# Application to Petroleum Engineering of Statistical Thermodynamics - Based Equations of State

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Oil & Gas Science and Technology – Rev. IFP, Vol. 61 (2006), No. 3, pp. 363-386 Copyright © 2006, Institut français du pétrole Dossier Petroleum Industry Applications of Thermodynamics Applications de la thermodynamique dans l'industrie pétrolière Application to Petroleum Engineering of Statistical Thermodynamics – Based Equations of State J.C. de Hemptinne1, P. Mougin1, A. Barreau1, L. Ruffine1, S. Tamouza1, 2 and R. Inchekel1 1 Institut français du pétrole, département Thermodynamique et Simulation moléculaire, 1 et 4, avenue de Bois-Préau, 92852 Rueil-Malmaison Cedex - France 2 Laboratoire d’Ingéniérie des matériaux et des hautes pressions, CNRS université Paris XIII, 99, avenue J.B. Clément, 93430 Villetaneuse - France e-mail: j-charles.de-hemptinne@ifp.fr - pascal.mougin@ifp.fr - alain.barreau@ifp.fr - livio.ruffine@ifp.fr - radia.inchekel@ifp.fr Résumé — Application des équations d'état basées sur la statistique thermodynamique, aux besoins pétroliers — L'utilité des équations d'état cubiques pour l'ingénieur pétrolier a été démontrée depuis de nombreuses années. Leur pouvoir prédictif reste cependant limité. De nouvelles équations d'état, basées sur des principes de mécanique statistique, ont été développées depuis environ 15 ans, et leur maturité est telle qu'elles sont maintenant de plus en plus utilisées par l'ingénieur. Dans cette publica- tion, nous nous concentrerons essentiellement sur les équations qui ont pour base la théorie de Wertheim. Nous aborderons trois thèmes : Les mélanges associatifs tels l'eau et les alcools en présence d'hydrocarbures ont toujours représenté un défi majeur pour les équations d'état. Nous démontrons ici que l'équation CPA (SRK avec un terme asso- ciatif), tout en conservant les caractéristiques de la cubique pour les molécules non-associatives, améliore significativement le pouvoir prédictif de ces équations pour les mélanges contenant de l'eau, du méthanol et des hydrocarbures. La signification physique des paramètres de l'équation SAFT rend possible l'élaboration d'une méthode de contribution de groupe pour la détermination des paramètres de cette équation. En utilisant cette approche, il est possible de prédire les tensions de vapeur et les volumes liquide des molécules lourdes avec une précision remarquable. Nous montrons cela sur les n-alcanes, les n-alcools et les 1-oléfines, ainsi que pour les isomères méthyls des alcanes. Finalement, nous présenterons les prédictions d'équilibres liquide-vapeur de mélanges non-idéaux, obtenu par l'équation SAFT en incluant des termes polaires. Nous montrons ainsi clairement l'effet du quadrupole pour les mélanges aromatique-non-aromatique. Le fait de tenir compte ou non des interac- tions dipole-dipole permet de différencier le comportement des isomères cis- et trans oléfiniques. Abstract — Application to Petroleum Engineering of Statistical Thermodynamics – Based Equations of State — Cubic equations of state (EOS) have proven their utility to the petroleum engineers for many decades. Their predictive power remains, however, limited. Statistical mechanical approaches have meanwhile grown allowing the development of powerful engineering equations of state. In particular, this paper investigates how equations that are based on the association term of Wertheim can improve significantly the predictive power in petroleum applications.

364 Oil & Gas Science and Technology – Rev. IFP, Vol. 61 (2006), No. 3 Three distinct issues are discussed. Mixtures of associating components such as water or methanol with hydrocarbons have so far represented one of the main challenges for the predictive power of equations of state. It is shown that the CPA equation, that combines the classical SRK EOS with the association term of Wertheim, provides a significant improvement for both methanol-hydrocarbon and water-hydrocarbon mixtures. The physical significance of the SAFT parameters makes it possible to develop a group contributions method for their determination. Using this approach, it becomes possible to predict vapour pressure and liquid volume of heavy molecules. This is shown for a number of families, such as n-alkanes, n-alcohols, 1-olefins as well as for isomers of methyl-alkanes. Finally, the improved predictions resulting from the use of polar terms with SAFT are illustrated. The effect of the quadrupole is clearly shown with an aromatic-non aromatic mixture. Including a dipolar interaction makes it possible to differentiate the behaviour of the cis and the trans 2-butene isomers. NOMENCLATURE R ideal gas constant = 8314.5 J/kmol/K T temperature K Latin lower case V total volume m3 XAi mole fraction of molecule i not bonded – a molar Helmholtz energy J/kmol at site A a energy parameter of the cubic eos J m3/kmol Z compressibility Factor – a0 energy parameter of the cubic eos J m3/kmol (= PV/NRT = Pv/RT) at T = Tc b volume parameter of the cubic equation m3/kmol Greek Letters of state c1 parameter of the cubic equation of state – α attractive function of a cubic EOS – d hard sphere diameter of the segments Å β association volume for CPA EOS m parameter of the SAFT equation – σ parameter of the SAFT EOS Å of state referring to the number expressing the segment diameter of segments in the molecule ε/κ parameter of the SAFT EOS expressing K f generalized function – the interaction energy between segments g radial distribution function – μ dipole moment, EOS parameter J/kmol kij binary interaction parameter ρ molar density kmol/m3 k Boltzman constant = R/Nav κ association volume for SAFT EOS – qN canonical Partition Function Δ association strength – s1 high pressure segment diameter Å ω acentric factor – for SAFT VR λ well width parameter in the SAFT-VR – v molar volume m3/kmol equation x mole fraction in liquid phase – η compacity (=v/v*), or non-dimensional – y mole fraction in vapor phase – volume z total mole fraction – Indices Latin Capitals 1,2,3 indices of the degree of perturbation used A Helmholtz energy J att attractive contribution Anm parameter in the Alder equation of state rep repulsive contribution E reduced energy density = e/RT 1/m3 hs hard sphere contribution (equivalent to repulsive G Gibbs energy J contribution) N number of moles kmol disp dispersive contribution (equivalent to attractive Nav Avogadro Number = 6.02.1023 contribution) P pressure Pa dipole- dipole-dipole interaction contribution Q quadrupole moment, eos parameter dipole

