Nonrandom Two-Liquid Segment Activity Coefficient Model with

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Cite This: Ind. Eng. Chem. Res. 2019, 58, 12773−12786

Nonrandom Two-Liquid Segment Activity Coefficient Model with Association Theory Yifan Hao† and Chau-Chyun Chen*,† †

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Department of Chemical Engineering, Texas Tech University, Lubbock, Texas 79409-3121, United States ABSTRACT: Association theory has been extensively applied to equations of state such as statistical associating fluid theory and cubic plus association. This work presents an integration of association theory with the nonrandom two-liquid segment activity coefficient (NRTL-SAC) model. Replacing the four conceptual segments (hydrophobic, hydrophilic, polar attractive, and polar repulsive) in the original NRTL-SAC model, the association NRTL-SAC model represents interacting molecules with two conceptual segments (nonpolar and polar) and two association site types (hydrogen bond acceptor and donor). Given the molecule-specific parameters, the model shows much improved predictive capability than the original NRTL-SAC model for phase equilibria of chemical mixtures in which associations take place.

1. INTRODUCTION Accurate correlation and prediction of thermophysical properties and fluid phase equilibria is a prerequisite for modern day

Table 2. Association NRTL-SAC Model Segment−Segment Binary Interaction Parameters segment 1

Table 1. NRTL-SAC Segment−Segment Binary Interaction Parameters17 segment 1

X −

segment 2

Y

τ12 τ21 α12=α21

1.643 1.834 0.2

X

Y−

Y+

X

Z

Z

Z

Y+

6.547 10.949 0.2

−2.000 1.787 0.3

2.000 1.787 0.3

1.643 1.834 0.2

segment 2

Y

τ12 τ21 α12 = α21

1.643 1.834 0.2

modeling is to explicitly and accurately model specific chemical interactions, or associations between molecules. Such specific chemical interactions strongly affect thermophysical properties and fluid phase equilibria. Classical thermodynamic models, like NRTL and UNIQUAC, capture the combined effects of physical and chemical interactions between molecules with binary interaction parameters. Although these binary interaction parameters are related to physically meaningful properties such as potentials of mean force and radii of first neighboring shells,3 they may fail to accurately model vapor− liquid equilibrium (VLE) and liquid−liquid equilibrium (LLE) simultaneously using the same set of binary interaction parameters.4 The associations have been successfully described with Wertheim’s perturbation theory.5−8 Several equations of state, such as variants9,10 of statistical associating fluid theory11 and cubic plus association,12 have been developed to model fluids or fluid mixtures involving association. Later, Fu et al.13 derived an activity coefficient expression from the Wertheim’s association Helmholtz energy term and integrated with

Figure 1. Molecular structures of (a) n-hexane, (b) water, (c) DMSO, (d) chloroform, (e) acetone, (f) dichloromethane, (g) ethanol.

Received: Revised: Accepted: Published:

development, design, simulation, and optimization of chemical processes.1,2 One of the challenges in the thermodynamic © 2019 American Chemical Society

X

12773

April 16, 2019 June 18, 2019 June 20, 2019 June 20, 2019 DOI: 10.1021/acs.iecr.9b02078 Ind. Eng. Chem. Res. 2019, 58, 12773−12786

Article

Industrial & Engineering Chemistry Research

solvents exhibiting hydrophobic, hydrophilic, polar attractive, and polar repulsive surface interaction characteristics. Once the conceptual segment numbers are identified, the phase behavior of the molecule in any solvent or solvent mixtures can then be reliably predicted. However, while the NRTL-SAC model proves to be a simple and effective model for correlating and predicting fluid phase behavior, the treatment of having two polar conceptual segments (polar attractive and polar repulsive) was arbitrary and reflects limited molecular insights.17 Also, following the nonrandom two liquid theory19 (NRTL), the NRTL-SAC model does not explicitly account for specific chemical interactions. The predictive capability of the NRTL-SAC model should be much enhanced if such shortcomings in the model formulation could be overcome by explicitly accounting for specific chemical interactions. In this work, the NRTL-SAC model is reformulated to explicitly account for the specific chemical interactions. The association NRTL-SAC model introduces two conceptual segments for physical interactions and two association sites for specific chemical interactions. To validate the applicability of the association model, we calculated phase behaviors of multiple associating systems and compared them to experimental measurements. In the following sections, the thermodynamic framework of NRTL-SAC and the association theory is presented first. Then, the NRTL-SAC model coupled with the association theory is used to correlate several typical association mixtures: mixtures with self-association only, mixtures with cross-association only, and mixtures with both self-association and cross-association. By analyzing results from these systems, conceptual segments for physical interactions and association sites for specific chemical interactions are redefined and their use in predicting phase behavior of mixtures involving associating molecules is discussed. Lastly, several ternary LLE systems are predicted with the association NRTL-SAC model to illustrate the improved performance of the association model over the original NRTL-SAC model and the NRTL model. 2. Thermodynamic Framework. Here the NRTL-SAC model and its conceptual segments are briefly explained. Also presented is the association activity coefficient expression proposed by Ferreira et al.16 from the association theory. 2.1. NRTL-SAC Model. The NRTL-SAC model uses two terms to calculate the activity coefficient for component I in the solution: the combinatorial term and the residual term:

Figure 2. Unbonded H fraction in water from Luck’s measurements27 and eq 17.

