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Equation of state dependency of thermodynamic consistency methods. Application to solubility data of gases in ionic liquids
Fluid Phase Equilibria 449 (2017) 76e82
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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Equation of state dependency of thermodynamic consistency methods. Application to solubility data of gases in ionic liquids O. Valderrama a, b, *, Claudio A. Faúndez c, Joaquín F. Díaz-Valde s c Jose a
Univ. of La Serena, Fac. of Engineering, Dept. of Mech. Eng., Casilla 554, La Serena, Chile n Tecnolo gica, Monsen ~ or Subercaseaux 667, La Serena, Chile Centro de Informacio c ticas, Casilla 160-C, Concepcio n, Fac. de Ciencias Físicas y Matema n, Chile Univ. de Concepcio b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 March 2017 Received in revised form 10 June 2017 Accepted 15 June 2017 Available online 17 June 2017
A thermodynamic method to check the consistency of phase equilibrium data developed by the authors and applied to several situations by different researchers is analyzed to determine the effect of the equation of state employed to correlate the data on the final response of the consistency test. The consistency test method, in fact, uses an equation of state to correlate the equilibrium data and to calculate some properties such as volume, fugacity and compressibility factor. After these properties are determined the Gibbs-Duhem equation is applied to determine data consistency. If the data fulfill the Gibbs-Duhem equation the data are considered to be consistent and if the data do not fulfill de GibbsDuhem equation the data are of doubtful thermodynamic consistency. The equation of state dependency of the method has been recognized by the authors and other researchers but discussion of the real effects of the equation of state used on the results of the consistency method has not been presented. This work demonstrates that after a good and accurate thermodynamic model is used to correlate the data, the consistency test can be considered equation-of-state independent. © 2017 Elsevier B.V. All rights reserved.
Keywords: Ionic-liquids Thermodynamic consistency Equation of state Gas solubility
1. Introduction During the last several years the authors have developed and applied a thermodynamic test to check the consistency of phase equilibrium data presented in the literature for many different types of systems. Cases studied by the authors include PTxy, PTy and PTx data, being P the pressure, T the temperature, y the gas phase mole fraction, and x the liquid mole fraction. The first work of the authors [1] analyzed gas-liquid mixtures at supercritical conditions. After that, several applications to solid-gas mixtures, gaswater systems and more recently to gas þ ionic liquid mixtures have been reported in the literature [2e6]. Initially and for simple mixtures, common cubic equations of state were successfully employed, such as the Peng-Robinson equation with classical mixing and combining rules. However, in several cases the equation of state cannot accurately correlate the phase equilibrium data and, therefore, the consistency test cannot be applied. This because the application of the consistency test
* Corresponding author. Univ. of La Serena, Fac. of Engineering, Dept. of Mech. Eng., Casilla 554, La Serena, Chile. E-mail address: [email protected] (J.O. Valderrama). http://dx.doi.org/10.1016/j.fluid.2017.06.013 0378-3812/© 2017 Elsevier B.V. All rights reserved.
needs values of the compressibility factor (Z) and of the fugacity coefficients (4) of the components in the mixture. As known, when equations of state are applied to mixtures, it is necessary to introduce mixing and combining rules to account for the concentration effect on the equation of state parameters. Several models for mixing and combining rules have been discussed in the literature [7] being the proposal of Kwak and Mansoori an interesting approach that has been several times explored [8]. Kwak and Mansoori state that van der Waals mixing rules were proposed for temperature-independent parameters and not for temperaturedependent parameters, as commonly done when cubic equations of state are applied to mixtures. Thus, they rewrote the PengRobinson expression obtaining an equation of state that includes three temperature-independent parameters: cm, bm and dm [9]. The three EoS parameters are expressed using the classical van der Waals mixing rules, each one including an adjustable parameter. The inclusion of the Kwak-Mansoori proposal has not been done for three parameter equations of state, a method that is also presented in this work. In particular, in this paper, the ValderramaPatel-Teja (VPT) equation of state, a generalization of the PatelTeja equation proposed by Valderrama in 1990 is employed for modeling the phase equilibrium data [10]. The VPT equation of state contains three parameters: two volume-type parameters (b and c)
J.O. Valderrama et al. / Fluid Phase Equilibria 449 (2017) 76e82
77
that are temperature independent and a force-type parameter (a) which is temperature dependent. When the VPT equation is transformed following the concepts of Kwak and Mansoori, a new parameter appears (d) but the force parameter becomes temperature independent. In this way, the VPT þ Kwak-Mansoori expression fulfills with a basic requirement of the van der Waals model. This means that the parameters of the EoS are constant and not functions of the temperature. One aspect that has been mentioned in the past by the authors is that their consistency test, as it happens with all consistency test methods, is model dependent. In the method analyzed in this paper, the model is the equation of state chosen for correlating the data and for calculating the properties needed in the consistency method (Z and 4). In a series of paper by the authors and by other researchers it is implicitly assumed that once the set of data is well represented by an equation of state, the results of the test are independent of the equation of state employed. However, this aspect has not been clearly proved and constitutes the novelty of this paper: to demonstrate that once an equation of state accurately correlates the phase equilibrium data, the consistency test is meaningful and the results about consistency or inconsistency of a set of experimental data can be accepted with confidence.
deviations [1]. To analyze this aspect of the equation of state influence on the consistency test results, this paper considers two models: the Peng-Robinson (PR) and the Valderrama-Patel-Teja (VPT) equations, both with the approach of Kwak and Mansoori for the equation of state parameters, and for the mixing and combining rules. Table 1 summarizes these models. In Table 1, xi is the mole fraction of component “i” in either phase: xi ¼ xi when the equations are applied to the liquid phase and xi ¼ yi when the equations are applied to the gas phase. Also, the PR/KM model contains three adjustable parameters (kij, bij, dij), one for each of the constants: kij for am, bij for bm, and dij for dm. The VPT/KM model contains four adjustable parameters (kij, lij, mij, nij), one for each of the constants: kij for am, lij for bm, mij for cm and nij for dm. It is also assumed that these adjustable parameters are the same for both the liquid and the gas phases. Additionally, the model requires the critical temperature Tc the critical pressure Pc, the acentric factor u and the critical compressibility factor for each component in the mixture. The average absolute deviations in the calculated bubble pressure in the binary system for each point “i” are used to determine the accuracy of the PR/KM and VPT/KM models. This deviation is defined as:
2. Thermodynamic analysis
cal exp 100 X P P i j%DPj ¼ N Pexp
The consistency test is based on the Gibbs-Duhem equation and has been presented with details in the literature [11]. In the test, the equation of state method is used for phase equilibrium correlation and prediction in complex systems. In this method, the same equation is used for the liquid and gas phases to apply the fundamental equation of phase equilibrium. That is the equality of fugacities of a given component in both phases, f Li ¼ f G i . In terms of the fugacity coefficients 4i, the equality of fugacities becomes L G xi 4Li ¼ yi 4G i . To evaluate the fugacity coefficients 4i and 4i an equation of state is used. To describe the concentration dependency of the equation of state parameters, appropriate mixing and combining rules are included. Therefore, as stated in the Introduction, if the consistency test give similar results when different equations of state are used to model the phase equilibrium data, the consistency test can be considered equation-of-state independent. This is the main and novel objective of this work. As detailed in the several papers published by the authors, the consistency test for isothermal data is summarized in an expression derived from the Gibbs-Duhem equation [1]:
Z
1 dP ¼ Px1
Z
1 d4 þ ðZ1Þ41 1
Z
ð1x1 Þ d42 x1 ðZ1Þ42
(1)
For simplicity of the explanation that follows the left-hand side is denoted by AP and the right-hand side is denoted by A4.
