8416
Ind. Eng. Chem. Res. 2008, 47, 8416–8422
Thermodynamic Consistency Test of Vapor-Liquid Equilibrium Data for Mixtures Containing Ionic Liquids Jose´ O. Valderrama,*,† Alfonso Rea´tegui,‡ and Wilson W. Sanga§ UniVersity of La Serena, Faculty of Engineering, Department of Mechanical Engineering, Casilla 554, La Serena, Chile, Centro de Informacio´n Tecnolo´gica, Casilla 724, La Serena - Chile, and UniVersidad Nacional del Altiplano, Ciudad UniVersitaria, AV. Floral 1153, Puno-Peru´
Vapor-liquid equilibrium of eight binary mixtures containing an ionic liquid and carbon dioxide have been tested for thermodynamic consistency. A method proposed by one of the authors has been modified to be applied to this special situation in which the gas phase contains practically one component (carbon dioxide) while the liquid phase contains both components in a wide concentration range. The method is based on the Gibbs-Duhem equation and on appropriate combination of equations of state, mixing rules, and combining rules. The Peng-Robinson equation of state with the Wong-Sandler mixing rules including the Van Laar model for the excess Gibbs free energy required in the mixing rules, are used. The experimental data were obtained from literature sources, and the adjustable parameters were found by minimizing the errors between predicted and experimental bubble pressure. It is shown that the proposed modified consistency test is accurate enough to decide about the thermodynamic consistency of this type of data. Introduction Ionic liquids are organic salts with some special characteristics that make them highly interesting for many applications, such as media for clean liquid-liquid extraction processes1 and as solvents for several types of reactions.2-8 Some ionic liquids can be used as biocatalysts with great advantages compared to conventional organic solvents9,10 and some others have liquid crystal and lubricant properties.11,12 Research work has also been done on the development of separation methods for solute recovery from ionic liquids,13 on extractions from azeotropic mixtures,14 and on hydrogen purification using room-temperature ionic liquids.15 Phase equilibrium data of mixtures containing ionic liquids are necessary for further development of some separation processes involving ionic liquids. On this line of research, Aki et al.,16 presented high-pressure phase equilibrium data of CO2 + ionic liquid mixtures. For all of the ionic liquids studied by these authors, large quantities of CO2 dissolve in the ionic liquid phase, but no appreciable amount of ionic liquid is found in the CO2 phase. In addition, the liquid phase volume expansion with the introduction of even large amounts of CO2 is negligible, in dramatic contrast to the large volume expansion observed for neutral organic liquids. Other studies on the phase equilibrium of mixtures containing a high pressure gas and an ionic liquid have been reported in the literature by Shariati and Peters,17 Lee and Outcalt,18 Kumelan et al.,19 and Shiflett and Yokozeki,20,21 among others. As in other areas of phase equilibrium thermodynamics, the data reported by several authors on mixtures containing an ionic liquid are rarely analyzed from the point of view of their thermodynamic consistency, that is, a method to discriminate among published data to use the correct ones in applications such as modeling, simulation, and design. Valderrama and co* To whom correspondence should be addressed. E-mail: jvalderr@ userena.cl. † University of La Serena, Faculty of Engineering, Department of Mechanical Engineering. ‡ Centro de Informacio´n Tecnolo´gica. § Universidad Nacional del Altiplano, Ciudad Universitaria.
