Hybridizing SAFT and Cubic EOS What Can Be Achieved

8/6/2019 Hybridizing SAFT and Cubic EOS What Can Be Achieved

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Published: March 04, 2011

r 2011 American Chemical Society 4183 dx.doi.org/10.1021/ie102420n| Ind. Eng. Chem. Res. 2011, 50, 4183–

4198

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 SAFTapproaches. The proposed idea is a revision of the Chapman’s et al. SAFT by addressing the problem of its numerical pitfalls and theissue of the space available for dispersive interactions with further attaching the SAFT part by the cubic EOS ’s cohesive term. It isdemonstrated that the resulting model on one hand preserves the characteristic for SAFT accuracy in estimating the liquidcompressibility, and on the other one - the characteristic for cubic equations capability of simultaneous modeling of critical andsubcritical data. Moreover, on the basis of the comprehensive set of thermodynamic properties of 8 challenging for modelingcompounds (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 entirethermodynamic phase space, and capable of accurate predictionof the entire set of thermodynamic properties is a challengingproblem that has not been satisfactorily solved yet.

The current progress in developing EOS models has beensurveyed in several comprehensive reviews and monographs (seefor example refs 1-7). It comes into view that the most widely used and the comprehensively investigated EOS models are thecubic ones. Cubic EOS models are empirical in nature rathertheoretically based. Nevertheless they have certain doubtlessadvantages which explain their success. In particular, the simpli-city of cubic EOSs allows generalizations, which might remove aneed for evaluating the particular compounds parameters. Itmeans 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. Anothersignificant advantage of these models is their capability of

yielding a mostly reasonable description the subcritical PVT with the parameters evaluated at the critical pressures andtemperatures. This feature typically allows a simultaneous con-sideration of the critical and the subcritical data both in purecompounds and their mixtures.

From the author’s viewpoint the major disadvantage of cubicEOS models is their tendency to generate a relatively close

proximity between their covolumes and the predicted saturatedliquidmolar volumes at low temperatures, which could be explained by the curvature of the van der Waals potential. As a result, cubicequations underestimate the liquid compressibility -(∂V/ ∂ P )T and, consequently, overestimate-(∂ P/ ∂V )T . The inaccurate pres-sure-volume interrelation established by cubic EOSs eventually aff ects the predictions of other properties as well.8

These facts outline the importance of the more realistic descrip-tion of the intermolecular potential provided by another family of thermodynamic models, namely the equations of state based on theStatistical Association FluidTheory (SAFT). Indeed, SAFT modelsmight describe the liquid phase compressibility and the relatedthermodynamic properties more accurately that cubic equations.8

However, often there is a priceto payfor the increased model’scomplexity. In particular, most versions of SAFT are not free of the undesirable numerical pitfalls responsible for inaccurate andsometimes even nonphysical predictions. One of these numericalpitfalls generated by the excessively complex dispersion terms isthe multiple phase equilibria for pure compounds. This problemaff ects several popular versions of SAFT, such as the Chen-Kreglewski’s9 SAFT of Huang and Radosz (CK-SAFT),10 thePerturbed Chain (PC-SAFT),11 and the Soft-SAFT.12 At thesame time, other dispersion terms, such as those implemented by the SAFT of Chapman et al.13 and by diff erent versions of VariableRange (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-

22Thus,theproblemof fictional phase splitscan beavoided by theappropriate selectionof SAFT’s dispersion term.Theadditional numerical pitfalls namely the negative heat capacities atthe extremely high pressures and the intersections of isothermsgenerated by the temperature dependencies attachedto the reduceddensities might be addressed as well.8

However there is another serious drawback characteristic forSAFT models, namely the wrong estimation of the pure com-pounds critical pressures and temperatures. It comes into view that SAFT models typically have a limited capability of simulta-neous description of the critical and subcritical PVT, comparingeven with simple cubic EOSs. Indeed, as demonstrated elsewhere(see for example ref 23), the SAFT parameters being fitted to theexperimental critical data might result in the particularly inaccu-rate 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 theseapproaches increase the complexity of the models, which mighthinder their implementation for industrial purposes.