JC de Hemptinne et al. / Application to Petroleum Engineering of Statistical Thermodynamics – Based Equations of State 365 quad-quad quadrupole - quadrupole interaction contribution In these relations, a, b, c and d may be constant or function of chain contribution due to chain formation the temperature and of some other properties of the fluid polar polar contribution, may be dipole-dipole, (acentric factor for example). These relations show the main quadrupole-quadrupole or dipole - quadrupole concept developed by van der Waals: the separation of repul- c critical property sive forces caused by molecular size from cohesive forces caused by molecular attraction. This approach is still used in ass contribution due to association recent modelling of fluid properties. This kind of equation i, k, j pure component index allows the continuous description of the vapour and the ij , ji parameter related to the interaction between liquid phase with the same tool. The most famous cubic components i and j equations of state are those of: r reduced property Soave-Redlich-Kwong (1972) Superscript RT a(T ,ω ) P= − v − b v ( v + b) * caracteristic property 0 segment property Peng and Robinson (1976) hs hard sphere property RT a(T ,ω ) hc hard chain property (in the PC-SAFT EOS) P= − v − b v ( v + b) + b ( v − b) sw square well property (in the SAFT-VR EOS) Aj refers to site A on molecule j These equations have two parameters: b is the covolume AjBi interaction parameter between sites A of and is a function only of critical properties (Tc and Pc) whilst molecule j and site B of molecule i the attractive parameter, a, is function of the temperature and of the acentric factor. These equations of state are included in all the computer process simulation packages. One of the INTRODUCTION main drawbacks of these equations is that the critical com- pressibility factor is constant regardless of the compound. The use of equations of state (EOS) has been the generally Equations of state with three or four parameters overcome accepted method for the calculation of many fluid physical this deficiency. properties since the famous van der Waals’ proposal (1873) The reason for the clear superiority of this kind of equa- in the 19th century. The cubic equations are the most used tions was not so much their accuracy, but rather their simplic- equations, but over the last fifteen years, many new equations ity of use, and the development of many ways to “tune” the have been proposed, based on statistical mechanical con- parameters for the specific application. cepts. Their use in the petroleum industry is not yet generally accepted. It is the purpose of this paper to show that they For Pure Component Vapour Pressure Improvement bring considerable advantages for a number of mixtures of The attraction term, a, has been adapted several time industrial interest. (Redlich and Kwong, 1949; Soave, 1972). The simple forms are a function only of temperature or of acentric factor whereas the more complex forms (Behar et al., 1985, 1986; 1 FROM CUBICS TO STATISTICAL MECHANICS Twu et al., 1995) increase the number of parameters. 1.1 The Cubic Equations For Volume Prediction An equation of state is a relation between the pressure P, the The cubic equations of state are known to give poor results temperature T and the molar volume v. In the last century, the for liquid molar volumes. However, one may add volume cubic equations of state have by far dominated in the descrip- translation (Péneloux et al., 1982; Ungerer and Batut, 1997; tion of such equations and their general form is: de Sant’Ana et al. 1999, etc.) to improve this information without modifying the phase equilibrium. RT P= − Patt ( v, T ) For Mixture Prediction v−b with One of the major advantages of the cubic equations is the wide variety of available mixing rules. These are more or less a Patt ( v ,T ) = complex, and thus require more or fewer parameters. The v (v + d ) + c (v − d ) classical mixing rules were proposed by van der Waals

366 Oil & Gas Science and Technology – Rev. IFP, Vol. 61 (2006), No. 3 (1873) and they should be applied to non-polar mixtures. The partition function is approximated using a potential Huron and Vidal (1979) have initiated a new type of mixing function that describes the potential energy of particles as a rule, which combines the power of Excess Gibbs energy function of their distance. A simple derivation makes it possi- models with the equation of state approach. Since then, many ble to calculate the compressibility factor: new models have been developed, the most known of which PV V ∂A are the MHV2 (Dahl & Michelsen, 1990), Wong- Sandler Z= =− (Wong et al., 1992) and LCVM model (Boukouvalas et al., NRT NRT ∂V T 1994) models. These more complex mixing laws extend the The best known of these models is the Carnahan-Starling use of the cubic EOS to mixtures with polar compounds such equation for hard spheres (Carnahan & Starling, 1972). as water or alcohols. But in these cases, the number of Using a derivation based on the interaction potential, they adjustable parameters is typically twice the number used with describe the compressibility factor as that of an ideal gas with the classical mixing rule (two per binary rather than one). a residual contribution (Fig. 1). The description of multiphase equilibria remains difficult Of course, this equation cannot describe a real fluid with this approach (unless specific parameters are used for behaviour, because it contains only a repulsive term. different liquid phases). However, it is of great use for describing the deviation from Many efforts have been made for making these equations the ideal gas behaviour at infinite temperature, i.e. when the more predictive (Coniglio et al., 2000; Chen et al., 2002; attraction between molecules is negligible. An empirical Péneloux et al., 1989), but the empirical nature of these equa- approach is generally used for describing the attraction in a tions results in a rather complex set of equations whose accu- square well fluid (Alder et al., 1972), (Fig. 2). racy remains limited. This conclusion shows that other ways Prigogine’s theory (1957) postulating how the rotational should be proposed to model the behaviour of fluids and and vibrational degrees of freedom could depend on density improve the predictive character of the EOS. has resulted in the development of many equations of state for chain-like molecules (COR - Chien et al., 1983; PHCT - 1.2 Equations Based on the Thermodynamic Donohue and Prausnitz, 1978, PACT - Vilmachand and Perturbation Theory - SAFT Donohue, 1985, APACT - Ikonomou and Donohue, 1986, SPHCT - Kim et al., 1986). These equations are developed With the advent of molecular simulation coupled with statis- using the so-called “Thermodynamic Perturbation Theory” tical physics, new approaches have been investigated for (TPT), because the fluid behaviour is described as a sum of a equations of state. These approaches develop an expression “reference” behaviour and a “perturbation”. The reference is for the Helmholtz free energy, A, using what is called the taken at infinite temperature: the hard sphere behaviour. “canonical partition function” (qN) for a system of N particles Using Wertheim’s theory (1986), Chapman et al. pub- (Atkins and de Paula, 2002): lished in 1988 and in 1990 the first papers applying the per- turbation theory to associating fluids (SAFT: Statistical- A = − ⎛⎝ 1+ ln N N ⎞⎠ q Associating Fluid Theory). We consider two major NkT contributions in their theory. ε ε 4 9 m ∑ ∑ m Anm (T * T ) n⎛ v* ⎞ 4η − 2η2 Z = Z rep + Z = Z rep = 1 + n =1 m =1 ⎝ v⎠ (1 − η) 3 v* With two pure component parameters: with η= and v* is the hars sphere volume. v T* et v*. The 36 coefficients Anm are universal. r r ν* ν* Figure 1 Figure 2 Carnahan and Starling's equation for hard spheres. The equation of Alder et al. for square well interactions.

