Novel Methodology for Analysis and Evaluation of SAFT-Type

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A novel methodology for analysis and evaluation of SAFT-type equations of state. Ilya Polishuk, Romain Privat, and Jean-Noeal Jaubert Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie4020155 • Publication Date (Web): 23 Aug 2013 Downloaded from http://pubs.acs.org on August 26, 2013

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Industrial & Engineering Chemistry Research

A novel methodology for analysis and evaluation of SAFT-type equations of state

Ilya Polishuk1, Romain Privat2, Jean-Noël Jaubert2 1

2

Department of Chemical Engineering, Ariel University, 40700, Ariel, Israel.

Université de Lorraine, ENSIC (École Nationale Supérieure des Industries Chimiques), LRGP (UMR CNRS 7274), 1 rue Grandville, BP 20451, Nancy cedex 9, France.

E-mails: [email protected], [email protected], [email protected]

Abstract. The current study demonstrates that the pure-component critical temperatures generated by SAFT-type equations of state (EoS) are independent of the hard-core diameter (σ). This remarkable feature leads to the development of a simple tool, called a global map, for a comprehensive investigation of the sophisticated SAFT-models under consideration. Practically, a net of critical isotherms in the m (chain length), u ° k or ε k (segment energy parameter) plane is plotted not only for the ordinary but also for the additional unrealistic critical temperatures. Such a map makes it possible to check whether unrealistic phase equilibria will be predicted by the considered SAFT model. In addition, the present study points out that the numerical pitfalls encountered at ambient temperature with the CK-SAFT and the PC-SAFT EoS can be eliminated by refitting the values of the universal constants.

Keywords: SAFT, equation of state, critical point, pure components, ionic liquids, pitfall, global map, segment diameter.

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I. Introduction. Comprehensive analysis of the SAFT-type equations of state (EoS) is a challenging problem that has not been satisfactorily solved yet. Such an analysis is particularly important in the context of the possible existence of an additional fictitious pure compound critical point, which is a well-identified pitfall of several SAFT-type models.1-6 Among the major obstacles for the development of methodologies for a comprehensive analysis are the complicated expressions and the multi-parameter nature of the SAFT-type equations. This multiplicity of parameters typically restricts the investigations to the particular combinations of molecular parameters. In addition, unlike most cubic equations7-10, in the case of SAFT models, evaluation of expressions relating values of the EoS parameters to the critical properties is very tedious and virtually impractical. Thus far, the critical properties have been empirically inter-related with SAFT parameters11, or sometimes considered in fitting procedures.12-21 At the same time, an advantage of examining the critical states for comprehensive and overall analysis of SAFTtype models does not seem to be widely recognized. The current study is one of the first steps in development of comprehensive and overall methodologies for analyzing and evaluation of SAFT-type EoS. In order to stress the necessity of developing efficient analysis tools for SAFT-type EoS, the first part of the paper shows how some particular combinations of SAFT parameters - obtained from data fitting can lead to unrealistic phase equilibria. Although this kind of results was extensively discussed throughout previous studies1-6, evidence is given for the first time that in the case of the PC-SAFT22 EoS these unrealistic phenomena may arise at ambient conditions, in temperature and pressure domains of industrial applications. The second part of the paper presents our main result. During the work with their various versions, we have observed that the predicted critical temperatures are dependent only on the


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effective number of segments (denoted m) and the segment energy (denoted ε), but not on σ (the temperature-independent segment diameter). In the following sections we will provide explanations of this remarkable phenomenon for particular versions of SAFT and then demonstrate how it can be implemented for the comprehensive analysis of these models.

II. On the unrealistic phenomena observed at ambient conditions. In a recent paper23, the PC-SAFT EoS has been implemented in various ion and molecularbased strategies for correlating the high-pressure densities of several ionic liquids and predicting phase equilibria of their mixtures with carbon dioxide. We have selected this work in order to show for the first time, a modeling of experimental data that is affected by the aforementioned pitfall (note that until now, the pitfall was only encountered outside classical temperature and pressure domains of interest). Examining the density figures of the paper by Ji et al.23, one could observe the characteristic overestimations of the low-temperature data at high pressures. Although this phenomenon might seem insignificant, it is in fact related to the fundamental numerical problem of the PCSAFT EoS, namely predicting of additional unrealistic saturation curve for pure compounds. In order to support this statement, let us choose as an example, the ionic liquid [C3MIM][BF4] modeled with the so-called “simplest strategy”23 (molecular approach without the association and polar contributions), and let us consider a wide pressure range. Fig. 1, in which the experimental data were reported from Ref. 24, demonstrates that the above mentioned overestimations of the high-pressure density data by the PC-SAFT EoS actually come from the unusual shape of the isotherms generated by this model. Indeed, after classically exhibiting a local maximum and a local minimum (see Fig. 1), the isotherms do not directly approach the infinity pressure (as in the case of the pitfall-free equations like GSAFT+Cubic25), but reach a second maximum, turn down, and then turn up again. This



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unrealistic shape is actually responsible for the appearance of an additional fictitious critical point at 393.7 K and 1,002 MPa ending an unrealistic pure-compound LLE curve (see Fig. 2). In addition, Fig. 1 highlights that a decrease of the temperature causes a decrease of the pressure of the second local maximum on the isotherm. Consequently, at certain low temperatures, the physically-meaningful part of the isotherms reduces to the vapor branch found at extremely low densities (for more details see references 1 and 3). This instance points out the great interest to be able to a priori identify the domains in the molecular parameters space where this characteristic pitfall of SAFT EoS occurs.

III. Theory. 1. General form of SAFT EoS The Statistical Associating Fluid Theory postulates that a molecule is a chain of m elementary segments (or monomers). If all the segments are identical, the chain is called homonuclear or, otherwise, heteronuclear.

There are mainly two general forms of the SAFT molar Helmholtz energy for associating molecules:  a a Ideal a HS a Disp seg a Chain of seg a Assoc Form 1: = + + + +  RT RT RT RT RT RT 144 244 3  Ref real seg  a /( RT )  Ideal HS Chain of HS a a a a a Disp chain a Assoc  Form 2 : = + + + +  RT RT RT 42444 RT 3 RT RT 144  Ref HS chain a /( RT ) 

(1)

where a Ideal denotes the molar Helmholtz energy of a pure ideal gas. The residual hard-sphere term a HS accounts for repulsive effects in a spherical-segment fluid. Residual means here departure from the ideal gas behavior.