workers have previously presented and applied a sound method to test the thermodynamic consistency of high pressure gas-liquid and gas-solid equilibrium data.22-24 The method is based on the Gibbs-Duhem equation, on the fundamental equation of phase equilibrium and on an appropriate combination between equations of state (EoS), mixing rules, and combining rules. Although the problem is different in some aspects, the treatment of mixtures containing an ionic liquid and high pressure carbon dioxide (including the near-critical region), present some similarities with those cases treated by the authors in previous publications: (i) the vapor-phase non-idealities can be important and a good model to evaluate the fugacity coefficients, φi, is needed; (ii) for isothermal data, the term involving the residual enthalpy (HR) vanishes; and (iii) the concentration of one of the components in one of the phases, the ionic liquid in the gas phase, is very low. Because of this last problem, the concentration of carbon dioxide is close to 1.0, but cannot be considered as a pure gas. For these reasons, the classical differential or integral methods described in standard books25-27 are not applicable. Therefore, a method specially designed for determining the thermodynamic consistency of high pressure mixtures containing an ionic liquid is proposed in this work. It should also be mentioned that in the case of gas + ionic liquid mixtures, only PTx equilibrium data are commonly available and, therefore, experimental data for only the liquid phase are used in the proposed test. This is fundamentally different from previous works by the authors who proposed a consistency test for PTy data.22,28 The thermodynamic relationship that is frequently used to analyze thermodynamic consistency of experimental phase equilibrium data is the fundamental Gibbs-Duhem equation.26 This equation, as usually presented in the literature, relates the activity coefficients of all components in a given mixture. If the equation is not obeyed, the data is declared to be thermodynamically inconsistent. The ways in which the Gibbs-Duhem equation is arranged and applied to the experimental data have given origin to several Consistency Test Methods, most of them designed for low-pressure data. Among these are the Slope Test, the Integral Test, the Differential Test and the Tangent-Intercept
10.1021/ie800763x CCC: $40.75 2008 American Chemical Society Published on Web 10/04/2008
Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8417
Test. Good reviews of these methods are found in the literature.25,27 Similar to the Van Ness-Byer-Gibbs test,29 the consistency method proposed in this work can be considered as a modeling procedure because a thermodynamic model that can accurately fit the experimental data must be used to apply the consistency test. It should be pointed out that, although the consistency method is model-dependent, good fitting does not guarantee that the data are consistent. The proposed area test, derived from the Gibbs-Duhem equation must also be fulfilled. The fitting of the experimental data requires the calculation of some model parameters using a defined objective function that must be optimized.30 The applications done in this work consider isothermal vapor liquid equilibrium data for eight CO2 + ionic liquid mixtures, including 16 isotherms and a total of 120 PTx data points. The pressure ranges from 1 to 13.2 MPa, the temperature from 313 to 333 K, and the mole fractions of carbon dioxide in the liquid phase from 0.06 to 0.8. All of the data were taken from the same literature source.16 Development of Equations The Gibbs-Duhem equation for a binary homogeneous liquid mixture at constant temperature can be written as follows:26
[ ]
VE dP ) x1d(Ln γ1) + x2d(Lnγ2) RT
(1)
being x1 and x2 the mole fraction of components 1 and 2 in the liquid phase, respectively. Equation 1 can be written in terms of the fugacity coefficients, as follows:31
[ (Z -P 1) ]dP ) ∑ x d(Ln φ ) i
[ Z -P 1 ]dP ) x dLn φ +(1 - x )dLn φ 2
2
(3)
1
This equation can be conveniently expressed in integral form, as follows: (1 - x2)
∫ Px1 dP ) ∫ (Z -11)φ d φ +∫ x (Z - 1)φ dφ 2
2
1
2
2
(4)
1
In this equation, P is the system pressure, x2 is mole fraction of the dissolved gas in the liquid mixture, φ1 and φ2 are the fugacity coefficients of components 1 and 2 in the liquid phase mixture, respectively, and Z is the compressibility factor of the liquid mixture. The properties φ1, φ2, and Z can be calculated using an appropriate equation of state and suitable mixing rules. The limits of the integrals are defined by the range of the experimental data. In eq 4, the left-hand side is designated by AP and the righthand side by Aφ, as follows: AP )
