A quasi-chemical nonrandom lattice fluid model for phase equilibria of associating systems

Fluid Phase Equilibria 256 (2007) 27–33

A quasi-chemical nonrandom lattice fluid model for phase equilibria of associating systems Moon Sam Shin, Hwayong Kim ∗ School of Chemical & Biological Engineering & Institute of Chemical Processes, Seoul National University, Seoul 151-744, Republic of Korea Received 1 July 2006; received in revised form 4 December 2006; accepted 5 December 2006 Available online 9 December 2006

Abstract The contribution from Veytsman statistics for associating is combined with the quasi-chemical nonrandom lattice fluid (QLF) equation of state developed by the present authors. The physical contribution is characterized by temperature independent molecular parameters, representing close packed volume of a mer, segment numbers and energy parameters of a pure fluid, and a binary interaction parameter for a mixture. The chemical part is represented by the internal energy and entropy of the associating system. The resulting quasi-chemical nonrandom associating lattice fluid equation of state was applied to describe thermodynamic properties of pure fluids and phase equilibria of mixtures. The absolute chemical potentials which are derived from the canonical partition function are presented by using the fundament thermodynamic relations to consistently calculate phase equilibrium properties. The results for alkane–alkanol and alcohol–alkanol mixtures were found satisfactory for most systems when compared with the quasi-chemical nonrandom lattice fluid model and the statistical associating fluid theory model. © 2006 Elsevier B.V. All rights reserved. Keywords: Associating; Quasi-chemical; Nonrandom; Lattice equation of state

1. Introduction The importance of contributions to real fluid properties by specific interactions has long been recognized. Hydrogen bonds that result from specific interactions have been studied experimentally and theoretically. Chapman et al. [1] and Huang and Radosz [2] derived the statistical associating fluid theory model based on Wertheim’s [3,4] cluster expansion theory. Ikonomou and Donohue [5] proposed the Associated Perturbed Anisotropic Chain Theory (APACT) EOS incorporating chemical equilibria between molecules. Economou and Donohue [6] reviewed the chemical, quasi-chemical and perturbation theories. A lattice-based equation of state was proposed by Sanchez and Lacombe [7] under the assumption of complete randomness in the distribution of molecules and holes upon the lattice. Smirnova and Victorov [8] presented the hole group-contribution model (HM) with the Guggenheim’s quasichemical approximation. However, the HM EOS includes on implicit nonrandomness factor for multi-component mixtures ∗

Corresponding author. Tel.: +82 2 880 7604; fax: +82 2 888 6695. E-mail address: [email protected] (H. Kim).

0378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2006.12.002

and this causes difficulties when one takes a derivative and use it at the cost of lengthy numerical calculations. Kumar et al. [9] and You et al. [10] obtained expressions for an explicit-form qausi-chemical nonrandom lattice fluid model. Shin and Kim [11] represented a quasi-chemical nonrandom fluid (QLF) model with temperature independence of pure component parameters. The lattice-fluid model offers an alternative basic framework for an EOS for associating systems. Sanchez and Panayiotou [12] reviewed equations of state for polymer solutions with hydrogen bonding effects. In their review, the hydrogen bonding effects were modeled by invoking association complexes [13] or by counting the number of arrangements of hydrogen bonds [14]. The latter approach, herein called the lattice fluid hydrogen bonding model (LFHB), yields less complicated solutions for complex systems. Veytsman [15] proposed a convenient method for the counting association complexes. This associating model is judged compatible with lattice fluid models. Yeom et al. [16] proposed a Nonrandom Lattice Hydrogen Bonding (NLF-HB) EOS by combining a nonrandom lattice fluid theory [10,17–19] with the Veytsman’s statistics. However, The LFHB model includes the assumptions that the nonrandom contribution is negligible and uses up the large

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M.S. Shin, H. Kim / Fluid Phase Equilibria 256 (2007) 27–33

coordination number limit known as the Flory approximation. Also, this model does not yield a consistent method for phase equilibrium calculations, as pointed out by Neau [20]. The NLF-HB model has strong a temperature dependence of energy parameters and segment numbers of pure systems, thus empirical correlations as functions of temperature were represented for the reliable and convenient use in engineering practice. In this study, we present a new approach combining the Veytsman statistics for the associating contribution and the quasi-chemical nonrandom lattice fluid model [11] for the physical description of fluids. Also, we present the absolute chemical potentials to consistently calculate phase equilibrium properties. The resulting QALF equation of state was applied to VLE of alkane–alkanol and alkanol–alkanol mixtures.

