Application of the corresponding-state law to the parameterization of

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Thermodynamics, Transport, and Fluid Mechanics

Application of the corresponding-state law to the parameterization of SAFT-type models: generation and use of “generalized charts” Romain Privat, Edouard Moine, Baptiste Sirjean, Rafiqul Gani, and Jean-Noel Jaubert Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b06083 • Publication Date (Web): 02 May 2019 Downloaded from http://pubs.acs.org on May 3, 2019

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Industrial & Engineering Chemistry Research

Application of the corresponding-state law to the parameterization of SAFT-type models: generation and use of “generalized charts” Romain Privat(a),*, Edouard Moine(a), Baptiste Sirjean(a), Rafiqul Gani(b), Jean-Noël Jaubert(a) (a) Laboratoire Réactions et Génie des Procédés, CNRS, Université de Lorraine, Ecole Nationale Supérieure des Industries Chimiques, 1 rue Grandville, 54000 Nancy, France. (b) PSE for SPEED, Skyttemosen 6, DK-3450 Allerod, Denmark (*) [email protected] Abstract Most of SAFT and cubic equations of state (EoS) for non-associating pure components obey the 3-parameter corresponding-state law meaning that the knowledge of 3 component-specific properties is a prerequisite to apply an EoS to a given pure species. The way used to attribute values to EoS parameters is called “parameterization procedure” here. In this article, it is shown that generalized charts, derived from the corresponding-state theory, can be used advantageously to parameterize SAFT models. SAFT EoS is a well-established class of models frequently used for process simulation. Depending on the EoS parameter set selected, most of SAFT EoS are likely to predict unrealistic phenomena (mainly, the presence of multiple critical points and 3-fluid phase equilibrium regions in pure-component phase diagrams). Generalized charts can be used to detect whether such unrealistic phenomena affect purecomponent phase diagrams in the temperature and pressure domains of interest. As a second application, it is proposed to parameterize SAFT EoS in the same spirit as cubic EoS: the 3 SAFT input parameters are fixed in order to exactly reproduce the experimental critical temperature, critical pressure and one point of the vaporization curve through the acentric factor. It is shown that the procedure is simple to implement and opens the door to a large industrialization of SAFT EoS (i.e., the possibility to determine SAFT input parameters for any compound using a universal and univocal method). Keywords: corresponding states; SAFT; cubic; equations of state; parameterization; generalized chart.

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Introduction The corresponding-state law is a general principle, deduced from empirical observations, postulating a universal (i.e., non component-specific) relationship between a set of dimensionless thermodynamic parameters (generally called “reduced variables”). In practice, the number of reduced parameters may vary depending on the expected accuracy. As a well-known result, the classical Van der Waals, Peng-Robinson (PR) and Soave-Redlich-Kwong (SRK) models obey the corresponding-state law. For a pure substance, these equations of state (EoS) can be written as a universal relationship between the state variables: reduced temperature Tr  T / Tc (where Tc is the critical temperature), reduced pressure Pr  P / Pc (where Pc is the critical pressure) and reduced molar volume vr  v / vc (where vc is the critical molar volume). For the SRK and PR EoS, an additional dimensionless size parameter (the acentric factor  ) is added in the universal relationship which can be written in the general form: Pr  f (Tr , vr ,  ) . It is said that the SRK and PR EoS obey the 3-parameter corresponding state law. To apply such models to a given pure species, one needs to switch from a space of dimensionless reduced state-variables Tr , Pr , vr  to a space involving dimensional state-variables T , P, v  . To do so, these EoS require the knowledge of 3 input parameters: the critical temperature Tc ,exp , critical pressure Pc ,exp and acentric factor exp . What about SAFT EoS? Derived from molecular-potential models, these equations of state consider as input variables, the ones used in the underlying molecular-potential models. For most of them (PCSAFT, CK-SAFT, SOFT-SAFT …), these are: m (segment number),  (segment diameter) and  (energy parameter), described below. Therefore, most of SAFT-type EoS also obey the 3-parameter corresponding-state law. When dealing with this class of models, the reduced state-variables involved in the corresponding-state law are normalized using combinations of the input model parameters ( m ,  ,  ). In this molecular approach, the reduced temperature, pressure and density are denoted: T * , P* and  . In their universal form, SAFT-type EoS for non-associating pure species obey the 3-parameter corresponding state law and therefore, they can be written as a universal relationship between the 3 aforementioned reduced state-variables and the size parameter m . As a consequence, it can be shown that: -

the reduced critical coordinates of a critical point only depend on the m parameter,

-

the reduced coordinates of a liquid-liquid-vapour triple point only depend on the m parameter (note that SAFT-type EoS are known to predict triple points although this behaviour is physically meaningless [1–3]),


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-

the acentric factor of a pure component only depends on m .