In this respect a relatively simple EOS (entitled as SAFT-CPor SAFT-BACK) proposed by Chen and Mi24 presents a particular

Received: December 1, 2010 Accepted: February 8, 2011Revised: February 1, 2011



4184 dx.doi.org/10.1021/ie102420n |Ind. Eng. Chem. Res. 2011, 50, 4183–4198

Industrial & Engineering Chemistry Research ARTICLE

interest.The latter model implements theBoublik ’s hard-convex- body repulsive term25 assuming that molecules consist of non-spherical segments. In addition, it modifies the CK-SAFT’s10

dispersive term proportionally to the ratio of the chain term tothe hard-convex-body term while assuming the factthat the chainformation reduces the space available for dispersive interactions.SAFT-CP has been lately modified by taking into account thedipole-dipole interaction26,27 for modeling the polar com-pounds data. In addition, it has been implemented for modelingdiff erent thermodynamic properties of pure compounds andmixtures.28-31

Unfortunately, the modifications proposed by Chen and Mi24

do not solve the problem of the CK-SAFT’s10

multiple isothermsmaxima phenomenon but extend its undesired impact. Inparticular, as demonstrated by Figure 1, which depicts thedensities of liquid n-octadecane, the CK-SAFT10 yields physically meaningful (although inaccurate) predictions of the data in theselected PVT range. In contrast, the SAFT-CP24 generates thefictional isotherms pressure maxima (γ22) and their furtherplunge to negative pressures. It can be seen that as the tempera-ture decreases, γ decreases as well. As a result the realistic liquidphase disappears below ∼350 K, and the model continuespredicting the existence of only the nonrealistic phase in the

vicinity of the covolume.Implementation of the Chen and Mi’s24 approachtotheSAFT

models free of the numerical pitfall above (SAFT of Chapman

et al.13 and SAFTs-VR 14-

16) performed by the author hasrevealed certain problems, which could probably be explained

by the fact that these robust versions of SAFT tend to over-estimate critical points more than the CK-SAFT. In particular, ithas been found that moving away from the spherical segmentsshape might aff ect the overall reliability of predictions. At thesame time, modification of the dispersive term proportionally tothe ratio of the chain term to the hard-sphere (HS) term might

yieldcertain improvement in simultaneousmodeling of saturatedand critical data. However this improvement still does notachieve the accuracy typically yielded by the popular cubic EOSmodels. Paradoxically, it comes therefore into view that in spite of their weak theoretical basis, van der Waals-type equations might in

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. Thedetails are discussed below.

’THEORY

The current study presents a concept rather than final

model, and the particular details could be reconsidered whilegaining more experience with the proposed approach. Itsmean idea is attaching SAFT by the attractive term of cubicEOS as follows

Ares ¼ Ares , SAFT -

a

V þ cð1Þ

where a and c are the cubic EOS’s parameters and

Ares , SAFT ¼ AHS þ Achain þ Adisp þ Aassoc ð2ÞSince

P ¼ RT

V -

D Ares

DV T

ð3Þ

the pressure is obtained as

P ¼ P SAFT -

a

ðV þ cÞ2 ð4Þ

In view of the fact that the proposed EOS is based on practicalobservations rather than molecular theory, the relatively simple

version of SAFT free of the multiple phase equilibria’s numericalpitfall,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 temperature-dependent reduced density η by making the EOS’s covolumetemperature independent (for more details see8) as follows

AHS ¼ mRT 4η- 3η2

ð1- ηÞ3=2 1-η

θðT Þ

1=2ð5Þ

where

η ¼ π N Av

6V mσ 3θðT Þ ð6Þ

N Av is the Avogadro’s number, m is the eff ective number of segments, σ is the Lennard-Jones’s segment diameter and