JC de Hemptinne et al. / Application to Petroleum Engineering of Statistical Thermodynamics – Based Equations of State 367 The Association Term Huang and Radosz (1990) proposes to simplify the set of implicit equations in a number of often—encountered simple The first contribution is based on Wertheim’s TPT and cases where a single association type exists in the fluid. They proposes an analytical expression for associating fluids: label these association types with a number (corresponding to the number of sites on the molecule) and a letter (A when ⎡ 1 1 ∂X A j ⎤ Z ass = ∑ x i ∑ ρ j ∑ ⎢ ( A − ) any site can associate with any other; B when one site is i j Aj ⎣ X j 2 ∂ρ i ⎥⎦ electropositive, and the other(s) electronegative; C when two sites are electropositive, etc.). This expression has since then been found to be equivalent The Chain Term to the much simpler form (Michelsen and Hendriks, 2001): The second major contribution of Chapman et al. (1988, 1⎛ ∂ ln g ⎞ 1990) is their proposal to consider a chain molecule as a Z ass = − ⎜1 + ρ ⎟ ∑ x i ∑ (1 − X Ai ) mixture of segments with infinite association strength 2⎝ ∂ρ ⎠ Ai (Fig. 3). Doing so, and using a Lennard-Jones type potential interaction, they write (for non-associating molecules): xi where ρ i = v is the molar density of component i, and XAj ( Z = 1 + m Z hs o + Zo ) disp + Z chain is the mole fraction of molecule i not bonded at site A. This last quantity must be computed based on following intrinsic o set of equations: where m is the number of segments in the molecule, Z hs and o Z disp are the segment hard sphere (repulsion) and dispersion −1 (attraction) terms and Zchain is the contribution of the ⎡ ⎛ ⎞⎤ XAj = ⎢1 + ∑ ⎜ ρ i ∑ X B i ΔA j B i ⎟ ⎥ chain-forming to the compressibility factor. We will show ⎢⎣ i ⎝ Bi ⎠ ⎥⎦ further in this article how the physical basis of this equation may lead to an unexpectedly good extrapolation of fluid The key parameter for calculating the amount of non- associating sites is the association strength which is defined properties to heavy components, by using a group contribu- by Chapman (1990) as: tion approach for the pure component parameters. Many authors have proposed improvements to the SAFT ⎡ ⎛ εA j Bi ⎞ ⎤ equation of state since it was initially proposed (Huang and ΔA j B i = d ij 3 g ij (d ij ) κ A j B i ⎢ exp⎜ ⎟ − 1⎥ Radosz, 1990; Banaszak et al., 1993: SW-SAFT; Kraska and ⎣ ⎝ kT ⎠ ⎦ Gubbins, 1996a, 1996b: LJ-SAFT; Fu and Sandler, 1995: S-SAFT; Galindo et al., 1996, 1997: HS-SAFT, Blas and The parameter εAjBi represents the association energy Vega, 1998: Soft-SAFT; von Solms et al. (2003): simplified; between the two sites Aj and Bi, and the parameter κAjBi char- PC-SAFT, etc.). Some reviews of these many versions and acterises the bonding volume. The radial distribution function their applications have been written (Wei and Sadus, 2000; gij(dij) represents the probability that two molecules of type i Muller and Gubbins, 2001; Economou, 2002). Further in this and j are found at a distance dij from each other. paper, we will discuss in some more details the original SAFT version (SAFT-0), SAFT-VR (Gil-Villegas et al., 1997) and PC-SAFT (Gross and Sadowski, 2000, 2001). 1.3 Hybrid Equations σ Monomer model Bonding site In the engineering community, hybrid equations of state have ex. Methane often been used. These equations combine energy interaction terms originating from different theories. As example, we can mention Carnahan and Starling (1972) who combine their own hard sphere repulsive equation with the attractive term 1 2 3 m of the cubic Redlich-Kwong equation. m-mer Model (n-alcane) Along these lines, Kontogeorgis et al. (1996) suggest using the association term proposed by Chapman Figure 3 et al. (1990) as an additional contribution to the classical Hard sphere (monomer) and chain molecules (m-mer) model, cubic equation of state. Thus, their equation is written with bonding sites (Chapman et al., 1990). as (using the Redlich-Kwong version of the attractive

368 Oil & Gas Science and Technology – Rev. IFP, Vol. 61 (2006), No. 3 term and the Michelsen and Hendricks formulation of the 2.1 The Main Contributions associative one): The two types of equations considered in this work are SAFT v a 1⎛ ∂ ln g ⎞ and CPA. Three different versions of SAFT are examined: Z= − − ⎜1 + ρ ⎟ ∑ x i ∑ (1 − X Ai ) SAFT-0, which is the original version of Chapman et al. v − b RT (v + b) 2 ⎝ ∂ρ ⎠ Ai (1988, 1990); SAFT-VR which has been proposed by Gil- Villegas et al (1997), and which seems to be quite promising, This equation is further discussed in this paper. The reason at the expense of an additional parameter, λ, that describes why it is of interest to petroleum engineers is that in the the width of the square well; and PC-SAFT (Gross and absence of associating compounds, the well-known cubic Sadowski, 2000, 2001), which benefits from a growing popu- equation is recovered, and all the existing correlations can be larity in the engineering world. used. The advantage of keeping the capability to work with complex hydrocarbon mixtures, including badly defined The common point between the SAFT equations and CPA pseudo-components is the main reason why, despite its dubi- is that the Helmholtz free energy expression (and therefore ous physical basis, this approach remains attractive in the the compressibility factor) is written as a sum of energetic petroleum engineering community. contributions. In both cases, a repulsive and an attractive con- Although association among molecules clearly results in tribution are identified. The former describes the repulsion non-ideal behaviour that can now be described explicitly, between molecules when they approach each other; the latter other causes may also result in non-ideal behaviour. Among is a combination of the long-range attractive forces. In addi- these, we can mention the presence of dipolar or quadrupolar tion, both equations take explicitly into account the short moments. For these contributions, a thermodynamic pertur- range association forces. Hence, they are written as: bation theory has also been developed (Kraska and Gubbins, 1996a, 1996b; Gubbins and Twu, 1978). The contribution of Z (T , v ) = Z rep (T , v ) + Z att (T , v ) + Z ass (T , v ) these physical phenomena can be included in the SAFT-type approach, resulting in a more complex hybrid equation: The association contribution is identical in both equations of state. The repulsive and the attractive part are expressed ( Z = 1 + m Z hs o + Zo ) disp + Z chain + Z ass + Z dipole − dipole differently in CPA or in the different SAFT versions. In + Z quad − quad CPA, they are directly taken from the cubic equation of Redlich-Kwong. In SAFT, the reference and the dispersive term can be considered equivalent to respectively the repul- As we will show below, these additional contributions can sive and attractive terms in CPA: improve significantly the predictions in the case of compo- nents of similar volatility. Z rep (T , v ) = 1 + mZ hs o (T , v ) = 1 + Z (T , v ) hs In this article, we will show how these equations of state can be of use to industrial problems in the petroleum industry. The added value of the association term, as used in the CPA Z att (T , v ) = mZ disp o (T , v ) = Z disp (T , v ) equation is clearly shown in the case of mixtures with polar compounds such as water and methanol. Binary interaction parameters for a number of binary mixtures are proposed. The Where the reference term is build starting from hard spheres power of the SAFT equation is its predictive capability for (segments). An additional term resulting from the chain for- polar or long-chain molecules. This is illustrated by two mation is added, yielding: additional examples. The first shows how group contributions can be used for predicting vapour pressures and liquid phase volume; the second uses polar terms for the prediction of the azeotropic behaviour of near-boiling mixtures. The expressions for each of these terms are compared in Table 1. The parameters are compared later in Table 2. Two major differences can be noted between the equations. 2 SHORT DESCRIPTION OF THE EQUATIONS The Radial Distribution Function It is not the purpose here to describe the equations in detail. The original articles can be consulted for that purpose. We The first difference concerns the radial distribution function only want to stress here some basic features that will help in that appears both in the association term and, as a derivative, understanding the strengths and weaknesses of these equations. in the chain term (except for CPA that has no chain term). It