Z AP ¼ Z A4 ¼
1 dP Px1 1 d4 þ ðZ1Þ41 1
Z
ð1x1 Þ d42 x1 ðZ1Þ42
The thermodynamic consistency test can be applied if these deviations, expressed as j%DPj are within pre-established ranges. Once the model is accepted, the consistency test model can be applied [11,12]. The equations that define the consistency test have been published in previous papers, so they are not repeated here. These are summarized in Eq. (1) in which (N-1) values for the area AP (Eq. (2)) and (N-1) values for the area A4 (Eq. (3)) are calculated if a set of N experimental data points at constant temperature is available [11]. One should notice that is in the calculations of A4 (Eq. (3)) where the equation of state plays its role. As observed in Eq. (3) the integral term A4 includes the compressibility factor (Z) and the fugacity coefficient (4) both determined using an equation of state. Accurate equations of state will give similar values of Z and 4 and the areas A4 should be the same. Simple equations of state with simple mixing rules cannot accurately calculate these values of Z and 4, the areas A4 will have different erroneous values, and the conclusion of the test may not be the correct one. As mentioned above, if a set of data is considered to be consistent, AP (from Eq. (2)) should be equal to A4 (from Eq. (3)) within acceptable defined deviations. To set the margins of errors, the individual relative percent area deviation %DAi and the individual absolute percent area deviation j%DAij between experimental and calculated values are defined as:
(2)
%DAi ¼ 100
A4 AP AP i
(6)
(3)
A4 AP j%DAi j¼ 100 AP i
(7)
Therefore, Eq. (1) becomes:
AP ¼ A4
(5)
i
(4)
The term AP is calculated from experimental data only (pressure P and solubility x1), while A4 is determined by calculating the compressibility factor Z and the fugacity coefficients (41 and 42) from an equation of state. If a set of data is considered to be consistent, AP should be equal to A4, within acceptable defined
In Eqs. (6) and (7), AP is determined using experimental phase equilibrium data at constant temperature, while different equations of state models can be employed to evaluate A4. If the data are correlated with deviations within established limits (as defined later here) and the individual area deviation %DAi are within defined margins of errors, then the data set is considered to be thermodynamically consistent. The criteria established to define if a set of data is consistent or
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Table 1 The Kwak-Mansoori concept applied to the PR and the VPT equations of state for mixtures. PR/KM
VPT/KM pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi RT am þRTdm 2 am dm RT P ¼ Vb VðVþbm Þþcm ðVbm Þ m PP pffiffiffiffiffiffiffiffiffi am ¼ xi xj aij aij ¼ ai aj ð1 kij Þ
a þRTd 2 a d RT
m m m RT m P ¼ Vb VðVþbm Þþbm ðVbm Þ Pm P am ¼ xi xj aij
i
aij ¼
j
i
pffiffiffiffiffiffiffiffiffi
ai aj ð1 kij Þ bm ¼ xi xj bij i
j
bm ¼
PP i
33
2 bij ¼ 4 dm ¼
1=3
bi
1=3
þbj 2
PP i
5 ð1bij Þ
i
xi xj dij
j
dm ¼ 33
2 1=3
di
1=3
þdj 2
j
PP
cm ¼
j
PP i
dij ¼ 4
j
ai ¼ aci ð1 þ Fi Þ2
PP
j
13
0 1=3
bi
xi xj bij bij ¼ @
1=3
þbj 2
A ð1 lij Þ
13
0
c1=3 þc1=3 i j A 2
xi xj cij cij ¼ @
13
0 1=3
di
xi xj dij dij ¼ @
ð1 mij Þ
1=3
þdj 2
2
A ð1 nij Þ
2
ai ¼ aci ð1 þ Fi Þ2 aci ¼ Uai RPTcici
5 ð1 dij Þ
ai ¼ aci ð1 þ Fi Þ2
RTci ci b ¼ Ubi RT Pci c ¼ Uci Pci d ¼
aci ¼ 0:4572
Uai ¼ 0:6612 0:7616Zci
b¼
R2 T2ci Pci RTci 0:0778 Pci
Ubi ¼ 0:0221 0:2087Zci
d¼
aci F2i RTci
aci F2i RTci
Fi ¼ 0:4628 þ 3:5823ðui Zci Þ
0:2699u2i
4. Results and discussion Table 4 presents the results of the consistency test for all mixtures considered in this study. In that table, the average absolute pressure deviations j%DPj, the maximum individual relative pressure deviations %Pmax, and the average absolute area deviationsj% DAij are given. The average absolute deviations j%DPj and j%DAij for a set of N data, are those defined by Eqs. (5) and (7) respectively. For clarity of the presentation, the results and their analyses are grouped in three cases: A) The gases O2, H2, and CO in [MDEA][CI]; B) The gases O2, H2, and CO in [Bmim][PF6]; and C) the gases O2, H2, and CO in [Hmim][BTI].
Uci ¼ 0:5777 1:8718Zci
F ¼ 0:3746 þ 1:5423ui
Table 2 shows pure component properties for all the substances considered in this study. In this Table, M is the molar mass, Tc is the critical temperature, Pc is the critical pressure, u is the acentric factor and Zc is the critical compressibility factor. The values of these properties were obtained from Reid et al. [14] for the gases (O2, H2 and CO) and from Valderrama et al. [15] for the ionic liquids ([Bmim][PF6], [Hmim][BTI] and [MDEA][CI]). Table 3 gives some details on the experimental data used in the study including the literature source. In these Tables, N is the number of data points, T is the temperature (expressed in kelvin), P is the pressure (expressed in bar), x1 is the solubility of the gas in the ionic liquids (expressed as mole fraction).
þ8:1942ðui Zci Þ2
inconsistent, require that the model accurately correlates the experimental PTx data. The correlation is considered acceptable if the deviation defined by Eq. (5) is lower than 10% for j%DPj. After the model is found appropriate, it is required that the deviations in the individual areas defined by Eq. (6) are all within the limits 20% to þ20% to declare the data as being thermodynamically consistent. These criteria have been used by the authors and other researchers for several years [11e13]. The distribution of the area deviations is also considered for determining consistency or inconsistency. This means that if some few points (up to one quarter of the original points) do not pass the area test but the rest of the data does pass the test, there is no reason to eliminate the whole set of data. In this case the original set of data is considered to be not fully consistent (NFC). If more than one quarter of the data does not pass the test, then the whole original set is declared to be thermodynamically inconsistent (TI).
Table 3 Details on the phase equilibrium data for the gases (O2, H2, CO) (1) þ ionic liquids ([MDEA][CI], [Bmim][PF6], [Hmim][BTI]) (2) considered in this study. Systems O2 þ [MDEA][CI]
H2 þ [MDEA][CI]
CO þ [MDEA][CI]
O2 þ[Bmim][PF6]
Ref. [16]
[16]
[16]
[17]
3. Experimental data used The study considers data of mixtures of three gases (O2, H2, CO) in three ionic liquids ([Bmim][PF6], [Hmim][BTI], [MDEA][CI]) for which experimental data have been published in the literature. The pressures range from 7.7 bar to 97.9 bar and the temperatures range from 293 K to 413 K. These sets include 40 isotherms with a total of 286 PTx data points.
Table 2 Properties for all substances involved in this study. Data for the gases are from Reid et al. [14] and the data for the ILs are from Valderrama et al. [15]. Components
M
Tc/K
Pc/MPa
u
Zc
O2 H2 CO [Bmim][PF6] [Hmim][(BTI] [MDEA][CI]
32.0 2.02 28.0 284.2 447.4 154.6
154.6 33.2 132.9 719.4 1298.8 720.7
5.04 1.31 3.50 1.73 2.39 3.61
0.025 0.216 0.066 0.792 0.389 1.125
0.288 0.305 0.295 0.220 0.245 0.264
H2 þ [Bmim][PF6]
COþ [Bmim][PF6]
O2 þ [Hmim][BTI]
H2 þ [Hmim][BTI]
CO þ [Hmim][BTI]
[18]
[19]
[20]
[21]
[20]
N
T/K
range of x1
range of P/Bar
7 7 7 7 7 8 8 8 8 8 7 7 7 7 7 9 8 8 8 8 8 8 8 8 8 8 6 6 8 6 6 6 7 6 6 6 6 6 6 6
313 318 323 328 333 313 318 323 328 333 313 318 323 328 333 293 313 333 353 373 313 333 353 373 293 314 334 354 373 293 313 373 293 333 373 413 293 333 373 413
0.0993e0.3342 0.0989e0.3170 0.0939e0.3163 0.0916e0.2912 0.0890e0.2909 0.0324e0.1306 0.0299e0.1254 0.0298e0.1100 0.0295e0.1054 0.0290e0.1001 0.0778e0.2330 0.0754e0.2151 0.0663e0.2050 0.0603e0.1901 0.0569e0.1755 0.0084e0.0433 0.0063e0.0435 0.0067e0.0441 0.0074e0.0443 0.0076e0.0446 0.0040e0.0216 0.0033e0.0237 0.0044e0.0265 0.0035e0.0293 0.0063e0.0407 0.0080e0.0386 0.0046e0.0388 0.0076e0.0404 0.0038e0.0419 0.0277e0.1154 0.0236e0.1210 0.0219e0.1197 0.0076e0.0490 0.0141e0.0555 0.0117e0.0624 0.0135e0.0739 0.0198e0.1088 0.0204e0.1050 0.0184e0.1053 0.0248e0.1103
19.50e74.50 20.10e72.30 19.50e74.40 20.30e69.20 20.60e75.90 13.20e54.40 13.40e57.70 15.90e58.50 16.40e58.80 16.50e58.70 26.10e82.20 27.10e83.90 25.10e82.10 25.90e85.20 26.20e86.20 15.68e90.55 11.64e90.86 12.29e91.16 13.43e89.62 13.71e88.22 16.95e90.85 12.31e90.14 14.56e91.00 10.84e90.21 12.47e91.67 16.11e85.15 8.87e86.12 15.33e88.20 7.68e90.23 16.18e78.77 15.02e89.98 14.51e91.04 14.76e98.19 22.64e93.05 15.91e90.77 16.20e95.08 14.59e92.61 15.78e92.84 14.64e93.90 19.71e97.85
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79
Table 4 Results obtained for the gases (O2, H2, CO) (1) þ ionic liquids ([MDEA][CI], [Bmim][PF6], [Hmim][BTI]) (2) with PR/KM model. Optimum binary interaction parameters (k12, b12, d12), area deviations, and results of the consistency test for all isotherms. Systems
T(K)
k12
b12
d12
rDP%r
DPmax%
rA%r
Results
O2 þ [MDEA][CI]
313 318 323 328 333 313 318 323 328 333 313 318 323 328 333 293 313 333 353 373 313 333 353 373 293 314 334 354 373 293 333 373 293 333 373 413 293 333 373 413
0.0808 0.8587 0.2789 0.9224 0.4326 2.0579 2.3894 0.8185 1.9652 1.4995 1.3526 0.3376 0.6496 0.5691 0.3929 0.8409 1.4471 0.4980 0.3340 0.1274 1.7985 1.2536 1.1503 1.5988 0.4804 0.3841 0.0903 1.0063 0.1252 0.4803 1.2355 0.2866 1.5573 0.9071 1.4231 0.2001 0.2632 1.2107 1.2077 1.5273
0.2761 0.2922 0.2841 0.2903 0.2814 0.3335 0.3369 0.3049 0.3427 0.3301 0.2045 0.2277 0.1922 0.2019 0.2264 0.1714 0.4607 0.0369 0.0876 0.1604 0.1122 0.2559 0.2343 0.0462 0.4938 0.5848 0.0422 0.3629 0.1431 0.5937 0.1884 0.6213 0.7478 0.6700 0.6550 0.6564 0.5320 0.3839 0.0759 0.0779
0.9629 0.0627 0.6302 0.0038 0.4704 0.61857 0.4527 1.0660 0.6324 0.8052 0.5424 0.5274 1.4014 1.3259 0.4481 1.4219 1.1578 1.3036 1.3825 1.2923 0.4137 1.1078 1.0747 1.7834 0.0296 0.3425 0.7408 0.7249 1.518 1.0557 4.2029 0.3686 0.6694 0.5454 0.2650 0.9230 0.8600 1.2999 2.9275 2.9936
1.3 1.7 0.7 0.3 0.5 0.6 0.7 3.3 1.2 0.9 0.4 0.2 0.4 0.8 0.8 0.4 2.3 0.3 0.4 0.3 0.4 0.2 0.4 0.2 0.2 0.7 0.2 0.5 0.4 1.8 5.6 4.6 3.0 0.2 4.1 0.2 0.9 1.9 0.2 4.4
3.2 4.3 1.2 1.0 0.9 1.9 1.7 6.3 2.7 3.0 0.9 0.3 0.7 1.5 1.6 0.8 4.7 0.5 0.9 0.5 1.6 0.5 1.5 0.5 0.3 1.2 0.5 1.2 1.5 3.3 8.5 9.6 6.3 0.6 4.5 0.5 1.8 4.7 0.3 8.3
9.7 14.7 7.4 7.9 5.5 8.0 12.0 16.1 22.5 13.6 5.6 2.6 7.5 5.3 7.3 51.8 4.4 4.1 3.1 2.4 28.3 5.8 5.4 3.2 25.2 4.8 5.9 6.4 4.9 71.0 6.4 3.2 43.2 65.8 30.3 19.9 41.6 32.3 39.3 2.3
TC NFC NFC TC TC TC TI NFC TI TI TC TC TC TC TC NFC TC TC TC TC NFC NFC TC TC NFC TC TC TC TC NFC TC TC TI TI NFC NFC TI NFC NFC TC
H2 þ [MDEA][CI]
CO þ [MDEA][CI]
O2 þ[Bmim][PF6]
H2 þ [Bmim][PF6]
COþ [Bmim][PF6]
O2 þ [Hmim][BTI]
H2 þ [Hmim][BTI]
CO þ [Hmim][BTI]
In the three cases (A, B and C), the criteria for the consistency test can be summarized in the following form: i) the set of data is found to be thermodynamically consistent (TC) when the modeling is acceptable and deviations in the areas are within the established limits. ii) the set of data is considered to be not fully consistent (NFC) if there are some few points in the original data set (less than a quarter of the data) that give high area deviations but the remaining areas give deviations within the established limits; and iii) the set of data is declared to be thermodynamically inconsistent (TI) if area deviations are very high for more than 25% of the points of the original data set, despite that the modeling is acceptable and within the limits established by the method.
