Sanchez Lacombe

Fluid Phase Equilibria 203 (2002) 133–140

A consistent method for phase equilibrium calculation using the Sanchez–Lacombe lattice–fluid equation-of-state Evelyne Neau∗ Laboratoire de Chimie Physique de Marseille, Faculté des Sciences de Luminy, Case 901, 163 Avenue de Luminy, 13288 Marseille Cedex 9, France Received 28 March 2002; accepted 5 June 2002

Abstract The Sanchez–Lacombe equation-of-state is known to describe thermodynamic properties of molecular fluids of arbitrary size, mainly polymer–solvent phase behaviour. On the basis of the Helmholtz energy obtained from the partition function of mixtures, chemical potentials are usually derived in order to compute phase equilibrium conditions at various temperatures and pressures. In this work, it is shown that, whatever the mixing rules considered, the chemical potentials derived in this way are thermodynamically inconsistent. The fugacity coefficients derived from the Sanchez–Lacombe equation-of-state are proposed for calculating consistent phase equilibrium conditions. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Lattice–fluid theory; Equation-of-state; Phase equilibria

1. Introduction The lattice–fluid model was proposed by Sanchez and Lacombe [1] to describe thermodynamic properties of molecular fluids of arbitrary size, mainly polymer–solvent phase behaviour. The basis of the model was, according to statistical mechanics, the estimation of thermodynamic properties from the partition function. As recalled in Table 1, the absolute Helmholtz energy A(T, V, n), chemical potentials µi (T, P, xj ) of mixture components and the equation-of-state P(T, V, n) may be derived from the canonical partition function Q(T, V, n). However, as usually for practical calculation of thermodynamic properties, Sanchez and Lacombe [1] used the more convenient configurational partition function Z(T, V, n):
 E Q(T , V , n) Z(T , V , n) = Ωc exp − = (1) RT λ(T , n) where Ω c is the number of configurations and E is the lattice energy of a system containing n molecules in a volume V, at temperature T; λ(T , n) corresponds to the internal and kinetic parts of the canonical partition ∗

Tel.: +33-491-829149; fax: +33-491-829152. E-mail address: [email protected] (E. Neau). 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 2 ) 0 0 1 7 6 - 0

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E. Neau / Fluid Phase Equilibria 203 (2002) 133–140

Table 1 Thermodynamic functions from statistical partition functions Thermodynamic functions

Canonical partition function Q(T, V, n)

Configuration partition function Z(T, V, n) = Q(T, V, n)/λ(T, n)

Helmholtz energy

A(T , V , n) = −RT ln Q
 ∂ ln Q µi (T , P , n) = −RT ∂ni T ,V ,nj
 ∂ ln Q P = RT ∂V T ,n

Aconf (T , V , n) = −RT ln Z
 ∂ ln Z µconf (T , P , n) = −RT i ∂ni T ,V ,nj
 ∂ ln Z P conf = RT =P ∂V T ,n

Chemical potentials (i = 1, . . . , c) Pressure

function, which are identical to those of an ideal gas (see Prausnitz et al. [2]). Thermodynamic properties obtained from this partition function are the relative configurational properties (Table 1). Obviously, the configurational pressure obtained in this way is the absolute pressure: P conf = P and the configurational chemical potentials µconf can be used for phase equilibrium calculations as long as the reference internal i and kinetic terms are independent on mixture mole fractions. In this work, it is pointed out that mixture configurational chemical potentials proposed in literature for the Sanchez–Lacombe equation-of-state do not verify this property, since their expressions correspond to a reference term function of phase composition. This problem is independent on mixing rules considered and is evidenced by the fact that literature chemical potentials do not satisfy the fundamental thermodynamic expressions of chemical potentials in the ideal gas state. Hence, whatever pressure, literature chemical potentials cannot be used for thermodynamically consistent phase equilibrium calculations. To overcome this problem, consistent expressions of fugacity coefficients derived directly from the equation-of-state P(T, V, n) are proposed for the most classical mixing rules. 2. Chemical potentials for the Sanchez–Lacombe equation-of-state According to the lattice–fluid model (Sanchez and Lacombe [1,3], Sanchez and Panayiotou [4]) the configurational Helmholtz energy can be expressed (Table 1) from the configurational partition function Z(T, V, n) (Eq. (1)) as:  
   xi φi  1 1 conf ∗ A (T , V , n) = nr −ρε (2) ˜ + RT − 1 ln(1 − ρ) ˜ + ln ρ˜ + RT ln ρ˜ r r ωi where the reduced density ρ˜ for a given molar volume v is defined as: ρ˜ =