UNIQUAC14 for the physical interactions. The resulting association UNIQUAC model requires two UNIQUAC binary interaction parameters and two self-association parameters to correlate binary VLE data, and the cross-association parameter is estimated as the geometric mean of self-association parameters. The association UNIQUAC model achieved significant improvement over UNIQUAC for mixtures containing self-associating systems such as alcohols or acids and delivered no improvements for cross-associating mixtures.13 Subsequently, Mengarelli et al.15 proposed a predictive association UNIFAC model that covered three functional groups for systems with water and alcohols. Ferreira et al.16 later extended the model to cover seven functional groups for acids, esters, and aromatics. However, the application of association UNIFAC model remains very limited because it is a daunting task to reparameterize the model parameters for the hundreds of the UNIFAC functional groups. NRTL-SAC model17 is a hybrid correlative and predictive activity coefficient model. Although UNIFAC18 captures the molecular surface interaction characteristics with hundreds of functional groups, NRTL-SAC uses only four conceptual segments to represent all types of molecular surface interaction characteristics. The four conceptual segments are hydrophobic, hydrophilic, polar attractive, and polar repulsive segments. The conceptual segment numbers for individual molecules are identified from regression of pertinent phase equilibrium data for binary or higher-order systems involving the molecule and

ln γI = ln γIC + ln γIR

(1)

here the combinatorial activity coefficient, γCI , is calculated using the Flory−Huggins (FH) equation presented in eq 2. To calculate the volume fraction, ϕI, the volume parameter, rI, is approximated as its total segment number. See eq 4.

Table 3. Association Parameters between HB Acceptors and Donors HB acceptor

HB donor

water DMSO DMSO water acetone acetone

water chloroform water chloroform water chloroform

κAD 0.034 0.381 0.080 0.029 0.038 0.330

± ± ± ± ±

0.054 0.003 0.010 0.006 0.297

ϵAD/k

ΔAD (298 K) from eq 20

ΔAD (298 K)from eq 24

1960 909 ± 20 2109 ± 148 1374 ± 108 1739 ± 48 590 ± 249

24 7.6 96 2.8 13 2.1

24 12 81 3.6 15 2.2

12774

DOI: 10.1021/acs.iecr.9b02078 Ind. Eng. Chem. Res. 2019, 58, 12773−12786

Article


Figure 3. n-Hexane−water binary system: (a) liquid−liquid equilibrium, experimental water solubility (circles), and n-hexane solubility (squares) from NIST database;28 (b) calculated activity coefficients at 298.15 K; (c) fractions of free HB acceptor and donor sites (which are equal) of water in the solution at 298.15 K.

Figure 4. DMSO−chloroform binary system: (a) vapor−liquid equilibria at 293.15 K (triangles) and 328.15 K (circles), experimental data from NIST database;28 (b) calculated activity coefficients at 298.15 K; (c) fractions of free HB acceptor site of DMSO (dashed line) and HB donor site of chloroform (solid line) in the mixture at 298.15 K.

ln γIC = ln γIFH = 1 −

xI

+ ln

rIxI ∑J rJxJ

ϕI = rI =

ϕI

molecules examined in this study, the difference between γFH and γ FH′is negligible. The residual activity coefficient, γRI , is calculated with the polymer NRTL local composition (lc) model.24 It first calculates the activity coefficient contributions to the constitutive segments of a molecule in solution and then sums up the contributions to all the segments that make up the molecule for the activity coefficient of the molecule in solution. The residual activity coefficient is expressed as

ϕI xI

(2)

(3)

∑ rm,I

(4)

m

where rm,I is the number of conceptual segment m in component I, m is the segment index, and J is the component index. This work uses a refined Flory−Huggins expression, γ FH′, proposed by Larsen et al.:20 ln γIFH′ = 1 −

ϕI′ =

ϕI′ xI

+ ln

ln γIR = ln γIlc =

m

ln Γ mlc =

ϕI′ xI

(5)

rI2/3xI ∑J r J2/3xJ

∑ rm,I[ln Γ mlc − ln Γ mlc,I]

xm Gmm ′ ′ ∑k xkGkm m′ ∑k xkGkm′ ij ∑ xG τ y jjτ − j j jm′ jm′ zzz jj mm z j ′ ∑k xkGkm zz k ′ {

(6)

ln Γ mlc, I =

here rI, calculated from the Bondi’s group-contribution method,21 is the van der Waals volume normalized against that of a standard segment as introduced by Abrams and Prausnitz.14 The volume parameters, rI’s, are available in several open source web-based databanks.22,23 For the small 12775

∑j xjGjmτjm

+

∑j xj , IGjmτjm

(7)

∑

xm , IGmm ′ ′ ∑k xk , IGkm m′ ∑k xk , I Gkm′ ij ∑x G τ y jjτ − j j , I jm′ jm′ zzz jj mm z j ′ ∑k xk , IGkm zz k ′ { +

(8)

∑

(9)


Article

Industrial & Engineering Chemistry Research Table 4. Association NRTL-SAC Model Parameters for 77 Organic Solvents solvents

X

acetic acid acetone acetonitrile aniline anisole benzene benzonitrile 1-butanol 2-butanol t-butanol n-butyl acetate carbon tetrachloride chlorobenzene chloroform cumene cyclohexane 1-decanol 1,2-dichloroethane 1,1-dichloroethylene 1,2-dichloroethylene dichloromethane diethyl ether diethylene glycol 1,2-dimethoxyethane N,N-dimethylacetamide N,N-dimethylformamide dimethyl sulfoxide 1,4-dioxane ethanol 2-ethoxyethanol ethyl acetate ethyl formate ethylbenzene ethylene glycol ethylene glycol n-butyl ether formamide formic acid n-heptane n-hexane 1-hexene 1-hexanol isobutyl acetate isobutyl alcohol isopropyl acetate isopropyl alcohol methanol 2-methoxyethanol N-methyl-2-pyrrolidone methyl acetate methyl butyl ketone methyl ethyl ketone N-methylformamide methyl isobutyl ketone methyl tert-butyl ether 3-methyl-1-butanol methylcyclohexane nitromethane i-octane n-octane 1-octanol n-pentane

0.100 0.202 ± 0.012

δA

Y

± 0.074

0.193 0.500 0.741 0.227 0.211 0.404 0.227 0.429 0.466 0.352 0.269 1.035 0.949 0.955 0.367