These observations can be of special interest when thinking about the parameterization of SAFT-type EoS. The present article aims at discussing two potential applications of such results in terms of EoS parameterization. First, as briefly sketched in a previous work by Polishuk et al. [3], it will be shown that it is possible to identify all the ‘dangerous’ combinations of SAFT EoS parameters (i.e., the ones leading to unrealistic predictions such as fictitious critical points or fictitious triple points) in comprehensive generalized charts deduced from the corresponding-state law. It has been previously shown that these unrealistic predictions affect generally very-low temperature regions of pure component phase diagrams or the phase behaviour of highly size-asymmetric systems (sometimes at ambient conditions) [1–3]. These charts are plotted for a wide range of m parameters, including the domains of very low and very high m values (typical for polymer compounds). As a second application, it is shown that SAFT-type EoS could be parameterized using an approach different from the one adopted by SAFT EoS users until now. For a pure component, the ( m ,  ,  ) parameters are generally fit to vapour-pressure and liquid-density data which are subcritical properties. This approach that was proposed originally by the first SAFT EoS developers, and was perpetuated until by subsequent developers and users, shows advantages and limitations. Surprisingly, while cubic EoS were parameterized since Van der Waals’ seminal work by imposing the reproduction of experimental critical coordinates ( Tc ,exp , Pc ,exp ), this procedure was rarely adopted by the SAFT community. As a few exceptions, let us mention the works by Galindo et al. [4], Blas and Galindo [5], Blas and Vega [6] and Pàmies and Vega [7]. Working on the reproduction of “global phase equilibrium diagrams” (as defined by Cismondi and Michelsen [8]), all these authors fixed first the values of purecomponent m parameters (using, e.g., molecular simulation values or values stemming from fitting procedures to macroscopic properties) and then, determined the values of  and  in order to exactly reproduce the critical coordinates ( Tc ,exp , Pc ,exp ). In the present paper, an alternative procedure, inspired from both the corresponding-state law and the classical parameterization of cubic EoS, is introduced: it is proposed to determine SAFT-EoS parameters ( m ,  ,  ) such that the three experimental properties ( Tc ,exp , Pc ,exp , exp ) are exactly reproduced by the EoS. It is shown that such a procedure is simple, robust ( m ,  ,  values satisfying such constraints always exist and are unique) and universal (i.e., can be applied to any non-associating compound provided Tc ,exp , Pc ,exp , exp are available). Advantages and limitations of this alternative way are briefly discussed.



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1. Corresponding-state law satisfied by SAFT-type equations of state for non-associating pure species The corresponding-state law (sometimes called ‘theorem of corresponding states’) indicates that two equally-sized pure fluids having the same reduced temperature Tr  T / Tc and the same reduced pressure Pr  P / Pc share a series of reduced properties such as the reduced molar volume vr  v / vc or reduced departure functions (from the perfect-gas law). As a consequence, this law postulates the existence of a universal relationship (i.e., non component-specific) between the three reduced statevariables Tr , Pr and vr :

Pr  f (Tr , vr )

(1)

As an example, the well-known Van der Waals EoS obeys the corresponding-state law: Pr (Tr , vr ) 

8Tr 3  2 3vr  1 vr

(2)

As highlighted by Eq. (2), two parameters among the three reduced variables Tr , Pr and vr must be specified to apply the corresponding-state law. Equivalently, it can be said that two input parameters, to be chosen among the critical temperature, pressure or molar volume, must be preliminarily known to apply Eq. (2) to a specific compound. For illustration, the pressure expression provided by the Van der Waals EoS is:

P(T , v, Tc ,exp , Pc ,exp ) 

RT

v



1 RTc ,exp 8 Pc ,exp



 

2 2 27 R Tc ,exp 64 Pc ,exp



v2

(3)

Consequently, experimental values of the critical temperature Tc ,exp and critical pressure Pc ,exp must be preliminary known to apply the Van der Waals EoS to a given pure compound. As a limitation, it is generally observed that two size-asymmetric pure fluids having the same Tr and Pr deviate significantly from the 2-parameter corresponding-state law. To overcome this issue, an additional parameter must be introduced. The 3-parameter corresponding-state law introduces frequently the acentric factor or the critical compressibility factor as additional parameter. The universal relationship between reduced variables becomes then:

Pr  f (Tr , vr ,  or zc )

(4)

As an example, the well-known Soave-Redlich-Kwong (SRK) EoS [9] obeys the 3-parameter corresponding-state law:


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



2  1  m(exp ) 1  Tr  3Tr     1/3  Pr (Tr , vr , exp )  1/3 1/3 3vr   2  1 vr  2  1 vr   2  1      2  with: m(exp )  0.480  1.574exp  0.176exp

(5)

Where the acentric factor exp is considered as an experimental property since it is a straightforward function of experimental quantities (i.e., the vapour-pressure at Tr  0.7 and the critical pressure). Consequently, knowing experimental values of 3 input parameters ( Tc ,exp , Pc ,exp , exp ) is a prerequisite to apply the 3-parameter corresponding-state law as illustrated with the SRK EoS.  1 R2Tc2,exp   9 21/3 1 Pc ,exp 1  m(exp ) 1  RT P(T , v, Tc ,exp , Pc ,exp , exp )   1/3 2  1   RT ,exp  21/3 1 RT ,exp  v  3 Pcc,exp v  v  3 Pcc,exp   



T

Tc ,exp



2    

(6)

The 3-parameter corresponding-state law entails that reduced vapour-liquid equilibrium (VLE) properties depend solely on the reduced temperature and acentric factor as well:

Prsat (Tr , exp ) ,

 vap H RTc

 f (Tr , exp ) , etc.

(7)

Where Prsat is the reduced vapour pressure and  vap H , the enthalpy of vaporization of pure species. What about SAFT-type EoS? Because SAFT-type EoS derive from intermolecular potential models, the input parameters used by these models are the same: these are microscopic quantities describing a pure fluid at a molecular scale. In some intermolecular potential models (e.g., Lennard-Jones) and in most of SAFT-type EoS (e.g., PC-SAFT), a molecule of pure fluid is seen as an assembly (or a chain) of m molecular segments (or monomers) characterized by a unique segment diameter  and a unique dispersion energy  . The corresponding-state law for such fluids is classically expressed by introducing variables reduced with respect to the input parameters m ,  ,  . For the temperature, pressure and molecular volume, these dimensionless parameters (denoted with superscript * ) are [10]:

T *  kT /   * 3  P  P /  v*  V / N  3 

(8)

Where k is the Boltzmann constant and V / N is the molecular volume. Similarly to Eq. (1), the molecular corresponding-state law applied to a monomer fluid can be written in the general form (see Eq. (4-77) of reference [10]):

P*  f (T * , v* )

