Phase Behavior of Linear Mixtures in the Context of Equation of State

Ind. Eng. Chem. Res. 2010, 49, 2943–2956

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Phase Behavior of Linear Mixtures in the Context of Equation of State Models J. M. Milanesio,† M. Cismondi,‡ L. Cardozo-Filho,§ L. M. Quinzani,† and M. S. Zabaloy*,† Planta Piloto de Ingenierıá Quı´mica (UniVersidad Nacional del Sur-CONICET) CC 717, 8000 Bahıá Blanca, Argentina, Facultad de Ciencias Exactas Fı´sicas y Naturales, UniVersidad Nacional de Co´rdoba, AVda. Velez Sarsfield 1611, Ciudad UniVersitaria, X5016GCA Co´rdoba, Argentina, and Departamento de Engenharia Quı´mica, UniVersidade Estadual de Maringa´, AV. Colombo 5790, Maringa´, PR 87020-900, Brazil

Models of the equation of state (EOS) type, based on the van der Waals (vdW) EOS, are able to deal with phase equilibrium properties of asymmetric systems over wide ranges of pressure, composition, and temperature. EOS type models are of great practical interest. In general, EOSs always give a nonideal behavior for mixtures, regardless the form of the set mixing rules. In other words, EOSs have no access to the “ideal solution” state. However, EOS type models do have access to the “linear system” (LS) state. A LS is a mixture whose EOS parameters depend linearly on composition. In a LS, the partial molar parameter of a given component within a multicomponent system equals the pure compound parameter. This could be seen as an alternative form of “ideality”. In this work, we study the phase equilibria of linear systems. We evaluate the phase behavior of binary linear (and also nonlinear) systems by generating different types of charts including pressure-temperature projections of phase equilibrium univariant lines (PTUL), which are defined by vapor-liquid and liquid-liquid (LL) critical lines, by liquid-liquid-vapor (LLV) lines, and by pure compound vapor-liquid saturation lines. The results of this work are useful to improve our understanding of models of the EOS type. Introduction Equations of state (EOSs) are very often used to represent the phase equilibria of pure compounds and mixtures of practical interest, over wide ranges of conditions. Valderrama1 has recently provided a review, for the case of cubic EOSs. Conventional EOSs are unable to meet the ideal solution limit,2,3 which has proven to be a useful reference for modeling the thermodynamic properties of real systems. The ideal solution state is characterized by a mixture molar volume which depends linearly on the mole fractions of the components of the system, i.e., the composition dependency of the ideal solution molar volume is the simplest possible. In contrast to the case of conventional EOSs, an unconventional treatment2,3 of the molar volume composition dependency of systems where we model the constituent components at pure state using an EOS does reproduce the ideal solution limit, if desired. However, such unconventional treatment is unable, in general, to describe vapor-liquid critical points for mixtures. This is because there is always a region in the pressure-temperature space where it is not possible to find, at given temperature and pressure, EOS roots (i.e., density values) of the same nature (e.g., liquid), for all pure compounds of the mixture. In such a region, the ideal solution system has property values that are meaningless. Therefore, the ideal solution reference finds limited applicability in the interpretation of the phase behavior of fluid systems at liquid-vapor equilibrium in wide ranges of conditions. On the other hand, vapor-liquid critical states are available to “linear” systems. We define a linear system (LS) as a mixture whose EOS parameters depend linearly on composition, i.e., an LS has the simplest possible composition dependency for the EOS parameters. This last feature, together with the access that LSs have to vapor-liquid critical states, makes LSs interesting * To whom correspondence should be addressed. Phone.: +54 291 4861700 ext. 232. Fax: +54 291 4861600. E-mail: mzabaloy@ plapiqui.edu.ar. † Universidad Nacional del Sur-CONICET. ‡ Universidad Nacional de Co´rdoba. § Universidade Estadual de Maringa´.