± ± ± ± ± ± ± ± ± ± ± ± ±

0.027 0.050 0.011 0.037 0.008 0.040 0.014 0.140 0.028 0.154 0.013 0.287 0.038

0.009 0.451 0.218 0.052 0.230 0.181 0.189

± ± ± ± ± ± ±

0.030 0.097 0.041 0.011 0.038 0.016 0.053

0.325 0.316 0.127 0.266 0.196 0.848

± ± ± ± ± ±

0.033 0.018 0.037 0.049 0.017 0.062

0.145 ± 0.083 0.193 ± 0.048

0.605 0.726 1.118 0.786 0.895 0.556 0.747 0.318 0.245 0.172 0.360 0.045 0.518 0.297 0.367

± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.184 0.014 0.170 0.105 0.123 0.024 0.116 0.027 0.047 0.010 0.048 0.003 0.252 0.017 0.095

0.500 0.623 0.576 1.261 0.138 0.093 0.735 1.000 1.000 1.000 0.598

0.014 0.737 0.116 0.344 0.620 0.113 0.338 0.151 1.000 0.696 1.766 0.635 0.249 0.243 0.472 0.535 0.611 0.794 0.177 0.700 1.422

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.129 0.096 0.100 0.240 0.080 0.024 0.047 0.043 0.028 0.068 0.095 0.028 0.027 0.036 0.041 0.022 0.086 0.283 0.048

1.000

0.097 0.324 0.136 0.633 0.248 0.064 0.160 1.293 0.369 0.600 0.607 0.700 0.482 0.101 0.037

± ± ± ± ± ± ± ± ±

± 0.138

± ± ± ± ± ±

0.011 0.034 0.695 0.060 0.006 0.035

δD

A

D

r

3.717 ± 0.332

1 2 1 1 2 1 1 2 2 2 2

1

2.20 2.57 1.87 3.72 4.17 3.19 4.21 3.45 3.60 6.42 4.83 3.45 3.81 2.87 5.27 4.05 7.50 2.88 2.73 2.73 2.29 3.39 4.00 3.64 3.76 3.09 2.83 3.07 2.11 3.70 3.48 2.82 4.60 2.41 10.55 1.68 1.50 5.17 4.50 4.27 4.80 4.83 3.45 2.91 2.91 1.43 3.02 3.98 2.80 4.60 3.25 2.40 4.60 4.07 4.27 4.64 2.01 5.85 5.85 6.15 3.83

0.075 ± 0.035

1.000 1.000 1.000 ± 0.014

0.145 ± 0.023 0.336 ± 0.330 1.000

± ± ± ± ± ± ±

0.048 0.017 0.013 0.094 0.084 0.697 0.006

± ± ± ±

0.022 0.014 0.033 0.063

± 0.076 ± 0.022 ± 0.058

1 1 1

1 1 2

0.005 ± 0.004 0.003 ± 0.002 0.021 ± 0.008 0.348 0.847 0.512 1.947 1.278 3.387 0.394 1.000 0.723 0.542 0.557 0.132 1.000 0.899 0.845 0.536

2

1.000

1.000 1.000

1.000 1.000 0.510 ± 0.240 7.938 ± 0.574

6 4 3 3 2 4 2 4 2 2 1 4 4 3 1

1 2 2 2 2 2

1 1

2 1 2 1

± 0.067

1.335 1.000 0.707 0.583 0.473 0.287 0.556 0.207 0.047 0.020 0.300 0.122 0.465 0.257 0.184 0.424 0.496 0.638 1.029

± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.030 0.050 0.157 0.008 0.124 0.011 0.003 0.018 0.074 0.024 0.197 0.005 0.149 0.027 0.006 0.058 0.462

1.296 1.399 1.135 0.890

± ± ± ±

0.071 0.075 0.013 0.080

0.054 0.065 0.010 0.076 0.009 0.004 0.030 0.054 0.043

± 0.007 ± 0.019 ± 0.006 ± 0.032

1.000 0.729 1.000 0.579 1.000 1.000 0.759 1.581 0.439 0.700 0.519 1.067 0.689 0.450 1.000

1.000 ± 0.061 1.000 ± 0.052

± 0.016 ± 0.056 ± 0.011 ± ± ± ±

0.053 0.277 0.026 0.080

0.734 ± 0.023

0.049 ± 0.005

0.124 ± 0.010

1.000

12776

1.000 1.000 1.000

0.493 ± 0.837

1.000

2 2 2 2 2 2 4 3 2 2 2 3 2 2 2

1 1 1 1 1

1

1

5

1.000

2

1


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Industrial & Engineering Chemistry Research Table 4. continued X

solvents 1-pentanol 1-propanol n-propyl acetate propylene glycol pyridine sulfolane tetraethylene glycol tetrahydrofuran 1,2,3,4-tetrahydronaphthalene toluene 1,1,1-trichloroethane trichloroethylene triethylamine triethylene glycol water m-xylene

δA

Y

0.705 0.373 0.435 0.053 0.071

± ± ± ± ±

0.080 0.015 0.008 0.077 0.028

0.028 0.442 1.157 0.618 0.487 0.351 1.010 0.043

± ± ± ± ± ± ± ±

0.017 0.016 0.186 0.028 0.038 0.134 0.047 0.034

0.712 ± 0.010

0.100 0.259 0.429 0.325 0.577 2.795 0.269 0.407 0.958 0.476 0.271 0.192 0.188 0.468 0.492 0.368

± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.020 0.007 0.197 0.062 3.116 0.083 0.009 0.116 0.036 0.033 0.081 0.023 0.106 0.015 0.012

1.000 1.000 0.468 1.000 3.794 0.437 0.810 0.719 0.134 0.100 0.007

δD 1.000 1.000

± 0.014 1.000 ± ± ± ± ± ± ±

0.206 0.202 0.015 0.012 0.078 0.022 0.002

1.000

A

D

r

2 2 2 4 1 4 10 2 1 1

1 1

4.13 2.78 4.15 3.08 3.00 4.04 7.19 2.87 5.34 3.92 3.54 3.27 5.01 5.59 0.76 4.66

1.000 1.000

2

3 1

0.089 ± 0.049 5.883 ± 0.366 0.508 ± 0.038 1.000 0.201 ± 0.041

2

1 8 2 1

2 2

Figure 5. Water−DMSO binary system: (a) vapor−liquid equilibria at 298.2 K (triangles) and 363.15 K (circles), experimental data from NIST database;28 (b) calculated activity coefficients at 298.15 K; (c) fractions of free water HB donor site (solid line), water HB acceptor site (dashed line), and DMSO HB acceptor site (dash-dotted line) in the mixture at 298.15 K.