(9)

If one considers now a chain of m monomers instead of free monomers, the molecular correspondingstate law becomes analogous to Eq. (4):



P*  f (T * , v* , m)

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(10)

In SAFT-type models, a reduced density variable denoted  is often introduced as volumetric state variable. Eq. (10) writes then:

P*  f (T * , , m)

(11)

According to Gibbs’ phase rule, one degree of liberty is lost when switching from a 1-phase to a VLE system. For the vapour pressure and reduced densities of the equilibrium phases, one has thus:  P sat ,*  f (T * , m)  sat * liq  f (T , m)  sat *  gas  f (T , m)

(12)

The critical point being a particular point lying on the vaporization curve, a critical fluid loses one degree of liberty with respect to VLE systems. Consequently and in agreement with the proof given in the study by Polishuk et al. [3], the reduced critical temperature, pressure and density solely depend on m :  Pc*  f (m)  c  f (m)  * Tc  f (m)

(13)

Note that the reduced coordinates defined for the classical (macroscopic) corresponding-state law are connected to molecular reduced coordinates through:

 T kT /  T * T   r T  kT /   T *  c c c  3 *  P  P  P /   P  r Pc Pc 3 /  Pc*

(14)

The acentric factor can be defined from reduced properties and is a function of m only:

 P*, sat T *  0.7Tc* (m), m    P sat Tr  0.7     f ( m)   1  log10    1  log10  * P P ( m )   c c   

(15)

To conclude this section, it has been emphasized that SAFT-type EoS applied to non-associating pure substances obey the 3-parameter corresponding-state law. This law can be alternatively expressed in terms of molecular reduced properties ( P* , T * , , m) or in terms of macroscopic reduced properties

( Pr , Tr , vr ,  ) . Both formulations are actually equivalent. This statement can be proved rather briefly. A pure fluid obeying the macroscopic corresponding-state law satisfies Eq. (4). By combining Eqs. (4), (14) and (15), one obtains: P*  Pc* (m)  f T *  Tc* (m), v*  vc* (m),  (m) 


(16)

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This equation is similar to Eq. (10): it proves that the molecular reduced pressure depends on the molecular reduced temperature, molar volume and the m parameter as well. As mentioned in the introduction, the results presented in this section can now be used to develop useful approaches for properly parameterizing SAFT-type EoS obeying the 3-parameter corresponding state law. In the next section, it is explained how exhaustive ‘generalized charts’ can be plotted to detect dangerous combinations of SAFT-EoS parameters (i.e., combinations of parameters leading to the prediction of unrealistic behaviors in temperature, pressure or density ranges of interest). 2. Use of generalized charts deduced from the 3-parameter corresponding-state law to check the safeness of SAFT-EoS input parameters 2.1 Examples of generalized charts in the literature Generalized charts or tables providing an overview of model predictions in a given parameter space were proposed numerous times in the past. This practice is quite frequent for authors developing universal correlations based on the corresponding-state law [11,12]. These charts or tables provide graphical representation of reduced state properties expressed as functions of Tr , Pr  (for the 2parameter corresponding-state law) or Tr , Pr ,  or zc  (for the 3-parameter corresponding-state law). As another example of generalized charts, “master diagrams” were introduced by Van Konynenburg and Scott [13,14] as diagrams enabling a global view of all possible model predictions, at a glance, in a well-chosen model-parameter space. Note that such representations are sometimes called “global phase diagrams” [13,15]. More precisely, working with the Van der Waals equation of state, Van Konynenburg and Scott proposed a classification of binary-system phase behaviours including 6 different types (a type is mainly defined by the arrangement of critical lines and 3-phase lines in the pressuretemperature projection) [16]. Master diagrams were then introduced as a synthetic tool making it possible to predict the system type from the mere knowledge of three master variables defined from a smart combination of Van der Waals parameters. While rather frequent for cubic EoS, master diagrams are almost never used to discuss SAFT EoS predictions. As a noticeable exception, Polishuk et al. presented figures playing the role of master diagrams and enabling the prediction of pure-component behaviours depending on the values of the parameter set  m;  / k  [3]. As this study prefigures ours, it is discussed with more details below.



2.2 What have we learnt from the study by Polishuk et al. [3]? If SAFT-type EoS for pure components only reproduced experimentally-observable phenomena in the fluid domain, the (pressure versus temperature) planes of pure species predicted by such models would be limited to a single vaporization curve ended by an (ordinary) vapour-liquid critical point. In practice, SAFT-type EoS are known to predict also unrealistic phenomena [1–3] in a systematic manner: an additional fluid-fluid saturation curve ended by an (additional) fluid-fluid critical point coexists with the ordinary ones. The fluid phases involved in this additional fluid-fluid equilibrium curve exhibit the characteristic of liquid-like phases justifying why the corresponding saturation curve was assimilated to a liquid-liquid equilibrium curve. The ordinary and additional saturation curves are connected through a triple point involving a vapour-liquid-liquid equilibrium. Polishuk et al. [3] showed that ordinary and unrealistic critical temperatures of pure components solely depend on the 2 input parameters m and  / k . They introduced the so-called “global maps” (a kind of master diagrams) for identifying, at a glance, ‘dangerous’ combinations of  m;  / k  leading to the prediction of unrealistic phenomena in temperature ranges of interest. In agreement with the 3parameter corresponding-state law mentioned above, they also showed that such predictions can be condensed advantageously in a simple 2-dimension representation space:  m; Tc*  .

