Ind. Eng. Chem. Res. 2010, 49, 7085–7092
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Phase Behavior Modeling of Alkyl-Amine + Water Mixtures and Prediction of Alkane Solubilities in Alkanolamine Aqueous Solutions with Group Contribution with Association Equation of State Francisco A. Sa´nchez,† Ticiana M. Soria,† Selva Pereda,*,† Amir H. Mohammadi,‡ Dominique Richon,‡ and Esteban A. Brignole† Planta Piloto de Ingenierı´a Quı´mica-PLAPIQUI, UniVersidad Nacional del Sur, CONICET, Camino La Carrindanga Km 7-CC 717, Bahı´a Blanca, Argentina, and MINES ParisTech, CEP/TEP-Centre Energe´tique et Proce´de´s, 35 Rue Saint Honore´, 77305 Fontainebleau, France
Precise knowledge of phase behavior of natural gas + aqueous alkanolamine solutions is of interest to design and optimize amine absorption units. In this work, we extend a group contribution with association equation of state to the modeling of phase behavior of mixtures containing water, alkanolamine, and light hydrocarbons (HC). The model parameters were estimated on the basis of binary vapor-liquid equilibrium and liquid-liquid equilibrium data of the following binary systems: HC + amines, HC + alcohol, water + amine, water + alcohol. The group contribution approach allows the use of an extended databank of experimental data which gives a solid foundation for predicting hydrocarbon solubility in aqueous alkanolamine solutions. 1. Introduction Aqueous solutions of alkanolamines are commonly used in the natural gas industry to strip acid gases (carbon dioxide and hydrogen sulfide) from natural gas and hydrocarbon liquids in order to avoid corrosion problems during transportation, and processing. A tool to predict phase behavior is therefore important for designing and optimizing amine absorption units. Moreover, to estimate hydrocarbon losses in stripping processes it is necessary to accurately predict the solubility of hydrocarbons in aqueous alkanolamine solutions. Association and solvation effects have a major role in the properties of pure compounds and mixtures that can form hydrogen bonds. The group contribution with association equation of state GCA-EOS1–3 explicitly takes into account these strong and highly directed attractive forces in its group contribution association term, which is based on Wertheim’s theory4 as applied in the statistical associating fluid theory (SAFT) equation.5 To describe the mixtures under study in this work, the following associating groups are needed: the alcohol group (OH) with two associating sites (2B), the water group (H2O) with four associating sites (4C), and the amine groups (NHx) with only one electronegative site (1A) that cannot self-associate but can cross-associate with H2O and alcohols. The effect of self-association is normally dominant in determining the value of the activity coefficient at infinite dilution. In a previous work,6 we showed that disregarding the self-association of the amine group does not hamper the predictions of vapor-liquid equilibria at low amine concentrations in amine + paraffin and amine + alcohol binary systems. These results are in agreement with the claim of Funke et al.7 that the self-association effect in alkyl amines is 2-3 orders of magnitude lower than in alcohols (characteristics association equilibrium constants are Kamin < 1 and 100 < Kalcohol < 1000). * To whom correspondence should be addressed. E-mail: spereda@ plapiqui.edu.ar. † Universidad Nacional del Sur. ‡ CEP/TEP-Centre Energe´tique et Proce´de´s.
Most amines show partial miscibility with water and generally have a lower critical end point at low temperature. Hydrogen bonding in this solvating system enhances miscibility. For example, water and triethylamine are completely miscible below 291 K. However, above 291-292 K, the increase of thermal energy breaks the hydrogen bonds and the immiscibility region expands with increasing temperature.8 To our knowledge, only Kaarsholm et al.9 and Grenner et 10 al. have modeled alkylamine + water systems using thermodynamic models that explicitly account for association. Kaarsholm et al.9 used the cubic plus association equation of state (CPA). The authors considered that primary and secondary amines self-associate and that tertiary amines do not. They also compare different combining rules for the cross-association parameters. The conclusion is that the overall performance of CPA for water-amines is not entirely satisfactory even if an additional binary parameter is included in the cross-association term. On the other hand, Grenner et al.10 modeled water + diethylamine and water + pyridine binaries with the nonrandom hydrogen bonding theory (NRHB), which is a lattice model, and compared the results with simplified perturbed chainstatistical associating fluid theory (sPC-SAFT). The authors concluded that both models fail to predict and hardly correlate the binaries under discussion (in contrast with the excellent performance that they achieve for many other systems). Quoting these authors in reference to amine + water binaries, “the prediction of phase equilibria in such mixtures is always a challenging task in the literature”. In a previous work, Sa´nchez et al.6 extended GCA-EoS to amine + paraffin and amine + alcohol; the authors showed that the model has good predictive capability for these systems. On the other hand, Soria et al.11 revised the model parameters for water, alcohol, and paraffin; also in this case, the model shows a good predictive capability. In this work, the GCA-EoS is extended to amine + water binary systems, and using the same set of parameters from this and previous works, the GCA-EoS is applied to predict phase behavior of alkanolamines + alkane + water solutions.