Figure 6. Water−chloroform binary system: (a) vapor−liquid equilibria at 101.3 kPa, experimental data (circles) from NIST database;28 (b) calculated activity coefficients at 298.15 K; (c) fractions of free water HB acceptor site (dashed line), water HB donor site (solid line), and chloroform HB donor site (dash-dotted line) in the mixture at 298.15 K

xj =

where i, j, k, m, and m′ are the segment indices, and J is the component index. xj and xj,I are the segment mole fractions in the solution and in component I. Γlcm and Γlc,I m are the segment activity coefficients in the solution and in component I, respectively. Gij is a local binary quantity calculated from the NRTL binary interaction parameters τij and the nonrandom-

∑J rj , JxJ ∑J ∑i ri , JxJ

xj , I =

(10)

rj , I ∑i ri , I

(11) 12777


ÄÅ Å


association NRTLSAC

NRTL-SAC key equations

γ (eq 2) γR (eq 7)

universal model parameters molecular surface descriptors

10 τij’s for pairs of X, Y−, Y+, Z X, Y−, Y+, Z

X, Y, δA, δD

Table 6. Predictions of Binary Isothermal VLE for the 77 Parameterized Molecules ARD% in Pa type of associations no association

self-association cross-association combined self- and cross-association total

no. of systems

association NRTL-SAC

NRTLSAC

inert + inert A + inert D + inert A+A D+D AD + inert A+D AD + A AD + D AD + AD

28

3.99

2.45

133 22 113 4 122 56 206 22 98 804

5.52 3.34 6.26 2.01 9.78 7.49 10.91 7.49 12.33 8.70

5.90 3.74 7.76 2.40 9.39 20.28 14.25 8.40 9.87 10.19

+

i

∑ jjjjj 1 A

(13)

k XA

−

1 zyz jij ∂XA yzz znj z 2 zz{ jjk ∂nI zz{T , P , n

J

(14)

here νA,I is the number of sites of site type A in component I, XA and XA,I are the unbonded site type A fractions in the solution and in pure component I, respectively. Later Ferreira et al.16 simplified the activity coefficient expression using an internal minimization procedure proposed by Michelsen and Hendriks:26 ÉÑ ÄÅ ÅÅ i X y XA , I − 1 ÑÑÑÑ ÅÅ jj A zz A zz + ÑÑ ln γI = ∑ νA , I ÅÅÅlnjjj ÑÑ ÅÅ k XA , I z{ 2 ÑÑÖ A ÅÇ − X 1 i y Az zz + rI∑ ρA jjj 2 k { (15) A

a

Average relative deviation error in total pressure.

here rI is the volume parameter of component I. XA is determined from the dimensionless association site molar density ρA and a dimensionless association strength ΔAB:

ness factor αij between the segments, and is expressed as followed: Gij = exp( −αijτij)

ÅÇ

where a is the residual association Helmholtz energy, R is the gas constant, T is the temperature, M is the number of association site types in the system, and A is the association site type index. To be defined later, XA is the fraction of the unbonded site type A. Fu et al.13 derived an activity coefficient expression from the residual association Helmholtz energy for the association contribution to activity coefficient, γAI : ÑÉ ÅÄÅ i ÅÅ j XA zy XA , I − XA ÑÑÑÑ A j z Å ÑÑ ln γI = ∑ νA , I ÅÅÅlnjjj zz + ÑÑ ÅÅ k XA , I z{ 2 A ÑÖÑ ÅÇ

C

type of molecules

A

É XA ÑÑÑ 1 ÑÑ + M 2 ÑÑÑÖ 2

assoc

γ (eq 5) γR (eq 7) γA (eq 15 and 24) AD τXY, τYX, κAD ref , ϵref

C

∑ ÅÅÅÅÅln XA −

aassoc = RT

Table 5. Comparison between NRTL-SAC Model and Association NRTL-SAC Model

Article

(12)

1 1 + ∑B ρB ΔABXB

XA =

To represent all different molecular surface interaction characteristics, Chen and Song17 introduced four molecular surface descriptors in the NRTL-SAC model: hydrophobic segment (X), polar attractive segment (Y−), polar repulsive segment (Y+), and hydrophilic segment (Z). The hydrophilic segment represents molecular surface with the tendency to form hydrogen bonding. Oppositely, the hydrophobic segment represents molecular surface adverse to form hydrogen bond. Polar segments represent molecular surfaces with either electron donor or acceptor. Reference molecules have been selected for these conceptual segments: water for hydrophilic, n-hexane for hydrophobic, and acetonitrile for both polar segments. The combined physical and chemical interactions between these segments are reflected in their respective NRTL binary interaction parameters. Shown in Table 1, these interaction parameters were identified from binary phase equilibrium data of the reference molecules. For example, the large, positive values for the interaction parameters between X and Z segments (6.547 and 10.949) reflect the strong repulsion between hydrophobic and hydrophilic segments. The polar segments have smaller repulsions with both X and Z segments. Additionally, one of the interaction parameters between Y− and Z is negative, indicative of attractive interactions between polar attractive segments and hydrophilic segments. 2.2. Association Theory. A generalized Helmholtz energy expression due to association contribution has been reported by Chapman:25

(16)

here B represents all site types that can bond to site type A. Similarly, the fraction of unbonded site A in component I is XA , I =

1 1 + ∑B ρB , I ΔABXB , I

(17)

here the molar density of associating site type A in the mixture and in component I are calculated from the following expressions: ρA =