4.1.2. VPT/KM model With this model also 9 of the 15 data sets were found to be thermodynamically consistent (TC) as it can be seen in Table 5. Four sets were found to be not fully consistent (NFC), being three of them the same ones that were found NFC with the PR/KM model. The fourth system is H2þ[MDEA][CI] at 318 K that was found to be TI with the PR/KM model and NFC with the VPT/KM model. The VPT/ KM model reproduces bubble pressures of these binary mixtures with mean absolute deviations below 3.3% for all temperatures. In 14 of 15 isotherms studied, pressure deviations are below 2.9%.
4.1. Case A: gases (O2, H2, and CO) þ [MDEA][CI]
4.2.1. PR/KM model For the gases (O2, H2, and CO) þ [Bmim][PF6] mixtures it can be seen in Table 4 that 10 of the 14 data sets were found to be thermodynamically consistent (TC). Four sets were found to be not fully consistent (NFC), for example the system H2(1) þ [Bmim][PF6] (2) at two temperatures. The PR/KM model reproduces bubble pressures of these binary mixtures with mean absolute deviations below 2.4% at all temperatures.
4.1.1. PR/KM model For the gases (O2, H2, and CO) þ [MDEA][CI] mixtures it can be seen in Table 4, that 9 of the 15 data sets were found to be thermodynamically consistent (TC). Three sets were found to be not fully consistent (NFC), for instance the system O2(1)þ[MDEA][CI] (2) at T ¼ 323 K, and three sets were found to be thermodynamically inconsistent (TI), for instance the system H2(1)þ[MDEA][CI] (2) at three temperatures. As indicated in Table 4, the PR/KM model reproduces bubble pressures of these binary mixtures with mean absolute deviations below 3.4% for all temperatures. In 11 of 15 isotherms studied, pressure deviations are below 1.0%.
4.2. Case B: gases (O2, H2, and CO) þ [Bmim][PF6]
4.2.2. VPT/KM model As observed in Table 5, the use of the VPT/KM model allows reproducing bubble pressures with absolute deviations lower than 1.1% for all isotherms. For this study case (Gases þ [Bmim][PF6]) all
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Table 5 Results obtained for the gases (O2, H2, CO) (1) þ ionic liquids ([MDEA][CI], [Bmim][PF6], [Hmim][BTI]) (2) with VPT/KM model. Optimum binary interaction parameters (k12, l12, m12, n12), area deviations, and results of the consistency test for all isotherms. Systems
T(K)
k12
l12
m12
n12
rDP%r
DPmax%
rA%r
Results
O2 þ [DMEA][CI]
313 318 323 328 333 313 318 323 328 333 313 318 323 328 333 293 313 333 353 373 313 333 353 373 293 313 334 354 373 293 333 373 293 333 373 413 293 333 373 413
0.0935 1.9379 1.4627 1.3057 0.9675 0.8914 1.0617 1.3801 1.2594 0.6554 0.1329 0.3208 0.6315 0.7043 0.0512 0.9370 0.1187 0.3721 0.4983 0.3822 0.4047 0.2782 1.1105 0.9477 0.4407 0.2501 0.6404 0.8075 0.3740 0.0655 0.5649 0.0822 0.7862 0.0850 0.8501 1.4295 0.6299 1.2862 0.9619 1.3618
0.2831 0.1677 0.1710 0.3855 0.2131 0.4110 0.3060 0.4279 0.2808 0.3194 0.2489 0.2794 0.0958 0.2612 0.2831 0.8163 0.3718 0.0417 0.5920 0.0141 0.0482 0.0433 0.1328 0.0350 0.6569 1.2953 0.3389 0.4370 0.0885 0.6700 0.0699 0.7106 0.5961 1.0845 1.0356 0.7772 0.2578 0.5464 0.5485 0.6958
0.9772 0.7994 0.3144 0.1885 0.1288 1.0074 0.9274 0.7854 0.8802 1.0469 0.8007 0.6323 1.3610 1.3864 0.9072 1.4508 1.3672 1.0753 1.4202 0.9040 1.4958 1.3495 0.8160 1.0469 1.2148 1.3716 0.8296 0.6824 0.9212 0.4426 0.5849 0.4930 0.5086 1.4951 1.0029 0.4772 0.5352 1.6584 0.5671 1.2675
0.2284 1.0891 1.3602 0.8291 1.0606 0.6583 0.8499 0.9863 1.5749 0.7202 0.0584 0.3438 1.5004 0.5223 0.4285 1.3471 0.7725 0.7178 0.3341 0.7361 0.9822 0.7720 0.9612 0.9507 0.5628 0.8582 1.0046 0.3793 0.6212 0.1842 3.4782 0.0886 1.8160 1.5627 1.3714 0.3354 2.1154 1.4298 0.0595 2.1638
1.3 2.8 0.8 1.3 0.5 0.7 1.4 3.2 1.2 1.0 0.4 0.2 0.4 0.4 2.0 0.6 0.2 0.3 0.4 0.3 0.4 0.2 0.4 0.2 0.8 0.7 0.3 1.0 0.4 2.0 3.8 2.8 0.5 0.2 0.2 0.2 2.0 0.7 1.1 0.5
3.3 4.6 1.9 1.6 0.9 2.1 4.3 6.9 2.7 3.0 0.9 0.4 0.7 0.7 3.5 1.5 0.4 0.5 0.8 0.5 1.5 0.4 0.9 0.4 1.6 1.4 0.7 2.4 1.5 3.4 4.3 3.4 1.2 0.6 0.5 0.5 3.0 1.1 2.5 1.9
9.7 15.2 8.1 8.0 5.5 8.0 12.7 15.1 22.5 13.5 5.8 2.6 7.5 5.4 7.9 5.1 2.0 2.8 2.1 2.2 4.2 2.4 2.9 2.3 4.6 7.3 2.9 4.3 3.6 28.7 8.7 6.1 36.4 34.3 31.8 8.5 79.6 38.6 14.7 12.0
TC NFC NFC TC TC TC NFC NFC TI TI TC TC TC TC TC TC TC TC TC TC TC TC TC TC TC TC TC TC TC NFC TC TC TI TI NFC NFC TI NFC NFC NFC
H2 þ [MDEA][CI]
CO þ [MDEA][CI]
O2 þ [Bmim][PF6]
H2 þ [Bmim][PF6]
CO þ [Bmim][PF6]
O2 þ [Hmim][BTI]
H2 þ [Hmim][BTI]
CO þ [Hmim][BTI]
the data, for the 14 isotherms, are considered to be thermodynamically consistent (TC) and absolute deviations in the areas are lower than 7.4%. The lowest absolute deviation in the areas is found for the system O2 þ [Bmim][PF6] at 313 K. 4.3. Case C: gases (O2, H2, and CO) þ [Hmim][BTI] 4.3.1. PR/KM model For the same gases, but now mixed with the ionic liquid [Hmim] [BTI], Table 4 shows that 3 of the 11 data sets were found to be thermodynamically consistent (TC). Five sets were found to be not fully consistent (NFC) and three sets were found to be thermodynamically inconsistent (TI), for example the system H2(1)þ[Hmim] [BTI] (2) at two temperatures. As seen in Table 4, the PR/KM model reproduces bubble pressures of these binary mixtures with mean absolute deviations below 5.7% for any temperature. In 6 of the 11 isotherms studied, pressure deviations are below 2.0%. 4.3.2. VPT/KM model For these systems the model VPT/KM allows reproducing bubble pressure with deviations lower than 3.9% for all temperatures, as observed in Table 5. For 6 of the 11 isotherms studied in this work, deviations in the bubble pressure are below 0.8%. Two of the data sets were found to be thermodynamically consistent (TC), six sets of data were found to be not fully consistent (NFC), and three cases were thermodynamically inconsistent (TI). These were the
mixtures COþ[Hmim][BTI] at 293 K and H2þ[Hmim][BTI] at 293 and 333 K. Some examples of the results obtained are presented in graphical form in Fig. 1. In this figure, the area deviation, a value that defines the consistency or inconsistency of the set of data, is presented for both models. Fig. (1a) and (1b) present the results found for the system COþ [MDEA][CI] at 318 K found to be TC using both models. The results with both models are practically the same. As observed in the figures there is a small, almost imperceptible, difference in the penultimate point for which the PR/KM gives a slightly higher deviation (0.8%) compared to the VPT/KM model which gives a deviation of 0.7%. Fig. (1c) and (1d) present the results found for the system H2þ[Bmim][PF6] at 333 K. In this case when the PR/KM model is used, the system is declared to be NFC since in this case there is an area deviation higher than 20%. As observed, figures are very similar in most cases, except for the last point (highest pressure) in which the PR/KM model gives a higher deviation, so the system is declared to be NFC in Table 4. With the VPT/KM model, represented in Fig. (1d), all individual deviations are below 6.1%, so all systems are considered to be TC. Fig. (1e) and (1f) show results for the system H2þ[Bmim][BTI] at 293 K. In this case, when both models are used, the system is declared to be TI, since in this case there are two points in which area deviations are higher than 20%. In fact, maximum deviations in
J.O. Valderrama et al. / Fluid Phase Equilibria 449 (2017) 76e82
81
Fig. 1. Deviations in the area for the liquid phase %DAi (Eq. (6)) for selected systems using the PR/KM and the VPT/KM models.