2.3 Comprehensive generalized charts for SAFT-type EoS aimed at helping their safe parameterization An extension of the work by Polishuk et al. is proposed here. The PC-SAFT EoS is chosen as reference EoS for proving our statements and illustrating our conclusions. As indicated above, only nonassociating pure species are concerned with the present study (the case of other species is discussed in the conclusion of this article). Let us note however that the proposed methodology is not limited to the PC-SAFT EoS and applies more generally to any non-associating SAFT-type EoS, mutatis mutandis. Following the spirit of Van Konynenburg and Scott’ approach, it is assumed that a purecomponent phase behaviour modelled by a SAFT-like EoS is fully determined by loci of variance equal to or lower than one [16]. In the present case, critical-point coordinates solely depend on the m parameter and are thus characterized by a variance equal to one. The same conclusion can be drawn for the triplepoint coordinates. Consequently, all SAFT EoS predictions, i.e., the ones corresponding to physically acceptable states and the others as well, can be advantageously gathered in a unique diagram (with m as abscissa and a property of interest such as, e.g., T * , P* or  , as ordinate) having all the feature of a master diagram. To emphasize the propinquity between such diagrams and the corresponding-state law concept, the term “generalized chart” is preferred to “master diagram” thereafter. As will be shown in the next sections, the PC-SAFT EoS predicts more unrealistic critical phenomena than those mentioned


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in the previous study by Polishuk et al., that are essentially found at very low m values (lower than one) and at high m values, typical for polymer systems. The main advantage of such a generalized diagram is the possibility to sketch a corresponding pure-component pressure-temperature or pressure-reduced density projection from the mere knowledge of m . For a given compound characterized by a given m value, it becomes easy to check whether unrealistic phenomena are likely to arise in a specified temperature, pressure or density range of interest. 2.4 Calculation of generalized charts for a given SAFT-type EoS Generalized charts contain pure-component critical loci and triple-point loci as well. The procedure used to calculate these charts is detailed below and illustrated with a simplified flowchart (see Figure 1). These calculations were performed using a home-made code written in Fortran. Calculation of critical points. The calculation of critical loci follows a 3-step procedure. 

Step 1. 25 discrete m values were generated in the range 0 ; 200 (following a non-uniform distribution favoring low m values). For each m value, all the corresponding critical points were computed. It is recalled that a critical point satisfies 2 equations (the so-called “critical constraints”):

 P*    2 P*  * (Tc* ,c )  0   (Tc ,c )   2    T *   T *

(17)

By solving these 2 equations using, e.g., a Newton-Raphson method, the 2 variables (Tc* ,c ) can be evaluated. The case where several critical points coexist at the same m value is more complex. The convergence of the method to a critical point or another strongly depends on the initial guesses chosen for the 2 variables (Tc* ,c ) . To ensure the detection of all the critical points existing at a given m value, it was decided to generate a high number of initial guesses in the variable ranges:

0  Tc*  10  step: Tc*  0.2   0.001  c  0.900  step: c  0.01

(18)

and to run a Newton-Raphson method for each possible combination of both variables. 

Step 2. The critical loci passing through each of the critical points calculated in step 1 can now be traced out. These loci are calculated following a procedure similar to the one proposed by Cismondi and Michelsen [8]. In short, each singular critical point (stemming from step 1) is considered as the first point of its locus and is characterized by coordinates  m(1) , Tc*,(1) ,c ,(1)  .



Page 10 of 28

The second point of the locus is calculated by specifying a m value equal to m(2)  m(1)  m , with m  0 . A sensitivity analysis is then performed to determine which variable among

 m, T

* c

,c  should be specified. The specified variable is then increased and the third critical

point is computed, and so on for all the other points of the critical locus. Critical line calculation stops when either the m value exceed 200, or the reduced temperature becomes negative, or the reduced density becomes negative or higher than 1. At this stage, only the first part of the locus was calculated: it is the one connected to the first point and directed to increasing values of m in the vicinity of the first point  m(1) , Tc*,(1) ,c ,(1)  denoted C(1) . Consequently, the missing part of the locus is the one directed to decreasing values of m in the vicinity of C(1) . To complete the critical locus calculation; C(1) is therefore considered again and the previous procedure is repeated with m  0 . Note that the stability of each critical point, characterized by variables (Tc* , Pc* ,c ) , is systematically tested: to do so, the EoS is solved at a fixed temperature and pressure set to the critical temperature and the critical pressure of the tested point, respectively. When several roots are found, the corresponding fugacity coefficients are computed. The root c is considered as stable if it shows the lowest fugacity-coefficient value. 

Step 3. From step 2, there are as many critical loci as singular critical points calculated in step 1. Most of them are the same. All these critical loci are thus numerically compared using an automatic procedure in order to remove the redundant ones. The remaining ones are reported on the generalized chart.

Calculation of liquid-liquid-vapour triple points. Triple points are characterized by a set of 4 equations: * * * ln  (Ttriple ,G )  ln  (Ttriple , L1 )  ln  (Ttriple , L 2 )  * * * * * *  P (Ttriple ,G )  P (Ttriple , L1 )  P (Ttriple , L 2 )

(19)

Where G ,  L1 and  L 2 are the gas root and liquid roots involved in the triple-point equilibrium;  denotes the pure-component fugacity coefficient. The complete triple-point locus is calculated by solving * Eq. (19) with respect to the four unknowns G , L1 , L 2 , Ttriple  by varying the m value.


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Figure 1: simplified flowchart of the procedure used to generate generalized charts for SAFT-type EoS.



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2.5 Results and discussion As previously mentioned; a “generalized chart” can be represented in various types of plane projections using m as abscissa. In our opinion, the projection in the T * , m  plane, represented in Figure 2, is certainly the most convenient representation. 10.0 T*

Critical line 1 (ordinary critical points)

UCEP

LCEP 3-phase lines

Critical line 2 0.0 0.0

Critical line 4

200.0

m 400.0

Figure 2: Generalized chart of PC-SAFT EoS predictions for non-associating compounds represented in the (reduced temperature; segment number) plane (full scale). Continuous lines: 3-phase lines and stable parts of critical lines. Dashed lines: non-stable parts of critical lines. (U/L)CEP = (upper/lower) critical end point.