θðT Þ ¼1

þ0:2977

k

ε T

1 þ 0:33163k

ε

T þ 0:0010477

k

ε

2

T 2

0BBBB@ 1CCCCA3

ð7Þ

2) Assuming the fact that the chain formation reduces thespace available for dispersive interactions and following the Chenand Mi’s24 approach, the dispersive term has been multiplied by the ratio of the chain term to the hard-sphere term as follows

Adisp ¼ mR ε

k

a

dispo1 þ

adispo1

ε

k

T

0BBB@

1CCCA 1 þ 2 Achain

AHS

!ð8Þ

Figure 1. Liquid densities of n-octadecane and its predictions by SAFT-CP24 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 - CK-

SAFT.

http://pubs.acs.org/action/showImage?doi=10.1021/ie102420n&iName=master.img-000.jpg&w=240&h=179





ε is the intersegment interaction’s dispersion energy, and k isthe Boltzmann’s constant. The original Chapman’s et al.13

expressions for ao1disp and ao1

disp might be reduced to

adispo1 ¼ 3

ffiffiffi2

p

π ½-8:5959η- 6:1344η2

- 3:87882η3 þ 25:3316η4�ð9Þ

adisp

o2 ¼3 ffiffiffi2p

π ½-1:9075ηþ 13:4675η2- 40:5171η

3

þ 39:1711η4

�ð10Þ

The original SAFT’s chain and the association terms have beenremained unchanged

Achain ¼ RT ð1-mÞln1- η=2

ð1- ηÞ3 ð11Þ

Aassoc ¼ RT ∑ A

ln X A - X A

2

!þ M

2ð12Þ

where M is the number of association sites on each molecule, X A

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). Diff erent rigorousassociation schemes are related to diff erent polar molecules. Forexample, the scheme 3B is typically assumed for alcohols, and thepertinent equations are given as

Aassoc3 B ¼ RT ðln½2ð X A3 BÞ3

- ð X A3 BÞ2�- 2 X A3 B þ 2Þ ð13Þand:

X A3 B ¼Δþ V

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ

2 þ 6ΔV þ V 2

V 2

r - 1

!

4Δð14Þ

The association scheme 4C is typically referred to water

Aassoc4C ¼ 4RT ln X A4C -

X A4C

2þ 1

2

!ð15Þ

where

X A4C ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ 8

Δ

V

r - 1

4Δ

V

ð16Þ

Δ is the “association strength” defined as

Δ ¼ N Avσ 3θðT Þ 1- η=2

ð1- ηÞ3 k AB exp

ε AB

k

!

T

2666664

3777775- 1

0BBBBB@

1CCCCCA ð17Þ

κ AB is the volume of interaction between the sites A and B, andε AB is their interaction’s association energy.

It can therefore be seen that the proposed model has thefollowing substance-dependent parameters: a , c , m ,σ , and ε/k . In

the case of polar compounds two other parameters, namely κ AB

and ε AB/k , are added. It is proposed to treat c , m and, if appropriate, the two additional polar interaction’s parametersas the adjustable ones. The remaining three parameters (a ,σ ,andε/k ) can be obtained by the procedure typically used for cubicequations, namely solving the system of three equations: the twocritical point conditions (setting the first and the second deriva-tives of P with respect to V equal to zero at the experimentalcritical pressure and temperature) and

V c , EOS ¼ ξV c , experimental ð18ÞIn the present study ξ = 1.1. For the heavy compoundsdecomposing below their critical temperatures it seems expedi-ent to simplify the model by taking c = 0 and removing the

adjustment to the imaginary experimental critical pressure. Insuch case the independent adjustable parameters are m and σ , while a and ε/k - the dependent calculated ones (see Tables 1and 2).

