An Empirical Modification of Classical Mixing Rule for the Cohesive

Apr 21, 2010 - xixjxkaijk, where kij(T) and κ are the system-dependent adjustable parameters. Regardless of its possible theoretical justification, t...
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Ind. Eng. Chem. Res. 2010, 49, 4989–4994

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An Empirical Modification of Classical Mixing Rule for the Cohesive Parameter: The Triple Interactions in Binary Systems Considered Ilya Polishuk* Department of Chemical Engineering & Biotechnology, Ariel UniVersity Center of Samaria, 40700, Ariel, Israel

The proposed mixing rule can be formally interpreted as an approach considering the triple cohesive interactions 2 2 2 2 2 of a part of the molecules in a mixture as follows: a ) k∑i)1 ∑j)1 xixjaij(1 - kij(T)) + (1 - κ)∑i)1 ∑j)1 ∑k)1 xixjxkaijk, where kij(T) and κ are the system-dependent adjustable parameters. Regardless of its possible theoretical justification, the proposed mixing rule is equivalent to the classical mixing rule with the composition-dependent adjustable parameter kij(T), and this dependence follows the regularities typically required for an accurate modeling of experimental data. This study demonstrates that the proposed mixing rule allows an accurate description of the entire thermodynamic phase space involving liquid-liquid-vapor equilibria, vapor-liquid equilibria, and the high pressure liquid-liquid equilibria. However in some complex cases it still may not cancel a need of evaluating the binary parameter l12. Introduction

N

Q)

Representing the properties of mixtures is a very important purpose of using equation of state (EOS) models. The advances in developing mixing rules are listed in the reviews.1-3 Although the classical mixing rules are unable to satisfy the basic criterion such as the ideal-solution limit,4 they are still robust and reliable in nature.5 Hence, it seems expedient to keep any mixing rules convertible to the classical ones. Unfortunately, evaluating the nonzero binary adjustable parameters for modeling mixtures is often unavoidable. Moreover, in many cases the accurate description of the entire phase diagrams can be achieved by using the temperature, composition, and even volume-dependent binary parameters. The temperature dependencies of certain adjustable parameters can be carefully introduced without violating the thermodynamic consistence of the models. However introducing other kinds of dependencies can be more problematic. In particular, the volumetric functionalities of binary parameters eliminate the third-order volume dependencies of cubic equations and impose the higher-order ones. The composition dependencies of binary parameters are a very effective tool for accurate modeling of data;6-10 however, they could be affected by the Michelsen-Kistenmacher syndrome.11,12 Mathias et al.13 have proposed to overcome the above problem by attaching the classical mixing rules by a term involving a cubic composition dependence. The idea of evaluating the cubic composition dependence by considering triple molecular interactions for the multicompound systems has been recently proposed by Zabaloy.14 However an idea of combining the pair and triple interactions has not been considered. The present study investigates the abilities and limitations of the cubic composition dependence, and it deals with binary mixtures. The details are given below. Theory. Conformal solution theory is applicable to mixtures in which only the pair interactions take place. Thus, according to the one-fluid theory the classical mixing rules are given as * To whom correspondence should be addressed. E-mail: polishuk@ ariel.ac.il; [email protected]. Tel.: +972-3-9066346. Fax +972-3-9066323.

N

∑ ∑xxQ i j

(1)

ij

i)1 j)1

In the case of the cohesive parameter a and binary mixtures eq 1 be expressed as a ) x2a11 + 2(1 - k12)x(1 - x)√a11a22 + (1 - x)2a22 (2) where k12 is the binary adjustable parameter. Let us consider the triple interaction, then N

Q)

N

N

∑ ∑ ∑xxx Q i j k

(3)

ijk

i)1 j)1 k)1

In the case of the cohesive parameter a and binary mixtures eq 3 becomes 3

3

a ) x3a11 + 3x2(1 - x)√a112a22 + 3x(1 - x)2 √a11a222 +

(1 - x)3a22 (4)

Equation 4 is equal to eq 2 if k12 ) χ12 + ψ12x

(5)

where 3

χ12

2a11a22 + √a11a22(a22 - 3√a11a222) ) 2a11a22 3

ψ12 )

3

a11 - a22 + 3√a11a222 - 3√a112a22 2√a11a22

(6)

(7)

In other words, eq 4 is identical to eq 2 with the concentration-dependent k12. It can be also seen that this concentration dependency is linear. Let us analyze the functions χ and Ψ. Figure 1 depicts the dependence of these functions on the mixture’s asymmetry. It can be seen that eq 5 always yields the positive k12, which decreases with x. Remarkably, such behavior of the binary adjustable parameter is often required for an accurate modeling of experimental data. Nevertheless,