the areas were 146% for the PR/KM model and 117% for the VPT/KM model. As observed in Tables 3 and 4, the VPT/KM model, in the overall, is slightly better in correlating the bubble pressure of all systems. From Tables 3 and 4, the average absolute deviation in the calculation of the bubble pressure for all systems is 0.95% while the relative deviation is 0.47. The PR/KM model gives slightly higher values (1.19% and 1.74%, respectively). With these results it can be
accepted that both models, PR/KM and VPT/KM correlate the data with sufficient accuracy for calculating the compressibility factor and the fugacity coefficients needed for the area test (Eq. (3)). 5. Conclusions Based on the results and the discussion presented above, the following conclusions can be drawn: i) good correlation of phase
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equilibrium data at medium-high pressure is found using the Kwak-Mansoori approach for both equations of state, PR and VPT; ii) modeling using the VPT equation is slightly better in the average for correlating the bubble pressure of all systems (0.9% deviation for VPT and 1.2% deviation for PR); iii) both models, PR/KM and VPT/ KM, correlate the data with sufficient accuracy for calculating the areas that define the consistency test (Eq. (3)); and iv) once the equation of state is appropriately accurate such as the ones used in this work, the thermodynamic consistency test can be considered to be independent of the equation of state.
Super/subscripts c critical cal calculated exp experimental i, j components i and j in a mixture m mixture
Acknowledgements
[1] J.O. Valderrama, V.H. Alvarez, A versatile thermodynamic consistency test for incomplete phase equilibrium data of high-pressure gaseliquid mixtures, Fluid Phase Equilib. 226 (2004) 149e159. s, J.O. Valderrama, Consistency test of solubility [2] C.A. Faúndez, J.F. Díaz-Valde data of ammonia in ionic liquids using the modified PengeRobinson equation of Kwak and Mansoori, Fluid Phase Equilib. 348 (2013) 33e38. €der, A.M. Gil, I.M. Marrucho, M. Aznar, [3] P.J. Carvalho, V.H. Alvarez, B. Schro L.M.N.B.F. Santos, J.A.P. Coutinho, Specific solvation interactions of CO2 on acetate and trifluoroacetate imidazolium based ionic liquids at high pressures, J. Phys. Chem. B 113 (2009) 6803e6812. [4] A. Eslamimanesh, A.H. Mohammadi, D. Richon, Thermodynamic consistency test for experimental data of water content of methane, AIChE J. 57 (2011) 2566e2573. [5] A. Eslamimanesh, A.H. Mohammadi, D. Richon, Thermodynamic consistency test for experimental solubility data in carbon dioxide/methane þ water system inside and outside gas hydrate formation region, J. Chem. Eng. Data 56 (2011) 1573e1586. [6] J.O. Valderrama, C.A. Faúndez, Thermodynamic consistency test of high pressure gas-liquid equilibrium data including both phases, Thermochim. Acta 499 (2010) 85e90. [7] J.O. Valderrama, The state of the cubic equations of state, Ind. Eng. Chem. Res. 42 (2003) 1603e1618. [8] T.Y. Kwak, G.A. Mansoori, Van der Waals mixing rules for cubic equations of state. Applications for supercritical fluid extraction modeling, Chem. Eng. Sci. 41 (1986) 1303e1309. [9] D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem. Fundam. 15 (1976) 59e64. [10] J.O. Valderrama, A generalized Patel-Teja equation of state for polar and nonpolar fluids and mixtures, J. Chem. Eng. Jpn. 23 (1990) 87e91. [11] J.O. Valderrama, A. Reategui, W. Sanga, Thermodynamic consistency test of vapor-liquid equilibrium data for mixtures containing ionic liquids, Ind. Eng. Chem. Res. 47 (2008) 8416e8422. s, J.O. Valderrama, Testing solubility data of H2S [12] C.A. Faúndez, J.F. Díaz-Valde and SO2 in ionic liquids for sulfur-removal processes, Fluid Phase Equilib. 375 (2014) 152e160. [13] S. Mattedi, P.J. Carvalho, J.A.P. Coutinho, V.H. Alvarez, M. Iglesias, High pressure CO2 solubility in N-methyl-2-hydroxyethylammonium protic ionic liquids, J. Supercrit. Fluids 56 (2011) 224e230. [14] R.C. Reid, J.M. Prausnitz, T.K. Sherwood, The Properties of Gases and Liquid, third ed., McGraw Hill Book Company, New-York, USA, 1977. [15] J.O. Valderrama, L.A. Forero, R.E. Rojas, Critical properties and normal boiling temperature of ionic liquids. Update and a new consistency test, Ind. Eng. Chem. Res. 51 (2012) 7838e7844. [16] Y. Zhao, X. Zhang, H. Dong, Y. Zhen, G. Li, S. Zeng, S. Zhang, Solubilities of gases in novel alcamines ionic liquid 2-[2-hydroxyethyl (methyl) amino] ethanol chloride, Fluid Phase Equilib. 302 (2011) 60e64. rez-Salado, I. Urukova, D. Tuma, G. Maurer, Solubility of [17] J. Kumelan, A. Pe oxygen in the ionic liquid [bmim][PF6]: experimental and molecular simulation results, J. Chem. Thermodyn. 37 (2005) 595e602. rez-Salado, D. Tuma, G. Maurer, Solubility of H2 in the ionic [18] J. Kumelan, A. Pe liquid [bmim][PF6], J. Chem. Eng. Data 51 (2006) 11e14. rez-Salado, D. Tuma, G. Maurer, Solubility of CO in the ionic [19] J. Kumełan, A. Pe liquid [bmim][PF6], Fluid Phase Equilib. 228e229 (2005) 207e211. rez-Salado, D. Tuma, G. Maurer, Solubility of the single gases [20] J. Kumelan, A. Pe carbon monoxide and oxygen in the ionic liquid [hmim][BTI], J. Chem. Eng. Data 54 (2009) 966e971. rez-Salado, D. Tuma, G. Maurer, Solubility of H2 in the ionic [21] J. Kumelan, A. Pe liquid [hmim][BTI], J. Chem. Eng. Data 51 (2006) 1364e1367.
