend Documents