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The dispersive term accounts for dispersion (or attraction) effects. In the first form of the SAFT EoS, this term, denoted a Disp seg , represents the amount of Helmholtz energy to add to the Helmholtz energy of repulsive hard-sphere segments to obtain the Helmholtz energy of real segments (or monomers) subject to both dispersive and repulsive forces. It should be

pointed out that real segment must be understood as opposed to hard-sphere segment. In the second form of the EoS, the dispersion term, denoted a Disp chain , represents the amount of Helmholtz energy to add to the Helmholtz energy of a hard-chain fluid (i.e., a fluid containing chains of hard spheres) to obtain the Helmholtz energy of a real non-associating compound. The chain terms a Chain are due to the formation of covalent bonds either between real segments (form 1 of SAFT EoS) or between hard-sphere segments (form 2). Regardless of the form of the SAFT EoS, a Assoc is the contribution due to intermolecular association between real chains of monomers. a Ref is a reference-fluid term resulting from an arbitrary arrangement of the different

Helmholtz energy terms in Eq. (1). Its definition is linked to how the particular SAFT EoS are built and to the common practices of SAFT developers. Considering form 1, the reference fluid is made up of real segments (or real monomers) not bonded together, i.e., of not yet formed molecules (molecules being chains of segments). Using the second form, the reference fluid contains chains of hard spheres not yet subject to dispersion effects. The pure fluid EoS written in terms of molar compressibility factor (or pressure) is obtained by simple derivation of the molar Helmholtz energy term with respect to the reduced density

 ∂ [ a / ( RT ) ]  η at a fixed temperature: z (T , η) = η   . Consequently, Eq. (1) leads to: ∂η  T Form 1: z = 1 + z HS + z Disp seg + z Chain of seg + z Assoc  HS Chain of HS + z Disp chain + z Assoc Form 2 : z = 1 + z + z


(2)


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The SAFT EoS for pure species are generally expressed as functions of the two following independent state variables: the absolute temperature T and either the reduced density (i.e. the packing fraction) η or the reduced molar density ρ* . Furthermore, application for pure components requires knowledge of several input parameters characterizing the segments and the intermolecular energies. In practice, the nonassociating terms of SAFT EoS generally involve three or four input parameters. In the present section, it is demonstrated that the critical temperatures and the critical reduced densities of a pure compound estimated with the SAFT-type models are completely independent of the value of the input parameter σ. The proof is divided into two parts: -

the first part is devoted to show that the residual Helmholtz energies of pure compounds calculated from SAFT models can be expressed as functions of T, η (or

ρ* ) and a series of input parameters except σ . -

in the second part, it is shown that the solutions of the system formed by the critical constraints, namely the critical temperature and the critical packing fraction ηc (or the critical reduced molar density ρ*c ) are also σ-independent.

However before proceeding to the proof two remarks should be made: 1) Unlike SAFT models, cubic EoS are often expressed as functions of T and v (molar volume). Reciprocally, it could be possible to express SAFT models as functions of T and v. The choice of the volumetric state variable (v, η, ρ* …) involved in the EoS generally

stems from the way used to derive the EoS, but also from EoS developers' habits. All the formulations of the EoS (using v, η or ρ*) are strictly equivalent since a simple change of variable makes it possible to switch from one variable to another one. Indeed,


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η=

π N A md 3 σ3 , where NA is the Avogadro constant, d is a and ρ* = ρσ3 = 6 v v

temperature-dependent segment diameter and σ is a temperature-independent segment diameter (characterizing the diameter of molecules colliding at a zero-temperature limit). 2) In this paper, SAFT EoS Helmholtz energies are classically expressed as functions of T and η (or T and ρ*) in agreement with the original formulations of these EoS and because the critical constraints can be simply written using these variables. Although less natural, it would have been also possible to perform the same analysis (i.e., to prove that Tc is σ-independent) by considering Helmholtz energies expressed as functions of T and v.

Four different SAFT models deriving either from the first or the second form of Eq. (1) are considered: -

the CK-SAFT EoS by Huang and Radosz26 which can be expressed using either form 1 or form 2 of Eq. (1).

-

the Soft-SAFT EoS by Blas and Vega27,28 relying on the 1st form of Eq. (1),

-

the SAFT-VR EoS by Gil-Villegas et al.29 relying on the 1st form of Eq. (1),

-

the PC-SAFT EoS by Gross and Sadowski22 relying on the 2nd form of Eq. (1).

The non-associating parts of these 4 models are briefly presented in a synthetic way in Table 1. Association terms which are more or less the same for all of these models are separately discussed thereafter.



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2. Proof that the residual Helmholtz energies of SAFT-type EoS, expressed as functions of (T,η) - or (T,ρ*) - do not depend on σ The residual Helmholtz energy

a a Ideal a res − is now denoted . Sections 2.1 to 2.4 deal RT RT RT

with SAFT EoS for non-associating components. Associating terms are discussed in section 2.5.

2.1 The CK-SAFT EoS A brief overview of

HS a Disp seg aChain of seg aCK -SAFT , CK-SAFT and CK-SAFT in Table 1 immediately shows RT RT RT

that these three terms only involve the three input parameters m, u°/k, e/k, the two state variables T and η but not σ (or equivalently v00): res aCK-SAFT (m; u ° / k ; e / k ;T ; η) RT

(3)

2.2 The Soft-SAFT EoS According to Table 1, the two terms

Ref real seg aSOFT-SAFT aChain of seg and SOFT-SAFT making up the nonRT RT

associating Soft–SAFT EoS only involve two reduced state variables: the reduced molar density ρ* = ρσ3 and the reduced temperature T * =

kT . Note in passing, that both these ε

parameters were introduced to express the molecular theory of corresponding states32 and are thus well-adapted to molecular EoS for the same reason that the two reduced variables Tr = T/Tc and vr = v/vc involved in the classical corresponding-state theory are convenient to

express cubic EoS. In addition to the two reduced state variables ( ρ* and T* ) and to the numerous universal constants (aij and the ones involved in the functions ai, bi and Gi31), the knowledge of the


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Chain of seg aSOFT-SAFT chain length input parameter m is required to evaluate the term (see Table 1). RT