A = Ndk − Nk0

J 

A Nkj ,

A N0l = Nal −

I 

j

NilA

(5)

i

A A AA kl = Ukl − TSkl

(6)

The Helmholtz energy is given by sum of a physical and a chemical contribution. The former is given by the QLF model. With NklA and NklA0 known, the configurational Helmholtz energy due to the associating contribution is readily written using the Stirling approximation. c βAcA = − ln ΩA

= (NA −NA0 )(ln Nr + 1)+

K 

A A A0 A0 (Nk0 ln Nk0 −Nk0 ln Nk0 )

k

2. The model and thermodynamic properties

+

We begin with the lattice configurational partition function for associating systems approximated as product of physical and chemical contributions. The physical term is written in the Guggenheim–Miller approximation. Veytsman statistics [15] allows us to write the partition function for the associating contribution in terms of the number of associating pair (NijA ) between donor group of type i and acceptor group of type j. The volume change on association was neglected. The physical term is taken from the QLF [11] model. The normalized configurational partition function of associating contribution [21] is c ΩA =

A0 ! L N A0 ! K L A0 N ! NrNA0 K Nk0 0l Π Π klA exp(−βNklA AA kl ) A! Π A! Π N Nr A k Nk0 l N0l k l Nkl !

(1)

For associating contribution, the maximization conditions lead to A A NklA Nr = Nk0 N0l exp(−βAA kl ) ⎞ ⎛  J I   k A l A Nkj ⎠ Na − Nil exp(−βAA = ⎝Nd − kl ) j=1

(k = 1, 2, . . . , K,

i=1

A0 A0 NklA0 Nr = Nk0 N0l = ⎝Ndk −

J 

(2) ⎞

A0 ⎠ Nal − Nkj

j=1

(k = 1, 2, . . . , K,

I 

 NilA0

i=1

l = 1, 2, . . . , L)

(3)

where k and l are indices of donor types and acceptor types. The numbers of donor type k, Ndk , and that of acceptor type l, Nal , are defined in terms of the number of donor type k in species i, dki , j and that of acceptor type l in species j, al Ndk =

c  i

dki Ni ,

Nal =

c  j

j

a l Nj ,

A A A0 A0 (N0l ln N0l − N0l ln N0l )

l

+

K  L  k

A A A A0 A0 (βAA kl Nkl + Nkl ln Nkl − Nkl ln Nkl )

(7)

l

The Helmholtz energy allows us to derive thermodynamic properties. 3. EOS and fugacity coefficients In this work, we present a quasi-chemical nonrandom associating lattice fluid (QALF) model with no temperature dependence of close packed volumes of a mer (v∗i ), segment numbers (ri ) and molecular interaction energy (εii ) of pure systems. The physical term is taken from the QLF model [11] by present authors and the associating term can be obtained from the configurational Helmholtz energy rein for the associating contribution (Eq. (7)), c

∂AA 1 PA = − = − ∗ (νA − νA0 )ρ˜ (8) ∂V T βv where

l = 1, 2, . . . , L) ⎛

L 

NA =

K  L  k

l

Nkl (4)

NA νA =  , i Ni ri

NA0 νA0 =  i N i ri

(9)

And the equation of state due to the physical contribution and the associating contribution is

q   P˜ z ˜ + ln 1 + = − ln(1 − ρ) − 1 ρ˜ 2 r T˜ θ2 (10) T˜ Here, all the quantities with the tilde (∼) denote reduced variables defined by  P T ρ r i Ni ˜ ˜ P = ∗, T = ∗, ρ˜ = ∗ = , P T ρ Nr − (νA − νA0 )ρ˜ −

ρ∗ =

1 rv∗

(11)

M.S. Shin, H. Kim / Fluid Phase Equilibria 256 (2007) 27–33

where the reducing parameters are related by z P ∗ v∗ = kT ∗ = εM 2 and εM , θ and θ i are defined by ⎡ ⎤ c c c c c c 1 ⎣ β  εM = 2 θi θj εij + θi θj θk θl εij (εij + 3εkl − 2εik − 2εjk )⎦ θ 2 i=0 j=0

 θ= θi =

29

(12)

(13)

i=0 j=0 k=0 l=0

Ni qi (q/r)ρ˜ = = 1 − θ0 Nq 1 + (q/r − 1)ρ˜

(14)

N i qi Nq

(15)

The configurational chemical potential, μci is derived from the configurational partition function,  

c c

∂ ln Ωphys ∂ ln Ωc ∂ ln ΩA c μi = −kT = −kT − kT ∂Ni ∂Ni ∂Ni T,V,Nj T,V,Nj

(16)