objects of study, which, to our knowledge, have not been considered previously in the literature for the case of highly asymmetric systems whose asymmetry is due both, to differences in attractive interactions and to differences in molecular size, i.e., in repulsive interactions. The purpose of this work is therefore to study the phase equilibria of some binary linear systems of varying degree of asymmetry when such asymmetry stems, simultaneously, from differences in pure compound attractive parameters and from differences in pure compound repulsive parameters. Here, we compare the behavior of LSs with that of EOSs coupled to conventional quadratic mixing rules (QMRs) (actually LSs correspond to QMRs used with special values for the interaction parameters). Such comparisons led us to gain a deeper understanding of the features of models of the EOS type, in particular of the effect of interaction parameters on predicted fluid phase equilibria. A previous contribution related to the present work is the seminal paper by van Konynenburg and Scott,4 which corresponds to binary Van Der Waals systems. In this paper,4 the linear mixture (a line at Λ ) 0 in the master chart on page 503 of ref 4) was studied under the restriction of equal size molecules. Ideal Solutions and Linear Systems The molar volume of an ideal solution (υideal_solution) is a linear function2,3 of the component mole fractions (xi). This implies that the partial molar volume of a given component in an ideal ) equals the molar volume of the component solution (υ j ideal_solution i in pure state (υ°) i at the system temperature T and pressure P. Ideal solutions cannot give liquid-liquid equilibrium (LLE). Models2,3 that recover the ideal solution behavior at zero values of their binary parameters cannot, in general, describe vapor-liquid critical points. Such models treat pure compounds using conventional equation of state (EOS) type models. Linear mixtures (LMs) have a mathematical behavior in some way analogous to ideal solutions, while being able to describe all kinds of critical points. In this work we use the words “linear mixtures” (LMs) and “linear systems” (LSs) as synonyms. LSs are defined within the context of EOS type models. An example

10.1021/ie900927f  2010 American Chemical Society Published on Web 02/17/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 6, 2010

of a conventional EOS is the SRK-EOS,5 which, for pure compounds, we write as follows, P)

a RT υ-b υ(υ + b)

(1)

where P is the absolute pressure, T is the absolute temperature, υ is the molar volume, R is the universal gas constant, b is the repulsive covolume parameter, and a is the attractive energy parameter, which is defined as follows: a ) Rac

(2)

where R is a temperature dependent parameter and ac is the attractive energy parameter at the critical temperature. Parameters a and b are positive. In the one-fluid approach, eq 1 applies also to mixtures, the a and b mixture parameters being functions of composition and of pure compound parameters. Typically, the a and b mixture parameters are described using quadratic mixing rules (QMRs). For a binary mixture, conventional QMRs lead to the following equations for the mixture attractive (aquad) and repulsive (bquad) parameters: a

) x1 a1 + 2x1x2√a1a2(1 - k12) + x2 a2

quad

b1 + b2 ) x1 b1 + 2x1x2 (1 - l12) + x22b2 2

quad

b

2

2

(

2

)

(4)

alin ) x1a1 + x2a2

(5)

blin ) x1b1 + x2b2

(6)

The attractive and repulsive partial molar parameters aji and bji are respectively defined as follows:6 )

b¯i )

( )

(7)

( )

(8)

∂na ∂ni ∂nb ∂ni

T,nj*i

T,nj*i

where “n” is the total number of moles in the mixture and ni is the number of moles of component i in the mixture. Combining eqs 5-8, we obtain (a1)lin ) a1

(a2)lin ) a2

(9)

(b2)lin ) b2

(10)

Therefore, in a linear mixture, the partial molar attractive parameter of component i equals the attractive energy parameter of the corresponding pure compound. An analogous statement is valid for the repulsive parameter. Thus, the relationships among pure compound parameters and partial parameters of linear mixtures (eqs 9 and 10) are in a way analogous to those connecting pure compound molar volumes and partial molar volumes of ideal solutions. Equations 9 and 10 make appealing the study of the phase behavior of LMs. To our knowledge, such an study on LMs is not available in the literature, for the case of highly asymmetric systems whose asymmetry is due both, to differences in attractive interactions and to differences in molecular size, i.e., in repulsive interactions. Notice that linear mixing rules, i.e., eqs 5 and 6, are particular cases of quadratic mixing rules. If we set l12 ) 0 in eq 4, we get eq 6. Similarly, we get eq 5 from eq 3 if we set the following expression for k12:

klin 12 ) -

(3)

where ai is the pure compound attractive energy parameter for component i, bi is the pure compound covolume repulsive parameter for component i, xi is the mole fraction of component i in the mixture, k12 is the attractive interaction parameter and l12 is the repulsive interaction parameter. In eqs 3 and 4, the dependence of the mixture parameters on composition is quadratic. In this work, we refer generically to arbitrary specifications of k12 and l12 as QMRs or “quadratic mixing”. If we set k12 ) 0 in eq 3, then, the second term within the righthand side (rhs) of eq 3 corresponds to the so-called geometric mean combining rule (gm4). If k12 differs from zero, we have a modified geometric mean. We define linear mixtures (LMs), which we also name linear systems (LSs), as those for which the mixture parameters depend linearly on composition:

a¯i

(b1)lin ) b1

a1 + a2 - 2√a1a2 2√a1a2

)1-

(

a1 + a2 2

√a1a2

)

(11)

In other words, when we use eq 3 coupled to eq 11, we obtain a linear mixing rule while working under the formalism of a quadratic mixing rule. Notice that if parameters a1 and a2 are lin must also be temperature dependent, then, from eq 11, k12 temperature dependent. Therefore, if we wanted to study the behavior of a linear mixing rule in a given temperature range, while working under the formalism of a quadratic mixing rule, lin temperature dependent (according we would need to make k12 to the prescription of eq 11), if parameters a1 and a2 were temperature dependent. Notice that if we use eq 3 coupled to eq 11, then, the second term within the rhs of eq 3 corresponds to the so-called arithmetic mean (am4) combining rule. Paramlin equals zero if a1 ) a2, and (it can be shown that) it is eter k12 negative otherwise. Some information on the physical significance associated to setting a linear mixing rule on an EOS type a model is obtained from the work by van Konynenburg and Scott.4 For the van der Waals EOS, they show,4 on page 499, an equation for the binary enthalpy change on mixing at low temperature. This equation4 shows that, if the molecular size is the same for all molecules in the binary system then, a linear mixing rule (i.e., an arithmetic mean combining rule) leads to a zero enthalpy change on mixing. In other words, a binary linear mixture of equal size molecules is athermal at low enough temperature, for the van der Waals EOS. Methodology In this work, we use, for the pure compound relationship between pressure P, temperature T, molar volume υ, attractive energy parameter “a”, and repulsive parameter “b”, the form prescribed by the SRK-EOS,5 i.e., eq 1, but setting the attractive parameter as independent from temperature and equal to its critical value “ac”, i.e., we set R ) 1 in eq 2. We label such modification of the SRK-EOS as mSRK-EOS (amSRK ) acRmSRK with RmSRK ) 1). The temperature dependence of the pure compound attractive energy parameter, which appears in the (not shown) expression of variable R in eq 2 (see ref 5), is of secondary importance in this work, in which we intend to identify behavioral patterns for linear mixtures. Setting a ) ac, i.e., R ) 1, implies that the only pure-compound experimental information quantitatively reproduced, by construction, by the


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Table 1. Combinations of Mixing Rules Used in This Work for the EOS a and b Parameters composition dependency of mixture parameters

expressions of mixture parameters case I (ab)

a lineara eq 5

alin ) x1a1 + x2a2

b linearb (eq 6)

blin ) x1b1 + x2b2

a lineara (eq 5)

alin ) x1a1 + x2a2

b quadratic (eq 4)

bquad ) x12b1 + 2x1x2

linear mixture

case II (abb)