1 rv ∗ = v˜ v

and the segment fraction φi is:  xi ri , r= xi ri φi = r

(3)

(i = 1, . . . , c)

(4)

where ωi is a constant depending on the segment number ri of component i and xi is its molar fraction.The characteristic parameters of the model are the segment number ri corresponding to each pure component,

E. Neau / Fluid Phase Equilibria 203 (2002) 133–140

135

i.e. the number of sites occupied in the lattice, the volume v ∗ occupied by one segment and the mer–mer pair interaction energy εii∗ . For mixtures appropriate mixing rules should be considered for the calculation of the characteristic parameters r, v ∗ and ε∗ . As pointed out by Prausnitz et al. [2], there is a fundamental difficulty since the segment sizes and volumes, ri and vi∗ , of different molecules are not the same. This leads to make some assumptions, and therefore, since the concept of mixing rules on v ∗ is somewhat arbitrary, different mixing rules were proposed in the literature [3,5–8]. By differentiation of the configurational Helmholtz energy (Eq. (2)) with respect to the amount of substance (Table 1):
conf  ∂A (T , V , n) conf µi (T , P , xj ) = RT (i = 1, . . . , c) (5) ∂ni T ,V ,nj we obtain the configurational chemical potentials:




  ˜ (T , P , x ) µconf 1 ρ ˜ r P v ˜ j i i = ln φi + 1 − + ln ρ˜ + ri − + − 1 ln(1 − ρ) ˜ + RT r ρ˜ T˜ T˜


 


  z nr ∂v ∗ ρ˜ nr ∂ε∗ + − (i = 1, . . . , c) r v ∗ ∂ni nj T˜ ε ∗ ∂ni nj

(6)

where the compressibility factor z is calculated from the equation-of-state (Eq. (17)) and the reduced properties are defined as: Pv ∗ P˜ = ∗ , ε

T RT T˜ = ∗ = ∗ , T ε

ρ˜ =

ρ rv ∗ = ∗ v ρ

(7)

In Table 2, the expressions of the partial derivatives of characteristic parameters v ∗ and ε∗ with respect to the amount of substance are given for classical literature mixing rules. Configurational chemical potentials given by (Eq. (6)) are related to the absolute chemical potentials µi (T , P , xj ) (Eq. (1) and Table 1) by the expression: 
∂ ln λ(T , n) ref ref µconf (T , P , x ) = µ (T , P , x ) − µ , µ = −RT (8) j i j i i i ∂ni T ,nj Hence, they can be used for phase equilibrium calculations, at given T and P, as long as the reference term µref i (Eq. (8)) is independent on molar fractions. In this case, the limiting value of configurational chemical potentials (Eq. (6)) at very low pressure, i.e. in the ideal gas state, should verify the fundamental thermodynamic relation: µconf(ideal) (T , P , xj ) = [µideal (T ) − µref i i (T )] + RT ln(Pxi ) i

(9)

However, it can be easily evidenced that at very low pressure, i.e. in the ideal gas state, literature configurational chemical potentials (Eq. (6)) do not verify the previous relation (9). Indeed, in this state characterised by P → 0, or ρ˜ → 0 and z → 1, it can be shown, thanks to the following relationships: 

z P˜ v˜ 1 lim = lim → (10) z→1 r r P˜ →0 T˜

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E. Neau / Fluid Phase Equilibria 203 (2002) 133–140

Table 2 Partial derivatives of v ∗ and ε ∗ for mixing rules proposed in literature v∗ =

∗

v =

∗

ε =



φi vii∗







Sanchez and Lacombe [3], Sanchez and Panayiotou [4]

φi φj vij∗

MacHugh and Krukonis [5], Sato et al. [6], Orbey et al. [7]: vij∗ = 21 (vii∗ + vjj∗ )(1 − lij ); ∗1/3 ∗1/3 West et al. [8]: vij∗ = [ 21 (vii + vjj )]3