10.1021/ie100421m 2010 American Chemical Society Published on Web 07/01/2010
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Table 1. GCA-EoS Binary Energy Interaction Parameters of Amines with H2O Group j group i
kij*
kij′
Rij
Rji
CH2NH2 CHNH2 CH2NH CH2N
1.212 1.269 1.264 1.450
0 0 -0.023 -0.095
2.5 2.5 1.5 3.5
2.5 2.5 2.0 1.3
Table 2. GCA-EoS Cross-Association Parameters of Amines with Water Electropositive Site associating group
ε (K)
κ (cm3 mol-1)
NH2 NH N
2531 2761 2794
0.1346 0.0900 0.0460
2. Thermodynamic Modeling The residual Helmholtz energy is described by three contributions in GCA-EoS: repulsive, attractive, and associative.2 Model equations and detailed description of each contribution are given in the Appendix. The free volume or repulsive term is described by the Mansoori et al.12 expression for hard spheres and has a single parameter, the critical diameter, dc. In the case of molecular description of compounds (for example: water, methanol, methylamine), this parameter is calculated by fulfilling the pure compound critical point conditions and null derivatives of pressure with respect to volume in this point (dP/dV ) d2P/ dV2 ) 0), as in classic cubic equations of state. In the case of compounds described by group contribution, dc in general is adjusted so that the model reproduces a given vapor pressure data point. None of these two procedures can be used for low volatile compounds, since neither critical properties nor vapor pressure experimental data are generally known. Bottini et al13 proposed to use infinite dilution activity coefficients experimental data to develop a correlation for dc of high molecular weight compounds (e.g., vegetable oils) by means of their van der Waals molecular volume. The association contribution, according to Wertheim’s theory,4 requires the estimation of the energy and volume of association (ε and κ). For self-associating compounds like water or alcohols these parameters were estimated on the basis of spectroscopic data.11 In this work, we consider that amines do
not self-associate, and only cross associate with those components that have electropositive sites. Given that primary, secondary, and tertiary amines present different chemical behavior, the cross association parameters for each family of amines is different and was estimated on the basis of binary vapor-liquid and liquid-liquid equilibrium data with water, similarly as it was performed with amines-alcohol crossassociation.6 Finally, the attractive term of the GCA-EoS Helmholtz energy is a group contribution version of the nonrandom two liquid (NRTL) model;14 in this case, the group energy (gii), the binary energy interaction (kij), and nonrandomness (Rij and Rji) parameters are required. All the groups used in this work are available in the model table of parameters; therefore, here, only binary interaction parameters were fitted for amine groups with water. Tables 1 and 2 report, respectively, GCA-EoS binary energy interaction and cross-association parameters adjusted in this work. Parameters for the attractive and association contribution for alcohol-water, alcohol-paraffin and paraffin-water binaries are from Soria.11 Moreover, parameters for amine-paraffin and amine-alcohols are from Sa´nchez et al.6 Binary Alkylamine + Water Systems. Table 3 summarizes GCA-EoS correlation of vapor-liquid (VLE) and liquid-liquid equilibrium (LLE) experimental data of water with primary, secondary, and tertiary amines. The table reports