∫ Px1 dP
(5)
2
Aφ )
(1 - x2)
∫ (Z -11)φ d φ +∫ x (Z - 1)φ dφ 2
2
1
2
[
|%∆A| ) 100Σ
|Aφ-AP| AP
]
(7) i
The maximum value accepted for this deviation has been discussed by Valderrama and Alvarez,22 although further analysis and discussion on the particular application being analyzed is presented in a separate section. In eq 5, AP is determined using the experimental P - x data at a fixed temperature, while a thermodynamic model, such as an equation of state, is employed to evaluate Aφ in eq 6. If the data are adequately correlated, meaning that the deviations in the calculated pressure are within acceptable margins of deviations and the individual area deviation %∆Ai are within defined margins of errors, then the data set is considered to be consistent. The deviation in the calculated bubble pressure for each point “i” is defined as follows:
[
]
Pcal - Pexp (8) Pexp i To evaluate the integrals in eq 6, the following must be defined: (i) an equation of state; (ii) a set of mixing rules; and (iii) a set of combining rules. In principle, any appropriate equation of state and any mixing and combining rules can be used to evaluate the pressure. It must be mentioned that for a set of N experimental data points, (N - 1) AP areas and (N 1) Aφ areas are calculated for a given mixture at a fixed temperature. %∆Pi ) 100
∑
(2)
i
For the situation of interest in this work, a mixture of an ionic liquid with a dissolved high pressure gas, this equation can be more conveniently written in terms of the properties of the dissolved gas in the liquid mixture. If the dissolved gas is component 2 in the binary mixture, then the above equation becomes: 2
Thus, if a set of data is considered to be consistent, then AP should be equal to Aφ within acceptable defined deviations. To set the margins of errors, an overall percent area deviation |%∆A| between experimental and calculated areas is defined as follows:
1
(6)
Vapor-Liquid Equilibrium Calculations Few reports are available on correlating high pressure VLE of mixtures containing ionic liquids. Simple equation of state models such as Redlich-Kwong or Peng-Robinson with van der Waals-type mixing rules give variable deviations.17,15 Deviations as high as 60% are found in some cases, errors that are not acceptable for the consistency purposes of this work. Our research group has been successfully applying the Peng-Robinson EoS with the Wong-Sandler mixing rules to model complex high pressure mixtures and to perform consistency tests.22-24 Although, some limitations of the Wong-Sandler mixing rule have been pointed out in the literature,32-34 the results of our previous work have clearly demonstrated that the Wong-Sandler mixing rule combined with the van Laar model (to express the excess Gibbs free energy included in the mixing rule), has the accuracy and necessary flexibility to correlate phase equilibrium variables in high-pressure systems. Therefore, we have chosen this combination, designated as PR/WS/VL, to perform the proposed consistency test. The fugacity coefficient is calculated from standard thermodynamic relations as follows:35 RT ln(φi) )
∫
∞
V
[( ) ∂P ∂nj
T,V,n
]
RT dV - RT ln Z V
(9)
The Peng-Robinson equation of state (PR) can be expressed as follows:36 P)
a RT V - b V(V + b) + b(V - b)
(10)
8418 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008
In this eqn., ND is the number of points in the experimental data set.
a ) 0.457235(R2T2C ⁄PC)R(Tr) b ) 0.077796(RTC ⁄PC) R(Tr) ) [1 + F(1 - T r0.5)]2
Consistency Criterion
F ) 0.7464 + 1.54226ω - 0.26992ω2 (11) For mixtures: P)
am RT V - bm V(V + bm) + bm(V - bm)
(12)
In this equation, am and bm are the equation of state constants to be calculated using defined mixing rules. The Wong-Sandler mixing rules (WS) for the Peng-Robinson EoS that are used in this work, can be summarized as follows:37 N
N
i
j
∑ ∑ x x (b - RTa ) i j
bm )
N
1-
∑ i
ij
(13)
A∞E (x)
xiai biRT ΩRT
(b - RTa ) ) 21 [b + b ] - √RT (1 - k ) aiaj
ij
i
(
am ) bm
N
∑ i
j
ij
xiai A∞E (x) + bi Ω
)
(14)
(15)
In these equations, kij is an interaction parameter, Ω ) E 0.34657 for the Peng-Robinson EoS, and A ∞ (x) is calculated E E assuming that A∞ (x) ≈ Ao (x) ≈ gOE. In this work, gOE has been calculated using the Van Laar model. The Van Laar model (VL) for a binary mixture is described by the following equation: (L12 ⁄ RT)x1x2 gEo ) RT x1(L12 ⁄ L21) + x2
(16)