The authors thank the support of the National Council for Scientific and Technological Research (CONICYT), through the research grant FONDECYT 1150802. JOV thanks the University of La Serena and the Center for Technological Information of La Serena-Chile, for permanent support. CAF and JFDV thank the University of Conn for continuous support. cepcio List of Symbols
Symbols AP, A4 integrals in the consistency test bm, cm, dm equation of state parameters for mixtures kij, lij, mij, nij binary interaction parameters f fugacity M molar mass P pressure Pc critical pressure R ideal gas constant T temperature Tc critical temperature V volume x1 experimental solubility of gas in the ionic liquid Zc critical compressibility factor Abbreviations EoS equation of state NFC not fully consistent PR Peng-Robinson PR/KM Peng-Robinson þ Kwak-Mansoori model TC thermodynamic consistent TI thermodynamic inconsistent VPT Valderrama-Patel-Teja VPT/KM Valderrama-Patel-Teja þ Kwak-Mansoori model Greek Letters am equation of state parameter for mixtures bij, dij binary interaction parameters 4 fugacity coefficient u acentric factor %DA area deviation %DP pressure deviation
References
Contents lists available at ScienceDirect
Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Equation of state dependency of thermodynamic consistency methods. Application to solubility data of gases in ionic liquids O. Valderrama a, b, *, Claudio A. Faúndez c, Joaquín F. Díaz-Valde s c Jose a
Univ. of La Serena, Fac. of Engineering, Dept. of Mech. Eng., Casilla 554, La Serena, Chile n Tecnolo gica, Monsen ~ or Subercaseaux 667, La Serena, Chile Centro de Informacio c ticas, Casilla 160-C, Concepcio n, Fac. de Ciencias Físicas y Matema n, Chile Univ. de Concepcio b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 March 2017 Received in revised form 10 June 2017 Accepted 15 June 2017 Available online 17 June 2017
A thermodynamic method to check the consistency of phase equilibrium data developed by the authors and applied to several situations by different researchers is analyzed to determine the effect of the equation of state employed to correlate the data on the final response of the consistency test. The consistency test method, in fact, uses an equation of state to correlate the equilibrium data and to calculate some properties such as volume, fugacity and compressibility factor. After these properties are determined the Gibbs-Duhem equation is applied to determine data consistency. If the data fulfill the Gibbs-Duhem equation the data are considered to be consistent and if the data do not fulfill de GibbsDuhem equation the data are of doubtful thermodynamic consistency. The equation of state dependency of the method has been recognized by the authors and other researchers but discussion of the real effects of the equation of state used on the results of the consistency method has not been presented. This work demonstrates that after a good and accurate thermodynamic model is used to correlate the data, the consistency test can be considered equation-of-state independent. © 2017 Elsevier B.V. All rights reserved.
Keywords: Ionic-liquids Thermodynamic consistency Equation of state Gas solubility
1. Introduction During the last several years the authors have developed and applied a thermodynamic test to check the consistency of phase equilibrium data presented in the literature for many different types of systems. Cases studied by the authors include PTxy, PTy and PTx data, being P the pressure, T the temperature, y the gas phase mole fraction, and x the liquid mole fraction. The first work of the authors [1] analyzed gas-liquid mixtures at supercritical conditions. After that, several applications to solid-gas mixtures, gaswater systems and more recently to gas þ ionic liquid mixtures have been reported in the literature [2e6]. Initially and for simple mixtures, common cubic equations of state were successfully employed, such as the Peng-Robinson equation with classical mixing and combining rules. However, in several cases the equation of state cannot accurately correlate the phase equilibrium data and, therefore, the consistency test cannot be applied. This because the application of the consistency test
* Corresponding author. Univ. of La Serena, Fac. of Engineering, Dept. of Mech. Eng., Casilla 554, La Serena, Chile. E-mail address: [email protected] (J.O. Valderrama). http://dx.doi.org/10.1016/j.fluid.2017.06.013 0378-3812/© 2017 Elsevier B.V. All rights reserved.
needs values of the compressibility factor (Z) and of the fugacity coefficients (4) of the components in the mixture. As known, when equations of state are applied to mixtures, it is necessary to introduce mixing and combining rules to account for the concentration effect on the equation of state parameters. Several models for mixing and combining rules have been discussed in the literature [7] being the proposal of Kwak and Mansoori an interesting approach that has been several times explored [8]. Kwak and Mansoori state that van der Waals mixing rules were proposed for temperature-independent parameters and not for temperaturedependent parameters, as commonly done when cubic equations of state are applied to mixtures. Thus, they rewrote the PengRobinson expression obtaining an equation of state that includes three temperature-independent parameters: cm, bm and dm [9]. The three EoS parameters are expressed using the classical van der Waals mixing rules, each one including an adjustable parameter. The inclusion of the Kwak-Mansoori proposal has not been done for three parameter equations of state, a method that is also presented in this work. In particular, in this paper, the ValderramaPatel-Teja (VPT) equation of state, a generalization of the PatelTeja equation proposed by Valderrama in 1990 is employed for modeling the phase equilibrium data [10]. The VPT equation of state contains three parameters: two volume-type parameters (b and c)
J.O. Valderrama et al. / Fluid Phase Equilibria 449 (2017) 76e82
77
that are temperature independent and a force-type parameter (a) which is temperature dependent. When the VPT equation is transformed following the concepts of Kwak and Mansoori, a new parameter appears (d) but the force parameter becomes temperature independent. In this way, the VPT þ Kwak-Mansoori expression fulfills with a basic requirement of the van der Waals model. This means that the parameters of the EoS are constant and not functions of the temperature. One aspect that has been mentioned in the past by the authors is that their consistency test, as it happens with all consistency test methods, is model dependent. In the method analyzed in this paper, the model is the equation of state chosen for correlating the data and for calculating the properties needed in the consistency method (Z and 4). In a series of paper by the authors and by other researchers it is implicitly assumed that once the set of data is well represented by an equation of state, the results of the test are independent of the equation of state employed. However, this aspect has not been clearly proved and constitutes the novelty of this paper: to demonstrate that once an equation of state accurately correlates the phase equilibrium data, the consistency test is meaningful and the results about consistency or inconsistency of a set of experimental data can be accepted with confidence.
deviations [1]. To analyze this aspect of the equation of state influence on the consistency test results, this paper considers two models: the Peng-Robinson (PR) and the Valderrama-Patel-Teja (VPT) equations, both with the approach of Kwak and Mansoori for the equation of state parameters, and for the mixing and combining rules. Table 1 summarizes these models. In Table 1, xi is the mole fraction of component “i” in either phase: xi ¼ xi when the equations are applied to the liquid phase and xi ¼ yi when the equations are applied to the gas phase. Also, the PR/KM model contains three adjustable parameters (kij, bij, dij), one for each of the constants: kij for am, bij for bm, and dij for dm. The VPT/KM model contains four adjustable parameters (kij, lij, mij, nij), one for each of the constants: kij for am, lij for bm, mij for cm and nij for dm. It is also assumed that these adjustable parameters are the same for both the liquid and the gas phases. Additionally, the model requires the critical temperature Tc the critical pressure Pc, the acentric factor u and the critical compressibility factor for each component in the mixture. The average absolute deviations in the calculated bubble pressure in the binary system for each point “i” are used to determine the accuracy of the PR/KM and VPT/KM models. This deviation is defined as:
2. Thermodynamic analysis
cal exp 100 X P P i j%DPj ¼ N Pexp
The consistency test is based on the Gibbs-Duhem equation and has been presented with details in the literature [11]. In the test, the equation of state method is used for phase equilibrium correlation and prediction in complex systems. In this method, the same equation is used for the liquid and gas phases to apply the fundamental equation of phase equilibrium. That is the equality of fugacities of a given component in both phases, f Li ¼ f G i . In terms of the fugacity coefficients 4i, the equality of fugacities becomes L G xi 4Li ¼ yi 4G i . To evaluate the fugacity coefficients 4i and 4i an equation of state is used. To describe the concentration dependency of the equation of state parameters, appropriate mixing and combining rules are included. Therefore, as stated in the Introduction, if the consistency test give similar results when different equations of state are used to model the phase equilibrium data, the consistency test can be considered equation-of-state independent. This is the main and novel objective of this work. As detailed in the several papers published by the authors, the consistency test for isothermal data is summarized in an expression derived from the Gibbs-Duhem equation [1]:
Z
1 dP ¼ Px1
Z
1 d4 þ ðZ1Þ41 1
Z
ð1x1 Þ d42 x1 ðZ1Þ42
(1)
For simplicity of the explanation that follows the left-hand side is denoted by AP and the right-hand side is denoted by A4.