(

)

b1 + b2 (1 - l12) + x22b2 2

case III (aab) a quadraticc (eq 3)

aquad ) x12a1 + 2x1x2√a1a2(1 - k12) + x22a2

b linearb (eq 6)

blin ) x1b1 + x2b2

a quadratic at k12 ) 0 a

gm

case IIIgm (aabgm) ) x12a1 + 2x1x2√a1a2 + x22a2

b linearb (eq 6)

blin ) x1b1 + x2b2

a quadraticc (eq 3)

case IV (aabb) a ) x12a1 + 2x1x2√a1a2(1 - k12) + x22a2

b quadratic (eq 4)

b ) x12b1 + 2x1x2

(

)

b1 + b2 (1 - l12) + x22b2 2

a This corresponds to the arithmetic mean (am) combining rule, i.e., to the introduction of eq 11 into eq 3. b This corresponds to setting l12 ) 0 in eq 4. c At k12 ) 0, this corresponds to the geometric mean (gm) combining rule.

mSRK-EOS is the critical temperature and pressure coordinates, which we took from the DIPPR database.7 Notice that the SRKEOS is to a certain extent representative of a large family of modified VdW EOSs. In the present work, we describe the properties of mixtures by using the one-fluid approach, contrasting LMs [i.e., l12 ) 0 and k12 from eq 11 (am)] to more conventional assumptions like k12 ) 0 (gm) and l12 ) 0, or like k12 and l12 having somehow arbitrary values. In other words, we use in this work the one fluid approach with quadratic and/or linear mixing rules (eqs 3-6). More specifically, we consider different combinations for the mixing rules that we use for the attractive and repulsive parameters. We summarize them in Table 1. For example, the label “abb” (Table 1/case II) implies a linear composition dependency for parameter a and a quadratic composition dependency for parameter b, while the label “aabb” (Table 1/case IV) implies quadratic mixing for both, a and b. The choices that Table 1 presents make it possible to better understand the properties of linear mixtures, as shown in the next section. The option “aabgm” (Table 1/case IIIgm) deserves special attention: it corresponds to quadratic mixing at k12 ) 0 for parameter a (gm) and to linear mixing for parameter b, i.e., to l12 ) 0 in eq 4. The aabgm option is a familiar standard

Figure 1. Composition dependence (eq 3) of the a parameter for the mSRK EOS, in a range of k12 values, for the system CO2-propane. The “k12 ) -0.11” curve is a straight line corresponding to eq 5.

reference case. Cases II, III, IIIgm, and IV in Table 1 are what we generally mean in this work by “nonlinear systems”. In this work, we generate pressure-temperature projections of phase equilibrium univariant lines (PTUL) for binary n-alkane/n-alkane systems and also for some binary carbon dioxide + n-alkane systems. The word “univariant” means that the number of degrees of freedom equals unity. A typical example of a univariant equilibrium is the liquid-liquid-vapor equilibrium of a binary system. For such a case, the Gibbs phase rule tells that the number of involved variables (e.g., component concentrations in different phases) minus the number of equilibrium equations (i.e., isofugacity conditions), i.e., the number of degrees of freedom, equals unity. Therefore, the calculation of a liquid-liquid-equilibrium point requires the specification of the value of a single variable such as the temperature or the pressure. PTUL diagrams are charts for binary mixtures where we plot together vapor-liquid and liquid-liquid critical lines, liquid-liquid-vapor lines, and pure compound vaporliquid saturation lines. The univariant nature of all such lines makes it possible to plot them all together on 2D (PTUL) charts, which become maps useful to grasp, at a glance, the main features of the phase behavior of the binary system under study, over a wide range of conditions. We calculate all the lines of PTUL diagrams by using numerical continuation methods, which are able to track highly curved lines.8-11 PTUL diagrams are also known as pressure-temperature projections of Bakhuis Roozeboom space diagrams.12 On the basis of the characteristics of PTUL diagrams, van Konynenburg and Scott4 established a convention for the classification of the phase behavior of binary systems. There are six major types of binary phase behavior. They are identified with Roman numerals (I, II, III, IV, V, and VI). The distinction among the types is based on the location of the critical points and critical loci.13 Besides PTUL diagrams, we also compute here some isothermal phase diagrams.9 Finally, we complete this work by studying diagrams similar in nature to the master diagrams of van Konynenburg and Scott.4 To fix ideas, we include here Figures 1 and 2. Figure 1 shows the a parameter of the mSRK model as a function of composition (eq 3) for the system CO2-propane, at varying values for the binary interaction parameter k12. The linear behavior happens at k12 ) -0.11 (eq 11), i.e., the k12 value that makes the mixing rule become linear with respect to composition is different from zero. At a given propane mole fraction, the mixture attractive parameter a decreases with the increase (from negative to positive values) of k12. Figure 2 shows the b parameter of the mSRK model as a function of composition (eq 4) for the system CO2-propane, for a number of l12 values. The linear behavior occurs at l12 )

