Evaluation of the Nonrandom Hydrogen Bonding (NRHB) Theory and

Evaluation of the Nonrandom Hydrogen Bonding (NRHB) Theory and the Simplified Perturbed-Chain−Statistical Associating Fluid Theory (sPC-SAFT). 2...
0 downloads 0 Views 184KB Size
Download PDF
Ind. Eng. Chem. Res. 2008, 47, 5651–5659

5651

Evaluation of the Nonrandom Hydrogen Bonding (NRHB) Theory and the Simplified Perturbed-Chain-Statistical Associating Fluid Theory (sPC-SAFT). 2. Liquid-Liquid Equilibria and Prediction of Monomer Fraction in Hydrogen Bonding Systems Ioannis Tsivintzelis,† Andreas Grenner,† Ioannis G. Economou,*,†,‡ and Georgios M. Kontogeorgis† Center for Phase Equilibria and Separation Processes (IVC-SEP), Department of Chemical and Biochemical Engineering, Technical UniVersity of Denmark, DK-2800 Lyngby, Denmark, and Molecular Thermodynamics and Modeling of Materials Laboratory, Institute of Physical Chemistry, National Center for Scientific Research “Demokritos”, GR-15310 Aghia ParaskeVi Attikis, Greece

Two statistical thermodynamic models, the nonrandom hydrogen bonding (NRHB) theory, which is a compressible lattice model, and the simplified perturbed-chain-statistical associating fluid theory (sPC-SAFT), which is based on Wertheim’s perturbation theory, were used to model liquid-liquid equilibria and predict the fraction of nonhydrogen bonded molecules in various hydrogen bonding mixtures. Carefully selected binary mixtures, which include water-hydrocarbon, 1-alkanol-hydrocarbon, water-1-alkanol, and glycol-hydrocarbon, were used to benchmark the accuracy of the models. Both models yielded satisfactory and often very similar results for the phase behavior of the investigated mixtures. sPC-SAFT yielded more accurate predictions, while NRHB yielded more accurate correlations, in mixtures of water with normal alkanes and cycloalkanes. In water-aromatic hydrocarbon mixtures, satisfactory correlations were obtained only when solvation was accounted for. Both models resulted in satisfactory correlations for all other mixtures, while for specific mixtures, one model may perform better than the other. Finally, both models, despite that they are based on totally different approaches for the treatment of hydrogen bonding, yielded similar predictions for the fraction of non-hydrogen bonded molecules (monomer fraction) in pure 1-alkanols and in 1-alkanol-n-hexane mixtures. 1. Introduction Advances in applied statistical mechanics in recent years have resulted in a number of equations of state (EoS’s) for real complex fluids with strong theoretical basis. Two of the most successful and widely used families of such models are based on Wertheim’s first-order thermodynamic perturbation theory1–4 and lattice theory.5 PC-SAFT6 is one of the most successful models of the first family, while a simplified version of the model, sPC-SAFT,7 reduces the computing time without compromising its performance. On the other hand, lattice models have been used in industry and academia since the 1970s. A recent development in this respect is the nonrandom hydrogen bonding theory (NRHB), which accounts explicitly for hydrogen bonding as well as for the nonrandom distribution of molecular segments and free volume.8,9 In this series of papers, we attempt to evaluate the performance of the sPC-SAFT and the NRHB model using complex fluid mixtures as benchmarks. In the first paper,10 we have tried to address capabilities and limitations of the models in the description of vapor-liquid equilibria (VLE) of several mixtures that consisted of nonpolar and also weakly, strongly, and very strongly polar fluids. A standard database of Danner and Guess,11 which is a collection of VLE experimental data for 104 mixtures, was used, and different approaches were adopted for the description of phase equilibrium. In this work, the models are applied to predict and correlate liquid-liquid equilibria (LLE) of carefully selected hydrogen bonding mixtures, such as water-hydrocarbon, 1-alkanolhydrocarbon, water-1-alkanol, and glycol-hydrocarbon. A * Corresponding author. Tel.: +302106503963. Fax: +302106511766. E-mail: [email protected]. † Technical University of Denmark. ‡ National Center for Scientific Research “Demokritos”.

thorough discussion of the two approaches for the treatment of hydrogen bonding is made through a detailed comparison of model predictions for the fraction of non-hydrogen bonded molecules in alcohol solutions. 2. Theory 2.1. Nonrandom Hydrogen Bonding (NRHB). The NRHB theory is a compressible lattice model, where holes are used to account for density variation as a result of temperature and pressure changes. NRHB accounts explicitly for the nonrandom distribution of molecular sites, while Veytsman’s statistics is used to calculate the contribution of hydrogen bonding to the thermodynamics of the system.8,9 Thus, the model is suitable for property calculations of highly nonideal fluids. According to NRHB theory, the molecules are assumed to be arranged on a quasi-lattice of Nr sites, N0 of which are empty, with a lattice coordination number, z. Each molecule of type i in the system occupies ri sites of the quasi-lattice. It is characterized by three scaling parameters and one geometric, or surface-to-volumeratio, factor, s. The first two scaling parameters, εh* and εs*, are used for the calculation of the mean interaction energy per molecular segment, ε*, according to the following expression: ε * ) ε/h + (T - 298.15)εs/

(1)

*

while the third scaling parameter, νsp,0, is used for the calculation * , as described by the of the close-packed density, F* ) 1/νsp following expression: / / / ) νsp,0 + (T - 298.15)νsp,1 νsp *

(2)

Parameter νsp,1 in eq 2 is treated to as a constant for a given homologous series.9,12 Furthermore, the hard-core volume per segment, ν*, is constant and equal to 9.75 cm3 mol-1 for all * /ν*. Finally, fluids. The following relation holds: r ) MW νsp

10.1021/ie071382l CCC: $40.75  2008 American Chemical Society Published on Web 06/25/2008