∑J νA , J xJ

ρA , I =

∑J rJxJ

(18)

νA , I rI

(19)

where xJ is the molar composition of component J. The dimensionless association strength between site type A and site type B, ΔAB, is calculated as follows: ÄÅ ÉÑ Å i ϵ AB y ÑÑ j z Å AB ABÅ zz − 1ÑÑÑ Δ = κ ÅÅÅexpjjj z ÑÑ ÅÅ k kT { ÑÑÖ (20) ÅÇ where κAB is the dimensionless association volume with radial distribution function incorporated as proposed by Mengarelli et al.,15 and ϵAB is the association energy. 12778


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Figure 7. Vapor−liquid equilibria of acetone with (a) n-hexane at 308.14 K, (b) DMSO at 308.13 K, (c) chloroform at 298.09 K, and (d) water at 297.99 K. Results from NRTL-SAC (red dashed lines) and association NRTL-SAC (blue solid lines) are compared with experimental data28 (markers).

3. Correlation of Association Systems. To understand the significance of association on phase behavior of association systems, we examine three different association scenarios: (1) water−n-hexane binary representing one self-associating component mixed with one nonassociating component, (2) DMSO−chloroform binary representing cross-association between two components, and (3) water−DMSO binary and water−chloroform binary representing simultaneous selfassociation and cross-association between two components. The structures of these molecules are shown in Figure 1. As mentioned earlier, NRTL-SAC captures the combined physical and chemical interactions between the interacting molecules with four conceptual segments: hydrophobic, hydrophilic, polar attractive and polar repulsive. From the association perspective, the hydrophilic segment represents the molecular surfaces that exhibit self-association and form hydrogen bonding, and the two polar segments represent the

molecular surfaces with electron donor or acceptor that may lead to cross-association. With the integration of association theory into NRTL-SAC, these conceptual segments designed to account for the association phenomena should be replaced with association sites. We show that the association NRTLSAC model is able to account for the physical and chemical interactions between molecules with two conceptual segments (nonpolar X and polar Y) and two association site types (hydrogen bond acceptor A and hydrogen bond donor D). We retain n-hexane and acetonitrile as the reference molecules for nonpolar and polar segments respectively. The only pair of segment−segment binary interaction parameters in the association model, τXY and τYX, is inherited from the original NRTL-SAC as listed in Table 2. 3.1. Self-Association. Water exhibits self-association and forms hydrogen bond (HB) networks in the solution. Shown in Figure 1b, each water molecule contains two hydrogen 12779


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interaction” between them as the water HB networks are broken when it is “diluted” with n-hexane. Figure 3c shows the HB acceptor and donor sites of water would be devoid of association when “diluted” with n-hexane. 3.2. Cross-Association. Molecules like DMSO and chloroform do not self-associate but they can cross-associate with each other. Shown in Figure 1c, DMSO has two pairs of electrons on the oxygen atom as two HB acceptor sites; the hydrogen on chloroform is an isolated proton that serves as a HB donor site, as all electrons on the molecule are attracted by the chlorines. Therefore, cross-association takes place in the DMSO−chloroform binary and the experimental liquid mixture vapor pressures are lower than that of ideal solution. See Figure 4a. To quantify the cross-association strength, ΔAD, between DMSO and chloroform, we first identify the conceptual segment numbers for DMSO and chloroform from their binary phase equilibrium data with nonassociating component (i.e., nhexane). See Table 4. Then the cross-association parameters, shown in Table 3, between DMSO and chloroform are identified from fitting their VLE and excess enthalpy data.28 Figure 4a shows that the combination of the NRTL-SAC model and the association model is able to describe the phase behavior of DMSO − chloroform binary, whereas the original NRTL-SAC predicts a trend opposite to the data. Figure 4b shows negative deviation from ideality suggesting attractive interactions between DMSO and chloroform due to their cross-association. Figure 4c shows that HB donor sites in chloroform and HB acceptor sites in DMSO are all unbonded in their pure component state, but largely bonded in the mixture. 3.3. Combined Self- And Cross-Associations. Selfassociation and cross-association may take place simultaneously in solutions. For example, in the water − DMSO binary, DMSO may cross-associate with water, whereas water also self-associates. As shown in Figures 5a and 6a, the binary systems of water − DMSO and water−chloroform can be well-described by combining the NRTL-SAC model and the association model. The cross-association parameters are regressed from binary phase equilibrium data. It is the same with the previous cases, the γA is the dominating contribution in activity coefficient calculations (see Figures 5b and 6b). However, the association contributions to activity coefficient suggest opposite molecular interactions between water−DMSO and between water− chloroform. This can be explained by their cross-association parameters, shown in Table 3, and the unbonded site fractions in the solutions, shown in Figure 5c and Figure 6c. The crossassociation strength between water and DMSO is stronger than the water self-association, therefore more water HB donor sites are bonded to DMSO HB acceptor sites than its own acceptor sites. It results in “attractive” interactions between water and DMSO (negative ln γA). On the contrary, the water−chloroform cross-association is weaker than the water self-association and it results in “repulsive” interactions between the two molecules (positive ln γA). 4. Association NRTL-SAC Model. The proposed association NRTL-SAC model calculates the activity coefficients in three contributions: γC from eq 5 accounts for the combinatorial contribution due to size difference of molecules, γR from eq 7 accounts for the energetic physical interactions, and γA from eq 15 accounts for the specific chemical associations:

Figure 8. Experimental28 (markers) and association NRTL-SAC model results (lines) for excess enthalpy of mixing acetone with nhexane (black), water (blue), DMSO (green), and chloroform (magenta) at 298.15K.