experimental data temperature and pressure range, number of points, source, and the relative deviations in the correlation. Similarly, Table 4 shows GCA-EoS predictive capability for several binary water + amine systems that were not included in the parametrization procedure. Figure 1 compares selected vapor-liquid equilibrium data of primary amines + water binary systems. As can be observed, GCA-EoS accurately predicts by group contribution (same set of parameters) the change of phase behavior between ethyl-, propyl-, and butyl-amine with water. Figure 2 depicts vapor-liquid equilibria of a diamine with water. GCA-EoS predicts the phase behavior in a wide temperature range. Regarding secondary amines, Figures 3 and 4 show the simultaneous simulation of vapor-liquid and liquid-liquid equilibria of di-n-propylamine and ethyl-n-butylamine with water, respectively. These two amines are configurational
Table 3. GCA-EoS Correlation of Water + Amine Binary Systems (VLE and LLE)a Vapor-Liquid Equilibria compound 1
2
T (K)
water
EDA
326-343
diethylamine di-n-propylamine
water water
330 283
methyldiethylamine triethylamine
water water
320 273
P (kPa)
∆z%b
∆y1%
no. data points
ref
0.1T
0.8
13
15
1.7P 1.9P
0.9
11 9
16 17
3P 2.7P
1.4 1.1
22 9
18 19
Primary Amine 13 Secondary Amines 52-104 1.5-2 Tertiary Amines 23-56 2-3
Liquid-Liquid Equilibria compound 1
2
T (K)
P (kPa)
AAD 2 in 1
ARD% 2 in 1
no. data points
ref
di-n-propylamine triethylamine
water water
301-326 308-333
101 101
0.08 0.03
15 19
4 3
20 20
Abbreviations: AAD ) average absolute deviation, ARD% ) percent average relative deviation. isothermal VLE data and in temperature for isobaric VLE. a
b
∆z% corresponds to errors in pressure for
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Table 4. GCA-EoS Prediction of Water + Amine Binary Systems (VLE and LLE)
a
Vapor-Liquid Equilibria compound 1
2
T (K)
P (kPa)
∆z%b
∆y1%
no. data points
ref
4.3
10 10 8 16 16 13
21 21 22 23 24 25
0.6
22 69 30 25
16 26 27 17
8.3P 5.2P 1.9P 8.9P 0.4T
3.5 1.2 5.2
24 17 18 30 8
18 28 19 28 29
0.3T 2.9P 0.4T
1.5 1.2 -
160 52 40
Primary Amines + Water etylamine
water
n-propylamine i-propylamine n-butylamine s-butylamine
water water water water
287-354 286-354 325-348 308-367 350-369 344-364
diethylamine methyl-n-butylamine water di-n-propylamine
water water EBA water
311,322 283-313 283-313 293-313
methyldiethylamine
water
triethylamine
water
308 313 283, 291 278, 288 350-371
water water water
EDA MAPA MEDA
325-461 333-373 324-389
1.1T 0.8T 0.4T 0.3T 0.5T 0.2T
80 93 101 101 101 101
1.9 1.6 17 5.5
Secondary Amines + Water 2.0P 3.7P 3.5P 4.1P
19-79 115 5-11 3-13
Tertiary Amines + Water 1-7 8-42 4-8 1-7 101 Diamines + Water 26-655 9-100 13-101
15, 30, 31 32 33
Liquid-Liquid Equilibria compound 1 water di-n-propylamine methyldiethylamine triethylamine
2 EBA water water water
T (K) 293-355 271-347 313-350 291-333
P (kPa) 101 101 101 101
AAD 2 in 1 0.15(30) 0.012(74) 0.01(42) 0.02(43)
ARD% 2 in 1 -3
1.5 × 10 (17) 0.06(10) 0.17(31) 0.34(71)
no. data points
ref
11 12 9 7
20, 27 34 20 20
a Abbreviations: AAD ) average absolute deviation, ARD% ) percent average relative deviation. b ∆z% corresponds to average relative deviation in pressure, P, for isothermal VLE data and in temperature, T, for isobaric.
Figure 1. Vapor-liquid equilibria of alkylamine (1) + water (2) binary system. Experimental data: (() ethylamine at 93.3 kPa;21 (9) propylamine at 101.3 kPa;22 (+) n-butylamine at 101.3 kPa.24 Solid lines: GCA-EoS prediction.