The expressions for the fugacity coefficient using the PR equation with the above-described WS mixing and combination rules can be obtained from eq 12. The problem is then reduced here to determine the parameters L12 and L21 in the Van Laar model and the k12 parameter included in the combining rule for (b - a/RT)12, using available high pressure vapor-liquid equilibrium data for the mixtures of interest. The optimum parameters are those that make minimum the difference between calculated pressure Pcal and experimental pressure Pexp. This difference is expressed through an objective function W, defined as follows: ND
W)
|
100 Pcalc - Pexp ND i)1 Pexp
∑
|
(17) i
Although the concept of consistency seems to be different from consistency tests applied to low pressure vapor-liquid equilibrium data, the situation is conceptually the same. At low pressures, the equilibrium equation is applied and the activity coefficients are determined and after the modeling the GibbsDuhem equation is applied. At high pressures, the equilibrium equation is applied and the fugacity coefficients are determined to then apply the Gibbs-Duhem equation. Since the gas phase has low concentration of the ionic liquid, and it is rarely measured, it seems more appropriate in this case to use the experimental data of the liquid phase to do the consistency analysis. Therefore, at high pressures, an appropriate model to evaluate the fugacity coefficients of each component in the liquid mixture and the compressibility factor of the mixture are needed. As explained above, the PR/WS/VL model is used to evaluate these properties. The model is accepted and the consistency test is then applied if the average absolute pressure deviations %∆P, defined by eq 8 is below 10%. After the model is found appropriate, it is required that the average absolute deviations in the individual areas AP and Aφ, defined by eqs 7 and 8, are below 20% to declare the data as being thermodynamically consistent. The data are considered to be thermodynamically inconsistent (TI) if the deviations in correlating the equilibrium bubble pressure are within the established limits but the individual deviations in the areas are outside the established limits, for more than 25% of the data points in the data set. The test cannot be applied if the equilibrium pressure is not well correlated, that means if deviations in the calculated pressure (eq 17) are greater than 10%. If the deviations in correlating the equilibrium pressure are within the established limits ((10%) but the individual deviations in the areas are outside the established limits for less than 25% of the points, then the data are considered to be not fully consistent (NFC). Data Selection Data for eight CO2+ ionic liquid mixtures presented by Aki et al.,16 have been considered in this study. The mixtures are carbon dioxide with each of the following ionic liquids: [bmim] [NO3], [bmim] [BF4], [bmim] [DCA], [bmim] [TfO], [bmim] [methide, [bmim] [Tf2N], [hmim] [Tf2N], and [omim] [Tf2N]. Table 1 shows the basic properties of the fluid substances involved in the study. In the Table, M is the molecular mass, Tc is the critical temperature, Pc is the critical pressure, Vc is the critical volume, and ω is the acentric factor. The data for carbon dioxide were obtained from the DIPPR database,38 while the values for the ionic liquids (critical properties and acentric factors) were taken from the literature.39,40
Table 1. Properties of the Substances Involved in the Vapor-Liquid Calculations compound CO2 [bmim] [bmim] [bmim] [bmim] [bmim] [bmim] [hmim] [omim]
[NO3] [BF4] [DCA] [TfO] [methide] [Tf2N] [Tf2N] [Tf2N]
IUPAC name
formula
M
TC (K)
PC (MPa)
ω
carbon dioxide 1-butyl-3-methylimidazolium nitrate 1-butyl-3-methylimidazolium tetrafluoroborate 1-butyl-3-methylimidiazolium dicyanamide. 1-butyl-3-methylimidiazolium triflate. 1-butyl-3-methylimidiazolium tris(trifluoromethylsulfonyl)methide 1-butyl-3-methylimidiazolium bis(trifluoromethylsulfonyl)imide 1-hexyl-3-methylimidiazolium bis(trifluoromethylsulfonyl)imide 1-octyl-3-methylimidiazolium bis(trifluoromethylsulfonyl)imide
CO2 C8H15N3O2 C8H15N2BF4 C10H15N5 C12H13N2O3SF3 C12H15 F9N2O6S3 C10H15N3F6S2O4 C12H19N3F6S2O4 C14H23N3F6S2O4
44.0 201.1 226.0 205.1 322.0 550.0 419.3 447.3 475.4
304.2 946.3 632.3 1035.8 1158.0 1571.4 1265.0 1287.3 1311.9
7.38 2.73 2.04 2.44 2.90 2.40 2.76 2.39 2.10
0.2240 0.6039 0.8489 0.8419 0.4118 0.1320 0.2656 0.3539 0.4453
Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8419 Table 2. Details on the Phase Equilibrium Data of Aki et al., the Eight Systems Considered in This Studya
16
for
range of data no.