5652 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008

the shape factor is defined as the ratio of molecular surface to molecular volume, s ) q/r, and is calculated from the UNIFAC group contribution method.13 As described in detail elsewhere,9 the equation of state for a fluid mixture assumes the following form,

[

P˜ + T˜ ln(1 - F˜ ) - F˜

(

li

∑φ r -ν i

)

H

i

i

q z - ln 1 - F˜ + F˜ + 2 r

(

)

]

z ln Γ00 ) 0 2

gas and the dispersion are identical to the PC-SAFT expressions. The modifications of von Solms et al.7 affect only the hardsphere chain and the association terms. First, it is assumed that all segments in the mixture have the same diameter. This diameter is calculated as follows: dav )

(

Σiximidi3 Σiximi

)

1⁄3

(3)

Using eq 7, ζn is calculated as follows:

while the chemical potential for the component i is given by

π ζn ) dnav FΣiximi 6

µi φi φjlj ) ln - riΣj + ln F˜ + ri(ν˜ - 1) ln(1 - F˜ ) RT ωiri rj qi zqi z q r ν˜ - 1 + ln 1 - F˜ + F˜ + ln Γii + 2 i ri r 2 ri P˜ν˜ qi µHi (ν ˜ - 1) ln Γ00 + ri - + qi RT T˜ T˜

[

]

[

]

]

(7)

(8)

By now setting the volume fraction η ≡ ζ3, a simpler expression for ghs (radial distribution function) compared to the original PC-SAFT is obtained,

[

ghs(η) )

1-η⁄2 (1 - η)3

(9)

(4)

and a˜hs (hard-sphere term of the Helmholtz energy) reduces to the Carnahan-Starling equation:14

where φi is the site fraction of component i, while li and ωi are characteristic quantities for each fluid. Parameters Γoo and Γii are nonrandom factors for the distribution of empty sites around an empty site and of molecular segments of component i around a molecular segment of component i, respectively. For associating fluids, νH denotes the number of hydrogen bonds per molecular segment, while the term µHi/RT is the hydrogen bonding contribution to the chemical potential of component i. Finally, parameters T˜ ) T/T*, P˜ ) P/P*, and v˜ ()1/F˜ ) F*/F) are the reduced temperature, pressure, and specific volume, respectively. The characteristic temperature, T*, and pressure, P*, are related to the mean intersegmental energy by

4η - 3η2 (10) (1 - η)2 In addition, sPC-SAFT7 employs a slightly different expression for the association strength ∆AiBj than in PC-SAFT,6 given by the following expression:

i

ε * ) RT * ) P * ν* (5) Detailed expressions for the calculation of all these parameters can be found elsewhere.9 For associating fluids, NRHB has three more pure-component parameters that are the energy, EijH, the volume, VijH, and the entropy change, SijH, for the formation of a hydrogen bond between proton donors of type i and proton acceptors of type j in different molecules. However, usually the volume change for the formation of the hydrogen bond, VH ij , is set equal to zero, so the number of hydrogen bonding parameters are reduced to two without compromising the performance of the model.9,12 2.2. Simplified Perturbed-Chain-Statistical Associating Fluid Theory (sPC-SAFT). The sPC-SAFT7 EoS was developed with the goal to reduce the computational and programming effort with respect to the original PC-SAFT without essentially changing the performance of the model. Thus, sPCSAFT7 is identical to PC-SAFT6 in the case of pure nonassociating compounds, but not for associating compounds and for multicomponent mixtures, in general. The sPC-SAFT7 EoS uses simpler mixing rules by taking into account that the segment diameters are usually similar for segments belonging to different molecules. The Helmholtz energy for a mixture of associating molecules is a˜ ≡

A ) a˜id + a˜hc + a˜disp + a˜assoc NkT

(6)

where a˜id is the ideal gas contribution, a˜hc is the contribution of the hard-sphere chain reference mixture, and a˜disp and a˜assoc are the contributions of dispersion forces and association, respectively. The expressions for the contribution from the ideal

hs a˜cs )

[ ( ) ]

π εAiBj -1 ∆AiBj ) NAV σ3ijghs(η)κAiBj exp 6 kT

(11)

Consistent with earlier simplifications, the radial distribution function with the simpler expression is used. Finally, the diameter cubed is replaced by the sphere volume. The replacement of gijhs by ghs (η) makes the composition dependence of ∆AiBj (required for evaluation of the fugacity coefficient) much simpler, since this dependence is contained only in η. 3. Pure-Fluid Parameters Pure-fluid parameters for the two models are summarized in Tables 1 and 2. Parameters for the majority of the fluids were adopted from literature. However, in some cases, new parameters were estimated by fitting the predictions of the theories to vapor-pressure and saturated liquid-density data of the Design Institute for Physical Property Data (DIPPR) correlation.15 In * the NRHB model, the parameter νsp,1 , which is used in eq 2, was treated as a characteristic parameter of a given homologous series. As before,9,12 it was set equal to -0.412 × 10-3 cm3 g-1 K-1 for non-aromatic hydrocarbons, -0.310 × 10-3 cm3 g-1 K-1 for alcohols, -0.300 × 10-3 cm3 g-1 K-1 for water, and 0.150 × 10-3 cm3 g-1 K-1 for all the other fluids. 4. Mixture Phase Equilibria The two models were applied in order to describe the LLE of various hydrogen bonding mixtures. A number of representative binary mixtures were examined. The mixture scaling parameters were calculated using appropriate mixing and combining rules.10 In both models, one binary interaction parameter, kij, was used in the combining rule for the dispersion energy between unlike molecules: εij ) (1 - kij)√εiiεjj

(12)

In mixtures of two self-associating fluids, the following combining rules were used for the NRHB model,10

Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5653 Table 1. NRHB Pure-Component Parameters and Deviation in the Correlation of Vapor-Pressure and Saturated Liquid-Density Data * εs* νsp,0 εh* (J mol-1) (J mol-1 K-1) (cm3 g-1)

fluid n-hexane n-heptane n-octane n-tetradecane n-hexadecane cyclohexane ethylcyclohexane benzene ethylbenzene methanol ethanol 1-butanol 1-pentanol water ethylene glycol (MEG) 1,2-propylene glycol (1,2-PG) diethylene glycol (DEG) triethylene glycol (TEG)

3957.1 4042.0 4105.3 4306.8 4339.0 4469.2 4402.3 5148.5 5024.8 4202.3 4378.5 4463.1 4471.7 5336.5 5800.9 5088.8 5518.4 5371.5

d ) donor and a ) acceptor. or saturated liquid density, Fliq. a

b

1.6580 1.7596 1.8889 2.2030 2.2657 1.8391 2.3146 -0.2889 0.3985 1.5269 0.7510 1.1911 1.55212 -6.5057 0.8653 1.0526 1.8177 2.0918

EH SH % AADb %AADb (J mol-1) (J mol-1 K-1) Nsitesassoca Psat Fliq

s

1.27753 1.25328 1.23687 1.18931 1.18135 1.19596 1.17802 1.06697 1.05355 1.15899 1.15867 1.13403 1.12869 0.97034 0.84239 0.84239 0.84161 0.81214

0.857 0.850 0.844 0.826 0.823 0.801 0.800 0.753 0.763 0.941 0.903 0.867 0.857 0.861 0.933 0.903 0.892 0.872

-25100 -24000 -24000 -24000 -16100 -22500 -22500 -22500 -22500

0d-2a 0d-2a 1d-1a 1d-1a 1d-1a 1d-1a 2d-2a 2d-2a 2d-2a 2d-2a 2d-2a

-26.5 -27.5 -27.5 -27.5 -14.7 -27.5 -27.5 -27.5 -27.5

1.1 0.5 0.7 1.3 1.6 0.6 0.8 1.7 1.3 2.2 1.8 0.2 1.4 1.3 2.8 5.0 0.4 1.6

0.5 0.5 0.4 0.3 0.3 1.9 1.6 0.5 1.3 2.6 1.1 0.5 0.4 2.0 1.8 0.7 0.3 0.9

Tr 0.51-0.97 0.52-0.97 0.53-0.97 0.58-0.97 0.54-0.97 0.53-0.97 0.50-0.98 0.52-0.97 0.50-0.87 0.52-0.97 0.54-0.97 0.55-0.97 0.55-0.97 0.45-0.99 0.52-0.92 0.52-0.92 0.52-0.92 0.52-0.92

ref. 12 12 12 12 12 12 this 12 10 12 10 10 10 10 this this this this

work

work work work work

n

%AAD ) (100/n) · Σ (Xical - Xiexp) ⁄ Xiexp where n is the number of data points and X represents vapor pressure, Psat, i)1

Table 2. sPC-SAFT Pure-Component Parameters and Deviation in the Correlation of Vapor-Pressure and Saturated Liquid-Density Data m

σ (Å)

ε/k (K)

3.0576 3.4831 3.8176 5.9002 6.6485 2.5303 2.8256 2.4653 3.0799 1.5255 1.2309 2.3983 2.6048 1.5000 1.9088 0.04491 3.0582 3.1809

3.7983 3.8049 3.8373 3.9396 3.9552 3.8499 4.1039 3.6478 3.7974 3.2300 4.1057 3.7852 3.9001 2.6273 3.5914 3.6351 3.6143 4.0186

236.77 238.40 242.78 254.21 254.70 278.11 294.04 287.35 287.35 188.90 316.91 276.90 282.31 180.30 325.23 284.62 310.29 333.17

fluid n-hexane n-heptane n-octane n-tetradecane n-hexadecane cyclohexane ethylcyclohexane benzene ethylbenzene methanol ethanol 1-butanol 1-pentanol water ethylene glycol (MEG) 1,2-propylene glycol (1,2-PG) diethylene glycol (DEG) triethylene glycol (TEG) a

For the % AAD definition, see the footnote of Table 1.

EHi + EHj , 2 and for sPC-SAFT,10 EHij )

SHij )

(

1⁄3

SHi + SHj 2

1⁄3

)

εAB/k (K)

2899.5 2811.02 2811.02 2811.02 1804.22 2080.03 2080.03 2080.03 2080.03 b

κAB c

0.06718 0.00633 0.00633 0.00633 0.18000 0.04491 0.04491 0.04491 0.04491

Nsitesassoc

%AADa Psat

%AADa VL,sat

Tr

1b 1b 2 2 2 2 4 4 4 4 4

0.3 0.3 0.8 4.8 4.9 0.5 2.4 0.6 0.4 2.4 1.1 2.2 2.9 0.9 0.6 2.1 0.7 1.5

0.8 2.1 1.6 1.3 0.7 3.1 1.0 1.4 1.1 2.0 2.2 0.5 0.3 2.6 1.6 1.4 0.2 0.4

0.35-0.99 0.34-1.15 0.38-1.00 0.40-1.00 0.40-1.00 0.50-1.00 0.35-1.00 0.49-1.00 0.29-1.00 0.39-0.99 0.50-0.90 0.50-0.90 0.50-0.90 0.50-0.90 0.50-0.89 0.45-0.80 0.49-0.63 0.48-0.70

Only solvation with water or glycols.

c

ref. 6 6 6 6 6 6 6 6 6 39 40 40 40 40 this this this this

work work work work

Note that κAB values in ref 40 are in fact κABπ/6.