Consequently, the residual molar Helmholtz energy of the Soft-SAFT EoS can be written as res aSOFT-SAFT (m; T * ; ρ* ) or, equivalently, as: RT res aSOFT-SAFT (m; ε / k ; T ; ρ* ) RT

(4)

2.3 The SAFT-VR EoS Although the SAFT-VR appears a bit more complicated than the two previous EoS, it res immediately appears that the expression of aSAFT-VR given in Table 1 does not involve the

hard-core diameter σ . res aSAFT-VR (m; λ; ε / k ; T ; η) RT

(5)

2.4 The PC-SAFT EoS

a) The hard-sphere term • According to the definition of the temperature-dependent diameter:

  3( ε / k )   d (T ; ε / k ) = σ 1 − 0.12exp  −  T    1444424444 3

(6)

f (T ;ε / k )

it can be seen that the quantity d (T ; ε / k ) / σ is σ -independent (but depends on ε/k and T). • The number density of molecules is defined by: ρ =

quantity ρ* = ρσ3 =

−1 6  η m ⋅ d 3  . As a consequence, the  π 

6  3 −1 η m [ f (T ; ε / k )]  only depends on the state variables T and η, on  π 

the parameters m and ε/k but is independent of σ:

ρ* (T ; η; m; ε / k ) • Similarly, from the definition of the zeta functions:


(7)


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ζn =

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π ρmd n 6

π 3 σn σ ρm 3 [ f (T ; ε / k )]n 6 σ π * = ρ (T ; η; m; ε / k ) ⋅ m ⋅ σn −3 [ f (T ; ε / k )]n 6 =

(8)

it is possible to conclude that the expression of the quantity ζ n σ3−n in the (T,η) base is independent of σ :

ζ n σ3−n  (T ; η; m; ε / k )  

(9)

In other words, one can claim that the quantities σ3ζ 0 , σ2ζ1 , σζ 2 and ζ3 are independent of σ in the (T,η) base. • It becomes now possible to express the residual hard-sphere term of the PC-SAFT EoS

(given in Table 1) with respect to quantities that are σ-independent in the (T,η) base.  ζ 3   3ζ1ζ 2 ζ 32 a HS  = m  2 2 − 1 ln (1 − ζ 3 ) + + 2  RT 1 ζ − ζ ( )  ζ 0 ζ3  ζ ζ 1 − ζ 0 3 ( )  3 0 3    3 3  3 σ2 ζ1 ( σζ 2 ) σζ 2 ) σζ 2 ) ( (   (10)   =m − 1 ln (1 − ζ 3 ) + + 2 3 3   σ3 ζ ζ 2  σ ζ 0 (1 − ζ 3 ) ζ 3 σ ζ 0 (1 − ζ 3 ) 0 3   

(

=

)

(

(

)

)

(

)

a HS (T ; η; m; ε / k ) RT

b) The chain term • The radial distribution function g hs of the hard-sphere fluid is defined by:

g HS =

1 3ζ 2 2ζ 22 2 + d (T ; ε / k ) + d ( T ; ε / k ) [ ] 1 − ζ3 (1 − ζ 3 )2 (1 − ζ3 )3

(11)

This function can be equivalently expressed in the (T,η) base, with respect to σ-independent parameters:

g HS =

2 1 3 f (T ; ε / k ) 2 2[ f (T ; ε / k )] + (σζ 2 ) + ( σζ ) = g HS (T ; η; m; ε / k ) (12) 2 2 3 1 − ζ3 (1 − ζ 3 ) (1 − ζ 3 )

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Chain of HS aPC-SAFT • According to the expression given in Table 1, = (1 − m) ⋅ ln  g HS (T ; η; m; ε / k )    RT

is σ-independent in the (T,η) base. c) The dispersion term

• According to their definitions22 the coefficients ai and bi are σ-independent in the (T,η) base. The same conclusions can be reached for I1 , I 2 and C1 . Note that all these quantities depend on (T ; η; m; ε / k ) . • According to the expression given in Table 1, Disp chain aPC-SAFT = −2π ρσ3 I1m 2 RT

( )

ε kT

( )

− π ρσ3 mC1I 2 m 2

= −2πρ* (T ; η; m; ε / k ) ⋅ I1m 2

ε kT

( kTε )

2

− πρ* (T ; η; m; ε / k ) ⋅ mC1I 2 m 2

( kTε )

2

(13)

Disp chain (T ; η; m; ε / k ) aPC-SAFT = RT

Consequently, it can be concluded that the residual molar Helmholtz energy of a fluid estimated from the PC-SAFT EoS is σ-independent in the (T,η) base. res aPC-SAFT (m; ε / k ; T ; η) RT

(14)

2.5 Association terms of SAFT EoS For pure components, the molar Helmholtz energy change due to association is more or less the same regardless of the SAFT EoS considered. In the present paper, the expression proposed by Huang and Radosz (for the CK-SAFT EoS) is adopted, i.e.: a assoc = RT

∑ A

A  A X  1 ln X −  + M 2  2 

(15)

Where M is the number of association sites on each molecule and XA is the mole fraction of molecules not bonded at site A. This fraction is determined by solving the following equation:



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−1    B AB  X A = 1 + N ρX ∆  Av   B       ε AB   3 AB AB ∆ = g ( d ) exp    − 1 σ κ     kT     g (d ) = g HS = 1 − 0.5η  (1 − η)3 

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∑

(16)

The ∆ AB parameter characterizes the association bonds and depends on the radial distribution function g. In the case of the CK-SAFT EoS, this function is approximated by the hard-sphere radial distribution function which is clearly σ-independent in the (T,η) base. κ AB and ε AB are additional constant input parameters characterizing the association. Eq. (16) shows that the quantity

∆ AB σ

3

= g ( d )  eε 

AB

/( kT )

− 1 κ AB is also σ-independent in the 

(T,η) base. Consequently, the mole fraction of molecules not bonded at site A can be expressed with respect to σ-independent quantities.