CO2+
1
[bmim] [NO3]
2
[bmim] [BF4]
3
[bmim] [DCA]
4
[bmim] [TfO]
5
[bmim] [methide]
6
[bmim] [Tf2N]
7
[hmim] [Tf2N]
8
[omim] [Tf2N]
a
T (K)
P(MPa)
xCO2
313 333 313 333 313 333 313 333 313 333 313 333 313 333 313 333
1.2-9.4 1.3-8.9 1.2-8.4 1.4-8.5 1.4-11.6 1.8-11.5 1.0-8.9 1.5-9.8 1.8-11.5 1.70-11.2 1.3-13.2 1.3-13.1 1.3-1.2 1.70-11.2 1.5-11.5 1.60-11.5
0.10-0.48 0.06-0.42 0.13-0.53 0.09-0.45 0.15-0.57 0.15-0.53 0.10-0.59 0.10-0.54 0.31-0.78 0.27-0.75 0.23-0.76 0.17-0.72 0.25-0.76 0.19-0.72 0.27-0.80 0.24-0.76
The temperature values have been rounded to the closest integer.
Table 2 gives details on the selected experimental vapor-liquid equilibrium data for the eight mixtures studied. As seen in the table, data for 16 isotherms were considered. The temperatures for which data are available are 313 and 333 K, while the pressure ranges from 1 to 13.2 MPa. Bubble pressure calculations for binary mixtures were performed using the PR/WS/ VL model. The adjustable parameters of the model (L12, L21, and k12), were determined by optimization of the objective function given by eq 17. To evaluate the integral in eqs 5 and 6 for a set of ND experimental points, two consecutive data points are used, obtaining ND - 1 values of the integrals. To evaluate AP (eq 5) only the experimental data are used, while the PR/WS/VL model is used to evaluate the fugacity coefficients and the compressibility factor included in Aφ (eq 6). Areas and the Limits of Deviations. The margins of acceptable error for the bubble pressure and for the calculated areas ((10% for the calculated pressure and ( 20% for the areas) need to be explained in more detail, as follows. The percentages defined for the consistency criterion are based on information presented in the literature related to the accuracy of experimental data in this type of mixture (for AP) and on the errors in the calculated pressure (for Aφ). Deviations found with the same model used here (PR/ WS/VL) for complex mixtures at high pressure have shown that this model has the required flexibility to correlate VLE data with low deviations.23,24 This proved to be equally applicable to the cases of mixtures containing high pressure CO2 and an ionic liquid, as the results show. To define the margins of errors for the calculated areas, error propagation on the measured experimental data and on the calculated data has been performed as elsewhere explained by the authors.24 The authors did bubble pressure calculations and therefore they considered pressure, temperature, and gas phase concentration as the independent variables. In the cases considered in this study, the gas phase concentration is unknown and therefore error propagation is done considering the pressure, temperature, and liquid phase concentration as the independent variables. This is done as follows: EA )
∂A ∆T [ ∂P ]∆P + [ ∂A∂x ]∆x + [ ∂A ∂T ]
(18)
For the errors in AP, which is determined using only experimental data, maximum uncertainties of ∆P ) 0.1 bar for the experimental pressure, ∆T ) 0.1 K for the experimental temperature, and ∆x
Figure 1. Error in the area Aφ (eq 18) for a deviation of 10% in calculated pressure for two mixtures s [bmim][NO3] + CO2 and --- [omim][ Tf2N] + CO2. Table 3. Calculated Parameters in the Wong-sandler Mixing Rules, Pressure Deviations for the Mixtures, And Consistency Tests for All the Systems Studied mixture, CO2+ T (K) ND [bmim][NO3] [bmim][BF4] [bmim][DCA] [bmim][TfO] [bmim][methide] [bmim][Tf2N]
[hmim][ Tf2N]
[omim][ Tf2N]
313 333 313 333 313 333 313 333 313 333 313 333 333 313 313 333 313 333
6 6 8 7 8 8 8 7 8 8 8 6 8 6 8 8 8 8
k12
A21
A12
0.4500 0.0130 0.7097 0.4428 0.5600 0.5970 0.2590 0.1088 0.4506 0.3359 0.3273 0.1939 0.2195 0.2852 0.3049 0.1749 0.2856 0.2235
2.3300 1.6466 1.5552 2.8820 0.2600 0.0362 1.6988 1.6990 -0.1330 -0.5001 -0.2908 -0.2706 -0.4558 -0.3102 -0.5142 0.0014 -0.4206 -0.7641