JC de Hemptinne et al. / Application to Petroleum Engineering of Statistical Thermodynamics – Based Equations of State 369 Table 1 Main expressions for the contributions to the compressibility factor of the three versions of the SAFT equation of state and CPA, used in this work Expressions Chapman et al., 1990 Gross and Sadowski, 2001 Gil-Villegas et al. 1997 Kontogeorgis et al., 1996 (SAFT-0) (PC-SAFT) (SAFT-VR) CPA 1 + η + η2 − η3 v Zhs Z hs = mZ hs 0 =m (Carnahan and Starling, 1969) Z rep = (1 − η) 3 v −b ∂ ln( g 0 (d )) Z chain = −( m − 1)ρ (Chapman et al. 1988, 1990) ∂ρ Zchain g0 = gsw : Radial distribution 2 −η g 0 (d ) = g hs (d ) = 2(1 − η) 3 (Carnahan and Starling, 1969) function developed up to the first order for a square- well potential ⎛ Z disp 01 02 ⎞ ∂ Z disp = mZ disp 0 = m⎜ Z disp + 2 ⎟ Z disp = ρ (a + a2 ) a(T ) ⎝ r ∂ρ 1 ⎡ ∂ ⎛ a ⎞⎤ T T r ⎠ Z att = − Z disp = m ⎢ρ a1 + 2 ⎥ RT (v + b) Zdisp a2 is a function of ⎣ ∂ρ ⎝ RT ⎠ ⎦ (Cotterman et al. 1986) Z hc = 1 + m( Z hs − 1) + Z chain (Soave, 1972) ⎡ 1 1 ⎤ ∂X A (Chapman et al. 1988, 1990) Z assoc = ρ∑ ⎢ A − ⎥ A ⎣X 2 ⎦ ∂ρ Zass 2 −η ghs (Carnahan and Starling, 1969) gsw (Gil-Villegas et al., 1997) g hs (d ) = 2(1 − η) 3 πN Av b η= ρd 3 m and ρ = 1/v η= 6 4v Comments d = σf1 (T) d = σf2 (T) d=σ b = σ3 is generally taken from the hard sphere expression of Soave (1972), with: Carnahan and Starling (1969): a = a 0 (1 + c1 (1 − T r )) 2 2 −η g seg (d ) = g hs (d ) = 2(1 − η) 3 where both a0 and c1 are considered as adjustable parameters. In the SAFT-VR equation, the radial distribution function In TPT based equations of state, the expression is usually is developed up to the first order and is taken from a square- obtained by fitting on molecular simulation results. For sin- well fluid. Its expression can be found in (Gil-Villegas, 1997) gle segments, the deviation from the reference can be made and is not given here. visible in a plot showing the energy potential as a function of The CPA equation is further used with a still simplified the distance between segments. This is done in Figures 4 radial distribution function, as was proposed by Elliot et al. through 6 for the three SAFT equations discussed here. (1990) and applied to CPA by Kontogeorgis et al. (1999): In most versions of the SAFT equation, the dispersion term contribution to the molecular Helmholtz energy is pro- 1 g ( η) = portional to the number of segments: a disp = ma disp o . This is (1 − 1 ⋅ 9η) the case for the SAFT-0 and the SAFT-VR equations. In PC- SAFT, the dispersion term is originally written for chains of The Dispersion Term segments. A second major difference between PC-SAFT and The dispersion expresses the attractive contribution that is the other SAFT versions is that the parameter fitting was generally not included in the reference term. The expressions done on physically measured data rather than molecular sim- are usually empirical in nature, as no exact theory exists. For ulation data. This is probably one of the reasons for its rather the cubic equation (CPA), the expression is that proposed by good description of hydrocarbon fluids.

370 Oil & Gas Science and Technology – Rev. IFP, Vol. 61 (2006), No. 3 TABLE 2 Parameters used in the equations considered Chapman et al., 1990 Gross and Sadowski, 2001 Gil-Villegas et al., 1997 Kontogeorgis et al., 1997 (SAFT-0) (PC-SAFT) (SAFT-VR) CPA Size parameter d3m d = σf1 (T) d = σf2 (T) d=σ d = σ3 Shape parameter m c1 Energy parameter ε/k a0 Well size parameter not used λ not used Association energy εA j Bi k Association volume κ 3 A j Bi d ij bβ A j B i Dipole moment μ not used Quadrupole moment Q not used u (r) u (r) Zone 1: reference Zone 2: pertubation : LJ potential : fitted potential σ r d σ -ε λσ -ε Figure 4 Figure 5 Lennard-Jones interaction potential described by the pertur- bation therory. Variable-range square-well potential model. SAFT-0 dispersion term attraction range width is characterised by an additional para- In the original version of SAFT, the Lennard-Jones meter, denoted by λ. potential is approximated as shown in Figure 4. The dispersion term is written as a development in The expression for the dispersion term has been obtained β = 1/kT, where the Helmholtz energy contributions, a1 and by Cotterman et al. (1986), using a correlation on molecular a2 are the first two perturbation terms associated with the simulation of Lennard-Jones spheres. The contributions attractive energy. disp disp Z 01 and Z 02 are simple polynomials as a function of PC-SAFT dispersion term reduced density. Figure 6 shows the pair potential used. The dispersion is SAFT-VR dispersion term modelled using a second order perturbation theory according Here, the dispersion term is developed according to a pertur- to Barker et Henderson (1967), on chain molecules rather bation development up to the first order, for a hard-sphere than spheres (the hc superscript is used rather than hs). As a reference fluid. The equation used in this work is based on a result, the dispersion terms a1 and a2 are functions of the square-well potential with a variable width (Fig. 5). The chain length m. The contributions to the Helmholtz energy,