Figure 2. Vapor-liquid equilibria of water (1) + ethylenediamine (2) binary system. Experimental data:15,30,31 ()) 53.3 kPa, (+) 101.3 kPa, (4) 273 kPa, (9) 464 kPa and (b) 655 kPa. Solid lines: GCA-EoS prediction.
isomers, which present practically the same vapor-liquid equilibria behavior, while the mutual solubility with water of di-n-propylamine is lower than the respective with ethyl-nbutylamine. For a group contribution approach, these compounds are indistinguishable. Consequently, the attractive and association contributions are identical. However, as it was mentioned
in the previous section, the repulsive contribution of the model is a molecular term. The model accurately predicts the VLE phase behavior of both compounds with water (Figures 3a and 4a). In the case of LLE the model correlation for di-npropylamine is accurate with a good agreement of the lower critical solution temperature (LCST). On the other hand, GCA-
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Figure 3. Binary di-n-propylamine (1) + water (2) system. (a) Vapor-liquid equilibria. Experimental data17 at (() 283.15, (+) 293.15, (b) 303.15, and (×) 313.15 K. (b) Liquid-liquid equilibria at 101 kPa: (symbols (9,()) experimental data;20,34 (dashed lines) GCA-EoS correlation; (solid lines) GCA-EoS predictions; (dotted line) predicted VLLE.
Figure 4. Binary ethyl-n-butylamine (1) + water (2) system. (a) Vapor-liquid equilibria: experimental data27 at (+) 283.15, (]) 303.15, and (×) 313.15 K. (b) Liquid-liquid equilibria at 101 kPa: (symbols (0,O)) experimental data;20,27 (solid lines) GCA-EoS prediction; (dotted line) predicted VLLE.
EoS adequately predicts the increase of water solubility in ethyln-butylamine; however, it underestimates the shrinking of the heterogeneous region. Finally, for tertiary amines, Figure 5 shows the results for triethylamine + water binary system, the LLE experimental data is part of the information included in the databank for model correlation, while VLE is predicted. The binary interaction parameters between the water and the amine groups (see Table 1) are rather high. In general, GCAEoS parameters are near the value of 1.3,6,11 Kaarsholm et al.9 also required high value interaction parameters for modeling this system with CPA. Although, they considered self-association for primary and secondary amines (not considered in our approach) the authors could not achieve a satisfactory correlation. Amines present unusual phase behavior. For instance, amines with four carbon atoms, like n-butylamine, do not present LLE with water, in contrast with the equivalent alcohol with four carbons, 1-butanol. Moreover, higher molecular weight amines that do depict LLE with water, present a high temperature LCEP, and there exists a wide temperature range of complete mutual solubility of water + amines before the solid phase appearance. Therefore, it is this high mutual solubility,
even in the presence of paraffinic groups, which requires high binary interaction parameters between water and amine groups. Hydrocarbon Solubility in Aqueous Alkanolamine Solutions. The alkanolamine molecules were assembled by group contribution; therefore, no extra parameters are required for the association or attractive term. The same set of binary energy interaction and association parameters fitted to binary data of alcohol, water, amines and hydrocarbon are used for these compounds. Figure 6 shows their group contribution description together with the active sites for association. Regarding the GCA-EoS repulsive contribution, since alkanolamines are low-volatile compounds, they required some procedure to adequately determine their free-volume contribution (dc parameter). As it was already mentioned, Bottini et al.13,35 developed a method to estimate the critical diameter of vegetable oils on the basis of infinite dilution activity coefficient. Unfortunately, this type of information is not available for alkanolamines. Later, Pereda et al.36 proposed the use of molar volume data at 298 K in combination with Bottini’s correlation35 to calculate dc for modeling gas solubility in ionic liquids (no fitting procedure). The molar volume is a natural choice to quantify the dc of compounds, as this property will take into
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Table 6. GCA-EoS Prediction of Alkanolamine + Water Binary Systems compound 1
2
T (K)
water MDEA 313-373 water DEA 311-450 365-473 water MEA 283-373 362-471 298-364
P (kPa) 7-101 6.7 0.6-1560 1-101 67, 101 0.2-69
∆z%a ∆y1% P
5.3 2.6T 16P 19P 2.8T 31P
0.02 43 3.8 10.6
no. data points
ref
39 8 76 163 27 62
32 39 40, 41 32, 42 39 43, 44
a ∆z% corresponds to average relative deviation in pressure, P, for isothermal VLE data and in temperature, T, for isobaric VLE.
