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Ind. Eng. Chem. Res. 2011, 50, 102–110
Vapor-Liquid Equilibrium Calculations of Aqueous and Nonaqueous Binary Systems Using the Mattedi-Tavares-Castier Equation of State Jose´ P. L. Santos,† Frederico W. Tavares,*,†,‡ and Marcelo Castier§,⊥ Departamento de Engenharia de Engenharia Quı´mica, Escola de Quı´mica, UniVersidade Federal do Rio de Janeiro, AV. Horacio Macedo, 30, CT-Bloco E, 21949-900, Rio de Janeiro, Brazil, Programa de Engenharia Quı´mica (PEQ/COPPE), UniVersidade Federal do Rio de Janeiro, AV. Horacio Macedo, 30, CT-Bloco E, 21949-900, Rio de Janeiro, Brazil, Department of Chemical and Petroleum Engineering, United Arab Emirates UniVersity, P.O. Box 17555, United Arab Emirates
The Mattedi-Castier-Tavares (MTC) equation of state (EOS) is used to model the phase behavior of mixtures whose components interact via strong specific interactions. This is the case of several aqueous and nonaqueous polar mixtures of industrial interest, whose phase behavior may be significantly different from that of ideal mixtures. The MTC EOS is based on the hole-lattice theory, whose partition function is derived from the generalized van der Waals theory. The model uses a region approach to characterize polar molecules and does not introduce terms based on other theories to account for hydrogen bonding. The results of the MTC EOS are generally in very good agreement with experimental results. 1. Introduction Mixtures that contain polar components with strong specific interactions often present phase behavior that is remarkably different from that of ideal solutions. The study of the behavior of such mixtures is of great interest from the industrial and theoretical viewpoints. Examples of mixtures whose modeling represents a theoretical challenge include aqueous solutions of alcohols (e.g., methanol + water and ethanol + water),1 aqueous solutions of alkanolamines (MEA (monoethanolamine), DEA (diethanolamine), MDEA (methyl diethanolamine), AMP (2amino-2-methyl-1-propanol)), and aqueous solutions of glycols (monoethylene glycol (MEG), diethylene glycol (DEG), and triethylene glycol (TEG)). Many of these mixtures are of interest to oil and gas industries, which are often associated with the processing of nonpolar mixtures. For example, aqueous solutions of alkanolamines are used to remove carbon dioxide and hydrogen sulfide from natural gas streams.2-5 Aqueous solutions of glycols are frequently utilized to prevent hydrate formation in the pipelines, to avoid physical damage to equipment.5 Furthermore, knowledge of the thermodynamic behavior of these systems is important in the design of separation equipment. Many models have been developed to address the challenges inherent to strong specific interactions.6 In the past, only models for the excess Gibbs free energy (GE) and models based on quasichemical theories were able to describe the behavior of such mixtures. Cubic equations of state (EOSs) with conventional mixing rules are largely unable to provide reliable correlations and predictions of physical properties for such systems. The use of mixing rules based on GE models in cubic EOSs enhances their ability to describe the behavior of highly nonideal mixtures.7,8 However, this type of hybrid approach * To whom correspondence should be addressed. Tel.: +55-2125627650. Fax: +55-21- 25627650. E-mail:
[email protected]. † Departamento de Engenharia de Engenharia Quı´mica, Escola de Quı´mica, Universidade Federal do Rio de Janeiro. ‡ Programa de Engenharia Quı´mica (PEQ/COPPE), Universidade Federal do Rio de Janeiro. § Department of Chemical and Petroleum Engineering, United Arab Emirates University. ⊥ Currently at Texas A&M University at Qatar, on leave from Universidade Federal do Rio de Janeiro, Brazil.