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Figure 2. Composition dependence (eq 4) of the b parameter for the mSRK EOS, in a range of l12 values, for the system CO2-propane. The “l12 ) 0” curve is a straight line corresponding to eq 6.

Figure 3. Effect of the k12 interaction parameter on the vapor-liquid critical locus: pressure-temperature projection of phase equilibrium univariant lines for the system methane + n-octadecane (high temperature range). Model: mSRK EOS with l12 ) 0 and different values of k12. The curve labeled k12 ) -2.7 corresponds to the linear mixture (Table 1/case I: ab mixture), and the other curves correspond to aab mixtures (Table 1/case III). More specifically, Table 1/case IIIgm corresponds to k12 ) 0 (aabgm mixture).

0. At a given propane mole fraction, the mixture repulsive parameter b decreases with the increase (from negative to positive values) of l12. Results and Discussion In this work, we studied the phase equilibrium behavior of linear mixtures by considering a series of binary systems where one of the components is methane and the other one is a heavier n-alkane. We have also considered CO2 + n-alkane systems. Figures 3 and 4 show, respectively, the high and low temperature portions of a series of PTUL diagrams for the asymmetric system methane + n-octadecane. In Figure 3, we see the vapor-liquid saturation lines of pure methane (Sat METHANE) and of pure n-octadecane (Sat C18), as described by the mSRK model. Figure 3 also presents the (mixture) vapor-liquid critical lines at “l12 ) 0” for a number of k12 values. Figure 4 displays the corresponding liquid-liquid critical lines, which arise at low temperature, for intermediate k12 values. The liquid-liquid-vapor (LLV) lines cannot be seen in Figures 3 and 4, because they are very close to the pure methane saturation line. A given critical line is a boundary between a homogeneous region and a heterogeneous region. In this last case, at set values of temperature and pressure, heterogeneity occurs, however, only for some concentration range or at a number of concentration ranges. The “k12 ) -2.7” (linear mixture) case in Figure 3 corresponds to the linear mixture (Table 1/case I: ab mixture) while all other cases correspond to

Figure 4. Effect of the k12 interaction parameter on the liquid-liquid critical locus: pressure-temperature projection of phase equilibrium univariant lines for the system methane + n-octadecane (low temperature range). Model: mSRK EOS with l12 ) 0 and different k12 values. All curves correspond to aab mixtures (Table 1/case III). More specifically, Table 1/case IIIgm corresponds to k12 ) 0 (aabgm mixture).