φi φj εij∗

Sanchez and Panayiotou [4]: εij∗ = 21 (εii∗ + εjj∗ ) − 21 RTkij ; West et al. [8]: εij∗ = (εii∗ εjj∗ )1/2 (1 − kij )

∗ ∗

εv= φi φj (εv)∗ij

MacHugh and Krukonis [5], Sato et al. [6], Orbey et al. [7]: (εv)∗ij = εij∗ vij∗ , εij∗ = (εii∗ εjj∗ )1/2 (1 − kij ); Sanchez and

1/2 εii∗ εjj∗ ∗ Lacombe [3]: (εv)ij = (1 − kij ) vii∗ vjj∗

nr v∗ nr v∗

nr ε∗

nr ε∗













∂v ∗ ∂ni ∂v ∗ ∂ni

∂ε ∗ ∂ni

∂ε ∗ ∂ni

  nj

=

1 [ri (−v ∗ + vii∗ )] v∗

=

 1 [2ri (−v ∗ + φj vij∗ )] ∗ v

=

 1 [2ri (−ε ∗ + φj εij∗ )] ∗ ε

  nj

  nj

 

nj


   nr ∂v ∗ 1 ∗ ∗ ∗ φj (εv)ij )]− ∗ = ∗ ∗ [2ri (−ε v + ε v v ∂ni nj

and


   Prv ∗ Prv ∗ lim ln ρ˜ = lim ln → ln z→1 ρ→0 ˜ zRT RT

(11)

that the limiting value of the configurational chemical potentials (Eq. (6)) is expressed by the following relation:


  n ∂v ∗ conf ∗ lim µi (T , P , xj ) = RT (ln ri + 1 − ri − ln RT) + ln v + ∗ + RT ln (Pxi ) (12) P →O v ∂ni nj Comparison of relations (9) and (12) clearly shows that, except for pure fluids or for special mixing rules assuming that the mixture characteristic parameter v ∗ is a constant, the reference term µref i in relation (8) is a function of both temperature and molar fractions which does not cancel when solving phase equilibrium conditions. Hence, whatever the pressure, literature chemical potentials with classical mixing rules will lead to erroneous calculations. 3. Fugacity coefficients from the equation-of-state Fugacity coefficients can be derived from the expression of the residual Helmholtz energy: Ares (T , V , n) = A(T , V , n) − Aideal (T , V , n)   V
 V RT dV =− P −n (z − 1) dV − nRT V V ∞ ∞

(13)

E. Neau / Fluid Phase Equilibria 203 (2002) 133–140

by the classical relation:
res  ∂A (T , V , n) ln ϕi = −ln z + ∂ni T ,V ,nj

137

(14)

since the limit of the compressibility factor, z = PV/nRT = 1 for the ideal gas when the volume V is infinite. The equation-of-state is obtained by differentiation of the configurational Helmholtz energy (Eq. (2)) with respect to the volume according to (Table 1):
conf  ∂A P =− (15) ∂V T ,n which leads to the Sanchez–Lacombe lattice–fluid equation-of-state: 
  1 P˜ = −T˜ ln(1 − ρ) ˜ + 1− ρ˜ − ρ˜ 2 r and the compressibility factor z:  
 1 P˜ v˜ 1 ρ˜ Pv r = r − ln(1 − ρ) = ˜ − 1− − z= RT ρ˜ r T˜ T˜ By integration of Eq. (13), we obtain:
   1 ρ˜ res − 1 ln(1 − ρ) ˜ +1 A (T , V , n) = nrRT − + ρ˜ T˜ and, according to Eq. (14), the fugacity coefficients are expressed by the following relation:  

∗  ρ˜ z−1 nr ∂v ln ϕi (T , P , xj ) = −ln z + ri −2 − ln(1 − ρ) ˜ + ∗ ˜ r v ∂ni nj T 


 ρ˜ nr ∂ε∗ (i = 1, . . . , c) − T˜ ε ∗ ∂ni nj

(16)

(17)

(18)

(19)