merges modelssan EOS plus a GE modelswhose original derivations are based on different assumptions at the molecular level. A different approach is to derive models from statistical mechanics that take the effect of strong specific interactions into account and are able to predict large deviations from ideal solution behavior. The statistical association fluid theory (SAFT) EOS9 and its variants (e.g., PC-SAFT10) are successful statistical-mechanical models that have stimulated other developments based on more-sophisticated theories.11 The SAFT model was obtained from perturbation theory as the sum of four Helmholtz function terms that account for hard-sphere repulsive forces, dispersion forces, chain formation, and association. To compute the association contribution in the SAFT and related EOS, it is necessary to solve a set of nonlinear equations. The SAFTfamily EOSs are substantially more complicated than cubic EOSs and require more computational time to evaluate physical properties. An intermediate approach is the cubic-plus-association (CPA) EOS. In this type of model, an association term is added to a conventional cubic EOS, such as the Soave or Peng-Robinson EOS. The cubic EOS accounts for the attractive and repulsive forces and the association term accounts for the formation of hydrogen bonds.12,13 The results of the CPA EOS rival those of the SAFT EOS and its variants but with a smaller computational load. Although these EOS are successful in predicting the phase behavior of polar mixtures, they require the numerical solution of a set of nonlinear equations at each specified condition before computing any thermodynamic property. Specific and efficient algorithms for this exist14 but a completely explicit model capable of accurate correlations and predictions of the phase behavior of polar mixtures would be useful for chemical process design. Other theories have also been used to account for hydrogen bonding, such as, for example, lattice theory.15-17 This work focuses on using the Mattedi-Castier-Tavares (MTC) EOS to model the behavior of mixtures whose components interact via strong specific interactions. The MTC model is based on the hole-lattice theory and its partition function is obtained from the generalized van der Waals theory.18 The MTC EOS does not contain extra terms based on other theories to account for hydrogen bonding and uses a region approach to represent associating molecules. Unlike the SAFT-family EOSs
10.1021/ie100791f 2011 American Chemical Society Published on Web 12/07/2010
Ind. Eng. Chem. Res., Vol. 50, No. 1, 2011
and the CPA EOSs, the model is completely explicit and there are no association equations to solve before computing any property.
(7)
i i
i)1 nc
∑ x zq
zq )
The MTC model is based on the hole-lattice theory, whose partition function is obtained from the generalized van der Waals (GVDW) theory.18,19 The GVDW partition function is the product of an athermal contribution and a residual contribution to the Helmholtz free energy. The athermal contribution, from the Staverman-Guggenheim lattice model, accounts for entropic effects that are due to size and geometry differences. The residual contribution is obtained from the configurational fluid energy, which is based on the multifluid and local composition concepts.20,21 From the partition function or from Helmholtz free energy, we derive thermodynamic properties.22 The compressibility factor (Z) and fugacity coefficient of a component i in a mixture (φˆ i), according to the MTC EOS, are PrV*V˜ V˜ z PrV*V˜ ) ) V˜ r ln + V˜ r × Z) RT kB T V˜ - 1 2 V˜ - 1 + (q/r) V˜ Ψ(q/r) × ln +lV˜ V˜ - 1 + (q/r)
(
[
∑∑
xiνai Qa
i)1 a)1
(Γa - 1) V˜ - 1 + (q/r)Γa
[
∑
]
]
V˜ - 1 + (q/r) Ψ r V˜ - 1 + (q/r)Γa
ng
νQγ ∑ ∑x ν Q ∑ nc
ng
(
a k)1 a)1 k k
e i
e ea
- ri)
a e)1
V˜ - 1 + (q/r)Γa (2)
respectively. In these equations, νai is the number of regions of type a in the molecule of type i, ng the total number of regions, Qa the superficial area of a region of type a, ri the number of segments of component i, V* the molar volume of a cell in the lattice, z the coordination number, and Ψ a characteristic constant of the lattice. We set z ) 10, V* ) 5.0 cm3/mol, and Ψ ) 1. The number of external contacts (zqi), the bulkiness factor (li), and reduced volume (V˜ ) are defined as ng
zqi )
∑ ν zQ a i
a
(3)
a)1
li )
( 2z )(r - q ) - (r - 1) i
i
(4)
V˜ )
V rV*
(5)
i