X

A

 = 1 + N Av 

∑ ( ρσ ) 3

B

 X  3   σ     B∆

AB

−1

(17)

XA is thus also σ-independent in the (T,η) base as well as aassoc: a assoc (m; ε AB ; κ AB ; M ; T ; η) RT

(18)

3. Application of the critical constraints to SAFT-type EoS. According to classical thermodynamics, the coordinates of a pure-component critical point are found by solving the critical constraints which can be expressed either in terms of reduced packing fraction:

(∂ 2 a res / ∂η2 )T ,Π  3 res 3  (∂ a / ∂η )T ,Π

= 0 = 0


(19)

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(e.g., for the CK-SAFT, SAFT-VR and PC-SAFT EoS) or in terms of reduced molar density:

(∂ 2 a res / ∂ρ*2 )T ,Π  3 res *3  (∂ a / ∂ρ )T ,Π

= 0 = 0

(20)

(e.g., for the Soft-SAFT EoS), where Π is a vector containing the set of input parameters related to the SAFT EoS considered. As previously detailed, the residual molar Helmholtz energy is σ-independent when it is expressed either as a function of (T ; η) or as a function of (T ; ρ* ) . Therefore, the solving of Eq. (19) or Eq. (20) allows determination of critical values (Tc ; ηc ) or (Tc ; ρ*c ) that are independent of σ.

IV. Results. 1. Global maps of critical temperatures This remarkable feature can substantially simplify the comprehensive analysis of SAFT EoS in general and in particular can be implemented for detecting and evaluating the extent of a characteristic numerical pitfall, namely the appearance of the additional fictitious pure compound critical points1-6. For the time being, the only way to know whether the calculation of the state of a pure component might return unrealistic predictions was to perform the pertinent calculation and to analyze a posteriori the results for particular combination of molecular parameters. However omitting σ allows drawing of the global maps of the generated unrealistic critical temperatures in the projection of ε/k (or u ° / k for CK-SAFT) and m. Such a global map for the CK-SAFT EoS is presented in Fig. 3 and highlights that the temperatures of the unrealistic critical points are mainly defined by u ° / k and few depend on m. In addition, in the range of values of u ° / k typically obtained for ordinary compounds



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(~200-300), the model predicts particularly high additional (and unrealistic) critical temperatures which vary between 100 and 400 K. It should be pointed out that appearance of the unrealistic critical points affect not only the pure compound phase diagrams, but also the phase diagrams of mixtures. For example, in a binary system, the proximity of the additional critical temperature of the heavy compound to the genuine critical temperature of the light compound can induce completely unrealistic phase diagrams as detailed below. Although the appearance of the unrealistic critical points can probably not be completely avoided, the SAFT models should be modified in a way that substantially decreases the unrealistic critical temperatures, far below the genuine critical points of the industriallyimportant gases such as carbon dioxide, methane or nitrogen. As an example1, such an effect can be obtained with the CK-SAFT EoS by deleting D29 in the Chen-Kreglewski’s parameter matrix.30 The corresponding global map shown in Fig. 4 demonstrates that in the range of values of u ° / k

typically obtained for ordinary compounds the unrealistic critical

temperatures appear far below 100 K. Fig. 5 compares the realistic critical temperatures yielded by the CK-SAFT EoS with the original universal parameters matrix30 and by the modified CK-SAFT EoS (D29 = 0). Such a figure highlights that both CK-SAFT versions predict nearly the same results and makes it possible to conclude that the modified version of the CK-SAFT EoS can be implemented with the molecular parameters derived for its original version, or, at least, a slight modification of their values. Fig. 6 depicts the global map of temperatures of the additional unrealistic pure compound critical points of non-polar compounds generated by the PC-SAFT EoS with the original universal parameters matrix.22 Several compounds have been placed in order to make the map more demonstrative. As discussed previously3,4, the unrealistic critical temperatures are usually low enough and may present danger mostly for the diagrams involving the solid phases rather than the fluid ones.4 However, in the case of certain heavy compounds the


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situation might be different. In particular, the present global map confirms the observations previously made with pure ionic liquids: their predicted additional pure compound critical points can be high enough for generating unrealistic fluid phase diagrams for their mixtures with light compounds such as CO2. This instance is detailed in section IV.3.

2. Simplified global maps for non-associating compounds The methodology previously explained can even be simplified using a change of variables. Indeed, as can be observed in Table 1 with any of the four non-associating SAFT-type EoS, introducing the reduced temperature T* instead of the temperature T (by using the linear relation T* = kT/ε) makes it possible to totally eliminate the segment-energy parameter ε/k from the EoS expressions. Consequently, one can claim that a unique relationship exists between the reduced critical temperature Tc* and the number of segments m. As an illustration, the reduced temperatures of the ordinary and additional critical points calculated by the original PC-SAFT EoS are shown in Fig. 7. This kind of representation has however two drawbacks: -

it is limited to non-associating compounds (since for associating compounds, additional energy parameters εΑΒ appear and consequently, the aforementioned change of variables does not totally eliminate ε in the association term of the EoS),

-

the global maps become less readable due to the use of an abstract reduced critical temperature instead of the critical temperature.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3. On the unrealistic global phase equilibrium diagrams of size-asymmetric binary systems predicted by SAFT EoS For the CO2 + [C3MIM][BF4] system, a Type III33,34 phase behavior is expected due to the large difference in sizes between the two pure compounds. Figs. 8 and 9 demonstrate that the unrealistic saturation pressure curve of pure [C3MIM][BF4] seriously affects the fluid-phase equilibria encountered in a temperature range where industrial applications are developed. Fig. 8 indeed highlights that for decreasing temperatures, the critical locus starting from the ordinary critical point of the heavy compound ([C3MIM][BF4]) does not reach infinite pressures (as should be observed in the case of a real Type III phase behavior) but falls down and connects a lower critical end point (LCEP), itself connected to a liquid-liquid-vapor threephase line. This three-phase line - which is nearly merged with the vaporization curve of the light compound - is bracketed by the aforementioned LCEP and an upper critical end point (UCEP), which is located in the extreme vicinity of the critical point of the light compound (CO2). Under high pressures, a small liquid-liquid critical line emerges from the fictitious liquid-liquid critical point of the heavy compound and ends at an UCEP. A second threephase line - involving three liquid phases in equilibrium - starts from this UCEP and stops at lower temperatures, at a LCEP. From this LCEP starts a second liquid-liquid critical line which increases with temperature increase and goes until infinite pressures. Finally, a third three-phase line is attached to the fictitious liquid-liquid-vapor triple point of the heavy compound. As a result, contrary to what would be observed in classical Type III phase diagrams, the LL critical curve going to infinite pressures does not continue the ordinary VL critical locus that starts from the ordinary VL critical point of the pure heavy compound. This LL critical curve that originates in the vicinity of the fictitious LL critical point of the pure heavy compound, is necessarily a locus of unrealistic equilibria. In other words, in these circumstances the model


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predicts an incongruous disappearance of the realistic LL phase equilibria at low temperatures that can be relevant for process design. Some unusual isothermal projections of the phase diagram shown in Fig. 8 around the LLLE domain are represented in Fig. 9. In particular, one observes: -

the co-existence of two three-phase equilibria at a same low temperature (310 K),

-

that the experimental points35 at 318 K are in total contradiction with the calculated phase diagram,

-

the appearance of an atypical liquid-liquid-liquid phase equilibrium under high pressure from a temperature of 370 K.