JC de Hemptinne et al. / Application to Petroleum Engineering of Statistical Thermodynamics – Based Equations of State 371 a1 and a2, are given by Gross and Sadowski (2000) using The first of these parameters is the hard sphere size of results of Chiew (1991). the molecules. For cubic equations, this size is equal to Modelling polar non ideal systems with classical the covolume b, expressed in molar volume units. The equations of state requires the use of additional parameters SAFT equations tend to use molecular parameters, and use that depends generally on the system conditions such as therefore: temperature. The alternative approach proposed here is to π N Av 3 explicitly take into account the polarity. The commonly used d m 6 expression for the polar terms is proposed by Twu and Gubbins (1978). It is written as a perturbation development. where d is expressed in Angstrom, and m is the number of segments in the molecule. This volume is independent of ⎡ ⎤ temperature for both SAFT-VR and CPA. It is calculated by ⎢ 1 ⎥ two different temperature-dependent expressions in the A polar = A2 ⎢ ⎥ ⎢ A3 ⎥ SAFT-0 and the PC-SAFT EOS (see the last line of Table 1). ⎢⎣ 1 − A2 ⎥⎦ SAFT-0: The second and third perturbation terms are explicitly cal- culated. The first term vanishes in the development and the ⎛ 1 + 0.2977 ( kT / ε) ⎞ d = σf1(T ) = σ⎜ ⎟ ⎝ 1 + 0.33163 ( kT / ε) + 0.0010477 ( kT / ε) ⎠ other terms are determined in the spirit of a Padé approxima- 2 tion by Stell et al. (1974). The expressions of A2 and A3 of the dipole - dipole and quadrupole - quadrupole terms for PC-SAFT: pure and non spherical compounds can be found in the papers of Gubbins and Twu (1978) and Kraska and Gubbins ⎡ ⎛ 3ε ⎞⎤ d = σf 2 (T ) = σ⎢1 − 0.12 exp⎜ − ⎟ (1996). The use of these terms implies one additional para- ⎣ ⎝ kT ⎠⎥⎦ meter for each contribution: the dipole moment for the dipole-dipole interaction, and the quadrupole moment for the The number of segments is a measure of the molecular quadrupole - quadrupole interaction. shape. In cubic equations, it is usually expressed using the acentric factor, that appears in the equation in the Soave func- 2.2 The Parameters tion of the energy parameter a(T). CPA uses exactly the same expression as that of Soave (1972) for non-associating com- As we will later discuss the pure component parameters, it ponents. However, while the temperature dependence of the a is useful to summarise them as in Table 2. Excluding the parameter is kept for non-associating components, the Soave association term, three parameters are generally needed for function of the acentric factor can no longer be used. The each pure component, except for SAFT-VR that uses four. parameter c1 is then considered as an adjustable parameter. u (r) ∞ r < (σ – s1) 3ε (σ – s1) ≤ r < σ u (r) = -ε σ ≤ r < λσ 3ε 0 r ≥ λσ σ λσ r u (r): pair potential σ-S1 r: radial distance between two segments -ε σ: segment diameter ε: well depth λ: reduced well depth Figure 6 Potential proposed by Chen and Kreglewski 1997 and used by PC-SAFT; s1/σ = 0.12.

372 Oil & Gas Science and Technology – Rev. IFP, Vol. 61 (2006), No. 3 Two parameters are required for describing each pair of has exactly the same shape as that for a pure component. The associating sites. Although the number of sites on each mole- mixture composition then appears in the mixing rules applied cule may vary from 1 to 4, only two types of sites are consid- to the pure component parameters. This can also be applied ered: electropositive or electronegative. Sites may only asso- to the SAFT equations, except for SAFT-VR. In Table 3, the ciate when they are of opposite sign. As only self-associating mixing rules for a number of parameters are shown. molecules are considered, at least one electropositive and one Note that in the combination rules, an adjustable kij para- electronegative site is found on each molecule (in the case a meter is employed for the energy parameter (ε or a), but not molecule contains one single site, no electrostatic sign is for the size parameter (σ or b). given to the site, and it can associate with any other). Two The one-fluid mixing rule can be used in SAFT-VR also, parameters describes each pair of sites (they have a double but in this work, we use the MX3B mixing rule (Galindo et superscript, AB): one energy parameter and one volume para- al., 1998), which introduces the mixture explicitly in the meter. The energy parameter is expressed in an identical SAFT-VR equation. However, cross-parameters (double fashion for all equations considered. The association volume index) are defined in a similar way as for SAFT-0 and PC- is a proportion of the average segment volumes of the two SAFT. The same is true for the association term and the associating molecules. This is why a double index is required terms describing the polar contributions. Combination rules for d in the case of SAFT, and for b in the case of CPA. for the association parameters are needed when two associat- These parameters are determined for self-associating mole- ing species are considered in the same mixture. These are dif- cules, and as a result they are considered as molecular para- ferent for SAFT and CPA. None of the combination rules meters. When considering cross-association, as in the case of considered here use a binary interaction parameter. methanol and water, a combination rule is required. It is For SAFT, we take: given in the next section. Each polar interaction comes with one additional pure ⎛ AB 1 3 ( )1 3 ⎞⎟ 3 ( κ ii ) + κ AB =⎜ component parameter. To our knowledge, the use of these addi- jj κ AB tional terms has only been applied to SAFT-type equations. ij ⎜ 2 ⎟ ⎝ ⎠ 2.3 Extension to Mixtures and The usual approach for mixture is the so-called “one-fluid” ε AB ij = ε ii ε jj AB AB approach that states that the equation of state for a mixture TABLE 3 Mixing rules for the one-fluid approach mixing rules Parameter Combination rules (cross parameters) SAFT-0 & PC-SAFT CPA n m mx = ∑ x i mi – – i =1 n n ε or a ∑ ∑ x i x j mi m j ε ij σ 3ij ( ) ε ij = ε ii ε jj 1 − k ij ε x σ 3x = i =1 j =1 a = ∑ ∑ x i x j a ij a ij = a i a j (1 − k ij ) ⎛n ⎞2 i j ⎜∑ x m ⎟ ⎜ i =1 i i ⎟ ⎝ ⎠ n n σ or b ∑ ∑ x i x j mi m j σ 3ij σ 3x = i =1 j =1 b = ∑ ∑ x i x j bij ( σ ij = σ ii + σ jj ) 2 ⎛n ⎞2 ⎜∑ x m ⎟ i j ( bij = bi + b j ) 2 ⎜ i =1 i i ⎟ ⎝ ⎠ λ iiσ ii + λ jj σ jj λ – – λ ij = σ ii + σ jj