where V is molar volume. The number of segments (ri) can be obtained assuming that the bulkiness factor is equal to zero (a linear chain); therefore, ri )
(zqi /2) - 1 (z/2)
-1
The average values of r and zq are
ng
∑S
Γa )
m ma
(9)
γ
m)1 nc
∑ν
m m i xiQ
i)1
Sm )
nc
q)
(10)
q ng
∑x ∑ν Q a i
i
a
(11)
a)1
(
uam uam 0 Bam ) 1+ R R T
)
(12)
3. Pure and Binary Parameters of the MTC EOS
]
[
Moreover, we define the following relations:
(1)
V˜ - 1 + (1 - li) × V˜ - 1 + (q/r) ng Ψ(q/r)(qi - ri) V˜ +Ψ + νai Qa × ln V˜ - 1 + (q/r) V˜ - 1 + (q/r) a)1
ln φˆ i ) -ri ln
(8)
i
Here, γma ) exp [ -uma/(RT)] and uma is the interaction energy between regions m and a, which is given as18
]
ng
i
i)1
i)1
)
nc
[
∑xr
r)
2. Model
ln
103
Nc
(6)
We assume that polar molecules are split into three regions: an electron-donor group (R), an electron-acceptor group (β), and a dispersion group (D), as reported by Ehlker and Pfennig23 (shown in Figure 1). An association (R-β) is a contact between an R-region and a β-region. To reduce the number of parameters, all other contacts (R-R, R-D, β-β, β-D, and D-D) are considered to be dispersion interactions. For hydrogen-bond formation, parameter BRβ (eq 12) is assumed to be equal to zero. Consistent with previous assumptions, the parameters of the interaction energies, other than hydrogen-bond formation, are uRD uββ uβD uDD uRR 0 0 0 0 0 ) ) ) ) R R R R R and BRR ) BRD ) Bββ ) BβD ) BDD Therefore, the MTC EOS uses six parameters to characterize pure polar compounds: QD, QR, Qβ, u0Rβ/R, u0DD/R, and BDD. However, for some polar compounds, we set QR ) Qβ, as discussed in the next section, reducing the number of fitted parameters to five. Notice that, for nonpolar substances, there are only three adjustable pure compound parameters in the MTC EOS: QD, u0DD/R, and BDD. The pure-compound parameters of the MTC EOS were fitted from vapor-pressure data generated by the highly accurate DIPPR correlation.24 The parameters were obtained by minimizing the following objective function: N
OF1 )
∑ j)1
(
Pexp - Pcal j j Pexp j
)
2
(13)
where Pexp and Pcal j j are the pseudo-experimental and calculated pressures at the temperature of point j, respectively, and N is the number of pseudo-experimental points. Details are available in Table 1. For mixtures constituted by two polar components, association (R-β) can occur either between molecules of the same
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Ind. Eng. Chem. Res., Vol. 50, No. 1, 2011 Table 2. MTC EOS Pure Compound Parameters
Figure 1. Schematic illustration of polar molecule. Molecule is split into three regions: an electron-donor group (R), an electron-acceptor group (β), and a dispersion group (D). Table 1. Pseudo-Experimental Information To Fit the MTC EOS Parameters compound
temperature range, TminsTmax (K)
N
acetone methanol ethanol MEG DEG TEG MEA MDEA DEA AMP water n-pentane
195.81s352.00 281.90s512.50 308.50s495.32 308.56s720.00 364.15s744.6 398.46s689.99 387.48s657.43 413.00s513.00 324.07s736.60 373.25s436.85 292.84s469.99 229.28s469.70
10 14 11 18 16 12 11 11 19 23 10 15
OF1 9.722 8.235 2.495 7.185 0.110 7.730 3.055 4.878 5.548 2.120 4.543 9.508
× × × ×
10-6 10-2 10-4 10-3
× × × × × × ×
10-3 10-2 10-4 10-2 10-3 10-5 10-3
compounds or between molecules of different compounds. We use a similar simplification to reduce the number of crossinteraction parameters: uR0 2β1 uassoc uR0 1β2 0 ) ) R R R BR1R2 ) BR1D2 ) BD1R2 ) Bβ1β2 ) Bβ1D2 ) BD1β2 ) BD1D2 ) Bdis
and u0R1D2 u0D1R2 u0β1β2 u0β1D2 u0D1β2 u0D1D2 u0dis u0R1R2 ) ) ) ) ) ) ) R R R R R R R R
Indexes “1” and “2” denote the type of molecule. Therefore, there are three cross-interaction parameters to fit: u0assoc/R, Bdis, and u0dis/R. Moreover, for binary mixtures constituted by a nonpolar molecule and a polar component, the parameters that need to fitted are u0D1D2/R ) u0D2R1/R ) u0D2β1/R ) u0dis/R, and Bdis. When using isothermal vapor-liquid equilibrium (VLE) data for parameter fitting, Bdis is set equal to zero. The binary parameters were fitted from VLE data of binary mixtures, by minimizing either objective function: N
N
OF2 )
∑ (y
cal 1i
2 - yexp 1i ) +
∑ i)1
i)1
(
Pcal i
)
2 - Pexp i Pexp i
(14)
or N
OF3 )
∑(
ycal 1i
i)1
N
-
2 yexp 1i
)
+
∑ i)1
(
exp Tcal i - Ti
Texp i
)
2
(15)