atoms (HB donors) and two pairs of electrons on the oxygen atom (HB acceptors). To model the self-association of water, the first step is to determine the association strength, ΔAD, between the HB acceptors and donors of water. Shown in Figure 2, Luck27 identified the unbonded H fraction in pure water at different temperatures from spectroscopic measurements. These unbonded H fraction data are used to identify ΔAD, reported in Table 3 with eq 17. The experimental data in the temperature range of 292 to 353 K are used in the regression. We focus on this temperature range because it is of interest to most phase equilibrium predictions with activity coefficient models and, given the narrow temperature range, we could conveniently ignore the temperature dependency of molecular volume in eq 19. With the water self-association strength identified, the activity coefficient from association contribution can be calculated for mixtures containing water and nonassociating molecules. Figure 3a shows the association contribution to activity coefficient alone nearly predicts the LLE phase behavior for the n-hexane−water binary. The deviation between the prediction and the data can be easily captured when the combinatorial and the residual contributions of activity coefficient are taken into account. To model the n-hexane−water binary with all three contributions, the residual contribution to activity coefficient, γR, requires the conceptual segment numbers of the two molecules. Since n-hexane is the reference molecule for the nonpolar (i.e., hydrophobic) segment with (X, Y)=(1, 0), the polar segment number, Y, of water is adjusted to fit the LLE data of n-hexane−water binary, resulting in (X, Y)=(0, 0.492) for water. Figure 3b shows the three contributions of activity coefficients for the n-hexane−water binary. The association term, γA, clearly is the dominant contributor. The large positive ln γ A′s of the two components suggest strong “repulsive 12780


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Figure 9. Vapor−liquid equilibria of dichloromethane (DCM) with (a) n-hexane at 298.14 K, (b) DMSO at 298.14 K, (c) chloroform at 101.3 kPa, and (d) water at 99.9 kPa. Results from NRTL-SAC (red dashed lines) and association NRTL-SAC (blue solid lines) are compared with experimental data28 (markers).

ln γI = ln γI C + ln γI R + ln γI A

4.1. Association Strengths. In this section, we show that the molecule−molecule association strength parameters, ΔAD’s, can be approximated as the product of two new site-specific parameters, δA and δD, association strengths of the HB acceptor site and the HB donor site, respectively.

(21)

The previous section shows that different association systems can be correlated by adjusting the association strength parameters, ΔAD. However, the association strength parameters are molecule−molecule specific and they cannot be used to predict association behavior of other associating mixtures. To overcome this limitation, we present a novel approach to calculate the association strength parameters from newly defined association strengths for HB acceptor and donor sites. We further illustrate the parametrization process and the model performance for mixtures associated with three representative molecules, i.e. acetone, dichloromethane, and ethanol. At last, the prediction performance of the association model is validated by comparing the average relative deviation (ARD%) of 804 binary systems to that of the original NRTLSAC model.

ΔAD = δ Aδ D

(22)

Equation 22 follows from observing the identified association strength parameters of different associating molecular pairs. Table 3 lists the association strength parameters between several HB donor and HB acceptor molecules, quantified by fitting pure component spectroscopy data27 or binary phase equilibrium data as discussed previously. By analyzing these association strength parameter values, one relationship of association strengths between different HB donor and HB acceptor molecules can be derived as following: ΔA ′ D′ΔA″D″ ≈ ΔA ′ D″ΔA″D′ 12781

(23) DOI: 10.1021/acs.iecr.9b02078 Ind. Eng. Chem. Res. 2019, 58, 12773−12786

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Figure 10. Experimental28 (markers) and association NRTL-SAC model results (lines) for excess enthalpy of mixing dichloromethane with n-hexane (black), DMSO (green), and chloroform (magenta) at 298.15K.

Figure 12. Experimental28 (markers) and association NRTL-SAC model results (lines) for excess enthalpy of mixing ethanol with nhexane (black), water (blue), DMSO (green), and chloroform (magenta) at 298.15K.

Equation 22 is a necessary and sufficient condition for eq 23. To facilitate the quantification of δA and δD, we define a AD reference association strength, Δref , as that of the selfassociation of water HB acceptor site and water HB donor site. We further define δAref and δDref as unity for water HB acceptor site and HB donor site. δA and δD of all other sites are scaled accordingly. Table 3 shows the association strength

parameters calculated from eq 24 closely approximate those determined from fitting experimental data. ΔAD =

δ A δ D AD Δ A D ref δref δref

(24)

The site-specific association strengths, δ and δ , of any given molecules can be regressed simultaneously with the conceptual A

D

Figure 11. Vapor−liquid equilibria of ethanol with (a) n-hexane at 298.14 K, (b) DMSO at 333.15 K, (c) chloroform at 318.13 K, and (d) water at 298.15 K. Results from NRTL-SAC (red dashed lines) and association NRTL-SAC (blue solid lines) are compared with experimental data28 (markers). Figures on the top are Pxy diagrams and figures on the bottom are y-x diagrams 12782


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Industrial & Engineering Chemistry Research Table 7. NRTL Binary Interaction Parametersa

a

I

J

aIJ

bIJ

water acetone water chloroform acetone chloroform water ethanol water ethyl acetate ethyl acetate ethanol water 1-butanol 1-butanol ethanol

acetone water chloroform water chloroform acetone ethanol water ethyl acetate water ethanol ethyl acetate 1-butanol water ethanol 1-butanol

0.05 6.40 8.84 −7.35 0.96 0.54 3.46 −0.80 9.46 −3.72 −0.24 −1.15 90.53 204.2

420 −1809 −1140 3241 −590 −106 −586 246 −1706 1284 283 524 −4983 −9292 129 −85.22

τIJ = aIJ +

bIJ T

eIJ

−12.06 −30.58

aIJ = aJI

ref

0.30

29

0.20

30

0.30

29

0.30

29

0.20

30

0.20

29

0.20

30

0.30

29

+ eIJ ln T

Figure 13. Water−acetone−chloroform ternary LLE diagram at 298.15 K in mole percentage. Predictions of NRTL (green dotted line), NRTL-SAC (red dashed line), and associaton NRTL-SAC (blue solid line) are compared with experimental data30 (circles).