Figure 5. Vapor-liquid and liquid-liquid equilibria of the triethylamine (1) + water (2) binary system at 101 kPa: (symbols) experimental data;20,29 (dashed line) GCA-EoS correlation; (solid line) GCA-EoS prediction; (dotted line) GCA-EoS prediction of liquid-liquid-vapor equilibria.
Figure 7. Vapor-liquid equilibria of the system water (1) + DEA (2). Experimental data40,41 at (×) 365.15, (b) 373.15, and (() 473.15 K. Lines: GCA-EoS prediction. Figure 6. Representation of the dispersive groups in alkanolamines and active sites for association: (a) monoethanolamine (MEA), (b) methyl diethanol amine (MDEA), and (c) diethanolamine (DEA). Table 5. Alkanolamines Critical Diameter and Deviation in Pure Compound Vapor Pressure Prediction alkanolamine
Rk
Tc (K)
dc (cm mol-1)
Tc source
MDEA DEA MEA
4.944 4.290 2.573
741.9 736.6 678.2
4.8894 4.5380 3.7426
37 38 38
account the different spatial configuration that a vegetable oil or an alkanolamine will have according to the polarity of their building groups. Table 5 reports the value of Tc and dc for the alkanolamines modeled in this work. Using this procedure to calculate dc, GCA-EoS predicts the pure alkanolamine vapor pressure with an average absolute deviation of 4, 0.2, and 0.1 kPa for MEA, MDEA, and DEA, respectively, in the temperature range 350-473 K (typical absorbers operating range). Table 6 shows GCA-EoS prediction of the binary water + alkanolamine systems. In some cases relative deviations in pressure are important. Figure 7 shows model prediction of vapor-liquid equilibria of the binary water + DEA system. Regarding hydrocarbon + alkanolamine binary systems, to our knowledge, only data on liquid-liquid equilibria of n-heptane + MEA has been measured.45 GCA-EoS cannot predict the mutual solubility of this binary system; it predicts 1 order of magnitude lower mutual solubility in both phases (average absolute deviation of 2.45 × 10-3 and average relative deviation of 93.5%). On the basis of previous results,6,11 the prediction of hydrocarbon solubility in aqueous alkanolamine solutions (AAS) is a big challenge. In general, there is at least 1 order of
Figure 8. Solubility of n-butane in aqueous MDEA solution. Experimental data:46,51,54 (O) Pure water; ()) 25 wt % MDEA; (4) 35 wt % MDEA; (0) 50 wt % MDEA. Solid lines: GCA-EoS prediction.
magnitude higher solubility in the AAS than in pure water. Moreover, the temperature dependence of this solubility when alkanolamine is present in the solution is much more important than in pure water. Except for the data measured by Mokuori et al.,46 all the data found in the literature reported AAS composition in molarity units. Excess volume information published by Zu´n˜iga-Moreno et al.47 and Lee et al.48 was used to convert it into molar fraction.