where Texp and Tcal i i are the experimental and calculated temexp cal and y1i are, respectively, the peratures, respectively; y1i experimental and calculated mole fractions of component 1 in the vapor phase. For either P-x or T-x data, the first terms on the right-hand side of eqs 14 and 15 are equal to zero, respectively. The optimization methods used to minimize the objective functions (eqs 13 and 14 or eq 15) include particle swarm optimization (PSO),25 followed by the simplex method.26
compound
QD
QR
Qβ
acetone methanol ethanol MEG DEG TEG MEA MDEA DEA AMP water n-pentane
3.4952 1.9591 2.8163 4.9253 3.0895 1.9811 2.8176 4.9247 4.5294 6.3581 1.4427 3.8942
0.0797 0.4280 0.2324 1.4126 1.6635 1.3042 1.0384 1.3882 1.4625 0.6963 0.4019
1.1987 0.3750 0.3603 0.0554 0.8355 1.1467 1.0384 1.3882 1.4625 0.6963 0.3200
assoc Bdis (K) udis /R (K) 〈∆P/P〉 0 /R (K) u0
46.0457 459.620 326.140 175.093 142.570 160.068 227.655 166.778 367.765 170.516 502.640 18.636
-574.157 -311.700 -415.410 -570.810 -570.733 -599.335 -428.709 -537.727 -228.077 -464.587 -512.547 -681.911
-1257.58 0.53 -2480.92 1.41 -2344.10 0.39 -2135.66 2.42 -1994.20 8.70 -2296.77 12.92 -1819.86 4.22 -1470.29 1.85 -2280.77 10.70 -1098.77 0.44 -2591.18 0.49 4.60
The simplex method is widely known and it will not be described here. PSO is an empirical global optimization algorithm inspired by bird flocking and fish schooling. The method does not provide a formal guarantee of finding the global minimum of an objective function, but it does increase the probability of locating it, compared to local optimization methods. It is a direct optimization method, i.e., it does not require derivatives of the objective function, with respect to the unknowns of the optimization problem (the EOS parameters in the case of this work). The solution procedure starts by defining a search region in parameter space and placing many “particles” in this region; their coordinates are the guessed values for the parameters. A systematic procedure, described elsewhere,25 displaces these “particles” inside the search region during the iterative procedure until convergence occurs. The parameters values found by PSO are then used as initial guesses for their final refinement using the simplex method. In this work, the following polar binary mixtures are studied: ethanol + water, methanol + ethanol, acetone + water, n-pentane + acetone, water + monoethylene glycol (MEG), water + monoethanolamine (MEA), water + methyl diethanolamine (MDEA), water + diethanolamine (DEA), and water + 2-amino-2-methyl-1-propanol (AMP). To assess the MTC EOS performance, we define the average relative and absolute deviations as follows:
〈 ∆PP 〉 ) N1 ∑ | 〈 〉
N
Pexp - Pcal i i
i)1
Pexp i
N
∆T 1 ) T N
∑
∆y )
1 N
i)1
|
Texp - Tcal i i Texp i
∑ |y
| |
× 100
(16)
× 100
(17)
|
(18)
N
exp i
i)1
- ycal i
4. Results (a). Pure Compounds. Table 2 shows the fitted MTC EOS pure-compound parameters for all compounds studied in this paper, with their respective relative pressure deviations. To reduce the number of parameters fitted, for each of the alkanolamines (MEA, MDEA, DEA, and AMP) and glycols (MEG, DEG, and TEG), the surfaces areas of the electronacceptor (Qβ) and electron-donor (QR) regions are assumed to be equal. For the other compounds, QR and Qβ are different. These strategies are similar to those used to represent association types between sites to form hydrogen bonds in the SAFT-family and CPA EOS.27 Table 2 shows that ethanol and water have the smallest relative pressure deviations. The alkanolamines MEA and DEA and the glycols DEG and TEG exhibit the highest relative pressure deviations, which result from the poor
Ind. Eng. Chem. Res., Vol. 50, No. 1, 2011
105
Table 3. Binary Parameters of the EOS MTC for Binary Aqueous Solutions of Alcohols and Acetone Methanol (1) + Water (2) System Isothermal VLE temperature (K) 323.15 328.15 333.15
uassoc /R (K) 0
udis 0 /R (K)
Bdis (K)
〈∆P/P〉
∆y
-2160.07
-514.265
312.925
0.47 0.57 0.43
0.0037 0.0067 0.0047
〈∆T/T〉 0.19 0.37 0.43
∆y 0.0128 0.0171 0.0164
Bdis (K)
〈∆P/P〉
∆y
431.313
0.80 0.72 0.87 1.65 3.89 4.84 4.75
0.0061 0.0070 0.0112 0.0110 0.0214 0.0185 0.0127
Bdis (K) 431.313
〈∆T/T〉 0.03 0.12
∆y 0.0113 0.0146
Isobaric VLE pressure (MPa) 0.1013 0.3040 0.5066
uassoc /R 0
(K)
udis 0 /R (K)
Bdis (K)
-2160.07
-514.265
312.925
Ethanol (1) + Water (2) System Figure 2. Vapor-pressure curves of pure compounds generated from DIPPR correlations for ([) methanol, (2) ethanol, (9) acetone, and (b) water. Solid lines represent the MTC EOS.