This figure corroborates some results introduced in reference [3]: at low to moderate m values (i.e., for

1  m  20 ), the PC-SAFT EoS predicts 2 stable critical points located on critical lines #1 and #2 assimilated to “ordinary” vapour-liquid critical points and to unrealistic liquid-liquid critical points, respectively. This figure highlights also the presence of a non-stable critical line (see critical line #4), regardless of the m value. In accordance with results shown previously [3], a 3-phase line (that can be considered as a vapor-liquid-liquid equilibrium line) starts at low m values and vanishes at m  35 (note that the 3-phase pressure tends to zero as m approaches this limiting value). As a consequence, because all ‘normal’ pure fluids (i.e., non-polymeric and non-associating pure components) are associated with segment numbers higher than 1 and lower than 20-30, the  P* , T *  projections of their phase diagrams contain systematically two critical points (one is liquid-liquid, the other is vapor-liquid) and one (vapor-liquid-liquid) triple point. As an illustration, Figure 3 shows the typical shape of a  P* , T *  phase-


Page 13 of 28 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


diagram projection for such pure species (in the present case, a segment number equal to 5 was selected).

0.0 0.0

-10.0

Critical point located on the critical line 2

-20.0

1.0

Saturation lines

2.0

3.0

Log10 P*


Triple point m=5 T*

-30.0

Figure 3: Typical (P*,T*) projection of an ordinary non-associating pure component predicted by the PC-SAFT EoS (this diagram was calculated for a segment number equal to 5).

Figure 2 also shows interesting results for people working with polymer systems. Following Gross and Sadowski [17], the segment number of a polymer compound is a linear function of the molar mass and consequently segment numbers higher than 100 are classically obtained for such species. For any m ranging in 88 ; 207  , the generalized chart shows the simultaneous presence of 3 stable critical points (located on critical lines #1, #2 and #3), two non-stable critical points (located on critical lines #3 and #4) and one triple point (located on the 3-phase line bounded by the LCEP and UCEP). The LCEP and UCEP denote lower and upper critical end points, respectively. These points are traditionally met in phasediagram types II to VI, according to the classification of Van Konynenburg and Scott for binary systems [16] and are associated with 3-phase equilibrium systems in which two phases among the three ones merge together to form a critical point. Although experimentally unlikely, these critical end points are found in pure-component phase diagrams predicted by the PC-SAFT EoS for the simple reason that this model predicts both unrealistic pure-component 3-phase systems and critical behaviors. Let us note also the presence of a cusp on the non-stable part of critical line #3; this phenomenon is highly similar to those observed in binary systems when computing global phase-equilibrium diagrams [8]. For illustrating PCSAFT EoS predictions for polymer-like compounds (i.e. compounds showing high segment number values), the pressure-temperature projection of a fictitious component having a segment number equal to 125 is represented in Figure 4.



Page 14 of 28

4.0

1.0

5.0 Log10 P*



-4.0

Saturation lines

0.0 0.7

0.8

0.9 -5.0


Triple point

Saturation line -1.0 -6.0

m = 125

m = 125

T* -7.0

-2.0

Figure 4: (P*,T*) projection of pure component having a segment number value equal to 125, as predicted by the PC-SAFT EoS. Left hand side: low-temperature region; right hand side: high-temperature region.

As another result of the present study, it was observed that: -

The PC-SAFT EoS predicts an additional critical line at very low segment-number values looking like an islet. This critical line #5 is represented in Figure 5 that highlights the low-temperature, low segment number part of the generalized chart shown in Figure 2.

-

The critical line #1 (containing “ordinary” critical points) goes to infinite critical temperatures as

m approaches zero (see the left hand side of Figure 5). 5.0

0.5

Focus

T*


T*

3-phase line Focus

Critical line 2 Critical line 4

0.0 0.0

Critical line 5

Critical line 5

m

m

0.0 10.0

0.0

2.0

Figure 5: Enlargement of the low-m, low T* region of the generalized chart represented in Figure 2. Continuous lines: 3-phase lines and stable parts of critical lines. Dashed lines: non-stable parts of critical lines.

Let us focus more specifically on the decreasing part of the critical line #1, represented in Figure 6(a). As a curiosity, for extremely small segment-number values (i.e., for m  0.3 ), the critical line exhibits an unusual behavior including 2 complex transitions from stable to non-stable states and from non-stable to stable states that reveal the presence of critical end points (LCEP and UCEP). A three phase line occurs


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between those 2 CEP. In addition, two cusps are found in the non-stable part of the critical line. Once again, the simultaneous existence of these phenomena (cusps and CEP bounding a 3-phase line) is an exact reproduction of phenomena observed in the (pressure-temperature) plane of binary systems showing type IV or type V phase behaviors [8,16,18]. For illustrating this behavior, the  P* , T *  phasediagram projection of a fictitious component associated with a segment number equal to 0.18 is shown in Figure 6(b). The value m  0.18 lies in the segment-number range of the 3-phase line. As a consequence, the  P* , T *  phase diagram contains one triple point and two critical points.

(a)

12.5

10.5

1.0

(b)

T*

Log10 P*

Saturation lines


Critical points located on the critical line 1

0.9

8.5

6.5

Triple point

0.8

UCEP

4.5

m = 0.18

LCEP 2.5 0.05

m

3-phase line

T* 0.7

0.25

3.1

3.3

3.5

Figure 6: (a) Enlargement of the very-low-m region of the generalized chart represented in Figure 2. Continuous lines: 3-phase line and stable parts of critical lines. Dashed lines: non-stable parts of critical lines. (U/L)CEP: upper/ lower critical end points. (b) (P*,T*) phase diagram projection for a component having a segment number equal to 0.18.