JC de Hemptinne et al. / Application to Petroleum Engineering of Statistical Thermodynamics – Based Equations of State 373 For CPA, several mixing rules (Derawi, 2002) can be non-ideal behaviour. Three types of phase behaviour can be used. Some of them are applied on the cross energy and vol- found: ume association (ε and β) and others are directly applied to – the binary systems which have a type I phase behaviour the cross association strength (Δ) as proposed by Suresh and do not show liquid-liquid equilibria. Examples are binary Elliot (1991). In this work, we have used the following mix- hydrocarbon systems containing methane ; ing rules: – those of type II are those where liquid-liquid equilibria do exist, but are unconnected to the liquid-vapour equilibria. ε ii + ε jj β ij = β iiβ jj and ε= An example is the propane-methanol system; 2 – finally, the type III such as the ethane-methanol binary and almost all water-hydrocarbon systems give a liquid- 3 APPLICATIONS liquid-vapour equilibrium. Each of these systems will be investigated in turn, and the The applications of this type of equation in the petroleum use of the CPA equation will be illustrated. In all cases, industry are potentially numerous. Depending on the type of methanol is considered as a component with two association application, different features may be of interest. We will dis- sites (Voutsas et al., 1997) and it is described with a 2B cuss some of these applications in some more detail: model. Water is considered as a 4C associating component. – for mixtures containing small associating molecules such The pure component parameters are taken from Kontogeorgis as water and alcohols, the association term makes a signif- et al. (1999) and are reported in Table 4. The parameters of icant difference, even combined with a simple cubic the non-associating components are calculated using the criti- equation. This will be discussed below; cal constraints (Soave, 1972). Their critical parameters are – when considering long chain molecules (polymers, long provided in Table 5. chain esters, heavy oils, paraffinic mixtures as in the case of Fischer-Tropsch synthesis), the SAFT approach allows 3.1.1 Binary Systems Containing Methanol the use of group contribution methods for the calculation and a Light Hydrocarbon of pure component parameters (Tamouza et al., 2004). As We present here two binary systems, ethane-methanol and we will see, it is even possible to differentiate between pentane-methanol. We choose them because they summarise isomers, which is usually one of the limitations of group the principal difficulties that may be found in fluid phase contribution methods; equilibrium modelling, i.e. phenomena such as liquid- liquid- – the possibility to add terms that represent specific polar vapour equilibrium, azeotropic point and critical point of liq- interactions opens up the possibility to use the SAFT EOS uid- liquid equilibrium. in a predictive manner for a number of applications that previously required a large amount of data in order to fit Methanol-ethane system empirical parameters (kij). We will illustrate this by two The investigation of the ethane-methanol system permits us to examples. In the first example, two isomers are differenti- emphasise the contribution of the association term by compar- ated by a dipolar contribution. In the second example, the ing the results of the CPA and the SRK equations of state with- use of a quadrupolar term is shown to be sufficient to out binary interaction parameter. Figure 7 shows the phase dia- describe non-idealities in mixtures containing aromatic gram of the binary system at 298.15 K. Although SRK molecules; correctly shows a liquid - liquid phase equilibrium split, the – obviously, when the above difficulties are combined, the three - phase pressure and the composition of the liquid phases new type of equations is expected to provide a significant are underestimated. On the contrary, CPA yields good results improvement with respect to the classical methods. particularly with regard to the three phase liquid- liquid- Beyond this list that describes the applications where it vapour condition. Comparing the CPA results with the experi- seems to us that the statistical mechanic EOS bring a signifi- mental data provided, we observe a good modelling of the cant improvement, the SAFT equation has also been used on VLE especially in vicinity of the liquid- liquid- vapour equilib- more exotic cases such as asphaltene precipitation (Ting et rium. The comparative performance between CPA and SRK, al., 2003) or hydrogen solubility (Florusse et al., 2003; Gosh based on the experimental results of Ishihara et al. (1998) at et al., 2003) . the three phase equilibrium is summarised in Table 6. Table 7 provides some information concerning the aver- 3.1 Phase Equilibria of Water age relative deviation between the two equations and the and Alcohol-Containing Mixtures experimental data. This additional piece of information will help evaluating the capacity of these equations using other Small molecules such as water and methanol have a strong binary systems. The compositions of the alcohol and of the hydrogen bonding faculty. As a result, they exhibit a strongly organic liquid phases are differentiated using respectively x

374 Oil & Gas Science and Technology – Rev. IFP, Vol. 61 (2006), No. 3 TABLE 4 CPA parameters for the associating components (from Derawi, 2002) a0 b (× 105) εOH/R c1 βOH (Pa.m3/mol) (m3/mol) (Pa.m3/mol) Methanol 0.40521 0.431 3.0978 2957 0.0161 Water 0.12274 0.67359 1.4515 2002.73 0.0692 TABLE 5 Critical parameters for the hydrocarbons TC (K) PC (MPa) ω Methane 190.55 4.6 0.0111 Ethane 305.43 4.884 0.097 Propane 369.82 4.25 0.1536 n-butane 425.13 3.8 0.2008 n-pentane 469.65 3.369 0.2506 TABLE 6 Three phase equilibrium pressure and composition for the ethane - methanol mixture at 298.15K experimental and predicted P (MPa) x (ethane) x' (ethane) y (ethane) Ishihara et al. (1998) 4.091 0.40 0.962 0.9950 CPA 4.223 0.39 0.9768 0.9936 SRK 3.525 0.30 0.8205 0.9902 TABLE 7 Correlation performance of the CPA and SRK EOS for the ethane (1) methanol (2) system: analysis in terms of the average relative deviation (%). ΔP/P Δx/x1 Δx/x2 Δx’/x1 Δx’/x2 Δy/y1 Δy/y2 CPA –3.13 2.56 1.64 –1.52 –63.79 0.14 21.87 SRK 16.06 33.33 1.43 17.25 78.83 0.48 48.98 and x’, and ethane is component 1. We can see that a weak equilibrium as well as liquid-liquid equilibrium. The deviation as shown in Figure 7 can yield an important rela- azeotropic point and the upper critical end point (UCEP) are tive deviation. When we compare both equations of state, the reproduced with errors on the pentane composition of deviations obtained for each confirm the important contribu- between 3% and 9%. tion of the association term. The use of this term improves Although a zero binary interaction parameter provides rea- significantly the phase equilibrium calculation. sonable results, it appears that a small non-zero value may Methanol-pentane system improve significantly the liquid-liquid equilibrium predic- The experimental data available for the binary pentane- tions. Table 8 shows the average relative deviations on the methanol system allows us to study simultaneously liquid- mole fraction of each component in each phase and on the vapour and liquid-liquid equilibrium at atmospheric pressure. equilibrium pressure, with and without kij. No real trend can In Figure 8 the predictions for both equilibrium lenses by be observed in the value of this parameter with molecular CPA is compared with experimental data of Bernabe et al. weight of the hydrocarbon. (1988) for the liquid-liquid equilibrium and Budantseva et al. In Table 8, number 1 refers to the hydrocarbon and x and (1975) for the vapour-liquid equilibrium. CPA has been used x’ are respectively the alcohol and organic phase composi- with no binary interaction parameter. This binary system tion. If we compare with the deviations shown in Table 7, we confirms the observations made previously, namely that the notice that the CPA model can provide reasonable results association term improves the modelling of the methanol- despite the important deviations. Generally speaking, with a hydrocarbon system. We can predict correctly liquid- vapour binary interaction parameter, CPA provides better results for