It comes therefore into view that the proposed approachrequires even less substance-dependent adjustable parametersthan 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 exam-ple, for many organic compounds c might be obtained as follows

c ¼ - 1:6049 þ 1:3440ðσ Ã109Þ- 0:3943ðσ Ã109Þ2

þ 0:0417ðσ Ã109Þ3 ð19Þ

Table 1. Parameters for the Compounds with the Experimentally Available Critical Points

independent adjustable parameters dependent calculated parameters

compound m c (L/mol) κ AB ε AB/k (K) 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 473.630CH3OH 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

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

independent

adjustable parameters

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





Implementation of eq 19 along with possible ideas for general-izing 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 completeset of thermodynamic properties in the entire thermodynamicphase space as achieved by the multiparameter expressions

implemented for evaluating the pure compound pseudoexperi-mental data (see for example refs 40-42). Its obvious and toughcompetitor could rather be SAFT-VR-Mie,14 which howeverpresents a more complex and deeper rooted in molecular theory approach. The doubtless advantages of the latter model inpredicting the auxiliary thermodynamic properties have beendemonstrated by Lafitte et al.15,16 The outstandingly accurate

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- theproposed EOS, red lines- SAFT-VR-Mie,blue lines-PR EOS.

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

Figure 5. Densities and sound velocities of liquid n-hexane. Experi-mental data:49O (black) - 293.15 K, O (blue) - 333.15 K, O (red) -373.15 K. 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 Shaw 43 should be noticed as well. Taking intoaccount the fact that SAFT-VR-Mie does not exhibit the non-realistic phase splits, it should be considered as one of the most

successful and promising versions of SAFT. The EOS of Pengand Robinson44 (PR) in its original form attached by the Soave’stemperature dependency has also been selected for the compar-ison as a reference model. The latter temperature dependence inspite of its inconsistency at high temperatures18 might be

Figure 6. Thermodynamic properties of liquid n-pentadecane. Experimental data: O (black)- 303.15 K,50O (blue) - 343.15 K,50O (red)- 383.15 K,50b(blue)-343.15K,51b (red)-383.15K,51b (green)-433.15K.51Solidlines- theproposedEOS,dashedlines-SAFT-VR-Mie,dot-dashed lines-PREOS.






F i g u r e

9 .

D e n s i t i e s a n d s o u n d v e l o c i t i e s o f l i q u i d n - h e x a t r i a c o n -

t a n e . E x p e r i m e n t a l d a t a :

5 2

O

( b l a c k ) -

3 7 3 . 1 5

K ,

O

( b l u e ) -

3 8 3 . 1 5 K , O

( r e d ) -

3 9 3 . 1 5 K , O

( g r e e n ) -

4 0 3 . 1 5 K . S o l i d l i n e s -

t h e p r o p o s e d E O S , d a s h e d l i n e s -

S A F T - V R - M i e , d o t - d a s h e d l i n e s

-

P R E O S .

F i g u r e 8 .

D e n s i t i e s

a n d s o u n d v e l o c i t i e s o f l i q u i d n - o c t a c o s a n e .

E x p e r i m e n t a l d a t a :

5

2

O

( b l a c k ) -

3 6 3 . 1 5 K , O

( b l u e ) -

3 8 3 . 1 5

K , O

( r e d ) -

4 0 3 . 1 5

K . S o l i d l i n e s -

t h e p r o p o s e d E O S , d a s h e d

l i n e s -

S A F T - V R - M

i e , d o t - d a s h e d l i n e s -

P R E O S .

F i g u r e

7 .

D e n s i t i e s a n d s o u n d v e l o c i t i e

s o f l i q u i d n - o c t a d e -

c a n e . E x p e r i m e n t a l d a t a :

3 2

O

( b l a c k ) - 3

2 3 . 1 5 K , O

( b l u e ) -

3 5 3 . 1 5

K , O

( r e d ) -

3 8 3 . 1 5 K . S o l i d l i n e s -

t h e p r o p o s e d E O S ,

d a s h e d l i n e s -

S A F T - V R - M i e , d o t - d a s h e d l i n e s -

P R E O S .








F i g u r e 1 0 .