Figs. 8 and 9 were plotted with the PC-SAFT EoS but similar results would be obtained by other versions of SAFT, e.g. the CK-SAFT or the soft-SAFT EoS, affected by the numerical pitfall under consideration.

4. Global maps of associating species It should be pointed out that consideration of the association interactions does not necessarily reduce the appearance of the undesired unrealistic phenomena36 and in most of the cases even increases their extent. The global maps of critical temperatures can be created if the values of the association parameters are the same for all the considered compounds, as in the approach proposed by Paduszyński and Domańska.37 (see Fig. 10). It can also be demonstrated that even higher unrealistic critical temperatures are yielded by the association scheme of Ji et al.23

5. Use of global maps to identify a way of improvement of the PC-SAFT EoS It becomes thus evident, that similarly to the CK-SAFT EoS, the revision of the universal parameters matrix is highly recommended also in the case of the PC-SAFT EoS. Recently



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Liang et al.38 have performed such a revision, while considering not only density and vapor pressure, but also sound velocity data. The results of the global analysis of this version of the PC-SAFT EoS are no less than surprising, while considering the fact that the issue of numerical pitfalls has not been mentioned by the authors (see Fig. 11). In particular, it can be seen that at the realistic values of ε k (below 1000 K) the numerical pitfall takes place only in the very near vicinity of absolute zero temperature, a range which cannot be relevant for the fluid phases of organic substances. At the same time, the higher unrealistic critical temperatures can be obtained only at the unrealistically high values of ε k , which cannot fit any experimental data. Thus it can be concluded that the novel PC-SAFT universal parameter matrix proposed by Liang et al.38 practically eliminates predicting of the additional unrealistic pure compound critical points. Consequently, we believe that in future applications this improved version of the model should replace its original version, especially in the cases of size-asymmetric systems containing heavy fluids. At the same time, it should be pointed out that the original and the revised parameter matrices yield detectable differences for the prediction of the ordinary critical temperatures (see Fig. 12). Consequently, the sets of molecular PC-SAFT parameters working with the original universal parameter matrix are not usable in the PCSAFT EoS working with Liang et al.’s universal parameter matrix. Therefore, the sets of molecular PC-SAFT parameters found in the open literature have to be necessarily reestimated before using Liang et al.’s version of the PC-SAFT EoS. As seen, the proposed global methodology can be useful not only for investigation of the undesired features of SAFT models, but also for the overall evaluation of their realistic predictions. Fig. 13 depicts a global map of the critical temperatures yielded by the pitfall-free SAFT-VR EoS29 for different values of λ. It can be observed that while λ = 1.5 provides results similar to other SAFT models, the limiting values of this parameter lead to the


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abnormal outcome of ε / k . Consequently, the current global analysis indicates that although the molecular approach considers 1.1 ≤ λ ≤ 1.829, the values in much narrower range around 1.5 can be assumed as applicable to the real substances.

V. Conclusions. The present work gives evidence that unrealistic predictions of the PC-SAFT EoS may be observed at ambient conditions in binary systems involving a heavy molecule (e.g., an ionic liquid) and a permanent gas (e.g., CO2). In this context, it becomes thus difficult for SAFT users to know whether a calculation can be safely performed. In order to help them, a simple tool is proposed taking advantage of the fact that the critical temperature is always independent of the hard-core diameter σ. Indeed, by tracing out a net of critical isotherms (called a global map), in the space of the molecular parameters of the EoS (σ excepted) (i) for the additional unrealistic pure compound critical points and (ii) for the ordinary critical temperatures it makes it possible to check whether unrealistic phenomena are expected. For a binary system, in which each component is characterized by a set of molecular parameters, it is advised to check that the value of the additional unrealistic critical temperature of the heavy component (e.g. a ionic liquid) does not approach the ordinary critical temperature of the light component (e.g. CO2). In addition, the present study points out that the pitfalls encountered in the CK-SAFT and PC-SAFT EoS are due to inadequate values of the universal constants. By refitting them, it becomes possible to substantially reduce the impact of the pitfall, practically eliminating it.



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Nomenclature a d EoS HS IL(s) k m NA P R SAFT SW T

Molar Helmholtz energy (J/mol) Temperature-dependent segment diameter (Å) Equation of State Hard sphere Ionic Liquid(s) Boltzmann constant (J/K) Number of segments in a molecule – input parameter of SAFT EoS Avogadro constant (mol–1) Pressure (bar) Gas constant (J/mol/K) Statistical Associating Fluid Theory Square well Thermodynamic temperature (K)

T* u°/k

Reduced temperature (adimensional quantity defined by T * = kT / ε ) Depth of square-well potential divided by the Boltzmann constant - input parameter of the CK-SAFT EoS26 (K) Molar volume (L/mol) Monomer mole fraction (mole fraction of molecules not bonded at site A)

v XA

Greek letters ∆AB Strength of interaction between sites A and B (Å3) ε/k Depth of square-well potential divided by the Boltzmann constant - input parameter of several SAFT EoS (K) εAB/k Association energy of interaction between sites A and B (K) Reduced density or packing fraction (adimensional quantity defined by η κAB λ

ρ* σ

Subscripts c disp ideal res seg

η = (π / 6)N A md 3 / v ) Volume of interaction between sites A and B Width of square-well potential - input parameter of SAFT EoS (adimensional input parameter of the SAFT-VR EoS)

Reduced molar density (adimensional quantity defined by ρ* = σ3 / v ) Temperature-independent segment diameter - input parameter of SAFT EoS (Å)

Critical Contribution due to the dispersive attraction Ideal gas Residual quantity (departure from the ideal gas behavior) Segment


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References (1) Polishuk, I. About the Numerical Pitfalls Characteristic for SAFT EOS Models. Fluid Phase Equilib. 2010, 298, 67.