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Table 7. GCA-EoS Prediction of Hydrocarbon Solubility in Aqueous Alkanolamine Solutionsa hydrocarbon solubility hydrocarbon
alkanolamine concentration
T (K)
P (MPa)
AAD
ARD %
no. of points
ref
2.9 × 10-4 1.2 × 10-4 1.6 × 10-4 7.1 × 10-5 1.1 × 10-4 1.9 × 10-5
22% 11% 22% 12% 34% 14%
51 49 34 58 3 19
46, 49 46, 50 50 46, 51 46 52
21% 40% 9.9%
37 11 43
51 53, 50 52
22% 14% 17% 28% 22% 10% 12% 26% 33% 52%
10 6 21 4 18 6 12 5 8 4
46 46 46, 51 51 46, 52 46 46 46 46 46
Vapor-Liquid Equilibria ethane propane n-butane
25-50 wt % MDEA 5-35 wt % DEA 18 wt % MEA 25-50 wt % MDEA 35 wt % DEA 35 wt % MDEA
283-403 283-398 298-398 273-423 298 298-423
35 wt % MDEA 0-100 wt % MEA 35 wt % MDEA
273-348 313 298-423
25, 50 wt % MDEA 35 wt % DEA 25-50 wt % MDEA 35 wt % DEA 25-50 wt % MDEA 35 wt % DEA 25, 50 wt % MDEA 35 wt % DEA 25, 50 wt % MDEA 35 wt % DEA
283-303 283-305 273-371 298-333 298-423 298-343 298-343 298-343 298-353 298-353
0.1-13 0.2-13 0.1-13 0.1-19 0.5-0.8 0.1-0.5
Liquid-Liquid Equilibria propane n-butane
1-19.6 1724 0.4-21
1.3 × 10-4 1.9 × 10-3 4.0 × 10-5
Vapor-Liquid-Liquid Equilibria ethane propane n-butane n-pentane n-hexane a
3-6 3-4.9 0.5-4.4 0.9-2.1 0.2-4.2 0.5-0.8 0.5 0.5 0.5 0.5
4.1 × 10-4 2.3 × 10-4 1.3 × 10-4 1.3 × 10-4 6.4 × 10-5 1.3 × 10-5 1.2 × 10-5 1.1 × 10-5 2.2 × 10-5 9.8 × 10-6
Abbreviations: AAD ) average absolute deviation, ARD% ) percent average relative deviation.
previous work for water + hydrocarbon, amine + hydrocarbon, and amine + alcohol, the phase behavior of aqueous alkanolamine solutions was predicted by a group contribution approach. GCA-EoS is able to predict the extremely low solubility of hydrocarbons in aqueous solutions of DEA, MDEA, and MEA. Acknowledgment The financial support by the ECOS-SUD program that provided the opportunity for this joint work is gratefully acknowledged. List of Symbols
Figure 9. Solubility of hydrocarbons (C2 to n-C6) in a 35 wt % DEA aqueous solution versus carbon number. Experimental data:46 (]) 298, (×) 313, (4) 333 K. Lines: GCA-EoS prediction.
Table 7 summarizes GCA-EoS predictions for hydrocarbon solubility in AAS. The model is able to predict with very low absolute deviation the tiny amount of hydrocarbon in AAS. It is important to highlight the model’s excellent accuracy to predict hydrocarbon solubility in pure water,11 giving a good basis for the model predictive capability in AAS. Figures 8 and 9 depict model accuracy for n-butane solubility in aqueous MDEA solution and solubility of hydrocarbons from C2 up to n-C6 in 35 wt % aqueous DEA solution, respectively. Interestingly, the model is also able to predict the change in temperature dependence of the solubility as alkanolamine concentration increases. 3. Conclusions GCA-EoS was applied to modeling phase equilibria of water + alkylamine, water + alkanolamine, and solubility of hydrocarbons in aqueous alkanolamine solutions. After correlation of model parameters, it showed a good predictive capability for alkylamine + water systems studied in this work. On the basis of the parameters fitted in this work for water + amine and the
AAD ) absolute average deviation ) 1/N∑i|Zi,exp - Zi,calc| AAS ) aqueous alkanolamine solution. ARD ) absolute relative deviation ) 1/N∑i|1 - Zi,calc/Zi,exp| dc ) critical diameter of rigid sphere. DEA ) diethanolamine EDA ) ethylenediamine EBA ) ethyl-n-butylamine gii ) pure group energy parameter kij ) binary interaction parameter LLE ) liquid-liquid equilibria MAPA ) 3-methylamino-n-propylamine MEDA ) methylethylenediamine MEA ) monoethanolamine MDEA ) methyldiethanolamine P ) pressure Rk ) van der Waals reduced volume T ) temperature V ) total volume VLE ) vapor-liquid equilibria x ) mole fraction in liquid phase y ) mole fraction in vapor phase z ) auxiliary variable Greek Characters Rij, Rji ) nonrandomness parameters ∆z ) relative deviation in variable z ε ) association energy κ ) association volume
Group-Contribution with Association Equation of State (GCA-EoS). The GCA-EoS1–3 is based on the group contribution expression for the configurational Helmholtz function, Ac. All ther-