mixtures. Mixtures of water with linear alkanes, cycloalkanes, and aromatic hydrocarbons were studied. Experimental data were taken from the literature.16,17 Results are summarized in Tables 3 and 4 for NRHB and sPC-SAFT, respectively. Two correlations were performed for each mixture. In the first, the binary interaction parameter was fitted to the experimental data for both phases. However, the solubility of hydrocarbons in water is very low, while in many industrial

3

(13)

εAiBi + εAjBj , κAiBj ) √κAiBiκAjBj (14) 2 4.1. Water-Hydrocarbon Mixtures. Initially, the two theories were applied to describe the LLE of water-hydrocarbon εAiBj )

Table 3. NRHB Calculations for LLE of Water-Hydrocarbon (HC) Mixtures prediction (kij ) 0) HC in water water in HC % AADa ref. % AADa

mixture

T (K)

n-hexane-water n-octane-water cyclohexane-water ethylcyclohexane-water benzene-water

280-480 280-520 280-510 280-540 280-540

16 17 16 17 16

>100 >100 >100 >100 83

97.5 86.7 >100 93.4 56.0

ethylbenzene-water

280-540 17

>100

33.3

>100

82.4

total a

For the % AAD definition, see the footnote of Table 1. mol-1, fitted to experimental data.

b

kij fitted to HC rich phase kij 0.1879 0.1769 0.2096 0.1717 -0.1512 0.0812b -0.0717 0.1013b

HC in water water in HC % AADa % AADa 53.4 47.9 75.2 70.2 >100 17.9 >100 17.5 47.0b

4.2 5.9 9.8 6.7 10.7 5.0 13.0 3.6 5.9b

kij fitted to both phases kij 0.1683 0.1641 0.1468 0.1359 0.0342 0.0812b 0.0576 0.1013b

HC in water water in HC % AADa % AADa 32.6 32.4 24.3 34.2 25.1 17.9 17.7 17.5 26.5b

6.6 7.2 23.8 13.1 62.1 5.0 47.7 3.6 9.9b

Solvation was accounted for water-aromatic hydrocarbon mixtures, E12HB ) -6900 J

5654 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 Table 4. sPC-SAFT Calculations for LLE of Water-Hydrocarbon (HC) Mixtures prediction (kij ) 0) HC in water water in HC % AADa ref. % AADa

mixture

T (K)

n-hexane-water n-octane-water cyclohexane-water ethylcyclohexane-water benzene-water

280-480 280-520 280-510 280-540 280-540

16 17 16 17 16

>100 >100 >100 >100 >100

12.3 22.8 25.8 10.7 67.8

ethylbenzene-water

280-540 17

>100

56.1

>100

32.6

total

kij fitted to HC rich phase kij

HC in water water in HC % AADa % AADa

0.025 0.037 0.044 -0.018 -0.15 0.048b,1 -0.04 0.039b,3

>100 >100 >100 >100 >100 37.5 >100 45.3 >100b

5.0 3.9 4.4 6.0 30.8 17.5 45.0 10.2 7.8b

kij fitted to both phases kij 0.0571 0.0609 0.0692 0.0533 0.0273 0.059b,2 0.025 0.046b,4

HC in water water in HC % AADa % AADa 32.1 31.5 19.0 36.3 26.6 22.6 19.1 17.8 26.6b

15.3 11.6 15.9 32.3 72.7 19.2 62.6 18.3 18.7b

For the % AAD definition, see the footnote of Table 1. b Solvation was accounted for water-aromatic hydrocarbon mixtures: (1) κ ) 0.1780, (2) κ ) 0.1728, (3) κ ) 0.1833, and (4) κ ) 0.1466, fitted to experimental data. a

Figure 1. n-Hexane-water LLE, experimental data16 (points) and model calculations (lines).

applications, the accurate correlation of the water solubility in hydrocarbons is needed.18 Subsequently, in the second correlation, the binary interaction parameter was fitted only to experimental data for the hydrocarbon rich phase (water solubility). Both models perform satisfactorily for n-alkane-water mixtures considering the complexity of such mixtures, using a binary interaction parameter. sPC-SAFT yields more accurate predictions (kij ) 0) in the LLE of water with linear alkanes and cycloalkanes, while NRHB yields more accurate correlations. Typical results for the n-hexane-water mixture are presented in Figure 1. The largest deviations between experimental data and correlationswereobservedforwater-benzeneandwater-ethylbenzene mixtures. The solubility of aromatic hydrocarbons in water is at least 1 order of magnitude higher compared to the solubility of normal alkanes. This should be attributed to the weak interaction of water molecules due to the π-electrons of the aromatic ring. In order to account for this effect, solvation was considered in both models and the aromatic ring was assumed to act as a protonacceptor site that could associate with the proton donors of water molecules. This assumption is based on experimental measurements and theoretical calculations showing that, for pure benzene at low temperatures, one of the most energetically favorable conformations consists of molecules lining up perpendicular to one another rather than parallel.19 A proton of a given molecule points directly to the center of another ring, indicating a weak hydrogen bonding interaction. For the application of the NRHB model, the association energy for the water-benzene cross-interaction, EijH, was calculated by fitting the theory to the experimental data. The corresponding association entropy, SijH, was set equal to one-half of water’s selfassociation entropy. In a similar way, for the application of sPC-

SAFT, the association volume, κ, was calculated by fitting the experimental data while the association energy, εAiBj, was set equal to one-half of water’s self-association energy. The results for the benzene-water mixture are illustrated in Figure 2. It is obvious that predictions and correlations fail if solvation is not accounted for. On the other hand, both models yield satisfactory correlations when solvation is taken into account. An alternative approach is the explicit introduction of polar interactions in the model to account for aromatic hydrocarbon π-electron-water interactions. Recently, Economou and coworkers proposed PC-Polar SAFT (PC-PSAFT) where dipole-dipole, quadrupole-quadrupole, and dipole-quadrupole interactions were calculated using appropriate terms.20,21 PCPSAFT was shown to be superior over SAFT and over PCSAFT (without solvation interactions) for the correlation of water-hydrocarbon mixtures.22 It remains to be seen whether

Figure 2. Benzene-water LLE, experimental data16 (points) and models calculations (lines): (a) NRHB and (b) sPC-SAFT.

Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5655 Table 5. Calculations for LLE of 1-Alkanol-Hydrocarbon Mixtures NRHB mixture

T (K)

ref.

kij

ethanol (1)-n-tetradecane (2) ethanol (1)-n-hexadecane (2) methanol (1)-n-hexane (2) methanol (1)-n-heptane (2) methanol (1)-cyclohexane (2) total

279.96-307.81 298.66-326.98 245.15-306.51 291.15-323.15 280.05-318.93

41 41 23 23 23, 24

0.0180 0.0207 0.0304 0.0320 0.0250

a

sPC-SAFT

alkanol in HC % AADa in x1

HC in alkanol % AADa in x1

8.3 8.2 13.3 42.0 23.7 19.1

5.4 6.5 6.0 1.4 11.4 6.2

kij -0.0065 -0.0040 0.0254 0.0255 0.0365

alkanol in HC % AADa in x1

HC in alkanol % AADa in x1

11.9 11.3 23.4 10.5 14.9 14.4

11.0 10. 9 4.6 2.7 7.6 7.4

For the % AAD definition, see the footnote of Table 1.

Figure 3. Ethanol-n-tetradecane LLE, experimental data41 (points) and model correlations (lines).

the explicit polar terms or a solvation term result in more accurate representation of these highly complex phase equilibria. 4.2. 1-Alkanol-Hydrocarbon Mixtures. Alkanols are among the most important self-associating fluids for industry. Subsequently, the accurate description of phase equilibria in such mixtures is of great importance. In this work, the models were applied to describe the LLE of 1-alkanol-hydrocarbon mixtures. Results are summarized in Table 5. Initially, the LLE of ethanol with higher n-alkanes, such as n-tetradecane and n-hexadecane, was investigated. In these mixtures, NRHB yields more accurate correlations, while both models overestimate the critical solution temperature. Characteristic results for the ethanol-n-tetradecane are illustrated in Figure 3. Furthermore, the models were used to correlate the phase equilibria of methanol-n-alkane mixtures. Experimental data were obtained from the literature.23–26 Such mixtures exhibit LLE at lower temperatures and VLE at higher temperatures. For both models, a binary interaction parameter was optimized to LLE data23,24 and then was used to predict the VLE data.25–27 For the LLE correlation, NRHB is more accurate for the methanol-n-hexane mixture, while sPC-SAFT is superior for methanol mixtures with n-heptane and cyclohexane. Furthermore, both models yield accurate predictions for the VLE. Results for the VLE calculations are presented in Table 6, while calculations for a characteristic mixture, namely, methanol-nhexane, are illustrated in Figure 4. 4.3. Water-1-Alkanol Mixtures. Cross-association occurs in water-alkanol mixtures, rendering the description of their phase behavior as a challenging task for thermodynamic models. In this work, the models were used to describe the phase equilibria of water-1-butanol and water-1-pentanol mixtures. Again, one binary interaction parameter was optimized to LLE data, and then it was used to predict the VLE data. Results are

Figure 4. Methanol-n-hexane LLE and VLE, experimental data23,25 (points) and model correlations (lines).

presented in Figures 5 and 6 for water-1-butanol and water-1pentanol, respectively. Both models exhibit similar behavior in the water-1-butanol mixture: if the 1-butanol rich phase is correlated with low deviations, then the deviations increase for the water rich phase and vice versa. In this mixture, NRHB performs better in LLE, while sPC-SAFT yields more accurate predictions for VLE. Lower deviations from experimental data were obtained for both models in the correlation of water-1pentanol phase equilibria. sPC-SAFT yields more accurate correlations for LLE, while NRHB yields more accurate predictions for VLE. 4.4. Glycol-Hydrocarbon Mixtures. Modeling of glycols is a challenging task because of the complexity of their molecules. Recently, sPC-SAFT was successfully used for the correlation of VLE and LLE of glycol-hydrocarbon mixtures.28 In that work, glycol molecules were modeled assuming that they have four association sites (two proton donors and two proton acceptors known as 4C scheme according to Huang and Radosz29). This approach does not account for the complex hydrogen bonding behavior that often is observed in such mixtures (i.e., intramolecular hydrogen bonding), but it could be very useful in the modeling of phase equilibria for the rational design of many chemical and petrochemical applications. Using a similar approach, in this work, the phase equilibria of n-heptane mixtures with monoethylene glycol (MEG), diethylene glycol (DEG), triethylene glycol (TEG), and 1,2-propylene glycol (1,2-PG) was investigated. Experimental data were obtained from Derawi et al.30 As is presented in Table 7, both models yield satisfactory correlations considering the complexity of these mixtures. However, NRHB yields better correlations than PC-SAFT in most cases. Using a small value for the binary interaction parameter, AAD lower than 20% can be obtained for both glycol and heptane rich phase for MEG, DEG, and 1,2-PG. However, relatively higher deviations occur for TEG. sPC-SAFT performs better in the DEG-n-heptane system, while NRHB yields more

5656 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008

Figure 7. MEG-n-heptane LLE, experimental data30 (points) and model correlations (lines).

Figure 5. Water-1-butanol LLE and VLE, experimental data23 (points) and model correlations (lines).