S a t u r a t e d p h a s e p r o p e r t i e s o f w

a t e r . O , b

-

P s e u d o e x p e r i m e n t a l d a t a .

4 1

B l a c k l i n e s -

t h e

p r o p o s e d E O S , r e d l i n e s -

S A F T - V R - M i e , b l u e l i n e s -

P R E O S .

F i g u r e 1 1 .

S a t u r a t e d p h a s e p r o

p e r t i e s o f m e t h a n o l . O , b

-


4 0


t h e p r o p o s e d E O S , r e d l i n e s -

S A F T - V R - M i e , b

l u e l i n e s -

P R E O S .







F i g u r e 1 2 .

S a t u r a t e d p h a s e p r o p e r t i e s o f c a r b o n d i o x i d e . O , b

-


4 2


t h e p r o p o s e d E O S , r e d l i n e

s

-


l u e l i n e s -

P R E O S .

F i g u r e

1 3 .

J o u l e - T h o m s o n i n v e r s i o n c u

r v e o f m e t h a n o l . O

-


4 0


t h e p r o p o s e d E O S , r e d

l i n e s -


l u e l i n e s -

P R E

O S .







F i g u r e 1 4 .

T h e r m o d y n a m i c p r o p e r t i e s o

f w a t e r i n o n e - p h a s e r e g i o n a s p r e d i c t e d b y t h

e p r o p o s e d

E O S . P s e u d o e x p e r i m e n t a l d a t a :

4 1

O

( b l a c

k ) -

3 0 0 b a r , O

( b l u e ) -

5 0 0 b a r , O

( d a r k g r e e n ) -

7 5 0

b a r , O

( p u r p l e ) -

1 0 0 0 b a r , O

( l i g h t g r e e n ) -

2 0 0 0 b a r , O

( r e d ) -

4 0 0 0 b a r , O

( b r o w n

) -

1 0 , 0 0 0

b a r . C a l c u l a t e d d a t a -

s o l i d l i n e s .

F i g u r e 1 5 .

T h e r m o d y n a m i c p r o p e r t i e s o f w a t e r i n o n e - p h a s e r e g i o n a s p r e d i c t e d b y S A F T - V R - M i e .

F o r l e g e n d s e e F i g u r e 1 4 .







F i g u r e

1 7 .

T h e r m o d y n a m i c

p r o p e r t i e s o f m e t h a n o l i n o n e - p h a s e r e g i o

n a s p r e d i c t e d b y t h e

p r o p o s e d E O S . P s e u d o e x p e r i m

e n t a l d a t a :

4 0

O

( b l a c k ) -

1 0 0 b a r ,

O

( b l u e ) -

4 0 0 b a r ,

O

( r e d ) -

7 0 0 b a r . C a l c u l a t e d d a t a -

s o l i d l i n e s .

F i g u r e

1 6 .

T h e r m o d y n a m i c p r o p e r t i e s o

f w a t e r i n o n e - p h a s e r e g i o n a s p r e d i c t e d b y P R E O S . F o r

l e g e n d s e e F i g u r e 1 4 .







F i g u r e 1 8 .

T h e r m o d y n a m i c p r o p e r t i e s o f m e t h a n o l i n o n e - p h a s e r e g i o n a s p r e d i c t e d b y

S A F T - V R -

M i e . F o r l e g e n d s e e F i g u r e 1 7 .

F i g u r e 1 9 .

T h e r m o d y n a m i c p r

o p e r t i e s o f m e t h a n o l i n o n e - p h a s e r e g i o n a s p r e d i c t e d b y P R E O S . F o r

l e g e n d s e e F i g u r e 1 7 .







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 theSAFT-VR-Mie’s parameters in the literature.15,16,47

Although both the proposed approach and SAFT-VR-Mie arethepolynomials of high order, their numerical behavior is analogous

to cubic equations. Hence, the similar procedures have been imple-mented for all the considered models. The calculations have beenperformedusing theMathematica7 software(the pertinent routinescan be obtained from the author by request).