0.5600 0.8440 0.4364 0.9144 0.0800 -0.9678 0.2545 0.2686 -3.6926 -2.8230 -0.2014 -0.3778 -1.0200 -0.5609 -1.6027 -0.0028 -0.2604 -0.4761
|%∆P| |%∆A| result 6.5 2.7 4.1 3.8 6.8 2.9 3.5 4.1 2.8 1.1 2.1 3.6 3.6 1.3 1.4 1.5 1.8 1.1
22.5 14.7 22.0 19.5 32.3 14.1 19.6 22.0 18.2 12.1 22.9 17.6 18.9 9.1 13.7 12.4 15.2 10.5
NFC TC NFC TC TI TC TC NFC TC TC NFC TC TC TC TC TC TC TC
) 0.001 for the experimental mole fraction were considered.41 The error EA for the experimental area AP is small, considering these small uncertainties (EP below 1%). For the errors in Aφ, the only calculated variable is the pressure and therefore maximum uncertainties are determined by the model accuracy (accepted up 10% for ∆P in eq 18). The results are similar to those previously found by the authors for other systems.42 An example of these calculations is presented in Figure 1. The error in the area Aφ (eq 18) for an error of 10% in calculated pressure (0.1Pcal) for two mixtures are presented. As seen in the figure, maximum deviations in the calculated areas up to 24% are found for errors in the pressure up to 10% (∆P ) 0.1Pcal). Results are similar for all other systems. Considering all of these results, it is established in the consistency test method that an error of 10% for the absolute pressure in the modeling step and 20% in the areas (eq 7), is acceptable for consistency purposes. Results and Discussion Table 3 presents the results of the consistency test for all the mixtures considered in this study. As seen in the table, 13 of the 18 data sets were found to be thermodynamically consistent (TC), four sets were found to be not fully consistent (NFC), and one single set was found to be thermodynamically inconsistent (TI). For the isotherms declared to be NFC, [bmim][NO3] + CO2, [bmim][BF4] + CO2 and [bmim][Tf2N] + CO2, all at 313 K, and
8420 Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008
Figure 2. Absolute pressure deviations between calculated and experimental values for three systems: s [bmim] [NO3]; --- [bmim] [DCA]; and - · - · - · [omim] [Tf2N]. (a) at 313 K and (b) at 333 K.
[bmim][TfO] + CO2 at 333 K, the modeling gave high pressure deviations for a single point, resulting average pressure deviations higher than the established limit of 10%. If that point is not considered in the analysis, then the remaining set gave pressure deviations and area deviations within the established limits. Therefore, the original set of data is declared to be NFC. It should be mentioned, however, that care must be taken with this, since the defined percentages have a frontier character only and the explanatory character of the eliminated data depends on the system, on the pressure and on the temperature of the data. For the case considered to be thermodynamically inconsistent, [bmim][DCA] + CO2, the average pressure deviation is not very high (6.8%) but there are three points with pressure deviations higher than 10% and area deviations much higher than the established limits (20%), with an average area deviation of 32.3%. A graphical description of the results is shown in Figures 2 and 3. Results for three representative systems at 313 and 333 K are shown: [bmim][NO3], [bmim][DCA] and [omim][Tf2N]. Figure 2 shows the absolute pressure deviation that varies between the margins established to accept the modeling for further analysis. For the three systems shown in the figure, and for all other cases studied, absolute deviations are below 10%, indicating the good accuracy of the PR/WS/VL model used in correlating the data. This fact, however, does not guarantee consistency of the data in the way defined in this work, as shown in Table 3. Conclusions A reasonable and flexible method to test the thermodynamic consistency of high-pressure phase equilibrium data of binary
Figure 3. Absolute area deviations for three systems: s [bmim] [NO3]; --[bmim] [DCA], - · - · - · - [omim] [Tf2N]. (a) at 313 K and (b) at 333 K.