10.1021/ie100138h  2010 American Chemical Society Published on Web 04/21/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 10, 2010 Table 1. The Values of the Binary Parameters Used in the Present Study system nitrogen (1)-ethane (2)

eq 9, set A eq 9, set B P-R EOS, set P-R EOS, set carbon dioxide (1)-n-hexanol (2) eq 9, set A eq 9, set B P-R EOS, set P-R EOS, set carbon dioxide (1)-n-decane (2) eq 9, set A eq 9, set B P-R EOS, set P-R EOS, set

Figure 1. Dependence of the functions χ and Ψ (eq 5) on the mixture’s asymmetry.

eq 5 typically overestimates the appropriate values of k12. In addition, a temperature dependence of k12 causing increase of this parameter with temperature may improve the accuracy of modeling. Considering all the above, as the preliminary approach, the following mixing rule with two adjustable parameters is proposed:

A B

A B

A B

k12

κ

l12

-0.260 -0.014 -0.235 -0.014 0.065 0.070 0.082 0.082 0.088 0.087 0.074 0.07

0.25 1 0.25 1 0.82 1 0.76 1 0.83 1 0.8 1

0 -0.06 0 -0.06 0 -0.05 0 -0.06 0 -0.06 0 -0.07

The mixing rule for a is given by eq 8. The mixing rules for b and c are b ) x2b11 + 2(1 - l12)x(1 - x)

b11 + b22 + (1 - x)2b22 2 (12)

c ) x2c11 + 2(1 - l12)x(1 - x)

c11 + c22 + (1 - x)2c22 2 (13)

(l12 is the same in eqs 12 and 13). In addition, the EOS of Peng and Robinson18 (P-R EOS) will be considered.

a ) κ(x2a11 + 2(1 - k12(T/Tc1))x(1 - x)√a11a22 + 3

(1 - x)2a22) + (1 - κ)(x3a11 + 3x2(1 - x)√a211a22 + 3x(1 - x)

2

3

√

a11a222

Results

+ (1 - x) a22) (8) 3

The adjustable parameter κ indicates the part of molecules that exhibit the triple cohesive interactions. Decreasing the value of κ strongly decreases the value of a at low concentrations, which in turn results in a decrease of the predicted liquid phase compositions without significant affection of the vapor phase. Remarkably, a similar effect can be achieved by decreasing the binary adjustable parameter l12 given for the volumetric parameters.15,15 Thus, fitting the experimental data with κ or l12 might be equivalent. In what follows let as evaluate the consequences of replacing l12 by κ using the previously proposed17 cubic EOS model: P)

a RT V-b (V + c)2

(9)

where a)

27(RTc)2 1 64Pc 1 + m(T 3/2 - 1) r b)

c)

Vc,experimental 2√3

3RTc - 4√3PcVc,experimental 24Pc

Vc,EOS )

3RTc + 2√3PcVc,experimental 12Pc

(10a)

(10b)

(10c)

(10d)

where m is generalized with the acentric factor ω defined as follows: m ) 0.3671 + 0.7697ω

(11)

The mixing rules with two or more adjustable parameters are typically capable of the exact local fitting of almost any datum. However in the entire thermodynamic phase space different approaches may have different degrees of success. Thus, especial interests present the systems for which the liquid-liquid-vapor equilibria (LLVE) and the vapor-liquid equilibria (VLE) data are available. Table 1 lists the systems selected for the present study and the pertinent values of the binary adjustable parameters. The methodology of the investigation was as follows: first, a fit of the LLVE data was performed with two adjustable parameters (the nonzero k12 and κ, set A, and the nonzero k12 and l12, set B). Selecting the best values for two adjustable parameters is a nontrivial procedure. However it is unlikely that significantly different values than those listed in Table 1 could result in a substantially better agreement with the experimental data. Afterward the predictions of other data with the sets A and B were compared. Usually eq 9 is more accurate that P-R EOS in correlating and predicting the phase equilibria data. However in some cases the P-R EOS is preferable. In addition, eq 9 has a clear advantage in predicting the volumetric properties and the virial coefficients. It should be also pointed out that the same computation time is required for implementation of both sets of binary adjustable coefficients and EOS models. The first system to be considered is nitrogen (1)-ethane (2). This system exhibits the phase behavior of type III. Figures 2 and 3 depict the LLVE and VLE yielded by both sets of the binary parameters. Figure 4 shows the critical loci, and Figure 5 shows the system’s virial coefficients. It can be seen that the results for the phase compositions are almost identical for the sets A and B. Both approaches describe very accurately the VLE and the ethane-rich phase of LLVE (L1), however they underestimate the content of nitrogen in the nitrogen-rich phase (L2). The difference between the approaches becomes evident in the volumetric projection. In particular, it can be seen that set A represents the data more successfully. It

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Figure 4. The critical locus of the system nitrogen (1)-ethane (2). Experimental data21,22 (O). Calculated data: (black solid lines) eq 9, set A; (black dotted lines) eq 9, set B; (red solid lines) P-R EOS, set A; (red dotted lines) P-R EOS, set B.