Figure 14. Water−ethyl acetate−ethanol ternary LLE diagram at 303.15 K in mole percentage. Predictions of NRTL (green dotted line), NRTL-SAC (red dashed line), and associatoin NRTL-SAC (blue solid line) are compared with experimental data31 (circles).

segment numbers, X and Y, from the VLE, LLE, or activity coefficient data for binary systems of the molecule mixed with solvents of different molecular characteristics (i.e., nonpolar, polar, HB acceptor, and HB donor). Including data from more binary systems will improve the parametrization, but usually data from four binary systems of different molecular characteristics is sufficient to identify an optimized parameter set for the molecule of interest. Table 4 lists the association model parameters for 77 common solvents: X and Y correspond to the segment numbers for conceptual nonpolar segment and polar segment respectively; δA and δD correspond to the association strengths of HB acceptors and donors, respectively. Also reported are A and D, the numbers of HB acceptor and donor sites in the molecule. r is the normalized Bondi’s volume parameter. During the parametrization of alcohol molecules, it is observed that the association strengths of the HB acceptor and donor sites for alcohols are very close to those of water. Therefore, for molecules that contain −OH groups, the

corresponding association site strengths, δA and δD, are fixed as unity. Table 5 summarizes the differences between the original NRTL-SAC model and the association NRTL-SAC model. While the association model uses an extra term (γA) to describe chemical associations, the number of universal model parameters associated with the association model is drastically reduced when compared to the original model. The original model requires ten NRTL binary interaction parameters for the four conceptual segments whereas the association model requires only two NRTL binary interaction parameters for the two conceptual segments (τXY and τYX) plus two reference AD association parameters (κAD ref and ϵref ) associated with selfassociation of water. In addition, the association model replaces the four conceptual segment numbers in the original model with two conceptual segment numbers (X and Y) plus association strengths for two association site types (δA and δD). 12783


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Industrial & Engineering Chemistry Research 4.2. VLE Prediction Performance. To illustrate the improvement in VLE prediction performance, Table 6 compares the prediction performance of the association NRTL-SAC model and the NRTL-SAC model in calculating isothermal VLE behavior for 804 binary systems. The average relative deviation (ARD%) error is calculated by comparing the calculated total pressure versus the experimental pressure as following: ARD% =

∑ i

|Picalcd − Piexp| Piexp

association. Both models predict similar results for the acetone−water binary. The original NRTL-SAC model predicts zero excess enthalpy because the binary interaction parameters are temperature independent. On the other hand, the association model does consider temperature dependence in the association term. Among the four binary systems examined, Figure 8 shows the association model qualitatively predicts the excess enthalpies for the two associating systems: the acetone− chloroform binary and the acetone−water binary. However, the association model predicts zero excess enthalpy for the two nonassociating systems: the acetone−n-hexane binary and the acetone−DMSO binary. 4.2.2. Binary Systems with Dichloromethane. Dichloromethane (DCM) is a polar molecule ([X, Y] = [0.451, 0.620]) with two weak HB donor sites (δD = 0.021). See Figure 1f. Figure 9a−d show the phase behaviors of DCM when mixed with n-hexane, DMSO, chloroform, and water, respectively. Both models predict similar phase behaviors for the DCM−nhexane binary, the DCM−chloroform binary, and the DCM− water binary because they exhibit no or weak cross association. In contrast, DMSO has two strong HB acceptor sites and the cross-association between DCM and DMSO in the DCM− DMSO binary can only be satisfactorily predicted with the association model. See Figure 9b. Figure 10 shows the association model qualitatively predicts the excess enthalpy for the cross-associating system: the DCM−DMSO binary. The model predicts zero excess enthalpy for the two nonassociating systems and for the DCM−water binary because of phase immiscibility. 4.2.3. Binary Systems with Ethanol. Ethanol is a weakly polar molecule ([X, Y] = [0.316, 0.249]) that self-associates with two HB acceptor sites (δA = 1.000) and one HB donor site (δD = 1.000). See Figure 1g. Figure 11a−d show the phase behaviors of ethanol-containing binary systems with n-hexane, DMSO, chloroform, and water, respectively. The association model yields improved predictions over the original model for all systems except for the ethanol−chloroform binary. See Figure 11c. The discrepancy is probably caused by the association calculations. Because of the bulkiness of chloroform molecule, the two HB acceptor sites of ethanol molecule mostly likely cannot simultaneously cross-associate with chloroform. In other words, the ethanol−chloroform crossassociation is overpredicted, and it results in lower predicted pressure for the ethanol−chloroform isotherm. The y−x diagrams show more clearly the improved predictions of the association model. For the ethanol − n-hexane binary, the original model incorrectly predicts immiscibility between the two solvents. The original model also fails to predict the azeotrope of the ethanol−water binary. Figure 12 shows the association model qualitatively predicts the excess enthalpies for the two largely self-associating systems: the ethanol−n-hexane binary and the ethanol−water binary. However, the model predicts the excess enthalpies incorrectly for the two combined self-associating and crossassociating systems: the ethanol−DMSO binary and the ethanol−chloroform binary. This model discrepancy highlights the direction to further refine the association model. 4.3. LLE Prediction Performance. To further examine the predictive capability of the association model, LLE of two ternary systems are predicted with the association model, the original NRTL-SAC model, and the NRTL model. Summarized in Table 7 are the binary interaction parameters required

(25)