At this step, let us comment briefly the potential use of m values lower than 1 for modeling compounds with a SAFT-type EoS. Remembering that the m parameter is defined as a number of Lennard-Jones segments involved in a molecular potential model and that simplest chains contain at least one segment, values lower than 1 are sometimes considered as physically meaningless by SAFT developers. However, some SAFT users (and even some SAFT developers) estimating SAFT EoS parameters from fitting procedure to macroscopic properties have proposed to consider m values lower than 1 (e.g., see argon parameters in reference [19]). For the sake of completeness, let us mention that although the T * , m  projection is a convenient representation of a generalized chart, other types of projections could be used for analyzing EoS predictions. The  P* , m  and  , m  projections are represented in Figure 7(a) and (b). Naturally, all the phenomena present in the T * , m  projection are visible in these projections. As an exception, the critical line #4, which only contains non-stable critical points, cannot be seen in Figure 7(b) which uses



Page 16 of 28

the decimal logarithm of reduced critical pressures as ordinate, for the reason that the corresponding critical pressures are all negative. 0.0

200.0

400.0

Critical line 2



Critical line 2

0.0

3-phase lines 0.1

200.0

Critical line 4 LCEP

400.0

Critical UCEP line 3

3-phase lines

UCEP

0.01

Log10 P*

0.0

-10.0

(b)

LCEP 0.001

(a)

m

0.0001

m -20.0

Figure 7: Generalized chart of PC-SAFT EoS predictions for non-associating compounds represented in (a) the (reduced density; segment number) plane and (b) the (reduced pressure; segment number) plane. Continuous lines: 3-phase lines and stable parts of critical lines. Dashed lines: non-stable parts of critical lines. (U/L)CEP = (upper/lower) critical end point

To conclude this section, it can be claimed that generalized charts provide an efficient tool for SAFT EoS users who can use them to check a priori whether safe calculations can be performed in the pressure and temperature ranges of interest. Although generated for the unique PC-SAFT EoS in the frame of the present study, the proposed method could be applied to any SAFT-type EoS likely to predict unrealistic phenomena. In particular, when input SAFT parameters are fit to experimental data, the use of generalized charts is highly recommended for checking the safeness of the parameter values returned by an optimization routine, thus ensuring a proper EoS parameterization. For a completely safe practice, it is also advised to check, when possible, that the obtained parameters - and especially, the segment number - have the right physical meaning (the m parameter value must reflect reasonably the structure of the molecule). 3. Use of generalized charts deduced from the 3-parameter corresponding-state law to parameterize SAFT-EoS like cubic EoS As mentioned in the introduction, people working with SAFT EoS are used to fit EoS input parameters on vapor-pressure and liquid-density data (and sometimes other properties such as heat capacities, speed of sounds etc.). As a main advantage, this procedure ensures an accurate reproduction of saturation properties and liquid-density data up to temperatures close to the critical temperature (provided data are available for both properties on a wide temperature range including near-critical temperatures). The limitations of this approach are:


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-

EoS parameters can only be estimated if a sufficient number of data is available,

-

EoS parameter values strongly depend on the temperature range covered by the data, the presence of both property data (vapour-pressure and liquid-density data) or of one of them only.

-

The EoS systematically overestimates the critical pressure of heavy compounds (which can be prejudicial to the prediction of supercritical properties for pure components and multicomponent systems involving these components). Note that SAFT EoS corrected with the crossover technique are beyond the scope of the present study.

In this section, an alternative procedure for parameterizing SAFT-type EoS obeying the 3-parameter corresponding-state law is proposed with the aim of counterbalancing the above limitations. This procedure is strongly inspired from parameterization methods classically used with cubic EoS and is fundamentally justified by the corresponding-state theory applied to SAFT EoS, as discussed above. Let us recalled that the attractive parameter and covolume of cubic EoS are always calibrated in order to: -

reproduce exactly experimental values of the critical temperature and pressure. This is the reason for which mathematical expressions of the attractive parameter and covolume are always [20]: a (T )   a R 2Tc2,exp (Tr ) / Pc ,exp  b  b RTc ,exp / Pc ,exp

(20)

Where  a , b are universal constants the values of which are fixed by the constraint to exactly reproduce critical coordinates;  (Tr ) is the so-called ‘alpha’ function (which is also constrained -

as discussed in reference [20]). reproduce accurately the experimental vaporization curve in the (pressure; temperature) plane. To do so, the experimental value of the acentric factor exp is generally used to calibrate the alpha function. It is recalled that exp must be considered as a vapour-pressure data at a reduced temperature Tr  0.7 . Thus, using exp as input parameter is roughly equivalent to constraint the EoS to reproduce the vapour-pressure data at Tr  0.7 .

In the same spirit, it is possible to parameterize SAFT EoS, i.e., to determine values of input parameters ( m ,  ,  ), in such a way that the EoS can exactly reproduce the Tc ,exp , Pc ,exp and exp data. This parameterization procedure for SAFT EoS is based on: -

the observation that Tc* , Pc* and  can be expressed as univocal functions of m as discussed earlier and as illustrated in Figure 8,

-

the high similarity between SAFT and cubic EoS parameters. Indeed, m is, somehow, the molecular equivalent of  : it explicitly takes into account the chain length, or non-sphericity of the molecules, which is the reason behind using the acentric factor in classical, macroscopic EoS.