Isothermal VLE temperature (K) 323.15 328.15 333.15 343.15 423.15 473.15 523.15
uassoc /R 0
(K)
-2764.45
udis 0 /R (K)
-320.901
Isobaric VLE pressure (MPa) 0.0133 0.1013
uassoc /R (K) 0 -2764.45
udis 0 /R (K) -320.901
Acetone (1) + Water (2) System Isothermal VLE temperature (K) uassoc /R (K) 0
Figure 3. Vapor pressures curves of pure compounds generated from DIPPR correlations for ([) MEG, (2) DEG, and (b) TEG. Solid lines represent the MTC EOS.
298.15 303.15 318.15 333.15
udis 0 /R (K)
-1857.01
Bdis (K)
-652.6120 -20.3675
〈∆P/P〉
∆y
4.27 3.19 1.69 3.27
0.0158 0.0173 0.0080 0.0085
Isobaric VLE pressure (MPa) 0.0267 0.0467 0.1013
uassoc /R 0
〈∆T/T〉 ∆y 0.17 0.0146 -652.6120 -20.3675 1.17 0.0506 0.57 0.0222 udis 0 /R (K)
(K)
-1857.01
Bdis (K)
n-Pentane (1) + Acetone (2) System Isothermal VLE temperature (K) 372.7 397.7 422.6
uassoc /R 0
(K)
0.0
udis 0 /R (K)
Bdis (K)
〈∆P/P〉
∆y
-622.859
-81.521
0.83 0.50 1.05
0.0126 0.0111 0.0139
Bdis (K) 218.479
〈∆T/T〉 0.04
∆y 0.0091
Isobaric VLE pressure (MPa) 0.10135 Figure 4. Vapor pressures curves of pure compounds generated from DIPPR correlations for ([) MEA, (2) DEA, (b) MDEA, and (9) AMP. Solid lines represent the MTC EOS.
description of their low vapor-pressure range. However, the MTC EOS describes vapor pressures at higher temperatures very well where we perform the VLE calculations for mixtures. The MTC EOS provides good correlations, compared with vapor pressures generated by the DIPPR correlation. Figures 2-4 summarize the results for pure compounds. Acetone, because of its chemical nature, can be treated either as inert or as an associating compound.13 Here, acetone is treated as an associat-
uassoc /R (K) 0
udis 0 /R (K) -373.383
ing compound. It is split into an electron-donor region (R), an electron-acceptor region (β), and a dispersion region (D), as given by Ehlker and Pfennig.23 The MTC EOS correlation of the acetone vapor pressure, shown in Figure 2, agrees very well with the pseudo-experimental DIPPR data. (b). Aqueous and Nonaqueous Solutions of Alcohols and Acetone. Table 3 presents the MTC EOS binary parameters for aqueous solutions of alcohols and acetone, their average relative pressure deviations, and average absolute vapor-phase mole fraction deviations. The small pressure and mole fraction deviations suggest that the MTC EOS is able to describe these systems. Figure 5 presents VLE diagrams for the ethanol (1) +
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Ind. Eng. Chem. Res., Vol. 50, No. 1, 2011
Figure 5. Ethanol (1) + water (2) system. Experimental data obtained at temperatures of ([) 323.15, (b) 328.15, and (9) 333.15 K (data taken from ref 28). Solid lines represent the MTC EOS.