The experimental data cover the temperature range from 266.51 to 567.16 K. The molecules are categorized by their association sites: no association site (inert), only HB acceptor site (A), only HB donor site (D), and both HB acceptor and HB donor sites (AD). For the 804 binary systems, the ARD% error in total pressure of the association model is 8.70%, compares to 10.19% for the original model. Specifically, Table 6 shows both models predict the VLE of nonassociating systems in similar accuracy. However, for associating systems, the predictions from the association model are significantly improved over those from the original model. Table 6 further shows the original model out-performs the association model for a few systems because of the larger number of universal model parameters. Of the possible 2926 binary systems from the 77 molecules reported in Table 4, only 804 have experimental VLE data in the literature. It highlights the importance of predictive models like the association NRTL-SAC model, which can be used to predict reliably phase behavior of mixtures when the experimental data are unavailable. The following sections examine the VLE predictions for binary systems with selected components with HB acceptor site, with HB donor site, and with both HB acceptor and HB donor sites. 4.2.1. Binary Systems with Acetone. Acetone is a polar molecule ([X, Y]=[0.202, 0.726]) with two HB acceptor sites (δA = 0.623). It has a molecular structure similar to that of DMSO, with a carbon atom replacing the sulfur atom in DMSO. See Figure 1e. This difference results in a much weaker HB acceptor association strength. Figure 7a−d show the nonideal phase behaviors when acetone is mixed with nhexane, DMSO, chloroform, and water, respectively. These four molecules are chosen to investigate how acetone interacts with nonassociating molecule (n-hexane), HB-acceptor molecule (DMSO), HB-donor molecule (chloroform), and selfassociating molecule (water). The nonideal phase behavior of the acetone − n-hexane binary is solely caused by the physical interactions between the nonpolar n-hexane molecule ([X, Y] = [1.000, 0.000]) and the polar acetone molecule. Both the association model and the original model predict the phase behavior satisfactorily. Both acetone and DMSO ([X, Y] = [0.000, 1.766]) are polar molecules and both have HB acceptor sites. No association occurs for the acetone − DMSO binary and both models predict similar phase behavior. Acetone has a HB acceptor site and chloroform has a HB donor site (δD = 0.145), cross-association exists in the acetone−chloroform binary and results in lower vapor pressures than ideal solution. The original model fails to describe the cross-association behavior and predicts pressure trends opposite to the observed data. Acetone cross-associates with water but its strength is much weaker than water self12784



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for the NRTL model as retrieved from ASPEN databases29 and the literature.30 4.3.1. Water−Acetone−Chloroform Ternary System. The three molecules in the ternary system contain different types of association sites: water has two HB donor sites and two acceptor sites; acetone has two HB acceptor sites; chloroform has one weak HB donor site. As discussed earlier, the crossassociation between the HB donor site in chloroform and the HB acceptor sites in water is too weak when compared to water self-association and that results in phase separation. However, by introducing acetone, another HB acceptor molecule, the immiscibility disappears as the acetone content is increased. Figure 13 shows the association model predicts the ternary LLE phase envelope very close to the experiment data.30 The NRTL model is capable of predicting the ternary LLE phase envelop with a similar accuracy with the three pairs of binary interaction parameters regressed from VLE or LLE data of the three binary systems. On the other hand, the original model significantly under-predicts the immiscible region of the ternary LLE because of its inadequacy in describing the cross-association between acetone and chloroform. 4.3.2. Water−Ethanol−Ethyl Acetate Ternary System. In this system, water has two HB acceptor sites and two donor sites; ethanol has two HB acceptor sites and one donor site; ethyl acetate has two HB acceptor sites. Ethyl acetate shows phase separation when mixed with water, but the phase separation disappears when 20 or higher mole percentages of ethanol is added to the system. Figure 14 shows the association model semiqualitatively predicts the phase separation region,31 whereas the original model under-predicts and the NRTL model overpredicts the phase separation region.

REFERENCES

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5. CONCLUSION The association NRTL-SAC model effectively describes the nonideal phase behavior of solvent mixtures with four molecular surface descriptors accounting for the relative abundance of segments responsible for physical interactions and chemical associations. Requiring only two moleculespecific parameters (X and Y) for physical interactions and two molecule-specific parameters (δA and δD) for chemical associations, the model yields improved VLE and LLE predictions over the original NRTL-SAC model for solvent mixtures with association. Future research should consider application of the association model for solubility modeling of pharmaceutical molecules and incorporation of the association term into the classical NRTL activity coefficient model to further enhance its versatility.

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AUTHOR INFORMATION

Corresponding Author

*Email: [email protected]. Tel.: +1 806.834.3098. ORCID

Chau-Chyun Chen: 0000-0003-0026-9176 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support of the Jack Maddox Distinguished Engineering Chair Professorship in Sustainable Energy sponsored by the J.F Maddox Foundation. 12785


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Industrial & Engineering Chemistry Research (23) Aspen Plus, version 8.4; Aspen Technology Inc.; Bedford, MA, 2013. (24) Chen, C.-C. A Segment-Based Local Composition Model for the Gibbs Energy of Polymer Solutions. Fluid Phase Equilib. 1993, 83, 301−312. (25) Chapman, W. G. Theory and Simulation of Associating Liquid Mixtures; Cornell University, Ithaca, NY, 1988. (26) Michelsen, M. L.; Hendriks, E. M. Physical Properties from Association Models. Fluid Phase Equilib. 2001, 180 (1−2), 165−174. (27) Luck, W. A. A Model of Hydrogen-Bonded Liquids. Angew. Chem., Int. Ed. Engl. 1980, 19 (1), 28−41. (28) Frenkel, M.; Chirico, R. D.; Diky, V.; Muzny, C. D.; Kazakov, A.; Magee, J.; Abdulagatov, I.; Kroenlein, K.; Diaz-Tovar, C.; Kang, J. NIST ThermoData Engine, NIST Standard Reference Database 103bPure Compounds, Binary Mixtures, and Chemical Reactions, version 8.0; NIST, 2013. (29) Aspen Technology, Inc. ASPEN PLUS V8.4 APV84 LLEASPEN. (30) Reinders, W.; De Minjer, C. Vapour-Liquid Equilibria in Ternary Systems. VI. The System Water-Acetone-Chloroform. Recueil des Travaux Chimiques des Pays-Bas 1947, 66 (9), 573−604. (31) Toikka, M.; Samarov, A.; Trofimova, M.; Golikova, A.; Tsvetov, N.; Toikka, A. Solubility, Liquid−Liquid Equilibrium and Critical States for the Quaternary System Acetic Acid−Ethanol−Ethyl Acetate−Water at 303.15 and 313.15 K. Fluid Phase Equilib. 2014, 373, 72−79.

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