mixtures containing an ionic liquid and a high pressure gas has been presented. On the basis of these results, the following three main conclusions can be drawn: (i) the proposed consistency test method allows to globally analyze vapor-liquid equilibrium data of mixtures containing an ionic liquid; (ii) the numerical technique used to find the optimum model parameters for the PR/WS/VL model shows to be efficient and accurate for modeling the solubility; and (iii) the method gives an answer about consistency or inconsistency of a set of experimental PTx data for all cases that are well correlated by a thermodynamic model. Acknowledgment The authors thank the Direction of Research of the University of La Serena for permanent support, the Center for Technological Information of La Serena-Chile for use of its library and computer facilities, and the National Council for Scientific and Technological Research (CONICYT), for its research grant FONDECYT 1070025. Nomenclature Symbols a ) Force constant in the PR equation of state am ) Force constant for a mixture A∞E(x) ) Helmholtz free energy at infinite pressure AoE(x) ) Helmholtz free energy at low pressure AP ) Integral for point x2i to x2i+1 using P - y experimental data Aφ ) Integral for point x2i to x2i+1 using a thermodynamic model %∆Ai ) Individual percent area deviation
Ind. Eng. Chem. Res., Vol. 47, No. 21, 2008 8421 %∆Aave ) Average relative percent area deviation for the (N-1) areas |%∆Aave| ) Average absolute percent area deviation for the (N-1) areas b ) Volume constant in the PR equation of state bm ) Volume constant for a mixture d ) Derivative operator EA ) Error in the calculated areas (eq 18) F ) Acentric factor parameter for the PR equation of state goE ) Gibbs free energy at low pressure HR ) Residual enthalpy kij ) Binary interaction parameters for the force constant in an EoS k12 ) Interaction parameters for the force constant in an EoS for a binary mixture L12, L21 ) Parameters in the van Laar model Ln ) Natural logarithm ni ) Number of moles of component “i” nj ) Number of moles of component “j” N ) Number of components in the mixture ND ) Number of data points in a data set P ) Pressure %∆Pi ) Percent deviation in the system pressure for a point “i” Pc ) Critical pressure of component R ) Ideal gas constant T ) Temperature Tc ) Critical temperature Tr ) Reduce temperature (Tr ) T/Tc) V ) Volume VE ) Excess volume W ) Objective function x ) Mole fraction in the liquid phase xi, xj ) Mole fraction of components i and j in the liquid phase x1, x2 ) Mole fraction of components 1 and 2 in the liquid phase Z ) Compressibility factor (Z ) PV/RT) AbbreViations VLE ) Liquid-Vapor equlibrium EoS ) Equation of state eq ) Equation NFC ) Not Fully Consistent PR ) Peng-Robinson EoS TC ) Thermodynamically Consistent TI ) Thermodynamically Inconsistent VL ) Van Laar WS ) Wong-Sandler Greek letters R ) Temperature function for the PR equation of state γ ) Activity coefficient ∆ ) Deviation |%∆A| ) Absolute deviation between AP and Aφ (eqn. 7) ∂ ) Partial derivative φ ) Fugacity coefficient ω ) Acentric factor Ω ) Constant in the WS mixing rule. Ω)0.34657 for the PR EoS Super/subscripts cal ) Calculated exp ) Experimental gas ) Gas i, j ) Component i or j
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ReceiVed for reView May 12, 2008 ReVised manuscript receiVed June 29, 2008 Accepted August 4, 2008 IE800763X