Figure 2. LLVE in the system nitrogen (1)-ethane (2). Experimental data19 (O,b). Calculated data: (black lines) eq 9, set A; (black dotted lines) eq 9, set B; (red lines) P-R EOS, set A; (red dotted lines) P-R EOS, set B.

Figure 3. VLE in the system nitrogen (1)-ethane (2). Experimental data:20 (9) 230; (O) 290 K. Calculated data: (black solid lines) eq 9, set A; (black dotted lines) eq 9, set B; (red solid lines) P-R EOS, set A; (red dotted lines) P-R EOS, set B.

should be pointed out that the cubic EOS models do not predict the pure compound volumetric data precisely, thus one cannot expect the exact results for mixtures as well. Nevertheless, it can be seen that decreasing l12 in order to fit the equilibrium compositions affects the accuracy in predicting volumes. Having the significant impact on the excess properties, l12 strongly influences the path of the LLE critical loci at high pressures. In particular, the positive values of this binary parameter direct the critical loci to the zero temperature and the negative values direct them to the high temperatures (see also ref 16). Figure 4 depicts the case of the system under

Figure 5. The 2nd and the 3rd virial coefficients of the system nitrogen (1)-ethane (2). Experimental data:23 (9) 270; (0) 350 K. Calculated data: (black solid lines) eq 9, set A; (black dotted lines) eq 9, set B; (red solid lines) P-R EOS, set A; (red dotted lines) P-R EOS, set B.

consideration. It can be seen that the set A predicts the data much more successfully than the set B. Thus, in this particular case, eq 8 has an obvious advantage over the classical mixing rule. In what follows let us check if the result above has a general validity. Figures 6-8 present a system exhibiting somewhat more complex interactions, namely carbon dioxide (1)-n-hexanol (2). The predictions of phase equilibria are similar for both sets of the binary adjustable parameters. However it can be seen that the set B has a clear advantage in predicting the high

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Figure 8. The critical locus of the system carbon dioxide (1)-n-hexanol (2). Experimental data27 (O). Calculated data: (black solid lines) eq 9, set A; (black dotted lines) eq 9, set B; (red solid lines) P-R EOS, set A; (red dotted lines) P-R EOS, set B.

Figure 6. LLVE in the system carbon dioxide (1)-n-hexanol (2). Experimental data24 (O,b). Calculated data: (black solid lines) eq 9, set A; (black dotted lines) eq 9, set B; (red solid lines) P-R EOS, set A; (red dotted lines) P-R EOS, set B.

Figure 7. VLE in the system carbon dioxide (1)-n-hexanol (2). Experimental data: (b) 313.1525 (O) 432.45 K.26 Calculated data: (black solid lines) eq 9, set A; (black dotted lines) eq 9, set B; (red solid lines) P-R EOS, set A; (red dotted lines) P-R EOS, set B.

pressure critical data. One could conclude that even better predictions of these data could be achieved by further decrease of l12 and taking k > 1. However such practice will result in further deterioration in accuracy of fitting LLVE. Anyways, it can be seen that in the particular case of the polar system under consideration set A does not have overall superiority over set B. Finally, let us consider the system exhibiting the type II of phase behavior. Figures 9-11 depict the results for carbon dioxide (1)-n-decane (2). Table 2 lists the values of the cross-

Figure 9. LLVE in the system carbon dioxide (1)-n-decane (2). Experimental data:28 O,b. Calculated data: (black solid lines) eq 9, set A; (black dotted lines) eq 9, set B; (red solid lines) P-R EOS, set A; (red dotted lines) P-R EOS, set B.

second virial coefficients. It can be seen that the set A is slightly more successful in fitting the LLVE data and predicting the VLE critical data of the system under consideration. Conclusions The proposed mixing rule can be formally interpreted as an approach considering the triple cohesive interactions of a part of the molecules in a mixture. However, according to the conformal solution theory only the pair interactions may take place. Thus, the theoretical background of the proposed approach can be questionable. Nevertheless, it can be considered as a

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adjustable parameter. Regardless of its possible theoretical justification, this composition dependence follows the regularities typically required for an accurate modeling of experimental data, namely the decrease of the binary adjustable parameter k12 with an increase of the concentration. This study demonstrates that the proposed mixing rule allows an accurate description of the entire thermodynamic phase space involving LLVE, VLE, and the high pressure LLE. However in some complex cases it still may not cancel a need of evaluating the binary parameter l12. Acknowledgment Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for support of this research, Grant No. PRF#47338-B6. Nomenclature