5. Monomer Fractions of Associating Fluids The sPC-SAFT model is based on Wertheim’s theory1–4 and explicitly accounts for hydrogen bonding. In the basic equations of the model, the fraction of molecules nonbonded at a given site A, XA, is incorporated. In general, this quantity is not identical to the monomer fraction. However, the latter can be easily calculated;31 e.g., for the 2B scheme (one proton-donor and one protonacceptor site following the terminology of Huang and Radosz29), it is equal to XA2, and for 4C, it is equal to XA4, if all sites are assumed to be identical. On the other hand, the NRHB model uses a hydrogen bonding term, which was proposed by Panayiotou and Sanchez32 and was used in the LFHB model.32,33 According to this approach, the number of ways of distributing the hydrogen bonds among the different proton donors and proton acceptors of the mixture is calculated. Nevertheless, even if the number of hydrogen bonds in the mixture is known, calculation of the number of bonded molecules is not a straightforward issue. In other words, for a certain number of hydrogen bonds, the number of bonded (or unbonded) molecules may vary. In order to obtain the

Figure 6. Water-1-pentanol LLE and VLE, experimental data23 (points) and model correlations (lines).

accurate correlations in all other investigated mixtures. Characteristic results for MEG-n-heptane are presented in Figure 7. Table 6. Calculations for VLE of 1-Alkanol-Hydrocarbon Mixtures mixture

ref.

T (K)

∆y a

∆T a (K)

kij

∆y a

∆T a (K)

0.1199 0.0601 0.0477 0.0759

5.25 5.53 3.59 4.79

0.0304 0.0320 0.0250

0.0512 0.0303 0.0137 0.0317

2.18 1.84 0.82 1.61

0.9500 0.0366 0.0607 0.3491

4.66 4.25 4.80 4.57

0.0254 0.0255 0.0365

0.0336 0.0112 0.0120 0.0189

1.93 1.43 1.50 1.62

kij NRHB

methanol (1)-n-hexane (2) methanol (1)-n-heptane (2) methanol (1)-cyclohexane (2) total

25 26 27

245.15-306.51 291.15-323.15 280.05-318.93

methanol (1)-n-hexane (2) methanol (1)-n-heptane (2) methanol (1)-cyclohexane (2) total

25 26 27

245.15-306.51 291.15-323.15 280.05-318.93

0 0 0 sPC-SAFT

a

0 0 0

∆z ) 1/n ∑|zicalc - ziexp| where n is the number of data points and z represents T or y.

Table 7. NRHB and sPC-SAFT Calculations for Glycol-Hydrocarbon LLE NRHB mixture

ref

T (K)

kij

MEG (1)-n-heptane (2) 1,2-PG (1)-n-heptane (2) DEG (1)-n-heptane (2) TEG (1)-n-heptane (2) total

30 30 30 30

315.95-351.85 308.05-342.05 312.75-353.05 309.35-350.95

0.0566 0.0166 0.0514 0.0573

a

For the % AAD definition, see the footnote of Table 1.

sPC-SAFT

HC in glycol % AADa in x2

glycol in HC % AADa in x1

7.7 6.5 7.2 3.7 6.3

16.1 8.5 10.6 45.9 20.3

kij 0.0400 0.0100 0.0320 0.0450

HC in glycol % AADa in x2

glycol in HC % AADa in x1

16.9 11.2 5.6 29.1 15.7

20.3 10.3 8.3 50.0 22.2

Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5657

Figure 9. Fraction of non-hydrogen bonded molecules in n-hexane-1alkanol mixtures; experimental data34,38 (points), NRHB (solid lines), and sPC-SAFT predictions (dashed lines) for mixtures of n-hexane with (a) methanol and (b) ethanol.

number of bonded (or unbonded) molecules, the type of oligomers that are present in the mixture must be known. In that direction, Economou and Donohue31 assumed the formation of linear oligomers for alkanols and obtained an equation for the number of unbonded molecules. They showed that the quasichemical approach followed by Panayiotou and Sanchez,32 which is incorporated in the NRHB model, and Wertheim’s theory, which is used in the sPC-SAFT model, lead to expressions that are of the same functional form for pure components and for mixtures.31 In the modeling of alkanols, usually two hydrogen bonding sites are assumed (i.e., one proton donor and one proton acceptor). According to the aforementioned approach, the fraction of unbonded molecules is obtained from the following equation,31 X1 )

2 1 + 2FK + √1 + 4FK

(15)

where F is the density and K ) ∆AiBj

(16)

for sPC-SAFT, while

(

H -∆GAB ν/ K ) exp r RT

Figure 8. Fraction of non-hydrogen bonded molecules in pure 1-alkanols; experimental data35–37 (points), NRHB (solid lines), and sPC-SAFT predictions (dashed lines) for (a) methanol, (b) ethanol, (c) 1-propanol, and (d) 1-octanol.

)

(17)

for NRHB. The fact that both models, despite that they are based on different approaches, result in similar equations for the extent of hydrogen bonding makes the comparison of their predictions an interesting task. In this work, we present the predictions of the models for the monomer (unboded molecules) fraction in