In this case, it can be observed that the limiting values of fugacity coefficients given by the above relation (Eq. (19)) are thermodynamically consistent in the ideal gas state, i.e. that ln ϕi → 0 when P → 0, whatever the mixing rules considered. 4. Comparison of liquid–vapour calculations by means of literature chemical potentials and fugacity coefficients In this section, the influence of the choice of procedures based on (1) literature chemical potentials (Eq. (6)) or on (2) fugacity coefficients (Eq. (19)) for solving liquid–vapour equilibrium conditions was investigated. The binary system CO2 –n-decane was considered since a wide range of isothermal VLE data were measured in literature. In order to compare the capabilities and limitations of the different methods we

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E. Neau / Fluid Phase Equilibria 203 (2002) 133–140

have compared, for each above procedures, results obtained when binary interaction parameters k12 and l12 were fitted: (I) to the whole range of VLE data or only (II) to lower temperature data. Thus, an erroneous method for VLE calculations should lead to very bad extrapolations form lower to higher temperatures. Calculations were performed using quadratic mixing rules on both parameters v ∗ and ε ∗ [5–7] which allow the best correlation of data with a restricted set of binary parameters:   φi φj vij∗ , (20) v∗ = ε∗ v ∗ = φi φj εij∗ vij∗ Table 3 System CO2 –n-decane: comparison of data correlation by means of chemical potentials and fugacity coefficients using quadratic mixing rules on vij∗ and εij∗ T (K)

Reference

From chemical potentials (k12 = 0.0573; l12 = 0.1745)

From fugacity coefficients (k12 = 0.1565; l12 = −0.0132) !x1V

!P/P%

!x1V

(i) k12 , l12 fitted to the whole set of experimental VLE data 310.93 [9] 35.70 342.90 [10] 44.15 344.26 [9] 26.71 344.30 [11] 52.01 377.60 [11] 22.43 377.59 [9] 66.71 477.59 [9] 40.02 510.93 [9] 37.08 520.40 [10] 54.25 542.95 [12] 23.35 583.65 [12] 15.24 594.20 [10] 58.11

0.0007 0.0100 0.0030 0.0085 0.0084 0.0262 0.0699 0.1238 0.1485 0.2470 0.2754 0.5082

16.34 5.57 5.50 5.12 2.50 3.47 14.64 13.17 14.47 15.55 9.36 20.64

0.0003 0.0144 0.0019 0.0146 0.0019 0.0187 0.0049 0.0147 0.0185 0.0275 0.0519 0.1241

Mean deviations

0.0623

8.60

0.0156

!P/P%

43.87 From chemical potentials (k12 = 0.0412; l12 = −0.1162)

From fugacity coefficients (k12 = 0.1657; l12 = −0.0304) !x1V

!P/P%

!x1V

(ii) k12 , l12 fitted to lower temperature VLE data (*) 310.93∗ [9] 24.47 342.90∗ [10] 11.05 344.26∗ [9] 7.78 344.30∗ [11] 10.62 377.59∗ [11] 15.45 377.60∗ [9] 12.88 477.59 [9] 89.27 510.93 [9] 97.99 520.40 [10] 139.78 542.95 [12] 82.49 583.65 [12] 51.65 594.20 [10] 62.73

0.0007 0.0190 0.0044 0.0206 0.0106 0.0398 0.0790 0.1455 0.1657 0.3082 0.3921 0.5949

9.74 2.41 1.89 3.09 8.30 4.09 19.56 17.44 19.09 20.30 11.88 23.14

0.0003 0.0140 0.0017 0.0139 0.0017 0.0160 0.0054 0.0155 0.0157 0.0336 0.0571 0.1260

Mean deviations

0.0794

9.26

0.0153

!P/P%

39.13

E. Neau / Fluid Phase Equilibria 203 (2002) 133–140

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with vij∗ = 21 (vii∗ + vjj∗ )(1 − lij ),

εij∗ = (εii∗ εjj∗ )1/2 (1 − kij )

(21)

The pure characteristic parameters of each component were taken as indicated by Sanchez and Panayiotou [4]: for CO2 : T ∗ = 283 K, P ∗ = 659 MPa, ρ ∗ = 1620 kg/m3 ∗ ∗ for n-decane : T = 530 K, P = 304 MPa, ρ ∗ = 837 kg/m3

(22)