Figure 6. Ethanol (1) + water (2) system. Experimental data obtained at pressures of (b) 0.0133 and (9) 0.1013 MPa (data taken from ref 29). Solid lines represent the MTC EOS.
water (2) system at 323.15, 328.15, and 333.15 K, for which the MTC EOS gives small pressure and mole fraction deviations (see Table 3). For this system, the MTC EOS binary parameters were simultaneously fitted from sets of isothermal VLE data at 323.15, 328.15, and 333.15 K. Mole fraction and pressure predictions at 343.15, 423.15, 473.15, and 523.15 K are rather good, with larger deviations in pressure occurring at the highest temperatures (see Table 3). The predictive capability of the MTC EOS is demonstrated in Table 3 and Figure 6 in the calculation of isobaric VLE data for the ethanol (1) + water (2) system at pressures of 0.0133 and 0.1013 MPa. Table 5 (presented later in this work) shows pressure and mole fraction deviations obtained from the MTC EOS. For the methanol (1) + water (2) system, the MTC EOS also gives good correlations of the experimental data. Table 3 shows that the relative pressure and composition deviations are very small at 323.15, 328.15, and 333.15 K. The MTC EOS, with the temperature-independent binary parameters fitted using VLE data at 323.15, 328.15, and 333.15 K, correlates very well with the behavior of this system. Figure 7 displays the corresponding VLE diagrams. To test the predictive capability, we have performed calculations under other conditions using the same set of parameters. Table 3 and Figure 8 show that the MTC EOS predicts the isobaric VLE behavior of the methanol (1) + water (2) system at 0.1013, 0.3040, and 0.5066 MPa very well. Table 3 also shows the binary parameters of the MTC EOS for the acetone (1) + water (2) system. The MTC EOS presents
Figure 7. Methanol (1) + water (2) system. Experimental data obtained at temperatures of (9) 323.15, (b) 328.15, and (2) 333.15 K (data taken from ref 28). Solid lines represent the MTC EOS.
Figure 8. Methanol (1) + water (2) system. Experimental data obtained at pressures of (9) 0.1013, (b) 0.3040, and (2) 0.5066 MPa (data taken fron ref 29). Solid lines represent the MTC EOS.
Figure 9. Acetone (1) + water (2) system. Experimental data obtained at temperatures of ([) 298.15, (b) 303.15, (9) 315.15, and (2) 330.15 K (data taken from ref 29). Solid lines represent the MTC EOS.
deviations that are similar in magnitude to those reported in the literature for the acetone (1) + water (2) system.13 At temperatures of 298.15, 303.15, 318.15, and 333.15 K, the MTC EOS describes the entire composition range very well, as Figure 9 shows. Here, the EOS with the same set of binary parameters was able to describe the isobaric VLE behavior of the acetone (1) + water (2) system at pressures of 0.0267, 0.0467, and 0.1013 MPa very well, as shown in Figure 10.
Ind. Eng. Chem. Res., Vol. 50, No. 1, 2011
Figure 10. Acetone (1) + water (2) system. Experimental data obtained at pressures of (2) 0.0267, (b) 0.0467, and ([) 0.1013 MPa (data taken from ref 29). Solid lines represent the MTC EOS.
107
Figure 12. n-Pentane (1) + acetone (2) system. Experimental data obtained at a pressure of (b) 0.101325 MPa (data taken from ref 31). Solid lines represent the MTC EOS. Table 4. Binary Parameters of the MTC EOS for the Methanol (1) + Ethanol (2) System pressure (MPa)
uassoc /R (K) 0
udis 0 /R (K)
Bdis (K)
〈∆T/T〉
0.040 0.053 0.067 0.080
-2473.66
-418.182
272.357
0.08 0.08 0.05 0.06
deviations generally decrease as the temperature increases. At 363.15 K, the vapor pressure of water is accurately predicted and the absolute composition deviations are very small. At lower temperatures, although the pressure deviations are higher, the MTC EOS describes the behavior this system very well, as
Figure 11. n-Pentane (1) + acetone (2) system. Experimental data obtained at temperatures of (9) 372.7, (b) 397.7, and ([) 422.6 K (data taken from ref 30). Solid lines represent the MTC EOS.