T,V,Nj





q   z ri v∗i P˜ θi + ri 1 − +Z ln 1 + − 1 ρ˜ − 1 + ln ∗ r v qi 2 r T˜   K   N N N N L A0 A0   2 j=0 θj εij + β j=0 k=0 l=0 θj θk θl εij (εij + 2εkl − 2εjk − εik ) Nk0 N0l qi θ i i + − dk ln A + al ln A θ− θεM T˜ Nk0 N0l k=1 l=1 (17)

βμci = γi (T ) + ri

partition function, Meanwhile, chemical potential, μ

i is derived from



the absolute the canonical ∂ ln Ωphys ∂ ln Ω ∂ ln ΩA μi = −kT = −kT − kT ∂Ni T,V,Nj ∂Ni ∂Ni T,V,Nj T,V,Nj Also, fugacity coefficients were derived from the configurational chemical potential using the fundamental relation,     id ) c /Ωc,id ) ∂ ln(Ω/Ω ∂ ln(Ω ln ϕi = βμi − βμid = = βμci − βμc,id i = i ∂Ni ∂Ni T,V,Nj

(18)

(19)

T,V,Nj



q   P˜ ri v∗i zqi ln ϕi = (Z − 1) − 1 + r − ln 1 + − 1 ρ˜ − ln Z i r v∗ 2 r T˜   K       L A0 A0   2 cj=0 θj εij + β cj=0 ck=0 cl=0 θj θk θl εij (εij + 2εkl − 2εjk − εik ) Nk0 N0l qi θ i i + − dk ln A + al ln A θ− θεM T˜ N N0l k0 k=1 l=1

(20)

The absolute and configurational chemical potentials for the ideal gas state were obtained from the fundamental thermodynamic relation [22] and Eq. (17), βμid i = ln(xi P) + λi (T )

ri v∗i = γ (T ) + − 1 + ln(xi P) + ln(βv∗ ) βμc,id i i r v∗ Therefore, the absolute chemical potentials, μi were derived from using the fundamental relation and Eq. (19),



q   z P˜ θi ri v∗i ∗ + r βμi = λi (T ) + ri + (Z − 1) 1 − ln 1 + − 1 ρ˜ − 1 − ln(βv ) + ln i r v∗ qi 2 r T˜   K   N N N  L A0 A0   2 N Nk0 N0l qi θ j=0 θj εij + β j=0 k=0 l=0 θj θk θl εij (εij + 2εkl − 2εjk − εik ) i i + − dk ln A + al ln A θ− θεM T˜ Nk0 N0l k=1 l=1

(21) (22)

(23)

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M.S. Shin, H. Kim / Fluid Phase Equilibria 256 (2007) 27–33

4. Results and discussion There are three molecular parameters for pure fluids; v∗i , ri , εii and the binary interaction parameter kij for a mixture in the present model. We set z = 10 as used in lattice fluid theories of the same genre [23]. Pure parameters are fitted to experimental liquid density and vapor pressure data [24]. Parameters of pure fluids for the QLF model are listed in Table 1. We use the associating energy parameters for 1-alkanols, UijA = −25.1 kJ/mol of Renon and Prausnitz [25] and SijA = −26.5 J/(mol K) as given by Panayiotou [26]. Pure parameters of associating fluids for QALF model are listed in Table 2 and the parameters of the model for non-associating components are the same as those defined previously [11]. While the QLF model has unreasonable values of segment number, ri in Table 1, the QALF model with the associating contribution has reasonable values of ri for alkonol systems in Table 2. For example, ri for methanol in the QLF model is 12.127 and ri in the QALF model is 2.377. The binary interaction parameter kij is determined in this calculation such that the deviation of calculated values from experimental vapor–liquid equilibria data [27]. Phase equilibrium properties can be calculated by the phase equilibrium conditions, not using configurational chemical potentials but using fugacity coefficients or absolute chemical potentials. The present model (QALF) is compared with the QLF model, the SAFT model, and the binary interaction parameter kij values for binary vapor–liquid systems are summarized in Table 3 together with percent average absolute deviation (AAD%) in pressure and vapor compositions. In Figs. 1–3, the calculated P–x VLE diagrams of the present model, the QLF model and the SAFT model for 1-hexane + ethanol, +1-propanol and +1-butanol system are compared with experimental data. The P–x diagrams for 1heptane + ethanol + 1-butanol system in Figs. 4 and 5 are compared with experimental data. The present model shows better calculated results than the QLF model and the SAFT model in alkane + alkanol systems.