(2) Polishuk, I.; Mulero, A. The Numerical Challenges of SAFT EoS Models. Rev. Chem. Eng. 2011, 27, 241.

(3) Privat, R.; Gani, R.; J.-N. 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. (4) Privat, R.; Conte, E.; Jaubert, J.-N.; Gani, R. Are Safe Results Obtained When SAFT Equations are Applied to Ordinary Chemicals? Part 2: Study of Solid-Liquid Equilibria in Binary Systems. Fluid Phase Equilib. 2012, 318, 61. (5) Yelash, L.; Müller, 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. (6) Yelash, L.; Müller, 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. (7) Anderko, A.; Cubic and Generalized van der Waals Equations. In: Equations of State for Fluids and Fluid Mixtures. Part I.; Elsevier: Amsterdam, 2000.

(8) Wei, Y. S.; Sadus, R. J. Equations of State for the Calculation of Fluid-Phase Equilibria. AIChE J. 2000, 46, 169.

(9) Valderrama, J. O.; The State of the Cubic Equations of State. Ind. Eng. Chem. Res. 2003, 42, 1603.

(10) 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.



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(11) Castro-Marcano, F.; Colina, C. M.; Olivera-Fuentes. C. Parametrization of MolecularBased Equations of State: The PC-SAFT, soft-SAFT, PHSC and PSCT Models. Polish J. Chem. 2006, 80, 37.

(12) Cismondi, M.; Brignole, E. A.; Mollerup, J. Rescaling of three-parameter equations of state: PC-SAFT and SPHCT. Fluid Phase Equilib. 2005, 234, 108. (13) Tan, S. P.; Adidharma, H.; Radosz, M. Recent Advances and Applications of Statistical Associating Fluid Theory. Ind. Eng. Chem. Res. 2008, 47, 8063. (14) Kiselev, S. B.; Ely, J. F. Crossover SAFT Equation of State: Application for Normal Alkanes. Ind. Eng. Chem. Res. 1999, 38, 4993. (15) Adidharma, H.; Radosz, M. Prototype of an Engineering Equation of State for Heterosegmented Polymers. Ind. Eng. Chem. Res. 1998, 37, 4453. (16) McCabe, C.; Kiselev, S. B. A Crossover SAFT-VR Equation of State for Pure fluids: Preliminary Results for Light hydrocarbons. Fluid Phase Equilib. 2004, 219, 3. (17) McCabe, C.; Kiselev, S. B. Application of Crossover Theory to the SAFT-VR Equation of State: SAFT-VRX for Pure Fluids. Ind. Eng. Chem. Res. 2004, 43, 2839. (18) Kiselev, S. B.; Ely, J. F. HRX-SAFT Equation of State for Fluid Mixtures: New Analytical Formulation. J. Phys. Chem. C 2007, 111, 15969. (19) Forte, E.; Llovell, F.; Vega, L. F.; Trusler, J. P. M.; Galindo, A. Application of a Renormalization-Group Treatment to the Statistical Associating Fluid Theory for Potentials of Variable Range (SAFT-VR), J. Chem. Phys. 2011, 134, 154102. (20) Forte, E.; Llovell, F.; Trusler, J. P. M.; Galindo, A. Application of the Statistical Associating Fluid Theory for Potentials of Variable Range (SAFT-VR) Coupled with Renormalisation-Group (RG) Theory to Model the Phase Equilibria and Second-Derivative Properties of Pure Fluids. Fluid Phase Equilib. 2013, 337, 274.


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(21) Polishuk, I. Hybridizing SAFT and Cubic EOS: What Can be Achieved? Ind. Eng. Chem. Res. 2011, 50, 4183.

(22) Gross, J. and Sadowski, G. Pertubed-chain SAFT: an Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (23) Ji, X.; Held, C.; Sadowski, G. Modeling Imidazolium-Based Ionic Liquids with ePCSAFT. Fluid Phase Equilib. 2012, 335, 64. (24) K. R. Harris, M. Kanakubo, L. A. Woolf. Temperature and Pressure Dependence of the Viscosity of the Ionic Liquid 1-Butyl-3-methylimidazolium Tetrafluoroborate: Viscosity and Density Relationships in Ionic Liquids. J. Chem. Eng. Data. 2007, 52, 2425-2430. (25) Polishuk, I. Generalization of SAFT + Cubic equation of state for predicting and correlating thermodynamic properties of heavy organic substances. J. Supercrit. Fluids. 2012, 67, 94-107. (26) Huang, S. H. and Radosz, M. Equation of State for Small, Large, Polydisperse and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (27) Blas, F. J. and Vega, L. F. Thermodynamic Behaviour of Homonuclear and Heteronuclear Lennard-Jones Chains with Association Sites from Simulation and Theory. Molecular Physics. 1997, 92(1), 135.

(28) Blas, F. J. and Vega, L. F. Prediction of Binary and Ternary Diagrams Using the Statistical Associating Fluid Theory (SAFT) Equation of State. Ind. Eng. Chem. Res. 1998, 37, 660. (29) Gil-Villegas, A., Galindo A., Whitehead, P. J., Mills, S. J. and Jackson, G. Statistical Associating Fluid Theory for Chain Molecules with Attractive Potentials of Variable Range. J. Chem. Phys. 1997, 106, 4168.

(30) 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.



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(31) Johnson, J. K., Zollweg, J. A., Gubbins, K. E. The Lennard-Jones Equation of State Revisited. Molecular Physics. 1993, 78(3), 591. (32) Prausnitz, J. M., Lichtenthaler, R. N., Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, third edition. 1999, Prentice-Hall, Upper Saddle

River. (33) van Konynenburg, P. H.; Scott, R. L. Critical Lines and Phase Equilibria in Binary Van Der Waals Mixtures. Philosophical Transactions of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences 1980, 298, 495.

(34) Privat, R.; Jaubert, J.N. Classification of global fluid-phase equilibrium behaviors in binary systems. Chem. Eng. Res. Des. 2013, http://dx.doi.org/10.1016/j.cherd.2013.06.026 (35) Kroon, M. C.; Shariati, A.; Costantini, M.; van Spronsen, J.; Witkamp, G.-J.; Sheldon, R. A.; Peters, C. J. High-Pressure Phase Behavior of Systems with Ionic Liquids: Part V. The Binary System Carbon Dioxide + 1-Butyl-3-methylimidazolium Tetrafluoroborate. J. Chem. Eng. Data 2005, 50, 173.