Z AP ¼ Z A4 ¼
1 dP Px1 1 d4 þ ðZ1Þ41 1
Z
ð1x1 Þ d42 x1 ðZ1Þ42
The thermodynamic consistency test can be applied if these deviations, expressed as j%DPj are within pre-established ranges. Once the model is accepted, the consistency test model can be applied [11,12]. The equations that define the consistency test have been published in previous papers, so they are not repeated here. These are summarized in Eq. (1) in which (N-1) values for the area AP (Eq. (2)) and (N-1) values for the area A4 (Eq. (3)) are calculated if a set of N experimental data points at constant temperature is available [11]. One should notice that is in the calculations of A4 (Eq. (3)) where the equation of state plays its role. As observed in Eq. (3) the integral term A4 includes the compressibility factor (Z) and the fugacity coefficient (4) both determined using an equation of state. Accurate equations of state will give similar values of Z and 4 and the areas A4 should be the same. Simple equations of state with simple mixing rules cannot accurately calculate these values of Z and 4, the areas A4 will have different erroneous values, and the conclusion of the test may not be the correct one. As mentioned above, if a set of data is considered to be consistent, AP (from Eq. (2)) should be equal to A4 (from Eq. (3)) within acceptable defined deviations. To set the margins of errors, the individual relative percent area deviation %DAi and the individual absolute percent area deviation j%DAij between experimental and calculated values are defined as:
(2)
%DAi ¼ 100
A4 AP AP i
(6)
(3)
A4 AP j%DAi j¼ 100 AP i
(7)
Therefore, Eq. (1) becomes:
AP ¼ A4
(5)
i
(4)
The term AP is calculated from experimental data only (pressure P and solubility x1), while A4 is determined by calculating the compressibility factor Z and the fugacity coefficients (41 and 42) from an equation of state. If a set of data is considered to be consistent, AP should be equal to A4, within acceptable defined
In Eqs. (6) and (7), AP is determined using experimental phase equilibrium data at constant temperature, while different equations of state models can be employed to evaluate A4. If the data are correlated with deviations within established limits (as defined later here) and the individual area deviation %DAi are within defined margins of errors, then the data set is considered to be thermodynamically consistent. The criteria established to define if a set of data is consistent or
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Table 1 The Kwak-Mansoori concept applied to the PR and the VPT equations of state for mixtures. PR/KM
VPT/KM pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi RT am þRTdm 2 am dm RT P ¼ Vb VðVþbm Þþcm ðVbm Þ m PP pffiffiffiffiffiffiffiffiffi am ¼ xi xj aij aij ¼ ai aj ð1 kij Þ
a þRTd 2 a d RT
m m m RT m P ¼ Vb VðVþbm Þþbm ðVbm Þ Pm P am ¼ xi xj aij
i
aij ¼
j
i
pffiffiffiffiffiffiffiffiffi
ai aj ð1 kij Þ bm ¼ xi xj bij i
j
bm ¼
PP i
33
2 bij ¼ 4 dm ¼
1=3
bi
1=3
þbj 2
PP i
5 ð1bij Þ
i
xi xj dij
j
dm ¼ 33
2 1=3
di
1=3
þdj 2
j
PP
cm ¼
j
PP i
dij ¼ 4
j
ai ¼ aci ð1 þ Fi Þ2
PP
j
13
0 1=3
bi
xi xj bij bij ¼ @
1=3
þbj 2
A ð1 lij Þ
13
0
c1=3 þc1=3 i j A 2
xi xj cij cij ¼ @
13
0 1=3
di
xi xj dij dij ¼ @
ð1 mij Þ
1=3
þdj 2
2
A ð1 nij Þ
2
ai ¼ aci ð1 þ Fi Þ2 aci ¼ Uai RPTcici
5 ð1 dij Þ
ai ¼ aci ð1 þ Fi Þ2
RTci ci b ¼ Ubi RT Pci c ¼ Uci Pci d ¼
aci ¼ 0:4572
Uai ¼ 0:6612 0:7616Zci
b¼
R2 T2ci Pci RTci 0:0778 Pci
Ubi ¼ 0:0221 0:2087Zci
d¼
aci F2i RTci
aci F2i RTci
Fi ¼ 0:4628 þ 3:5823ðui Zci Þ
0:2699u2i
4. Results and discussion Table 4 presents the results of the consistency test for all mixtures considered in this study. In that table, the average absolute pressure deviations j%DPj, the maximum individual relative pressure deviations %Pmax, and the average absolute area deviationsj% DAij are given. The average absolute deviations j%DPj and j%DAij for a set of N data, are those defined by Eqs. (5) and (7) respectively. For clarity of the presentation, the results and their analyses are grouped in three cases: A) The gases O2, H2, and CO in [MDEA][CI]; B) The gases O2, H2, and CO in [Bmim][PF6]; and C) the gases O2, H2, and CO in [Hmim][BTI].
Uci ¼ 0:5777 1:8718Zci
F ¼ 0:3746 þ 1:5423ui
Table 2 shows pure component properties for all the substances considered in this study. In this Table, M is the molar mass, Tc is the critical temperature, Pc is the critical pressure, u is the acentric factor and Zc is the critical compressibility factor. The values of these properties were obtained from Reid et al. [14] for the gases (O2, H2 and CO) and from Valderrama et al. [15] for the ionic liquids ([Bmim][PF6], [Hmim][BTI] and [MDEA][CI]). Table 3 gives some details on the experimental data used in the study including the literature source. In these Tables, N is the number of data points, T is the temperature (expressed in kelvin), P is the pressure (expressed in bar), x1 is the solubility of the gas in the ionic liquids (expressed as mole fraction).
þ8:1942ðui Zci Þ2
inconsistent, require that the model accurately correlates the experimental PTx data. The correlation is considered acceptable if the deviation defined by Eq. (5) is lower than 10% for j%DPj. After the model is found appropriate, it is required that the deviations in the individual areas defined by Eq. (6) are all within the limits 20% to þ20% to declare the data as being thermodynamically consistent. These criteria have been used by the authors and other researchers for several years [11e13]. The distribution of the area deviations is also considered for determining consistency or inconsistency. This means that if some few points (up to one quarter of the original points) do not pass the area test but the rest of the data does pass the test, there is no reason to eliminate the whole set of data. In this case the original set of data is considered to be not fully consistent (NFC). If more than one quarter of the data does not pass the test, then the whole original set is declared to be thermodynamically inconsistent (TI).
Table 3 Details on the phase equilibrium data for the gases (O2, H2, CO) (1) þ ionic liquids ([MDEA][CI], [Bmim][PF6], [Hmim][BTI]) (2) considered in this study. Systems O2 þ [MDEA][CI]
H2 þ [MDEA][CI]
CO þ [MDEA][CI]
O2 þ[Bmim][PF6]
Ref. [16]
[16]
[16]
[17]
3. Experimental data used The study considers data of mixtures of three gases (O2, H2, CO) in three ionic liquids ([Bmim][PF6], [Hmim][BTI], [MDEA][CI]) for which experimental data have been published in the literature. The pressures range from 7.7 bar to 97.9 bar and the temperatures range from 293 K to 413 K. These sets include 40 isotherms with a total of 286 PTx data points.
Table 2 Properties for all substances involved in this study. Data for the gases are from Reid et al. [14] and the data for the ILs are from Valderrama et al. [15]. Components
M
Tc/K
Pc/MPa
u
Zc
O2 H2 CO [Bmim][PF6] [Hmim][(BTI] [MDEA][CI]
32.0 2.02 28.0 284.2 447.4 154.6
154.6 33.2 132.9 719.4 1298.8 720.7
5.04 1.31 3.50 1.73 2.39 3.61
0.025 0.216 0.066 0.792 0.389 1.125
0.288 0.305 0.295 0.220 0.245 0.264
H2 þ [Bmim][PF6]
COþ [Bmim][PF6]
O2 þ [Hmim][BTI]
H2 þ [Hmim][BTI]
CO þ [Hmim][BTI]
[18]
[19]
[20]
[21]
[20]
N
T/K
range of x1
range of P/Bar
7 7 7 7 7 8 8 8 8 8 7 7 7 7 7 9 8 8 8 8 8 8 8 8 8 8 6 6 8 6 6 6 7 6 6 6 6 6 6 6
313 318 323 328 333 313 318 323 328 333 313 318 323 328 333 293 313 333 353 373 313 333 353 373 293 314 334 354 373 293 313 373 293 333 373 413 293 333 373 413
0.0993e0.3342 0.0989e0.3170 0.0939e0.3163 0.0916e0.2912 0.0890e0.2909 0.0324e0.1306 0.0299e0.1254 0.0298e0.1100 0.0295e0.1054 0.0290e0.1001 0.0778e0.2330 0.0754e0.2151 0.0663e0.2050 0.0603e0.1901 0.0569e0.1755 0.0084e0.0433 0.0063e0.0435 0.0067e0.0441 0.0074e0.0443 0.0076e0.0446 0.0040e0.0216 0.0033e0.0237 0.0044e0.0265 0.0035e0.0293 0.0063e0.0407 0.0080e0.0386 0.0046e0.0388 0.0076e0.0404 0.0038e0.0419 0.0277e0.1154 0.0236e0.1210 0.0219e0.1197 0.0076e0.0490 0.0141e0.0555 0.0117e0.0624 0.0135e0.0739 0.0198e0.1088 0.0204e0.1050 0.0184e0.1053 0.0248e0.1103
19.50e74.50 20.10e72.30 19.50e74.40 20.30e69.20 20.60e75.90 13.20e54.40 13.40e57.70 15.90e58.50 16.40e58.80 16.50e58.70 26.10e82.20 27.10e83.90 25.10e82.10 25.90e85.20 26.20e86.20 15.68e90.55 11.64e90.86 12.29e91.16 13.43e89.62 13.71e88.22 16.95e90.85 12.31e90.14 14.56e91.00 10.84e90.21 12.47e91.67 16.11e85.15 8.87e86.12 15.33e88.20 7.68e90.23 16.18e78.77 15.02e89.98 14.51e91.04 14.76e98.19 22.64e93.05 15.91e90.77 16.20e95.08 14.59e92.61 15.78e92.84 14.64e93.90 19.71e97.85
J.O. Valderrama et al. / Fluid Phase Equilibria 449 (2017) 76e82
79
Table 4 Results obtained for the gases (O2, H2, CO) (1) þ ionic liquids ([MDEA][CI], [Bmim][PF6], [Hmim][BTI]) (2) with PR/KM model. Optimum binary interaction parameters (k12, b12, d12), area deviations, and results of the consistency test for all isotherms. Systems
T(K)
k12
b12
d12
rDP%r
DPmax%
rA%r
Results
O2 þ [MDEA][CI]
313 318 323 328 333 313 318 323 328 333 313 318 323 328 333 293 313 333 353 373 313 333 353 373 293 314 334 354 373 293 333 373 293 333 373 413 293 333 373 413
0.0808 0.8587 0.2789 0.9224 0.4326 2.0579 2.3894 0.8185 1.9652 1.4995 1.3526 0.3376 0.6496 0.5691 0.3929 0.8409 1.4471 0.4980 0.3340 0.1274 1.7985 1.2536 1.1503 1.5988 0.4804 0.3841 0.0903 1.0063 0.1252 0.4803 1.2355 0.2866 1.5573 0.9071 1.4231 0.2001 0.2632 1.2107 1.2077 1.5273
0.2761 0.2922 0.2841 0.2903 0.2814 0.3335 0.3369 0.3049 0.3427 0.3301 0.2045 0.2277 0.1922 0.2019 0.2264 0.1714 0.4607 0.0369 0.0876 0.1604 0.1122 0.2559 0.2343 0.0462 0.4938 0.5848 0.0422 0.3629 0.1431 0.5937 0.1884 0.6213 0.7478 0.6700 0.6550 0.6564 0.5320 0.3839 0.0759 0.0779
0.9629 0.0627 0.6302 0.0038 0.4704 0.61857 0.4527 1.0660 0.6324 0.8052 0.5424 0.5274 1.4014 1.3259 0.4481 1.4219 1.1578 1.3036 1.3825 1.2923 0.4137 1.1078 1.0747 1.7834 0.0296 0.3425 0.7408 0.7249 1.518 1.0557 4.2029 0.3686 0.6694 0.5454 0.2650 0.9230 0.8600 1.2999 2.9275 2.9936
1.3 1.7 0.7 0.3 0.5 0.6 0.7 3.3 1.2 0.9 0.4 0.2 0.4 0.8 0.8 0.4 2.3 0.3 0.4 0.3 0.4 0.2 0.4 0.2 0.2 0.7 0.2 0.5 0.4 1.8 5.6 4.6 3.0 0.2 4.1 0.2 0.9 1.9 0.2 4.4
3.2 4.3 1.2 1.0 0.9 1.9 1.7 6.3 2.7 3.0 0.9 0.3 0.7 1.5 1.6 0.8 4.7 0.5 0.9 0.5 1.6 0.5 1.5 0.5 0.3 1.2 0.5 1.2 1.5 3.3 8.5 9.6 6.3 0.6 4.5 0.5 1.8 4.7 0.3 8.3
9.7 14.7 7.4 7.9 5.5 8.0 12.0 16.1 22.5 13.6 5.6 2.6 7.5 5.3 7.3 51.8 4.4 4.1 3.1 2.4 28.3 5.8 5.4 3.2 25.2 4.8 5.9 6.4 4.9 71.0 6.4 3.2 43.2 65.8 30.3 19.9 41.6 32.3 39.3 2.3
TC NFC NFC TC TC TC TI NFC TI TI TC TC TC TC TC NFC TC TC TC TC NFC NFC TC TC NFC TC TC TC TC NFC TC TC TI TI NFC NFC TI NFC NFC TC
H2 þ [MDEA][CI]
CO þ [MDEA][CI]
O2 þ[Bmim][PF6]
H2 þ [Bmim][PF6]
COþ [Bmim][PF6]
O2 þ [Hmim][BTI]
H2 þ [Hmim][BTI]
CO þ [Hmim][BTI]
In the three cases (A, B and C), the criteria for the consistency test can be summarized in the following form: i) the set of data is found to be thermodynamically consistent (TC) when the modeling is acceptable and deviations in the areas are within the established limits. ii) the set of data is considered to be not fully consistent (NFC) if there are some few points in the original data set (less than a quarter of the data) that give high area deviations but the remaining areas give deviations within the established limits; and iii) the set of data is declared to be thermodynamically inconsistent (TI) if area deviations are very high for more than 25% of the points of the original data set, despite that the modeling is acceptable and within the limits established by the method.