JC de Hemptinne et al. / Application to Petroleum Engineering of Statistical Thermodynamics – Based Equations of State 375 7 340 Beranbe et al. CPA without kij Budantseva et al. 330 6 320 5 310 Pression (MPa) 4 T (k) 300 3 290 Ishihara et al. Zeck and Knapp 280 2 Ma and Kohn CPA 270 1 SRK 260 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 x (C2), y (C2) x (C5), y (C5) Figure 7 Figure 8 Comparison of the SRK and CPA performance compared to Phase diagram of pentane-methanol system at the atmospheric the literature data for the ethane-methanol system at 298.15 K. pressure. The points represent data; the line is the prediction from CPA (kij = 0). TABLE 8 Correlation performance of the CPA model (deviations in %) Vapour-Liquid-Equilibrium (1 is the hydrocarbon) T range Systems NP CPA without kij CPA with kij from Table 9 (K) ΔP/P Δy/ y1 Δy/ y2 ΔP/P Δy/ y1 Δy/ y2 C1 - methanol 220-340 56 –11.8 0.1 9.3 –6.5 0.19 0.19 C2 - methanol 298-323 30 –17.0 –1.7 –14.9 –19.4 –1.3 –1.3 C3 - methanol 213-474 46 –11.3 –7.9 –1.69 –14.3 –9.6 –9.0 nC4 - methanol 273-373 33 -4.1 0.7 –9.2 –2.9 0.3 0.3 nC5 - methanol 303-422 90 -4.5 2.7 2.6 –8.7 0.7 0.7 Liquid-Liquid-Equilibrium T range Systems NP CPA without kij CPA with kij from Table 9 (K) Δx/ x1 Δx/ x2 Δx’/x 1 Δx’/x 2 Δx/ x1 Δx/ x2 Δx’/x 1 Δx’/ x2 C2 - methanol 233-300 10 17.1 10.1 -3.6 –49.8 -6.6 -4.8 -4.5 –62.7 C3 - methanol 323-250 10 52.3 36.2 2.8 32.4 -40.1 –28.7 –10.7 –72.6 nC4 - methanol 263-272 31 23.8 22.1 8.9 39.8 1.8 10.0 –16.9 22.9 nC5 - methanol 269-277 15 –6.8 –18.0 –6.3 -4.3 –5.0 9.8 6.5 6.4

376 Oil & Gas Science and Technology – Rev. IFP, Vol. 61 (2006), No. 3 all systems and it remains difficult to calculate the composi- liquid-liquid equilibrium calculation. We observe that the tion of the minority component in each phase. Although we best results are obtained with a temperature-dependent kij. use kij fitted on liquid-liquid equilibrium data, the results are The parameters are listed on Table 9, when the kij is obtained also improved for vapour-liquid equilibrium. from the linear equation: For each binary system, the vapour-liquid and liquid-liq- uid equilibrium have been calculated with the same binary k ij = k ij (1) ⋅T + k ij ( 2) interaction parameter (kij). These parameters have been fitted only on liquid-liquid equilibrium data by minimising the fol- 3.1.2 Binary water - light hydrocarbon systems lowing objective function: Water-hydrocarbon mixtures are frequently encountered in NP ⎛ 2⎛ x exp − x cal ⎞2 ⎛ x ' exp − x ' cal ⎞2 ⎞ the petroleum industry. Due to the strong polarity of water, ⎜ OF = ∑ ∑ ⎜⎜ i exp i ⎟⎟ + ⎜⎜ i exp i ⎟⎟ ⎟ the mutual solubilities are very small. This is the reason why ⎜ ⎠ ⎟⎠ n =1 ⎝i =1 ⎝ xi ⎠ ⎝ x 'i the relative uncertainties on the experimental data are signifi- cant and large scatter is observed. where x1 is the mole fraction of methanol and x2 that of the Several authors have attempted to model these systems hydrocarbon. (Soreide and Whitson, 1992; Dhima, 1998; Daridon et al., 1993) but all came to the conclusion that a different model TABLE 9 (or different parameters) is required for a correct description Alcohol- hydrocarbon binary parameters used, of the solubilities. with kij = kij(1).T + kij(2) Figures 9 to 12 show how the CPA model is able to pre- Binaries systems dict the hydrocarbon solubility in water. The figures show kij(1) kij(2) both the predictions in the absence of a kij, and with a kij that methanol + has been fitted on the data. The optimal kij parameters are Methane 0.00 1.00E-02 generally temperature-dependent. Ethane –2.77E-04 8.31E-02 Propane –2.86E-04 8.03E-02 The pressure and temperature range of the water-hydro- n-butane –1.36E-05 8.71E-03 carbon experimental data are summarised in Table 10. The n-pentane –1.52E-05 5.73E-03 values of the binary parameters in function of temperature are provided in Table 11. The water- methanol mixture has similarly been tested, The vapour-liquid equilibrium is almost insensitive to the and a single binary interaction parameter of –0.03 has been value of the kij, while a small kij may significantly affect the found to best reproduce the data. 1 0.01 0.001 Calculated mole fraction 0.01 Calculated mole fraction 0.001 0.0001 0.0001 0.00001 CPA CPA CPA + kij (T) CPA + kij (T) 0.00001 0.000001 0.00001 0.0001 0.001 0.01 0.1 0.000001 0.00001 0.0001 0.001 0.01 Experimental mole fraction Experimental mole fraction Figure 9 Figure 10 Experimental vs calculated methane solubility in water. Experimental vs calculated ethane solubility in water.

JC de Hemptinne et al. / Application to Petroleum Engineering of Statistical Thermodynamics – Based Equations of State 377 0.01 0.01 0.001 0.001 Calculated mole fraction Calculated mole fraction 0.0001 0.0001 0.00001 CPA 0.00001 CPA + kij (T) Serie 4 CPA CPA + kij 0.000001 0.000001 0.000001 0.00001 0.0001 0.001 0.01 0.000001 0.00001 0.0001 0.001 0.01 Experimental mole fraction Experimental mole fraction Figure 11 Figure 12 Experimental vs calculated propane solubility in water. Experimental vs calculated n-Butane solubility in water. 3.1.3 Multicomponent Systems Hydrocarbons-Methanol Tables 12 and 13 show both the CPA calculation results and Water and the experimental values for the three phase VLLE of a four component mixture at 6.7 MPa, for two temperatures In conclusion of our CPA study, we examine a quaternary (253.15 K and 263.15 K). CPA correctly predicts a three system containing methane, propane, water and methanol. A phase equilibrium. The mole fraction prediction of the minor- study of this system has been published by Rossilhol (1995). ity components (the hydrocarbons) in the aqueous phase, The purpose is here to test the predictive capacities of CPA where the association phenomena are most important, is very for multicomponent mixtures when we use the defined binary accurate. The vapour phase also is accurately reproduced. In parameters. the organic phase, CPA reproduces well the molar fraction of the majority components, but there are significant deviations TABLE 10 from experiment concerning methanol and water. Never- The data sets used to evaluate CPA theless, we notice that the order of magnitude is correct. The results on this quaternary system show that CPA is able to Binary T(K) P(MPa) Points number model in a very satisfactory way the complex systems con- Water - methane 283.15-573.15 0.097-75 448 taining hydrocarbons and associating components. We can Water - ethane 298.15-510.93 0.06-50 115 notice that these good results are obtained with the same Water - propane 273.15-422 0.01-19.32 46 equation of state for each phase and that the same binary Water - n-butane 298.15-477.59 0.1-48.26 230 parameters are used for each compound. To perform the same kind of results with classical EOS, we would need to use more complicated mixing rules such as Huron-Vidal or MHV2. TABLE 11 Water - hydrocarbon binary parameters used, 3.2 Vapour Pressure of Heavy Molecules with kij = kij(1).T + kij(2) In this second section, we illustrate the use of the SAFT Binaries systems kij(1) kij(2) equation of state coupled with a group contribution method. water + Heavy hydrocarbon molecules are often present in petro- Methane 1.76E-03 -0.539 leum fluids, and their properties must also be adequately Ethane 9.22E-04 -0.230 described. When cubic equations are used, their properties Propane 7.65E-04 -0.229 are predicted through a double assumption. First, the cubic n-butane 0 0.0057 equation parameters are calculated using the corresponding