In the same manner,  the diameter of the groups making the molecules in SAFT is clearly related to b , the parameter in cubic EoS related to the total volume occupied by the molecules; and,  , the van der Waals energy of interaction between groups making the molecules, is the molecular equivalent of a the parameter taking into account the overall interactions between molecules in classical EoS, as originally introduced by Van der Waals. Eventually, we considered this high similarity as an invitation to use similar procedures for parameterizing SAFT and cubic EoS. Discrete evaluations of the Tc* (m) , Pc* (m) and  (m) functions for m ranging from 1 to 20 are provided in


Page 18 of 28

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Table 1. Alternatively, the following polynomial correlations can be used to approximate the relation between Tc* , Pc* and  and m :  (m)  8.5775 104 m 2  0.13079m  0.12138  * 6 5 4 4 3 3 Tc (m)  4.7968 10 m  3.0895 10 m  7.8649 10 m  1.0215 101 m 2  0.75358m  0.63659   Log P* (m)  1.6345 107 m6  1.1346 105 m5  3.1389 104 m 4 10 c   4.4618 103 m3  3.6282 102 m 2  2.2498 101 m  7.7655 101 



(a)

0.8

Tc*

(b)

4.0

(21)

0.3

2.0 -0.2

0.0

4.0

8.0

m -0.7

m

0.0 0.0

2.0

4.0

6.0

8.0

10.0

2.0

(c)

Log10 Pc*

0.0 0.0

4.0

8.0

m -2.0

Figure 8: Graphical representation of the universal functions (m), Tc*(m) and Log10 Pc*(m) as predicted by the PC-SAFT EoS for m < 10 (most of ‘normal’ compounds are located within this range). Note that panels (b) and (c) are enlargements of the general charts introduced in Figure 2 and Figure 7(b).



Page 20 of 28

Table 1: Discrete evaluations of the universal functions represented in Figure 8 for m in [1; 20]

m

Tc*

Pc*



1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0

0.00507 0.06672 0.13260 0.19805 0.26275 0.32671 0.38997 0.45254 0.51443 0.57566 0.63625 0.69622 0.75562 0.81445 0.87274 0.93053 0.98783 1.04466 1.10104 1.15701 1.21256 1.26772 1.32251 1.37694 1.43103 1.48478 1.53822 1.59134 1.64417 1.69671 1.74898 1.80097 1.85270 1.90418 1.95542 2.00642 2.05718 2.10772 2.15804

1.27575 1.55830 1.80172 2.00494 2.17553 2.32040 2.44499 2.55338 2.64868 2.73325 2.80893 2.87713 2.93899 2.99543 3.04720 3.09488 3.13900 3.17998 3.21817 3.25387 3.28735 3.31882 3.34848 3.37651 3.40304 3.42821 3.45213 3.47490 3.49662 3.51736 3.53720 3.55621 3.57443 3.59193 3.60875 3.62494 3.64054 3.65558 3.67010

0.11468 0.08843 0.07601 0.06720 0.06012 0.05419 0.04914 0.04478 0.04100 0.03770 0.03479 0.03222 0.02994 0.02790 0.02607 0.02442 0.02293 0.02158 0.02034 0.01921 0.01818 0.01723 0.01635 0.01554 0.01479 0.01409 0.01345 0.01284 0.01228 0.01175 0.01126 0.01079 0.01036 0.00995 0.00956 0.00920 0.00886 0.00853 0.00822

The leading idea is simply to identify  m,  ,  / k  from Tc ,exp , Pc ,exp , exp  by imposing the following constraints:


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 exp  predicted by   (m) the SAFT EoS    * Tc ,exp  Tc,predicted by   Tc (m) k the SAFT EoS    *  Pc ,exp  Pc,predicted by  3  Pc (m)  the SAFT EoS 

(22)

In practice, knowing exp , Figure 8(a) makes it possible to estimate univocally the m parameter. Once m known, Figure 8(b) provides the value of Tc* . According to Eq. (22), parameter  / k is given by :

 k



Tc ,exp

(23)

Tc* (m)

Then, knowing m , Figure 8(c) provides the value of Pc* . Knowing both the values of m and  / k , the last input parameter  is obtained from the following formula also deduced from Eq. (22): 1/3

  k  Pc* (m)      Pc ,exp   k

(24)

This parameterization method offers as main advantages: -

its simplicity,

-

its robustness: as can be observed in Figure 8, functions  (m) , Tc* (m) and Pc* (m) are bijective functions on the practical range of interest m   0.4; 20 ,

-

its wide applicability: by using this parameterization method, the actual input variables of the PCSAFT EoS are no longer  m,  ,  / k  but Tc ,exp , Pc ,exp , exp  . Not only, experimental values of these 3 latter properties are available for a large number of ordinary compounds, but in addition, numerous predictive methods exist for their estimation (e.g., group-contribution methods for welldefined compounds and characterization methods for ill-defined pseudo components).

For these reasons, the proposed parameterization method opens the door to an intensive industrialization of SAFT type EoS (i.e., to apply this EoS to any non-associating pure species following a robust and systematic parameterization procedure). To illustrate this procedure, let us consider 14 pure compounds: 12 pure n-alkanes, carbon dioxide and benzene. For all of them, the

 m,  ,  / k 

parameters were determined following the proposed

parameterization. Results and deviations between experimental and calculated properties are reported in Table 2. Experimental data are all issued from the DIPPR database following the procedure described in previous papers [21].