Figure 10. VLE in the system carbon dioxide (1)-n-decane (2). Experimental data:29 (9) 310.93; (O) 510.93 K. Calculated data: (black solid lines) eq 9, set A; (black dotted lines) eq 9, set B; (red solid lines) P-R EOS, set A; (red dotted lines) P-R EOS, set B.

a ) cohesive parameter in eq 9 b ) covolume c ) volumetric parameter in eq 9 k12 ) binary adjustable parameter for the mixture’s cohesive parameter l12 ) binary adjustable parameter for the mixture’s covolume m ) compound-dependent parameter in eq 10a P ) pressure R ) universal gas constant T ) temperature x ) concentration and equilibrium composition in the liquid phase of the compound (1) y ) equilibrium composition in the vapor phase of the compound (1) V ) molar volume Greek Letters κ ) adjustable parameter in eq 8 ω ) acentric factor χ,Ψ ) parameters in eq 5 Subscripts c ) critical state 11 ) 1-1 interaction 22 ) 2-2 interaction 12 ) 1-2 cross-interaction r ) reduced property

Literature Cited

Figure 11. The critical loci of the system carbon dioxide (1)-n-decane (2). Experimental data28,29 (b). Calculated data: (black solid lines) eq 9, set A; (black dotted lines) eq 9, set B; (red solid lines) P-R EOS, set A; (red dotted lines) P-R EOS, set B. Table 2. The Values (in cm3/mol) of the Cross Second Virial Coefficient B12 in the System Carbon Dioxide (1)-n-Decane (2) temperature (K) experimental23 323.15 348.15

-417 ( 7 -321 ( 8

eq 9, set A

eq 9, set B

P-R EOS, P-R EOS, set A set B

-523.8 -518.521 -546.168 -541.624 -439.644 -434.321 -464.88 - 460.185

simple empirical modification of the classical mixing rule that implicitly introduces composition-dependence for the binary

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(9) Schwartzentruber, J.; Galivel-Solastiouk, F.; Renon, H. Representation of the Vapor-Liquid Equilibrium of the Ternary System Carbon Dioxide-Propane-Methanol and Its Binaries with a Cubic Equation of State: A New Mixing Rule. Fluid Phase Equilib. 1987, 38, 217. (10) Sandoval, R.; Wilczek-Vera, G.; Vera, J. H. Prediction of Ternary Vapour-Liquid Equilibria with the PRSV Equation of State. Fluid Phase Equilib. 1989, 52, 119. (11) Michelsen, M. L.; Kistenmacher, H. On CompositionsDependent Interaction Coefficients. Fluid Phase Equilib. 1990, 58, 229. (12) Zabaloy, M. S.; Vera, J. H. Identification of Variant and Invariant Properties in the Thermodynamics of Mixtures: Tests for Models and Computer Codes. Fluid Phase Equilib. 1996, 119, 27. (13) Mathias, P. M.; Klotz, H. C.; Prausnitz, J. M. Equation-of-State Mixing Rules for Multicomponent Mixtures: The Problem of Invariance. Fluid Phase Equilib. 1991, 67, 31. (14) Zabaloy, M. S. Cubic Mixing Rules. Ind. Eng. Chem. Res. 2008, 47, 5063. (15) Polishuk, I.; Wisniak, J.; Segura, H. Simultaneous Prediction of the Critical and Subcritical Phase Behavior in Mixtures Using Equation of State I. Carbon Dioxide-Alkanols. Chem. Eng. Sci. 2001, 56, 6485. (16) Milanesio, J. M.; Cismondi, M.; Cardozo-Filho, L.; Quinzani, L. M.; Zabaloy, M. S. Phase Behavior of Linear Mixtures in the Context of Equation of State Models. Ind. Eng. Chem. Res. 2010, 49, 2943. (17) Polishuk, I. Generalized Cubic Equation of State Adjusted to the Virial Coefficients of Real Gases and Its Prediction of Auxiliary Thermodynamic Properties. Ind. Eng. Chem. Res. 2009, 48, 10708. (18) Peng, D. Y.; Robinson, D. B. A New Two Constant Equation of State. Ind. Eng. Chem. Fund. 1976, 15, 59–64. (19) Llave, F. M.; Luks, K. D.; Kohn, J. P. Three-Phase Liquid-LiquidVapor Equilibria in the Binary Systems Nitrogen + Ethane and Nitrogen + Propane. J. Chem. Eng. Data 1985, 30, 435. (20) GrausØ, L.; Fredenskund, A.; Mollerup, J. Vapor-Liquid Equilibrium Data for the Systems C2H6 + N2, C2H4 + N2, C3H8 + N2, and C3H6 + N2. Fluid Phase Equilib. 1977, 1, 13.

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ReceiVed for reView January 20, 2010 ReVised manuscript receiVed March 30, 2010 Accepted April 9, 2010 IE100138H