’RESULTS

Figure 2 illustrates the numerical contributions of the partsconstituting the models considered in the present study. As

Figure 20. Thermodynamicproperties of carbon dioxide in one-phase region as predictedby theproposedEOS. Pseudoexperimental data:42O (black)- 100 bar,O (blue)- 200 bar,O (dark green)- 500 bar,O (purple)- 1000bar,O (light green)- 2000 bar,O (red)- 4000 bar,O (brown)- 8000

bar. Calculated data- solid lines.






F i g u r e

2 2 .

T h e r m o d y n a m i c p

r o p e r t i e s o f c a r b o n d i o x i d e i n o n e - p h a s e r e g

i o n a s p r e d i c t e d b y P R

E O S . F o r l e g e n d s e e F i g u r e 2 0

.

F i g u r e

2 1 .

T h e r m o d y n a m i c p r o p e r t i e s o f c a r b o n d i o x i d e i n o n e - p h a s e r e g i o n a s p r e d i c t e d b y

S A F T - V R - M i e . F o r l e g e n d s e e F i g u r e 2 0 .







follows from thefigure, the cubic EOS’s cohesive term hasa smallnumerical input comparing to the SAFT’s parts. It comes there-fore into view that the latter parts play a dominant role in theproposed approach.

Figure 3 depicts the vapor pressure lines of several compoundsconsidered in the present study. It can be seen that all threemodels can be characterized by a comparable accuracy in modeling

vapor pressures. The proposed EOS is the best estimator of thecarbon dioxide’s vapor pressure data while it is the less successfulone in the case of methanol. If necessary, developing a cubicEOS-type temperature dependency for the parameter a in eq 1that significantly improves its precision might be considered.

Figure 4 shows the saturated phase densities of three n-alkanes. 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 bediscussed henceforth). Neglecting the problem of wrong estima-tion of critical data, SAFT-VR-Mie is a sufficiently preciseestimator of thesaturatedphase densities.However the proposedEOS has yet a doubtless advantage.

Figure 5 presents the densities and sound velocities of liquid n-

hexane. It can be seen that both PR EOS and SAFT-VR-Mie donot succeed in precise description of these data. PR EOS is aparticularly inaccurate estimator of the sound velocities. Incontrast, the proposed EOS predicts the data in a very accuratemanner. The same should be concluded considering the resultsfor heavier homologues (see Figures 6-9). In particular, itappears that PR EOS becomes especially imprecise, which makesthis model irrelevant for design of high pressure processesimplementing these compounds.

One could consider replacing the generalized expressions of the PR EOS’s parameters by the compound-specific ones as inthe case of SAFT models. Indeed, this practice could improve theaccuracy in modeling the saturated and the low-pressure liquiddensities. However it could not address the problems of high

pressures and auxiliary properties due to the principal difficulty indescription the liquid compressibility characteristic for cubicequations.8 These facts outline the fundamental advantage of SAFT models. It particular, it can be seen that SAFT-VR-Mieexhibits stably satisfactorily, although imprecise predictions for

both densities and sound velocities regardlessof the chain length.It comes into view that SAFT-VR-Mie establishes not entirely exact slopes of both isobars and isotherms. These facts could raisequeries regarding the obviousness of the superior considerationof advanced molecular approaches and simulated data instead of tracking the trends established by the real compound’s experi-mental results. Indeed, it appears that in spite of its weakertheoretical basis, the proposed EOS is capable of the almostprecise modeling of both densities and sound velocities in the

entire experimentally available range for all the heavy n-alkanesconsidered. Thus, one could assume that this model could be areliable estimator of the currently unavailable data, for example athigher pressures and temperatures. The proposed EOS has alsoan apparent advantage over PR EOS and SAFT-VR-Mie inpredicting 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 formodeling than normal paraffins. Figure 10 depicts the entire setof the saturated phase properties of water. It can be seen that theproposed model has the doubtless superiority over PR EOS andSAFT-VR-Mie; however, yet its results are not always precise.Similar conclusions could be reached also in the cases of methanol

and carbon dioxide (Figures 11 and 12). It can be also seen thatthe overestimation of critical points might significantly aff ect theaccuracy of SAFT-VR-Mie. In addition, the proposed EOS is nota particularly successful estimator of the carbon dioxide ’s C V .