Table 3 and Figure 11 show results for the n-pentane (1) + acetone (2) system at 372.7, 397.7, and 422.6 K. Azeotropy occurs at these three temperatures. This phenomenon, which is related to large deviations from ideal behavior, was correlated very well with the MTC EOS, as indicated by the small pressure and mole fraction deviations in Table 3. The pressure deviation is higher at 422.6 K than at the two other temperatures. The most likely reason is that 422.6 K is outside the temperature range used to fit the pure-compound acetone parameters (see Table 1). The MTC EOS results for an isobaric n-pentane (1) + acetone (2) dataset (see Table 3 and Figure 12) are also in excellent agreement with the experimental information. Table 4 shows binary parameters for the methanol (1) + ethanol (2) system. The VLE data used are of type T-x, at pressures of 0.04, 0.053, 0.067, and 0.08 MPa. The relative temperature deviations are very small at all these pressures, as shown in Table 4. However, small discrepancies appear when the compositions of the binary mixtures approach pure methanol, as shown in Figure 13. (c). Aqueous and Nonaqueous Solutions of Alkanolamines and Glycols. Table 5 shows binary parameters for aqueous solutions of alkanolamines and glycols and for the ethanol + MEA system. For the water + MEG system, there are either T-P-x or T-P-x-y data at temperatures of 338.25, 350.85, and 363.15 K. According to Table 5, relative pressure
Figure 13. Methanol (1) + ethanol (2) system. Experimental data obtained at pressures of ([) 0.040, (9) 0.053, (b) 0.067, and (2) 0.080 MPa (data taken from ref 32). Solid lines represent the MTC EOS. Table 5. Binary Parameters of the MTC EOS To Represent Isothermal VLE Data temperature, T (K) uassoc /R (K) 0
udis 0 /R (K)
Bdis (K)
〈∆P/P〉
∆y
4.37 2.68 0.57
0.0050
Water (1) + MEG (2) System 338.65 350.85 363.15
-2013.09
-404.815 472.509
Water (1) + MEA (2) System 351.15 363.15 364.85
-2302.65
-441.252 347.151
3.50 2.19 3.53
Ethanol (1) + MEA (2) System 338.15 348.15 358.15
-2090.43
-427.554 267.932
3.60 2.50 2.70
0.0232
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Ind. Eng. Chem. Res., Vol. 50, No. 1, 2011
Figure 14. Water (1) + MEG (2) system. Experimental data obtained at temperatures of (9) 338.25, (2) 350.85, and (b) 363.15 K (data taken from ref 33). Solid lines represent the MTC EOS.
Figure 16. Ethanol (1) + MEA (2) system. Experimental data obtained at temperatures of ([) 338.15, (9) 348.15, and (2) 358.15 K (data taken from ref 33). Solid lines represent the MTC EOS. Table 6. Binary Parameters of the MTC EOS To Represent Isobaric VLE Data pressure, P (MPa)
uassoc /R (K) 0
udis 0 /R (K)
Bdis (K)
〈∆T/T〉
∆y
0.34 0.57
0.0092 0.0130
Water (1) + MEA (2) System 0.0667 0.1013
-2302.65
0.0400 0.0667
-2052.00
0.00667
-2981.06
-441.252
347.151
MDEA(1) + Water (2) System -404.815
-7.21497
0.20 0.27
Water (1) + DEA (2) System Figure 15. Water (1) + MEA (2) system. Experimental data obtained at temperatures of ([) 351.15, (b) 363.15, and (2) 364.15 K (data taken from refs 33 and 34). Solid lines represent the MTC EOS.