378 Oil & Gas Science and Technology – Rev. IFP, Vol. 61 (2006), No. 3 TABLE 12 Three phase equilibrium for quaternary systems at 253.15 K and 6.7 MPA Experimental data CPA predictions Component Feed Aqueous Organic liquid Vapour Aqueous Organic liquid Vapour phase phase phase phase phase phase Methanol 0.153 0.536 0.0007 0.528 0.00165 0.000219 Water 0.128 0.447 0.444 0.000039 0.000017 Methane 0.532 0.0105 0.540 0.874 0.0201 0.522 0.908 Propane 0.187 0.0071 0.459 0.126 0.0078 0.477 0.092 Mole fractions 1 0.286 0.286 0.429 0.288 0.310 0.401 TABLE 13 Three phase equilibrium for quaternary systems at 263.15 K and 6.7 MPA Experimental data CPA predictions Component Feed Aqueous Organic liquid Vapour Aqueous Organic liquid Vapour phase phase phase phase phase phase Methanol 0.156 0.545 0.001 0.539 0.0030 0.00042 Water 0.124 0.435 0.432 0.000081 0.000034 Methane 0.520 0.0105 0.516 0.861 0.0190 0.474 0.880 Propane 0.200 0.0098 0.483 0.139 0.0095 0.523 0.119 Mole fractions 1 0.28557 0.286 0.429 0.287 0.279 0.434 states principle. Here, it is assumed that if the critical point is The three SAFT parameters are then calculated as follows correctly represented, all properties, including at very low (four parameters if we include λ for SAFT -VR): reduced temperatures, are correctly represented. The second assumption lies in the prediction method of these critical ngroupes ⎛ n groupes ⎞ ∑ ni ⎜ ∏ εni ⎟ properties that is impossible to verify with physical ε molécule = i =1 ⎜ i =1 i ⎟ experiments, since the molecules are unstable at high temper- ⎝ ⎠ ature. Many methods exist, but their extrapolations to very n groupes long chains differ significantly. Probably the best of them is based on theoretical concepts that provide a limit for these ∑ n iσ i i =1 properties for infinite chain length (Elliott et al., 1990). σ molécule = n groupes The industrial need, however, concerns a moderate tem- ∑ ni i =1 perature range, that usually corresponds to very low reduced temperatures, and very low vapour pressures. Some data n groupes exist, which will allow us to validate the proposed method, but much more data would be necessary considering the ∑ n iλ i i =1 λ molécule = n groupes number of isomers and the large experimental errors that are observed in these extremely low pressure ranges. Hence, ∑ ni there is a need for predictive calculation methods. i =1 In their work, Tamouza et al. (2004) propose a group con- n groupes tribution method for calculating SAFT pure component para- m molécule = ∑ n i Ri i =1 meters. It is believed that because the SAFT equation is based on physical concepts, the use of group contributions for determining its parameters is justified. However, they where ni, is the number of groups of type i in the molecule. remain adjustable parameters, and the use of experimental Figure 13 shows an example of how a molecule is split into data is needed for regressing the group parameters. groups.

JC de Hemptinne et al. / Application to Petroleum Engineering of Statistical Thermodynamics – Based Equations of State 379 CH2 CH2 CH3 CH2 CH2 OH (ε, σ, λ, R)CH3 { (ε, σ, λ, R)CH2 (ε, σ, λ, R)OH (ε, σ, λ, m)molecule Figure 13 Group contribution and SAFT models for 1-pentanol, withe three associating sites. TABLE 14 Group contribution parameters for SAFT-0, SAFT-VR and PC-SAFT EOS Equation Group ε/k · K σ·Å λ R εAB/k · K(1) kAB(1) –CH2- 208.1 3.42 0.51 –CH3 167.9 3.51 0.86 SAFT-0 –OH 279.9 2.87 0.93 2212.0 0.01382 (C)α-oléf. 186.6 3.41 0.64 (C)Bz 243.5 3.45 0.42 (C)Ch 239.9 3.69 0.41 –CH2- 136.4 3.42 1.90 0.47 –CH3 202.9 3.54 1.47 0.80 SAFT-VR –OH 328.2 2.97 1.55 0.82 2170.0 0.01122 (C)α-oléf. 280.7 3.95 1.47 0.41 (C)Bz 165.6 3.43 1.88 0.39 (C)Ch 233.2 3.96 1.70 0.30 –CH2- 261.1 3.93 0.38 –CH3 190.0 3.49 0.79 PC-SAFT –OH 391.8 3.15 0.67 2281.6 0.00554 (C)α-oléf. 226.1 3.63 0.54 (C)Bz 306.7 3.83 0.32 (C)Ch 305.3 4.09 0.31 (1)Here, the association is considered with a three sites (3B) model. The parameters that have been proposed for the equations volumes are predicted with a deviation of less than 5%. The SAFT-0 and SAFT-VR (Tamouza et al., 2004) have been results are compared with another group contribution extended here with those more recently determined for PC- method, reported by Mattedi et al (1998). Note that no polar SAFT and they are reported in Table 14. term was used in the predictions reported here. In Tables 15 and 16, and Figures 14 and 15, the method is More recently Tamouza et al. (2005) have adapted their used in a predictive mode, i.e. no additional regression was approach by including methyl-alkanes in their group contri- performed. The vapour pressures and liquid molar volumes bution method. Here, the branched methyl group is consid- results are shown for several n-alkanes, n-alkanols, an olefin ered as an additional group that has the same contribution to and a heavy alkyl-benzene. None of these compounds belong the σ and ε/k parameters, but that is differentiated from the to the fitting data bank. It can be seen that the deviations regular –CH3 group in its R parameter. The reason for this between calculations and experimental data increase with choice is that it may be considered that the contribution of a chain length. Not surprisingly, the vapour pressure data are branched methyl group does not change the global alkane the most difficult to predict. Despite this, the VR-SAFT energy contribution, nor the segment size, but that the chain method is clearly superior, probably because of the additional length is now no longer linear with the number of carbon parameter. The results are impressively accurate as the aver- atoms. Depending on whether the branched methyl group is age deviation remains below 15%, which is of the same order at position 2, 3 or beyond, the contribution is found to change of magnitude as the experimental uncertainties. The molar according to the Table 17, and the graph shown in Figure 16.

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