It is obvious that, due to the large temperature range of VLE data considered (Table 3), improved results could be obtained by using interaction parameters depending on temperature (especially for kij ). More satisfactory results could also be obtained by correlating these VLE data with improved characteristic parameters T ∗ , P ∗ and ρ ∗ (Eq. (22)), especially for carbon dioxide. But the purpose of this study was only the comparison between two procedures of phase equilibrium calculations, independently of mixing rules and pure component characterisation. Results obtained for VLE correlation are reported in Table 3. Comparison of deviations on both vapour pressures and vapour mole fractions shows that, in case (I), when parameters were fitted to the whole set of temperatures, method (1) derived from chemical potentials is not satisfactory and leads to poor results when compared to method (2). This default becomes more evident in case (II), where only a reliable procedure for equilibrium calculations can provide a correct extrapolation at higher temperatures and pressures. The gap between results obtained with the two methods illustrates the dangers of using literature chemical potentials from a practical point of view, especially in process design. 5. Conclusion Fugacity coefficients were calculated by derivation of the residual Helmholtz energy Ares (T, V, n) obtained by integration of the Sanchez–Lacombe equation-of-state. It was also shown that classical chemical potentials of mixtures found in literature fail to satisfy the fundamental expression of chemical potentials in the ideal gas state, because their expressions correspond to a reference term function of phase composition. Since this behaviour is observed whatever the pressure and the mixing rules considered, erroneous calculation of phase equilibria will always be obtained in this way. This result was illustrated by liquid–vapour equilibrium calculations in the case of a binary mixture well studied in literature in a large range of temperatures and pressures. The comparison of data correlation using both chemical potentials and consistent fugacity coefficients proposed in this work gave evidence of the dangers of using literature methods for chemical potential estimation. List of symbols A Helmholtz energy c number of components E lattice energy n number of molecules P pressure Q canonical partition function r number of lattice sites occupied by one molecule

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R T T∗ v v∗ V x z Z

E. Neau / Fluid Phase Equilibria 203 (2002) 133–140

ideal gas constant temperature characteristic temperature molar volume characteristic volume total volume molar fraction compressibility factor configutational partition function

Greek letters ε∗ characteristic energy φ segment fraction ϕ fugacity coefficients λ internal and kinetic partition function µ chemical potential ρ density ω reference constant Ωc number of configurations Subscript i component of a mixture Superscripts ∼ reduced property conf configurational property ideal ideal gas state ref reference property res residual property References [1] I.C. Sanchez, R.H. Lacombe, J. Phys. Chem. 80 (21) (1976) 2352–2362. [2] J.M. Prausnitz, R.N. Lichtenhalter, E.G. De Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, Prentice-Hall, Englewood Cliffs, NJ, 1999. [3] I.C. Sanchez, R.H. Lacombe, Macromolecules 11 (6) (1978) 1145–1156. [4] I.C. Sanchez, C. Panayiotou, Equation-of-state thermodynamics of polymers and related solutions, in: Models for Thermodynamic and Phase Equilibria Calculations, S.I. Sandler, New York, 1994. [5] M.A. MacHugh, V.J. Krukonis, Supercritical Fluid Extraction: Principles and Practice, 2nd Edition, H. Brenner, Boston, 1993. [6] Y. Sato, M. Yuguri, K. Fujiwara, S. Takishima, H. Masuoka, Fluid Phase Equilib. 125 (1996) 129–138. [7] H. Orbey, C.P. Bokis, C.C. Chen, Ind. Eng. Chem. Res. 37 (1998) 4481–4491. [8] B.L. West, D. Bush, N.H. Brandley, M.F. Vincent, S.G. Kazarian, C.A. Eckert, Ind. Eng. Chem. Res. 37 (1998) 3305–3311. [9] H.H. Reamer, B.H. Sage, J. Chem. Eng. Data 8 (1963) 508–513. [10] H. Inomata, K. Tuchiya, K. Arai, S. Saito, J. Chem. Eng. Jpn. 19 (5) (1986) 386–391. [11] N. Nagarajan, R.L. Robinson, J. Chem. Eng. Data 31 (2) (1986) 168–171. [12] H.M. Sebastian, J.J. Simnick, H.M. Lin, K.C. Chao, J. Chem. Eng. Data 25 (1980) 138–141.