(36) Polishuk, I. Implementation of Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT), Generalized (G)SAFT+Cubic, and Cubic-Plus-Association (CPA) for Modeling Thermophysical Properties of Selected 1-Alkyl-3-methylimidazolium Ionic Liquids in a Wide Pressure RangeJ. Phys. Chem. A 2013, 117, 2223−2232. (37) Paduszyński, K.; Domańska, U. Thermodynamic Modeling of Ionic Liquid Systems: Development and Detailed Overview of Novel Methodology Based on the PC-SAFT. J. Phys. Chem. B 2012, 116, 5002.

(38) Liang, X.; Maribo-Mogensen, B.; Thomsen, K.; Yan, W; Kontogeorgis, G. M. Approach to Improve Speed of Sound Calculation within PC-SAFT Framework. Ind. Eng. Chem. Res.

2012, 51, 14903.


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Table 1. Presentation of four SAFT EoS for pure components Soft-SAFT SAFT-VR

SAFT EoS Reference papers

CK-SAFT

PC-SAFT

26,30

27,28,31

29

22

Model input parameters characterizing a pure component

m, σ(1), u°/k and e/k (generally set to 10)

m, σ, ε/k

m, λ, σ, ε/k

m, σ, ε/k (Boublik ; Mansoori et al.)

HS

a RT

(Carnahan & Starling) 4η − 3η2 m (1 − η)2 Eqs. (12), (13), (14) in ref. 26

Residual term of the Lennard-Jones EoS (see Eq. (5) in ref 31) 8

(Carnahan & Starling) 4η − 3η2 m (1 − η)2 Eqs. (2), (10), (18) in ref. 29

6

a i (T* )ρ*i + bi (T* )G i (ρ* ) i =1 = i144444 i 1 42444444 3

∑ m

∑∑ i

a Disp seg RT

j

i

 u   η Dij      kT   τ 

j

u u°  e 1  = 1+ ⋅ k k  k T  Eqs. (8), (12); (13), (15) in ref. 26 Dij values are given in ref. 30 with:

∑

a Ref real seg RT

*

3

where ρ = ρσ , and T* = kT/ε

  m  ζ32 3ζ ζ ζ 32  − ζ 0  ln (1 − ζ3 ) + 1 2 +   ζ 0  ζ32 1 − ζ 3 ζ3 (1 − ζ 3 )2    

 πρmd n ζ n = 6 with  d = σ 1 − 0.12 exp −3ε  kT    Eqs. (A.4), (A.6), (A.8) and (A.9) in ref. 22

( )

mβ a1SW + mβ2 a SW 2 with β = 1/ (kT) and: a SW = −4ηε(λ3 − 1)g HS (1; η ) eff  1 1 − 0.5ηeff  HS g (1; ηeff ) = (1 − η )3 eff   2 3 ηeff = c1 (λ)η + c2 (λ)η + c3 (λ)η  SW 1 HS SW a 2 = 2 εK η(∂a1 / ∂η)  4 K HS = (1 − η)  1 + 4η + 4η2

-

Eqs. (10), (18), (22), (33-36), (38) in ref. 29

Continued on next page

– 25 – ACS Paragon Plus Environment


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Table 2. (continued) SAFT EoS

a Chain of seg RT

CK-SAFT

1 − 0.5η  (1 − m) ln  3  (1 − η)  Eq. (16) in ref. 16

Soft-SAFT (1 − m) ln y R (σ) with: LJ  yR = g LJ R exp [ φLJ (σ) / (kT) ] = g R  5 5  LJ = + g 1 a ijρ*i T*j  R  i =1 j=1  φLJ (r = σ) = 0

∑∑

Eqs. (2), (6), (7) in ref. 27 aij values are given in ref. 31 a

PC-SAFT

(1 − m) ln yM (σ) with:  yM = gSW exp [ −ε / (kT)]  SW  SW λ ∂a1SW  HS 1  ∂a1 −  g = g + 4 β  3η ∂λ   ∂η   HS 3 g = (1 − 0.5η) / (1 − η)

-

Eqs. (13), (77), (78) in ref. 29 (1 − m) ln g HS with:

Chain of HS

RT

SAFT-VR

See

a Chain of seg RT

-

-

1 3dζ 2 2d 2ζ 22 + + 1 − ζ 3 (1 − ζ 3 )2 (1 − ζ 3 )3 Eqs. (A.4) and (A.7) in ref. 22

g HS =

−2πρI1m 2

ε σ3 kT

− πρmC1I2 m 2

( kTε )

2

σ3

with: 6  I1 = a i (m)ηi  i=0  6  bi (m)ηi I 2 =  i =0  −1   8η − 2η2 20η − 27η2 + 12η3 − 2η4  + − (1 m)  C1 =  1 + m  (1 − η)4 (1 − η)2 (2 − η)2   

∑

a

Disp chain

RT

See

a

Disp seg

RT

-

-

∑

Eqs. (A.10-A.13), (A.16-A.19) in ref. 22 (1) according to ref. 26, the second input parameter is actually v00. Since a linear relationship exists between v00 and σ3, see Eq. 4 in Ref. 26: v00 = πN Av σ3 / (6τ) with τ = 0.74048 , σ can alternatively be considered as the second input parameter, instead of v00.


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Figures P (bar) 6000 348.15 K

ENLARGEMENT 4000 10

TK

-1

298.15 K

2000 10-2 283.15 K

0 10-1

1

101

1450.

2150.

ρ (g/L)

Figure 1. High-pressure density of the ionic liquid [C3MIM][BF4]. Points - experimental data24, solid lines - PC-SAFT EoS23, dashed lines - GSAFT+Cubic 25.



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105 sat

P (bar) (a) 10

fictitious LL critical point

fictitious LL saturation line (LLE)

0

ordinary VL critical point

10–5 ordinary LV saturation line (VLE 1)

10–10 VLE 2 –15

10

fictitious triple point

320.

360.

400.

500.

700.

Regular scale 105

1300.

T(K)

Log scale

Psat(bar) (b) 100

900.

fictitious LL critical point

ordinary VL critical point

10–5

LLE

VLE 1 10–10 3-phase equilibrium line (triple point)

10–15

VLE 2 0.