Page 22 of 28

Table 2: Illustration of the results obtained from the proposed parameterization method

 /k

Mean average percent error (MAPE) between experimental and EoS properties (in %) Saturatedliquid heat capactity Vapour Vaporization cP ,liq Molar volume pressure enthalpy 1.77 0.74 7.71 1.83 Non 5.72 0.83 1.80 available 8.78 0.24 11.73 1.48 11.59 0.23 2.90 1.73

Compou nd Methane

m

 (Å)

1.058

3.633

(K) 145.5

Ethane

1.749

3.473

181.2

Propane n-butane npentane nhexane nheptane n-octane nnonane ndecane nundecan e ndodecan e Carbon dioxide Benzene

2.150 2.516

3.611 3.720

198.1 211.4

2.913

3.800

218.7

14.51

0.07

2.84

1.16

3.300

3.865

224.1

17.26

0.35

2.34

1.44

3.679

3.918

228.2

19.00

0.94

2.53

2.07

4.076

3.958

230.9

21.05

0.92

0.69

2.46

4.427

4.002

234.2

21.91

1.43

0.32

2.99

4.821

4.028

236.1

22.83

1.18

0.30

2.97

5.129

4.080

239.2

24.16

1.40

0.15

4.05

5.506

4.097

240.7

24.54

1.45

2.19

4.15

2.697

2.602

146.6

13.57

0.66

8.56

1.57

2.594

3.711

275.6

14.47

0.72

3.81

1.80

As a main limitation of the proposed method and as highlighted in Table 2, the reproduction of the Tc ,exp and Pc ,exp critical coordinates entails inevitably a less accurate prediction of liquid densities at temperatures far from the critical point (as liquid density is not a target property of the new parameterization method). It is also mentioned that in the vicinity of the critical point, the correct density behaviour cannot be captured. This limitation is not specific to the proposed parameterization and is always observed, regardless of the parameterization technique used. The fundamental reason is that the EoS is based on a mean field theory that does not account for density fluctuations characterizing a near-critical fluid. To overcome such an issue, more advanced ‘patches’ such as the addition of a crossover treatment should be considered.


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With the aim of understanding how PC-SAFT parameters vary from a parameterization method to another one, a comparison between different n-alkane parameter sets is proposed in Figure 9. Three different parameterization methods are compared: the one we proposed here, the one proposed by Gross and Sadowski for original PC-SAFT [19] and the one used by Galliero for parameterizing a Lennard-Jones chain model used for molecular dynamics simulations [22]. It is worth noting that both PC-SAFT EoS parameterizations (ours and the one by Gross and Sadowski) lead to rather-similar trends and values for the chain parameter ( m ) and the energy parameter (  / k ), while the parameter set used for molecular simulations deviate more and more with increasing values of carbon atom numbers. For the segment diameter, while global trends are similar for both PC-SAFT parameter sets, it is interesting to note that  values obtained by our parameterization method are very close to Gross and Sadowski’s  values at low carbon atom numbers and are close to Galliero’s values at elevated carbon atom numbers. It thus appears that the use of a parameterization based on the reproduction of the critical point coordinates drives  values to those used in molecular model.

Figure 9: Evolution of PC-SAFT parameters with respect of atom carbon numbers for n-alkanes from methane to n-dodecane.



The application of the proposed methodology to hundreds of compounds, the evaluation of the model performances parameterized as proposed, the discussion of the limitations and the proposition of solutions for overcoming these limitations will be the subject of a next study. Eventually, let us mention that the proposed methodology can be immediately applied to non-associating components. Let us note first that a large part of pure species encountered in chemical-engineering processes are non-associating. Its application to associating component is however not straightforward as for this class of compounds, the EoS is augmented by an association term that involves additional input parameters. The modification of the proposed methodology for such systems will be addressed in a next study.


Page 24 of 28

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Conclusion SAFT and cubic models obey the corresponding-state theory, a fundamental law based on experimental observations. While Van der Waals initially introduced a 2-parameter corresponding-state law, meaning that the sole knowledge of 2 component-specific properties makes it possible to apply cubic EoS to a given pure substance, further developments of SAFT and cubic EoS have shown that a third parameter is required to gain accuracy. It is recalled that the parameterization of a model is the selection of appropriate values for the model input parameters. When dealing with cubic EoS, the parameterization method is unambiguous and nearly-universally admitted: EoS parameters are determined in order to exactly reproduce the experimental critical coordinates Tc ,exp , Pc ,exp  and to describe accurately vapor-pressure data. The parameterization of SAFT-type EoS is however not well established and depend on both the availability of properties used for fitting input parameters and SAFT-user habits. From our viewpoint, SAFT developers often concentrate their efforts on the modification of model terms (addition of terms to better account for specific types of interactions, e.g., use of advanced association schemes, addition of dipolar, quadrupolar terms …) to the detriment of parameterization issues which influence yet in a large extent the model performances. In the present study, the relationship between the corresponding-state law and SAFT type EoS have been highlighted and used to parameterize these EoS. It was shown that: -

Generalized charts based on the corresponding-state law can be generated and provide a comprehensive overview of model predictions (both physically-meaningful and unrealistic predictions). They can be used to detect the domains of parameter spaces where the EoS can be safely used. For illustration, this methodology was applied to the PC-SAFT EoS for non-associating compounds. In particular, it was shown that unrealistic prediction are likely to arise when working with polymer compounds.

-

Generalized charts also inspired the development of an alternative parameterization method for SAFT EoS. Indeed, it was proposed to parameterize SAFT EoS using the same spirit as the parameterization method used with cubic EoS. It was shown that the 3 input parameters

 m,  ,  / k 

T

c ,exp

can be determined using an univocal way in order to exactly reproduce the data

, Pc ,exp , exp  . This proposal shows advantages and limitations. The main advantage lies

in the possibility to industrialize SAFT versions by proposing a robust, systematic and widely applicable parameterization method. The main limitation is the less-accurate reproduction of liquid-density data (although few discussed in this article, this issue will be extensively discussed in the next study).



To conclude, we consider this article as the first stone of an ambitious project aimed at proposing an industrialized version of a SAFT EoS. The next study will be dedicated to a fair and extended comparison between SAFT and cubic EoS performances. In this study, SAFT EoS will be parameterized as proposed here and will be modified in order to overcome liquid-density limitations. Parameterization methods for associating compounds will be proposed in the future.


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Application of the corresponding-state law to the parameterization of

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