Figure 13 shows the Joule-Thomson inversion curve of methanol. Concerning this figure it should be pointed out thatthe pertinent steam table keeps going only until 700 bar. Thus,

the evaluation of the Joule-Thomson inversion data at the higherpressures might be considered as questionable. Anyways, theadvantage of the proposed EOS over PR EOS and SAFT-VR-Mieand its particular accuracy in the confident data range should bepointed out.

The doubtless superiority of the proposed EOS persists also inpredicting the high pressure single phase data (see Figures 14-22). However its tendency of underestimating the high-density heat capacity data might raise concerns. It appears that thesuccess in predicting densities and the property meanly basedon the first derivates such as sound velocities does not alwaysguarantee the same accuracy in estimating the property based onthe second derivatives, such as the isochoric heat capacity. Inaddition, it can be seen that the association terms attached to

both SAFT models do not adequately address the complicatedtrends established by the auxiliary properties of water andmethanol. Moreover, it comes into view that the proposedEOS does not have significant advantage over PR EOS inpredicting the data carbon dioxide’s, while the trends established

by SAFT-VR-Mie for this compound at very high pressures seemunacceptable.

’CONCLUSIONS

The aim of the present study was creating a concept of thehybrid model gathering the strong sides of both cubic EOS andSAFT approaches. Whereas the popular Cubic Plus Associationequation7 considers a cubic EOS as a basis model and attaches it

by the SAFT’s association term, the proposed idea was attachinga revised SAFT of Chapman et al.13 by the cubic EOS’s cohesiveterm. Such a selection is explained by the fact that the latter

version of SAFT does not exhibit the nonrealistic phase splitscharacteristic for several othersversions.17-22 In order to addressthe issue of the numerical pitfalls generated by the temperature-dependent reduced densities, the Carnahan-Starling’s HS termhas been modified by making the covolumes temperature-independent.8 Assuming the fact that the chain formationreduces the space available for dispersive interactions andfollowing the proposal of Chen and Mi,24 the Chapman’s et al.13

dispersive term has been multiplied by the ratio of the chain termto the hard-sphere term. The proposed model contains 5substance-characteristic parameters: two of them are indepen-

dent and should be acquired by fitting experimental data andthree others are dependent and be obtained by the proceduretypically used for cubic equations, namely solving the criticalpoint conditions. Modeling polar compounds implies the asso-ciation term and its two additional adjustable parameters. It isdemonstrated that the resulting model on one hand preserves thecharacteristic for SAFT accuracy in estimating the liquid com-pressibility, and on the other one - the characteristic for cubicequations capability of simultaneous modeling of critical andsubcritical data. Moreover, on the basis of the comprehensive setof thermodynamic properties it has been demonstrated that theproposed concept has a superiority comparing even to one of themost successful versions of SAFT such as the SAFT-VR-Mie.





’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 AmericanChemical Society Petroleum Research Fund for support of thisresearch, grant N° PRF#47338-B6.

’ LIST OF SYMBOLS A Helmholtz free energy C V isochoric heat capacity C P isobaric heat capacity m number of segments M the number of association sites on each molecule N Av Avogadro’s number P pressureR universal gas constant

T temperatureV molar volumeW speed of sound X A the mole fraction of molecules not bonded at site A

Greek letters

Δ association strengthκ AB the volume of interaction between the sites A and Bε AB interaction of association energy η reduced density ε /k segment energy parameter divided by Boltzmann’s constantθ(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 stateHS hard sphereSAFT statistical association fluid theory

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Hybridizing SAFT and Cubic EOS What Can Be Achieved

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