0.2

0.4

0.6

0.8

η (reduced density)

Figure 2. General overview of the phase behavior of pure [C3MIM][BF4] modeled with the PC-SAFT EoS. (a) pressure - temperature plane and (b) pressure - reduced density plane.


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700

uo/k

800 K

550 700 K 600 K

400 500 K 400 K

250 300 K 200 K 100 K

100 1

5

m

9

13

Figure 3. A global map of temperatures of the additional unrealistic pure compound critical points of non-polar compounds generated by the CK-SAFT EoS with the original universal parameters matrix26,30 and e/k = 10.



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800

uo/k

90 K

600

80 K 70 K 60 K

400

50 K 40 K

200

30 K 20 K 10 K

0 1

5

m

9

13

Figure 4. A global map of temperatures of the additional unrealistic pure compound critical points of non-polar compounds generated by the modified CK-SAFT EoS with D29 = 0. As in Fig. 3, e/k was set to 10.


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800

uo/k

e/k=10

600

400

200

11 10 00 K 0 90 0 K 0 80 K 70 0 K 60 0 K 50 0 K 40 0 K 0 30 K 0 200 K K

0 1

5

m

9

13

Figure 5. A global map of the ordinary critical temperatures yielded by the CK-SAFT EoS with the original universal parameters matrix30 (black solid lines) and by the modified CKSAFT EoS with D29 = 0 (red dashed lines).



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900

ε/k [PF6] - & [BF ]4

700

[Tf2N] -

500

300

C11 C12C13C14 C15C16 C17C18 C19 C20

C5 C6 C7 C2

C3 C4

C8 C9 C10

410 K 370 K 330 K 290 K 250 K 210 K 170 K 130 K

100 1.5

4.5

m

7.5

10.5

Figure 6. A global map of temperatures of the additional unrealistic pure compound critical points of non-polar compounds generated by the PC-SAFT EoS with the original universal parameters matrix.22 Circles - n-alkanes and squares - ionic liquids treated as inert substances by Ji et al.23


Page 33 of 39

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3.

Tc* =

Tc (ε / k) 2.

Ordinary critical points

Additional (unrealistic) critical points

1.

1.5

4.5

7.5

10.5

m Figure 7. A global map of the reduced critical temperatures generated by the PC-SAFT EoS with the original universal parameters matrix.22


371,2 K

P(bar)

LLLE line

LL critical line

UCEP

LL critical line LCEP

 C3MIM   BF4  − 2 (additional non-physical saturation curve)

P(bar) L

10000

l ica rit c L

e lin

Fictitious LL critical point [C3MIM][BF4]

LLLE

LL critical line

Ordinary VL critical line

VL

1000

VL critical point CO2

 C3MIM   BF4  − 1 (ordinary vapor pressure curve)

Ordinary VL critical point [C3MIM][BF4]

100

VLE

VLLE lines

CO 1519 K

T(K)

2

Unrealistic VLL triple point [C3MIM][BF4] 317,4 K

LCEP

UCEP

[C4MIM][BF 4]-2

LL critical line

311,8 K

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Page 34 of 39

393,8 K 393,9 K


crit ica

l lin

e

M [C 4

IM

][B

]-1 F4

10 200

700

T (K)

1200

1700

Figure 8. Critical locus of the CO2(1) + [C3MIM][BF4](2) system as predicted by the PC-SAFT EoS (using the inert substance parameters of Ji et al.23 and k12 = 0). Left hand side: schematic view. Right hand side: rigorous calculation. Solid lines: critical locus; black dotted line: realistic vapor pressure of [C3MIM][BF4]; red dotted line: unrealistic saturation pressure of [C3MIM][BF4]; green dotted line: realistic vapor pressure of CO2.


Page 35 of 39

P(bar)

P(bar)

600.

500

T = 318 K T = 310 K

400.

LLE-2

LLE-2

LLE-1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


VLL line 1

200. VLL line 2

VLE

LLE-1

VLE

0

0. 0

0.4

x1,y1

0.8

0.

0.4

7400

P(bar)

P(bar) 8800

LLE-2

7000

8400

LLE-3

LLLE line

LLE-4

LLE-2

T = 385 K

T = 375 K

LLE-2 T = 370 K (temperature just below the temperature at which the LLLE line appears)

0.8

9200

8000

P(bar)

x1,y1

LLE-3

8000

7600 6600

7000 0

0.4

x1,y1

0

0.4

x1,y1

0.8

0

0.4

x1,y1

0.8

Figure 9. Five isothermal Pxy phase diagrams predicted by the PC-SAFT EoS (using the inert substance parameters of Ji et al.23 and k12 = 0) for the CO2(1) + [C3MIM][BF4](2) system. Circles: experimental data35.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 39

900

ε/k 700 410

K 370 K 330 K 290 K 250 K 210 K 170 K 130 K

500

300

100 1.5

5.5

m

9.5

13.5

Figure 10. A global map of temperatures of the additional unrealistic pure compound critical points of ionic liquids treated as associating compounds by the PC-SAFT EoS with the molecular parameters of Paduszyński and Domańska.37 Circles - particular ionic liquids.


Page 37 of 39

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


100000

ε/k

K 290 K 170 90 K

10000

50 K 25 K 12 K 6K

1000

3K C1

100 1.00

C3

C2

1 .5

K

0 .7

1.66

m

C4

C5

K

2.32

2.98

Figure 11. A global map of temperatures of the additional unrealistic pure compound critical points of non-polar compounds generated by the PC-SAFT EoS with the universal parameters matrix of Liang et al.38 Circles - the n-alkanes.38



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 39

800

ε/k 600 11 10 00 K 0 90 0 K 400 0 80 K 70 0 K 60 0 K 50 0 K 0 40 K 0 200 30 K 0 200 K K

0 1

5

m

9

13

Figure 12. A global map of the ordinary critical temperatures yielded by the PC-SAFT EoS with the original universal parameters matrix22 (black solid lines) and with parameters matrix of Liang et al.38 (red dashed lines).


Page 39 of 39

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1800

ε/k

1000 K

1200 600 K 10

600

00

K

60 0K

200 K 1000 K

200 K

600 K 200 K

0 1

5

m

9

13

Figure 13. A global map of the critical temperatures yielded by the SAFT-VR EoS29 with λ = 1.5 (black solid lines), λ = 1.1 (blue dashed lines) and λ = 1.8 (red dotted lines).


Novel Methodology for Analysis and Evaluation of SAFT-Type

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