shown in Figure 14. The EOS temperature-independent binary parameters were obtained using experimental data at 338.25 and 350.85 K. The MTC EOS predicts the VLE behavior at 363.15 K very well, showing extrapolation capability. Table 5 also provides a single set of binary parameters for the water + MEA system fitted from data at three different temperatures. The MTC EOS provides a good description of the phase behavior of the entire composition range for this system under the conditions studied. The MTC EOS temperature-independent binary parameters were obtained using experimental VLE data at 351.15 and 364.85 K. At 363.15 K, experimental vapor-phase composition information is available and the MTC EOS is able to predict this new experimental information. Figure 15 shows that the MTC EOS gives good correlation for this system, compared with the experimental data. Here, we also analyze the ethanol (1) + MEA (2) system. Table 5 shows the set of MTC EOS binary parameters for this system and pressure deviations. Figure 16 shows that the MTC EOS results agree with the experimental data. Table 6 presents binary parameters for systems whose data are of types T-x-y and T-x. The MTC EOS provides average relative temperature deviations below 0.34% for all alkanolamine systems, and average absolute mole fraction deviations in the vapor phase below 0.0219. For water + MEA system, the MTC EOS predicts the isobaric VLE behavior very well, using the temperature-independent binary parameters obtained from data obtained at 0.0667 and 0.1013 MPa (see Table 5). Figure 17 shows the isobaric VLE diagram for the water + MEA system. Our EOS is also able to represent the isobaric
-2133.974
-353.554
0.14
0.0219
0.15 0.10 0.18
0.0081 0.0048 0.0056
AMP (1) + Water (2) System 0.0667 0.0800 0.1013
-2063.94
-543.671
134.981
binary for the MDEA + water system very well. From Table 6 and Figure 18, we observe small temperature and composition deviations, compared to the experimental data. For the water + DEA system, the MTC EOS gives good correlations, compared to the experimental data, as shown in Figure 19. The proposed EOS presents also good correlations for the AMP + water system at pressures of 0.0667, 0.0800, and 0.1013 MPa (see Figure 20). We observe that a single set of binary parameters (temperature-independent parameters) is able to describe all the isobaric conditions.
Figure 17. MDEA (1) + water (2) system. Experimental data obtained at pressures of ([) 0.0400 and (9) 0.0667 MPa (data taken from ref 35). Solid lines represent the MTC EOS.
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Acknowledgment The authors acknowledge the financial support of PRH-13/ ANP, FAPERJ, and CNPq. Nomenclature
Figure 18. Water (1) + MEA (2) system. Experimental data obtained at pressures of ([) 0.0667 and (9) 0.1013 MPa (data taken from ref 36). Solid lines represent the MTC EOS.
Figure 19. Water (1) + DEA (2) system. Experimental data obtained at a pressure of (b) 0.00666 MPa (data taken from ref 36). Solid lines represent the MTC EOS.
Figure 20. AMP (1) + water (2) system. Experimental data obtained at pressures of (9) 0.0667, ([) 0.08, and (b) 0.1013 MPa (data taken from ref 37). Solid lines represent the MTC EOS.
5. Conclusions The results show that the Mattedi-Tavares-Castier (MTC) equation of state (EOS) successfully describes highly polar mixtures and also presents a good predictive capacity. Moreover, the idea of splitting a polar molecule into three regions can be considered to be a good strategy to represent polar molecules. This approach allows a very good representation of the phase behavior of highly polar mixtures without the need to set up and solve a system of nonlinear equations to account for association, which is necessary in the statistical association fluid theory (SAFT)-family and cubic-plus-association (CPA) EOSs.
nc ) number of components ng ) number of regions N ) number of experimental points P ) pressure Qa ) surface area of a group of type a ri ) number of segments of component i R ) universal gas constant kB ) Boltzmann’s constant T ) temperature (K) Tmin ) minimum temperature (K) Tmax ) maximum temperature (K) uma ) molar interaction energy between groups of types m and a u0ma ) temperature-independent molar interaction energy between groups of types m and a Sm ) area fraction of group of type m on a void-free basis V* ) volume occupied by one mol of lattice cells (cm3/mol) V* ) volume of a lattice cell (cm3) V˜ ) reduced molar volume V ) total volume x ) mole fraction in liquid phase y ) mole fraction in vapor phase Z ) compressibility factor z ) lattice coordination number zq ) mean number of nearest neighbors OF ) objective functions, defined by eqs 13, 14, and 15 〈 · 〉 ) percentage Greek Letters R ) electron donor β ) electron acceptor γma ) exponential term in the expression of the most probable distribution Γa ) auxiliary symbol defined as eq 9 ∆P/P ) relative pressure deviation, defined as eq 16 ∆T/T ) relative temperature deviation, defined as eq 17 ∆y ) absolute mole fraction, defined as eq 18 νia ) number of groups of type a in the molecule of type i φ ) fugacity coefficient Ψ ) characteristic constant of the lattice Superscripts cal ) calculated * ) hard core exp ) experimental D ) dispersive interactions dis ) dispersive assoc ) association ∧ ) mixture Subscripts i ) refers to component i j ) refers to component j
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ReceiVed for reView April 1, 2010 ReVised manuscript receiVed October 21, 2010 Accepted October 25, 2010 IE100791F