4.1.2. VPT/KM model With this model also 9 of the 15 data sets were found to be thermodynamically consistent (TC) as it can be seen in Table 5. Four sets were found to be not fully consistent (NFC), being three of them the same ones that were found NFC with the PR/KM model. The fourth system is H2þ[MDEA][CI] at 318 K that was found to be TI with the PR/KM model and NFC with the VPT/KM model. The VPT/ KM model reproduces bubble pressures of these binary mixtures with mean absolute deviations below 3.3% for all temperatures. In 14 of 15 isotherms studied, pressure deviations are below 2.9%.
4.1. Case A: gases (O2, H2, and CO) þ [MDEA][CI]
4.2.1. PR/KM model For the gases (O2, H2, and CO) þ [Bmim][PF6] mixtures it can be seen in Table 4 that 10 of the 14 data sets were found to be thermodynamically consistent (TC). Four sets were found to be not fully consistent (NFC), for example the system H2(1) þ [Bmim][PF6] (2) at two temperatures. The PR/KM model reproduces bubble pressures of these binary mixtures with mean absolute deviations below 2.4% at all temperatures.
4.1.1. PR/KM model For the gases (O2, H2, and CO) þ [MDEA][CI] mixtures it can be seen in Table 4, that 9 of the 15 data sets were found to be thermodynamically consistent (TC). Three sets were found to be not fully consistent (NFC), for instance the system O2(1)þ[MDEA][CI] (2) at T ¼ 323 K, and three sets were found to be thermodynamically inconsistent (TI), for instance the system H2(1)þ[MDEA][CI] (2) at three temperatures. As indicated in Table 4, the PR/KM model reproduces bubble pressures of these binary mixtures with mean absolute deviations below 3.4% for all temperatures. In 11 of 15 isotherms studied, pressure deviations are below 1.0%.
4.2. Case B: gases (O2, H2, and CO) þ [Bmim][PF6]
4.2.2. VPT/KM model As observed in Table 5, the use of the VPT/KM model allows reproducing bubble pressures with absolute deviations lower than 1.1% for all isotherms. For this study case (Gases þ [Bmim][PF6]) all
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Table 5 Results obtained for the gases (O2, H2, CO) (1) þ ionic liquids ([MDEA][CI], [Bmim][PF6], [Hmim][BTI]) (2) with VPT/KM model. Optimum binary interaction parameters (k12, l12, m12, n12), area deviations, and results of the consistency test for all isotherms. Systems
T(K)
k12
l12
m12
n12
rDP%r
DPmax%
rA%r
Results
O2 þ [DMEA][CI]
313 318 323 328 333 313 318 323 328 333 313 318 323 328 333 293 313 333 353 373 313 333 353 373 293 313 334 354 373 293 333 373 293 333 373 413 293 333 373 413
0.0935 1.9379 1.4627 1.3057 0.9675 0.8914 1.0617 1.3801 1.2594 0.6554 0.1329 0.3208 0.6315 0.7043 0.0512 0.9370 0.1187 0.3721 0.4983 0.3822 0.4047 0.2782 1.1105 0.9477 0.4407 0.2501 0.6404 0.8075 0.3740 0.0655 0.5649 0.0822 0.7862 0.0850 0.8501 1.4295 0.6299 1.2862 0.9619 1.3618
0.2831 0.1677 0.1710 0.3855 0.2131 0.4110 0.3060 0.4279 0.2808 0.3194 0.2489 0.2794 0.0958 0.2612 0.2831 0.8163 0.3718 0.0417 0.5920 0.0141 0.0482 0.0433 0.1328 0.0350 0.6569 1.2953 0.3389 0.4370 0.0885 0.6700 0.0699 0.7106 0.5961 1.0845 1.0356 0.7772 0.2578 0.5464 0.5485 0.6958
0.9772 0.7994 0.3144 0.1885 0.1288 1.0074 0.9274 0.7854 0.8802 1.0469 0.8007 0.6323 1.3610 1.3864 0.9072 1.4508 1.3672 1.0753 1.4202 0.9040 1.4958 1.3495 0.8160 1.0469 1.2148 1.3716 0.8296 0.6824 0.9212 0.4426 0.5849 0.4930 0.5086 1.4951 1.0029 0.4772 0.5352 1.6584 0.5671 1.2675
0.2284 1.0891 1.3602 0.8291 1.0606 0.6583 0.8499 0.9863 1.5749 0.7202 0.0584 0.3438 1.5004 0.5223 0.4285 1.3471 0.7725 0.7178 0.3341 0.7361 0.9822 0.7720 0.9612 0.9507 0.5628 0.8582 1.0046 0.3793 0.6212 0.1842 3.4782 0.0886 1.8160 1.5627 1.3714 0.3354 2.1154 1.4298 0.0595 2.1638
1.3 2.8 0.8 1.3 0.5 0.7 1.4 3.2 1.2 1.0 0.4 0.2 0.4 0.4 2.0 0.6 0.2 0.3 0.4 0.3 0.4 0.2 0.4 0.2 0.8 0.7 0.3 1.0 0.4 2.0 3.8 2.8 0.5 0.2 0.2 0.2 2.0 0.7 1.1 0.5
3.3 4.6 1.9 1.6 0.9 2.1 4.3 6.9 2.7 3.0 0.9 0.4 0.7 0.7 3.5 1.5 0.4 0.5 0.8 0.5 1.5 0.4 0.9 0.4 1.6 1.4 0.7 2.4 1.5 3.4 4.3 3.4 1.2 0.6 0.5 0.5 3.0 1.1 2.5 1.9
9.7 15.2 8.1 8.0 5.5 8.0 12.7 15.1 22.5 13.5 5.8 2.6 7.5 5.4 7.9 5.1 2.0 2.8 2.1 2.2 4.2 2.4 2.9 2.3 4.6 7.3 2.9 4.3 3.6 28.7 8.7 6.1 36.4 34.3 31.8 8.5 79.6 38.6 14.7 12.0
TC NFC NFC TC TC TC NFC NFC TI TI TC TC TC TC TC TC TC TC TC TC TC TC TC TC TC TC TC TC TC NFC TC TC TI TI NFC NFC TI NFC NFC NFC
H2 þ [MDEA][CI]
CO þ [MDEA][CI]
O2 þ [Bmim][PF6]
H2 þ [Bmim][PF6]
CO þ [Bmim][PF6]
O2 þ [Hmim][BTI]
H2 þ [Hmim][BTI]
CO þ [Hmim][BTI]
the data, for the 14 isotherms, are considered to be thermodynamically consistent (TC) and absolute deviations in the areas are lower than 7.4%. The lowest absolute deviation in the areas is found for the system O2 þ [Bmim][PF6] at 313 K. 4.3. Case C: gases (O2, H2, and CO) þ [Hmim][BTI] 4.3.1. PR/KM model For the same gases, but now mixed with the ionic liquid [Hmim] [BTI], Table 4 shows that 3 of the 11 data sets were found to be thermodynamically consistent (TC). Five sets were found to be not fully consistent (NFC) and three sets were found to be thermodynamically inconsistent (TI), for example the system H2(1)þ[Hmim] [BTI] (2) at two temperatures. As seen in Table 4, the PR/KM model reproduces bubble pressures of these binary mixtures with mean absolute deviations below 5.7% for any temperature. In 6 of the 11 isotherms studied, pressure deviations are below 2.0%. 4.3.2. VPT/KM model For these systems the model VPT/KM allows reproducing bubble pressure with deviations lower than 3.9% for all temperatures, as observed in Table 5. For 6 of the 11 isotherms studied in this work, deviations in the bubble pressure are below 0.8%. Two of the data sets were found to be thermodynamically consistent (TC), six sets of data were found to be not fully consistent (NFC), and three cases were thermodynamically inconsistent (TI). These were the
mixtures COþ[Hmim][BTI] at 293 K and H2þ[Hmim][BTI] at 293 and 333 K. Some examples of the results obtained are presented in graphical form in Fig. 1. In this figure, the area deviation, a value that defines the consistency or inconsistency of the set of data, is presented for both models. Fig. (1a) and (1b) present the results found for the system COþ [MDEA][CI] at 318 K found to be TC using both models. The results with both models are practically the same. As observed in the figures there is a small, almost imperceptible, difference in the penultimate point for which the PR/KM gives a slightly higher deviation (0.8%) compared to the VPT/KM model which gives a deviation of 0.7%. Fig. (1c) and (1d) present the results found for the system H2þ[Bmim][PF6] at 333 K. In this case when the PR/KM model is used, the system is declared to be NFC since in this case there is an area deviation higher than 20%. As observed, figures are very similar in most cases, except for the last point (highest pressure) in which the PR/KM model gives a higher deviation, so the system is declared to be NFC in Table 4. With the VPT/KM model, represented in Fig. (1d), all individual deviations are below 6.1%, so all systems are considered to be TC. Fig. (1e) and (1f) show results for the system H2þ[Bmim][BTI] at 293 K. In this case, when both models are used, the system is declared to be TI, since in this case there are two points in which area deviations are higher than 20%. In fact, maximum deviations in
J.O. Valderrama et al. / Fluid Phase Equilibria 449 (2017) 76e82
81
Fig. 1. Deviations in the area for the liquid phase %DAi (Eq. (6)) for selected systems using the PR/KM and the VPT/KM models.