Ind. Eng. Chem. Res., Vol. 49, No. 15, 2010
modynamic phase equilibrium properties may be derived from Ac by differentiation with respect to composition or volume. The Helmholtz energy is considered as composed of two parts, the first describes the ideal gas behavior, Aideal, and the second part takes into account the intermolecular forces, which can be evaluated by a repulsive or free volume term, Afv, a contribution from attractive intermolecular forces, Aatt, and an associative term,Aassoc: A ) Aideal + (Afv + Aatt + Aassoc)
(A/RT)fv ) 3(λ1λ2 /λ3)(Y - 1) + (λ23 /λ32)(Y2 - Y - ln Y) + n ln Y (A.2) with
(
πλ3 6V
Y) 1-
)
attraction energy parameter for interactions between groups i and j, and Rij is the NRTL nonrandomness parameter. The interactions between unlike groups are calculated from gij ) kij(giigjj)1/2
∑nd
k
(A.4)
j j
j
where ni is the number of moles of component i, NC stands for the number of components, V represents the total volume, R stands for universal gas constant, and T is temperature. The following generalized expression is assumed for the hard sphere diameter temperature dependence: di ) 1.065655dci{1 - 0.12 exp[-2Tci /(3T)]}
[
kij ) k*ij 1 + kij′ ln
(A.5)
att
(A/RT)
z )2
∑n ∑ i
i)1
∑
2T T*i + T*j
)]
(A.13)
where g*jj is the interaction parameter for reference temperature T*i . The Helmholtz function due to association is calculated with a modified form of the expression used in the SAFT equation, and is formulated in terms of associating groups: Aassoc ) RT
[
∑ n* ∑ (ln X
NGA
i
i)1
Mi
(k,i)
-
k)1
)
]
X(k,i) 1 + Mi (A.14) 2 2
where NGA represents the number of associating groups, ni is the total number of moles of associating group i, X(k,i) stands for the mole fraction of group i not bonded at site k,and Mi is the number of associating sites assigned to group i. The number of moles of the associating group is NC
n*j )
∑ν
(i,m) assocnm
(A.15)
(i,m) where νassoc represents the number of associating group i in molecule m and nm is the total number of moles of molecules m; the summation includes all the NC components in the mixture. The mole fraction of group i not bonded at site k is determined by
NGA Mj
NG
NG
(
m)1
where dc is the value of the hard sphere diameter at the critical temperature, Tc, for the pure component. For the evaluation of the attractive contribution to the Helmholtz energy, a group contribution version of a densitydependent NRTL-type expression14 is derived:
NC
(A.12)
and
NC
λk )
(A.11)
+ gjj′ (T/T*j - 1) + gjj′′ ln(T/T*)] gjj ) g*[1 jj j
-1
(A.3)
(kij ) kji)
with the following temperature dependencies for the interaction parameters:
(A.1)
The free volume contribution is modeled by assuming a hard sphere behavior for the molecules, characterizing each substance i by a hard sphere diameter di. A Carnahan-Starling12 type of hard-sphere expression for mixtures is adopted:
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X(k,i) ) [1 +
θkgkjq˜τkj /(RTV)
k)1 νijqj
j)1
∑θτ
∆(k,i,l,j)]-1
(l,j)
j
(A.16)
j)1 l)1
(A.6)
NG
∑ ∑FX
X(k,i) depends on the molar density of the associating group j, Fj ) n*j /V, and on the association strength between site k of group i and site l of group j:
l lj
l)1
where
∆(k,i,l,j) ) κ(k,i,l,j)[exp(ε(k,i,l,j) /kT) - 1]
(A.17)
NC
θj ) (qj /q˜)
∑nν
i i j
(A.7)
The associating strength is a function of the temperature and characteristic association parameters ε (association energy) and κ (association volume, cm3 · mol-1).
(A.8)
Literature Cited
i
NC
q˜ )
NG
∑n ∑νq
i j j
i
i
j
τij ) exp(Rij∆gijq˜ /(RTV))
(A.9)
∆gij ) gij - gjj
(A.10)
where z is the number of nearest neighbors to any segment (set to 10), NG is the number of groups, νji is the number of groups type j in molecule i, qj stands for the number of surface segments assigned to group j, θk represents the surface fraction of group k, q˜ is the total number of surface segments, gij stands for the
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ReceiVed for reView February 25, 2010 ReVised manuscript receiVed May 10, 2010 Accepted June 7, 2010 IE100421M