Fig. 1. Isothermal vapor–liquid equilibria for 1-hexane + ethanol system at 308.15, 313.15, 318.15 K.

Fig. 2. Isothermal vapor–liquid equilibria for 1-hexane + 1-propanol system at 298.15, 318.15 K.

Fig. 3. Isothermal vapor–liquid equilibria for 1-hexane + 1-butanol system at 332.53, 348.15 K.

In Figs. 6 and 7, the calculated P–x VLE of the present model, the QLF model and the SAFT model for methanol + 1propanol system and methanol + 1-butanol system are compared with experimental data. The calculated results of QALF, HM

Fig. 4. Isothermal vapor–liquid for 1-heptane + ethanol system at 303.27, 333.15, 343.15 K.

M.S. Shin, H. Kim / Fluid Phase Equilibria 256 (2007) 27–33

31

Table 1 Pure parameters of pure fluids for QLF model Components

νi∗ (cm3 /mol)

ri

εii /k (K)

AADP

AADρ

T range (K)

Methanol Ethanol 1-Propanol 1-Butanol 1-Pentanol 1-Hexanol 1-Heptanol 1-Octanol

2.912 3.235 3.905 4.513 5.542 6.375 6.930 7.693

12.127 14.631 16.106 16.741 17.043 17.162 17.960 18.258

104.46 98.600 98.575 101.344 104.230 108.441 110.228 113.453

0.978 0.883 0.745 0.924 0.846 0.936 0.853 0.785

1.483 1.325 0.667 0.954 0.843 0.954 0.847 0.737

273–473 273–513 273–453 313–453 295–415 308–418 333–443 333–453

AADP(%) =

1 N

N  N       P exp −P cal   ρexp −ρcal   P exp  × 100, AADρ(%) = N1  ρexp  × 100. i=1

i=1

Table 2 Pure parameters of pure fluids for QALF model Components

νi∗ (cm3 /mol)

ri

εii /k (K)

AADP

AADρ

T range (K)

Methanol Ethanol 1-Propanol 1-Butanol 1-Pentanol 1-Hexanol 1-Heptanol 1-Octanol

16.388 12.633 11.099 11.560 12.155 13.425 13.744 13.843

2.377 4.236 6.144 7.294 8.242 8.671 9.725 10.807

132.747 116.024 112.405 114.703 117.268 123.380 124.818 126.243

0.636 0.385 0.535 0.817 0.642 0.538 0.757 0.639

1.046 1.108 0.448 0.816 0.578 0.462 0.579 0.685

273–473 273–513 273–453 313–453 295–415 308–418 333–443 333–453

AADP(%) =

1 N

N  N       P exp −P cal   ρexp −ρcal   P exp  × 100, AADρ(%) = N1  ρexp  × 100. i=1

i=1

Table 3 Binary interaction parameters, kij of the QLF model, the QALF model, the SAFT model regressed from the experimental phase-equilibrium data and their average fitting error percent System

T (K)

1-Butane + methanol 1-Butane + methanol 1-Pentane + 1-propanol 1-Pentane + 1-butanol 1-Hexane + ethanol 1-Hexane + ethanol 1-Hexane + ethanol 1-Hexane + 1-propanol 1-Hexane + 1-propanol 1-Hexane + 1-butanol 1-Hexane + 1-butanol 1-Heptane + ethanol 1-Heptane + ethanol 1-Heptane + ethanol 1-Heptane + 1-butanol 1-Heptane + 1-butanol 1-Heptane + 1-pentanol 1-Heptane + 1-pentanol 1-Heptane + 1-pentanol Methanol + ethanol Methanol + 1-propanol Methanol + 1-butanol

323.15 373.15 317.15 303.15 308.15 313.15 318.15 298.15 318.15 332.53 348.15 303.27 333.15 343.17 333.15 363.15 348.15 358.15 368.15 298.15 333.17 298.15

QLF

i

SAFT

kij

AADP

AADy

kij

AADP

AADy

kij

AADP

AADy

0.1471 0.1665 0.1034 0.0738 0.0948 0.0994 0.1054 0.0716 0.0752 0.0721 0.0795 0.0920 0.1084 0.1074 0.0753 0.0714 0.0620 0.0625 0.0627 0.0037 0.0008 0.0148

10.971 8.836 7.726 4.063 4.988 5.740 7.196 5.414 2.594 2.892 4.002 5.743 7.226 5.851 5.628 2.696 2.624 2.417 2.428 1.436 1.550 2.606