5658 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008

pure alkanols and alkanol-n-alkane mixtures. Experimental data based on spectroscopic studies were taken from the literature.34–38 For the application of sPC-SAFT, we assumed for all alkanols the 2B association scheme (terminology that refers to the work of Huang and Radosz29). In a similar manner, for the application of the NRHB theory, we assumed one proton-donor and one proton-acceptor site per alkanol molecule. As presented in Figure 8a, both models underestimate the monomer fraction for pure methanol, while NRHB performs better than sPC-SAFT. On the other hand, both models perform satisfactorily in predicting the monomer fraction for ethanol, 1-propanol, and 1-octanol (Figure 8 parts b-d, respectively). However, a trend to underestimate the fraction of monomers is observed. The models were also applied to predict the monomer fractions in 1-alkanol-n-hexane mixtures. Predictions were compared with the experimental data of Asprion et al.,34 who published a systematic study for this kind of mixtures. In methanol-n-hexane (Figure 9a), both models underestimate the monomer fraction, in a similar way as observed in pure methanol. However, NRHB clearly yields better predictions than sPC-SAFT, a result that is in line with the superiority of the former model over the latter for the LLE prediction of these mixtures presented in Section 4.2 (see Figure 4). Finally, both models yield very similar and accurate predictions for the binary mixtures of n-hexane with ethanol, 1-propanol, 1-butanol, 1-pentanol, and 1-hexanol. A representative plot for n-hexane-ethanol is presented in Figure 9b. Results for the other mixtures are very similar and are omitted. 6. Conclusions We evaluated the performance of NRHB and sPC-SAFT for the description of LLE in various hydrogen bonding mixtures, such as water-hydrocarbon, 1-alkanol-hydrocarbon, water-1alkanol, and glycol-hydrocarbon. sPC-SAFT yields more accurate predictions while NRHB yields more accurate correlations in mixtures of water with normal alkanes and cycloalkanes. In water-aromatic hydrocarbon mixtures, satisfactory correlations were obtained only when solvation was accounted for, assuming that the aromatic ring acts as proton-acceptor site able to interact with water’s proton-donor sites. NRHB provides a more accurate correlation, while both models overestimate the critical solution temperature in mixtures of ethanol with higher alkanes such as n-tetradecane and n-hexadecane. In methanol-nalkane and water-1-alkanol mixtures, a binary interaction parameter was optimized to LLE data and then was used to predict the VLE data. In this manner, both models successfully predict the VLE, while some differences between the two models are observed concerning the LLE. In addition, both models describe in a satisfactory way the LLE of glycol-nheptane mixtures, considering also the complexity of such systems, but in most cases NRHB yields more accurate correlations than sPC-SAFT. Finally, the predictions of the models for the monomer fractions in pure 1-alkanols and in alknanol-hydrocarbon mixtures were compared to experimental spectroscopic data. Both models, despite that they are based in totally different approaches for the treatment of hydrogen bonding, lead to expressions for monomer fractions that are of the same functional form. In this direction, it was shown that both models result in similar predictions for the fraction of unbonded molecules in most pure 1-alkanols and in 1-alkanol-n-hexane mixtures. However, for pure methanol and its n-hexane mixtures, NRHB clearly predicts the monomer fractions more accurately

than sPC-SAFT, and this should be related to the better performance of this model for the LLE of the same system. The results presented here (and also in the first paper of this series10) show clearly that the two advanced models can be used successfully for a wide range of fluid-phase equilibria calculations. Despite their different initial points (i.e., lattice fluid theory for NRHB and Wertheim’s first-order thermodynamic perturbation theory for sPC-SAFT), they yield satisfactory and often very similar results for various hydrogen bonding mixtures. However, for specific mixtures, one model may perform better than the other. Work is underway in, among others, evaluating the models in the description of the phase behavior in polymer systems. Acknowledgment The authors are grateful to the Danish Research Council for Technology and Production Sciences for financial support of this work as part of the project “Advanced Thermodynamic Tools for Computer-Aided Product Design”. Literature Cited (1) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. 1. Statistical Thermodynamics. J. Stat. Phys. 1984, 35, 19. (2) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. 2. Thermodynamic Perturbation-Theory and Integral-Equations. J. Stat. Phys. 1984, 35, 35. (3) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. 3. Multiple Attraction Sites. J. Stat. Phys. 1986, 42, 459. (4) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. 4. Equilibrium Polymerization. J. Stat. Phys. 1986, 42, 477. (5) Guggenheim, E. A. Mixtures; Oxford University Press: Oxford, U.K., 1952. (6) Gross, J.; Sadowski, G. Perturbed-chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (7) von Solms, N.; Michelsen, M. L.; Kontogeorgis, G. M. Computational and Physical Performance of a Modified PC-SAFT Equation of State for Highly Asymmetric and Associating Mixtures. Ind. Eng. Chem. Res. 2003, 42, 1098. (8) Panayiotou, C.; Pantoula, M.; Stefanis, E.; Tsivintzelis, I.; Economou, I. G. Nonrandom Hydrogen-Bonding Model of Fluids and their Mixtures. 1. Pure Fluids. Ind. Eng. Chem. Res. 2004, 43, 6592. (9) Panayiotou, C.; Tsivintzelis, I.; Economou, I. G. Nonrandom Hydrogen-Bonding Model of Fluids and their Mixtures. 2. Multicomponent Mixtures. Ind. Eng. Chem. Res. 2007, 46, 2628. (10) Grenner, A.; Tsivintzelis, I.; Kontogeorgis, G. M.; Economou, I. G. Evaluation of the Nonrandom Hydrogen Bonding (NRHB) Theory and the Simplified Perturbed-Chain-Statistical Associating Fluid Theory (sPCSAFT). 1. Vapor-Liquid Equilibria. Ind. Eng. Chem. Res. 2008, 47, 56365650. (11) Danner, R. P.; Gess, M. A. A Data-Base Standard for the Evaluation of Vapor-Liquid-Equilibrium Models. Fluid Phase Equilib. 1990, 56, 285. (12) Tsivintzelis, I.; Spyriouni, T.; Economou, I. G. Modeling of Fluid Phase Equilibria with two Thermodynamic Theories: Non-random Hydrogen Bonding (NRHB) and Statistical Associating Fluid Theory (SAFT). Fluid Phase Equilib. 2007, 253, 19. (13) Fredenslund, A.; Sorensen, M. J. Group Contribution Estimation Methods. In Models for Thermodynamic and Phase Equilibria Calculations; Sandler, S., Ed.; Marcel Dekker Inc.: New York, 1994. (14) Carnahan, N. F.; Starling, K. E. Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys. 1969, 51, 635. (15) Daubert, T. E.; Danner, R. P. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation; Hemisphere: New York, 2003. (16) Tsonopoulos, C.; Wilson, G. M. High-Temperature Mutual Solubilities of Hydrocarbons and Water. 1. Benzene, Cyclohexane and NormalHexane. AIChE J. 1983, 29, 990. (17) Heidman, J. L.; Tsonopoulos, C.; Brady, C. J.; Wilson, G. M. HighTemperature Mutual Solubilities of Hydrocarbons and Water. 2. Ethylbenzene, Ethylcyclohexane, and Normal-Octane. AIChE J. 1985, 31, 376.

Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5659 (18) Economou, I. G.; Tsonopoulos, C. Associating Models and Mixing Rules in Equations of State for Water/Hydrocarbon Mixtures. Chem. Eng. Sci. 1997, 52, 511. (19) Lee, C. E. K. D.; Jurecka, P.; Tarakeshwar, P.; Hobza, P.; Kim, K. S. Understanding of Assembly Phenomena by Aromatic-Aromatic Interactions: Benzene Dimer and the Substituted Systems. J. Phys. Chem. A 2007, 111, 3446. (20) Karakatsani, E. K.; Spyriouni, T.; Economou, I. G. Extended SAFT Equations of State for Dipolar Fluids. AIChE J. 2005, 51, 2328. (21) Karakatsani, E. K.; Economou, I. G. Perturbed Chain-Statistical Associating Fluid Theory Extended to Dipolar and Quadrupolar Molecular Fluids. J. Phys. Chem. B 2006, 110, 9252. (22) Karakatsani, E. K.; Kontogeorgis, G. M.; Economou, I. G. Evaluation of the Truncated Perturbed Chain-Polar Statistical Associating Fluid Theory to Complex Mixture Phase Equilibria. Ind. Eng. Chem. Res. 2006, 45, 6063. (23) Sørensen, J. M.; Arlt, W. Liquid-liquid equilibrium data collection (Binary Systems); DECHEMA Chemistry Data Series: Frankfurt/Main, Germany, 1995; Vol V, Part 1. (24) Marhold, H.; Waldner, P.; Gamsjager, H. The Phase Diagram of Cyclohexane-Methanol: A Challenge in Chemical Education. Thermochim. Acta 1998, 321, 127. (25) Raal, J. D.; Best, D. A.; Code, R. K. Examination of EthanolHeptane, Methanol-Hexane Systems Using New Vapor-Liquid Equilibrium Still. J. Chem. Eng. Data 1972, 17, 211. (26) Budantseva, L. S.; Lesteva, T. M.; Nemtsov, M. S. Imperfect Systems Formed by Methanol with C5-C8 Hydrocarbons of Various Classes. Zh. Fiz. Khim. 1977, 51, 1048. (27) Marinichev, A. N. S.; Susarev, M. P. Zh. Prikl. Khim. 1965, 38, 1619. Data taken from Gmehling, J.; Onken, U. Vapor-Liquid Equilibrium Data Collection; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, Germany, 1977; Vol. I, Part 2a, 1965. (28) Grenner, A.; Kontogeorgis, G. M.; von Solms, N.; Michelsen, M. L. Application of PC-SAFT to Glycol Containing SystemssPC-SAFT Towards A Predictive Approach. Fluid Phase Equilib. 2007, 261, 248. (29) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284.

(30) Derawi, S. O.; Kontogeorgis, G. M.; Stenby, E. H.; Haugum, T.; Fredheim, A. O. Liquid-Liquid Equilibria for Glycols plus Hydrocarbons: Data and Correlation. J. Chem. Eng. Data 2002, 47, 169. (31) Economou, I. G.; Donohue, M. D. Chemical, Quasi-Chemical and Perturbation Theories for Associating Fluids. AIChE J. 1991, 37, 1875. (32) Panayiotou, C.; Sanchez, I. C. Hydrogen-Bonding in FluidssAn Equation-of-State Approach. J. Phys. Chem. 1991, 95, 10090. (33) Gupta, R. B.; Panayiotou, C. G.; Sanchez, I. C.; Johnston, K. P. Theory of Hydrogen-Bonding in Supercritical Fluids. AIChE J. 1992, 38, 1243. (34) Asprion, N.; Hasse, H.; Maurer, G. FT-IR Spectroscopic Investigations of Hydrogen bonding in Alcohol-Hydrocarbon Solutions. Fluid Phase Equilib. 2001, 186, 1. (35) Fletcher, A. N.; Heller, C. A. Self-Association of Alcohols in Nonpolar Solvents. J. Phys. Chem. 1967, 71, 3742. (36) Lien, T. R. A Study of the Thermodynamic Excess Functions of Alcohol Solutions by IR Spectroscopy. Applications to Chemical Solution Theory. Thesis, University of Toronto, Toronto, Canada, 1972. (37) Luck, W. A. P. A Model of Hydrogen-Bonded Liquids. Angew. Chem., Int. Ed. Engl. 1980, 19, 28. (38) Martinez, S. Methanol/n-Hexane System. I. Infrared studies. Spectrochim. Acta, Part A 1986, 42, 531. (39) Gross, J.; Sadowski, G. Application of the Perturbed-Chain SAFT Equation of State to Associating Systems. Ind. Eng. Chem. Res. 2002, 41 (22), 5510–5515. (40) Grenner, A.; Kontogeorgis, G. M.; von Solms, N.; Michelsen, M. L. Modeling Phase Equilibria of Alkanols with the Simplified PC-SAFT Equation of State and Generalized Pure Compound Parameters. Fluid Phase Equilib. 2007, 258, 83. (41) Matsuda, H.; Ochi, K. Liquid-Liquid Equilibrium Data for Binary Alcohol plus n-Alkane (C-10-C-16) Systems: Methanol plus Decane, Ethanol plus Tetradecane, and Ethanol plus Hexadecane. Fluid Phase Equilib. 2004, 224, 31.

ReceiVed for reView October 15, 2007 ReVised manuscript receiVed March 1, 2008 Accepted March 11, 2008 IE071382L