the areas were 146% for the PR/KM model and 117% for the VPT/KM model. As observed in Tables 3 and 4, the VPT/KM model, in the overall, is slightly better in correlating the bubble pressure of all systems. From Tables 3 and 4, the average absolute deviation in the calculation of the bubble pressure for all systems is 0.95% while the relative deviation is 0.47. The PR/KM model gives slightly higher values (1.19% and 1.74%, respectively). With these results it can be
accepted that both models, PR/KM and VPT/KM correlate the data with sufficient accuracy for calculating the compressibility factor and the fugacity coefficients needed for the area test (Eq. (3)). 5. Conclusions Based on the results and the discussion presented above, the following conclusions can be drawn: i) good correlation of phase
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equilibrium data at medium-high pressure is found using the Kwak-Mansoori approach for both equations of state, PR and VPT; ii) modeling using the VPT equation is slightly better in the average for correlating the bubble pressure of all systems (0.9% deviation for VPT and 1.2% deviation for PR); iii) both models, PR/KM and VPT/ KM, correlate the data with sufficient accuracy for calculating the areas that define the consistency test (Eq. (3)); and iv) once the equation of state is appropriately accurate such as the ones used in this work, the thermodynamic consistency test can be considered to be independent of the equation of state.
Super/subscripts c critical cal calculated exp experimental i, j components i and j in a mixture m mixture
Acknowledgements
[1] J.O. Valderrama, V.H. Alvarez, A versatile thermodynamic consistency test for incomplete phase equilibrium data of high-pressure gaseliquid mixtures, Fluid Phase Equilib. 226 (2004) 149e159. s, J.O. Valderrama, Consistency test of solubility [2] C.A. Faúndez, J.F. Díaz-Valde data of ammonia in ionic liquids using the modified PengeRobinson equation of Kwak and Mansoori, Fluid Phase Equilib. 348 (2013) 33e38. €der, A.M. Gil, I.M. Marrucho, M. Aznar, [3] P.J. Carvalho, V.H. Alvarez, B. Schro L.M.N.B.F. Santos, J.A.P. Coutinho, Specific solvation interactions of CO2 on acetate and trifluoroacetate imidazolium based ionic liquids at high pressures, J. Phys. Chem. B 113 (2009) 6803e6812. [4] A. Eslamimanesh, A.H. Mohammadi, D. Richon, Thermodynamic consistency test for experimental data of water content of methane, AIChE J. 57 (2011) 2566e2573. [5] A. Eslamimanesh, A.H. Mohammadi, D. Richon, Thermodynamic consistency test for experimental solubility data in carbon dioxide/methane þ water system inside and outside gas hydrate formation region, J. Chem. Eng. Data 56 (2011) 1573e1586. [6] J.O. Valderrama, C.A. Faúndez, Thermodynamic consistency test of high pressure gas-liquid equilibrium data including both phases, Thermochim. Acta 499 (2010) 85e90. [7] J.O. Valderrama, The state of the cubic equations of state, Ind. Eng. Chem. Res. 42 (2003) 1603e1618. [8] T.Y. Kwak, G.A. Mansoori, Van der Waals mixing rules for cubic equations of state. Applications for supercritical fluid extraction modeling, Chem. Eng. Sci. 41 (1986) 1303e1309. [9] D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem. Fundam. 15 (1976) 59e64. [10] J.O. Valderrama, A generalized Patel-Teja equation of state for polar and nonpolar fluids and mixtures, J. Chem. Eng. Jpn. 23 (1990) 87e91. [11] J.O. Valderrama, A. Reategui, W. Sanga, Thermodynamic consistency test of vapor-liquid equilibrium data for mixtures containing ionic liquids, Ind. Eng. Chem. Res. 47 (2008) 8416e8422. s, J.O. Valderrama, Testing solubility data of H2S [12] C.A. Faúndez, J.F. Díaz-Valde and SO2 in ionic liquids for sulfur-removal processes, Fluid Phase Equilib. 375 (2014) 152e160. [13] S. Mattedi, P.J. Carvalho, J.A.P. Coutinho, V.H. Alvarez, M. Iglesias, High pressure CO2 solubility in N-methyl-2-hydroxyethylammonium protic ionic liquids, J. Supercrit. Fluids 56 (2011) 224e230. [14] R.C. Reid, J.M. Prausnitz, T.K. Sherwood, The Properties of Gases and Liquid, third ed., McGraw Hill Book Company, New-York, USA, 1977. [15] J.O. Valderrama, L.A. Forero, R.E. Rojas, Critical properties and normal boiling temperature of ionic liquids. Update and a new consistency test, Ind. Eng. Chem. Res. 51 (2012) 7838e7844. [16] Y. Zhao, X. Zhang, H. Dong, Y. Zhen, G. Li, S. Zeng, S. Zhang, Solubilities of gases in novel alcamines ionic liquid 2-[2-hydroxyethyl (methyl) amino] ethanol chloride, Fluid Phase Equilib. 302 (2011) 60e64. rez-Salado, I. Urukova, D. Tuma, G. Maurer, Solubility of [17] J. Kumelan, A. Pe oxygen in the ionic liquid [bmim][PF6]: experimental and molecular simulation results, J. Chem. Thermodyn. 37 (2005) 595e602. rez-Salado, D. Tuma, G. Maurer, Solubility of H2 in the ionic [18] J. Kumelan, A. Pe liquid [bmim][PF6], J. Chem. Eng. Data 51 (2006) 11e14. rez-Salado, D. Tuma, G. Maurer, Solubility of CO in the ionic [19] J. Kumełan, A. Pe liquid [bmim][PF6], Fluid Phase Equilib. 228e229 (2005) 207e211. rez-Salado, D. Tuma, G. Maurer, Solubility of the single gases [20] J. Kumelan, A. Pe carbon monoxide and oxygen in the ionic liquid [hmim][BTI], J. Chem. Eng. Data 54 (2009) 966e971. rez-Salado, D. Tuma, G. Maurer, Solubility of H2 in the ionic [21] J. Kumelan, A. Pe liquid [hmim][BTI], J. Chem. Eng. Data 51 (2006) 1364e1367.
The authors thank the support of the National Council for Scientific and Technological Research (CONICYT), through the research grant FONDECYT 1150802. JOV thanks the University of La Serena and the Center for Technological Information of La Serena-Chile, for permanent support. CAF and JFDV thank the University of Conn for continuous support. cepcio List of Symbols
Symbols AP, A4 integrals in the consistency test bm, cm, dm equation of state parameters for mixtures kij, lij, mij, nij binary interaction parameters f fugacity M molar mass P pressure Pc critical pressure R ideal gas constant T temperature Tc critical temperature V volume x1 experimental solubility of gas in the ionic liquid Zc critical compressibility factor Abbreviations EoS equation of state NFC not fully consistent PR Peng-Robinson PR/KM Peng-Robinson þ Kwak-Mansoori model TC thermodynamic consistent TI thermodynamic inconsistent VPT Valderrama-Patel-Teja VPT/KM Valderrama-Patel-Teja þ Kwak-Mansoori model Greek Letters am equation of state parameter for mixtures bij, dij binary interaction parameters 4 fugacity coefficient u acentric factor %DA area deviation %DP pressure deviation
References