13.315 9.188 6.302 5.045 7.570 10.148 13.644 6.673 7.468 4.436 4.271 14.586 14.749 13.585 8.166 5.745 6.216 6.211 6.610 2.220 2.673 1.667

0.0220 0.0249 0.0476 0.0252 0.0473 0.0506 0.0513 0.0348 0.0376 0.0326 0.0338 0.0478 0.0574 0.0605 0.0274 0.0305 0.0473 0.0506 0.0513 −0.0044 −0.0008 0.0073

2.174 3.082 1.757 0.430 0.237 0.617 0.482 1.79 0.680 0.75 1.177 0.384 1.009 0.464 0.294 0.382 0.237 0.617 0.482 0.359 1.388 0.301

5.950 5.389 1.642 0.804 2.695 1.031 1.145 2.865 3.665 3.756 2.767 0.871 2.036 2.436 0.800 1.563 2.695 1.031 1.145 2.352 1.726 0.909

0.0404 0.0429 0.0219 0.0082 0.0209 0.0243 0.0232 0.0113 0.0148 0.0159 0.0187 0.0216 0.0256 0.0267 0.0130 0.0155 0.0209 0.0243 0.0232 0.0048 0.0235 0.0292

5.141 3.284 4.561 2.117 1.440 1.588 1.068 3.391 2.324 2.697 2.581 0.776 2.602 0.887 1.347 1.302 1.440 1.588 1.068 1.351 1.164 2.450

6.492 5.319 5.818 3.548 5.258 2.406 1.994 2.772 2.656 2.972 6.884 1.416 3.535 1.466 2.226 2.542 5.258 2.406 1.994 3.076 2.570 3.803

NDT     P cal −P exp  AADP = NDT  P exp  × 100, AADy = 1

QALF

1

NDT 

NDT i

1 NC

NC     yical −yiexp   yexp  × 100. j

NDT : Number of experimental points, NC : Number of components.

i

32

M.S. Shin, H. Kim / Fluid Phase Equilibria 256 (2007) 27–33

Fig. 5. Isothermal vapor–liquid for 1-heptane + 1-butanol system at 333.15, 363.15 K.

Fig. 8. Comparison of isothermal VLE data with the results of QALF, HM, PRWS model for ethanol and n-butane system (kij = 0.0216 for QALF model).

5. Conclusions

Fig. 6. Isothermal vapor–liquid equilibria for methanol + 1-propanol system at 333.17 K.

and PRWS [28] model show for ethanol and n-butane system at 345.65 K and 373.15 K [29] in Fig. 8. The present model has slightly better calculated results than the QLF model and the SAFT model in alkanol + alkanol systems.

Fig. 7. Isothermal vapor–liquid equilibria for methanol + 1-butanol system at 298.15 K.

The Veytsman statistics for associating contribution is combined with the approximate quasi-chemical nonrandom lattice fluid model developed recently by the present authors. Also, the absolute chemical potentials are presented by using the fundament thermodynamic relations to consistently calculate phase equilibrium properties. The resulting QALF equation of state was applied to describe thermodynamic properties of pure fluids and phase equilibrium of mixtures. We have tested the present model (QALF) on 22 phase equilibrium data sets of vapor–liquid equilibria and the results for alkane–alcohol and alcohol–alcohol mixtures were found satisfactory for most systems when compared with the QLF and the SAFT model. List of symbols A Helmholtz energy c number of the component f fugacity k Boltzmann’s constant N number of molecule Nij the number of associating pair Nq Nq = N0 + qN Nr total number of lattice sites, Nr = N0 + rN q surface area parameter Q canonical partition function r number of segments per molecule T temperature v molar volume v∗ close packed volume of a mer V volume x liquid mole fraction y vapor mole fraction z lattice coordination number Z compressibility factor Greek letters β reciprocal temperature (1/kT)

M.S. Shin, H. Kim / Fluid Phase Equilibria 256 (2007) 27–33

εij εM ϕ μ νA νA0 θ ρ ρ*

molecular interaction energy defined by Eq. (13) fugacity coefficient chemical potential defined by Eq. (9) defined by Eq. (9) surface area fraction molar density close packed molar density

Subscript A Association contribution A0 zero association free energy contribution a acceptor d donor i component i phys physical contribution Superscript c configurational properties id ideal gas state r residual properties ∼ reduced properties * characteristic properties

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13] [14] [15] [16] [17] [18] [19] [20] [21]

Acknowledgements

[22]

This work was supported by the BK21 project of Korea Ministry of Education and the National Research Laboratory (NRL) Program of Korea Institute of Science & Technology Evaluation and Planning.

[23] [24]

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33

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