aab mixtures (Table 1/case III). More specifically, for k12 ) 0, we have in Figure 3 an aabgm mixture (Table 1/case IIIgm). In Figure 3, we observe the evolution of the liquid-vapor critical locus as we increase the value of k12 from k12 ) -2.7 to k12 ) 0.2, i.e., the evolution that takes place when the a mixture parameter, at given composition, decreases (see Figure 1). According to the classification of van Konynenburg and Scott,4 the evolution of the phase equilibrium behavior, in Figures 3 and 4, is as follows: [at k12 ) -2.7, type V], [at k12 ) -1.5, type V], [at k12 ) 0, type II], and [at k12 ) 0.2, type III]. Notice that the Roman numerals of the classification of Konynenburg and Scott4 (which identify “types” of phase behavior) should not be confused with the Roman numerals of Table 1 (which identify “cases” of combinations of mixing rules). It is clear that the linear mixture [k12 ) -2.7] presents a wider hightemperature (Figure 3) vapor-liquid immiscibility region in the P-T plane than the aab mixtures of higher k12 values (the size of the vapor-liquid immiscibility region in the P-T plane is roughly taken as the area below the vapor-liquid critical line). Notice that the difference between maximum critical pressures for the linear system and the system with k12 ) 0 is large, i.e., of the order of 150 bar. The critical line of the system at k12 ) 0.2 has a vapor-liquid part at higher temperature, and a steep liquid-liquid part at lower temperature. The vapor-liquid part bounds a vapor-liquid immiscibility region also narrower than that of the linear mixture [k12 ) -2.7]. Figure 4 illustrates the evolution of the liquid-liquid critical line. Heterogeneity occurs to the left of such line. We observe in Figure 4 that when the value of the interaction parameter k12 decreases, the liquid-liquid critical locus is shifted to lower temperatures. At k12 ) -0.15, the temperature range of such line corresponds to very low temperatures. At k12 ) -2.7, no liquid-liquid critical line is detected. Therefore, a low temperature heterogeneous zone does not exist for the linear mixture (Table 1/case I: ab mixture). Notice that the mSRK model, as most EOSs models, has no access to the solid state. Therefore, Figures 3 and 4 do not account for the possible interference of solids with the critical and liquid-liquid-vapor lines and with the pure compound vapor-liquid lines. We focus in this work on the behavior of models for fluid mixtures, rather than on the extent to which such models reproduce the actual phase behavior that would be measured in the laboratory, which could eventually involve the presence of solid phases. However, in a further subsection, i.e., the one in which we discuss the results in Figures 13-23, we consider ranges of variation for the


Figure 5. Effect of the k12 interaction parameter on the critical loci: pressure-temperature projection of phase equilibrium univariant lines for the system methane + n-heptane. Model: mSRK EOS with l12 ) 0 and different values of k12. The curve labeled k12 ) -0.972 corresponds to the linear mixture (Table 1/case I: ab mixture), and the other curves correspond to aab mixtures (Table 1/case III). More specifically, Table 1/case IIIgm corresponds to k12 ) 0 (aabgm mixture).

interaction parameters which include values typically found in the practical correlation of fluid phase equilibria experimental data. A behavior analogous to that of methane + n-octadecane (Figures 3 and 4) is found for other binary systems such as methane + n-heptane. This is shown in Figure 5, where the evolution of the phase equilibrium behavior is as follows: [at k12 ) -0.97, type V], [at k12 ) -0.5, type I], [at k12 ) 0, type II], and [at k12 ) 0.2, type III]. Again we see in Figure 5 that the vapor-liquid immiscibility region is wider, in the pressure-temperature plane, for the linear mixture (k12 ) -0.97, Table 1/case I: ab mixture) in comparison with the aab mixtures (Table 1/case III). Notice that for the cases of Figures 3-5, the ab mixture (or linear mixture, which is based on the am [Table 1/case I]) corresponds to type V while the aabgm mixture (which is based on the gm [Table 1/case IIIgm]) corresponds to type II. The master chart on page 503 of the paper by van Konynenburg and Scott,4 which corresponds to binary van der Waals mixtures with equal size molecules, shows that the am system (Λ ) 0) falls on the I/II, V/IV, or V/III boundaries and that the gm system corresponds either to type II or III. Such a master chart4 shows also that, at constant values of the pure compound parameters, the am to gm transition corresponds to the following possibilities: [a] I/II boundary to II, [b] I/II boundary to III, [c] V/IV boundary to III and, [d] V/III boundary to III. We observe none of those transitions for the cases of Figures 3-5, which correspond to binaries where components 1 and 2 have differing molecular sizes. In Figures 3-5, the ab (LM) to aabgm transition is from type V to II. For the same system of Figure 5, Figure 6 presents a couple of over imposed vapor-liquid equilibrium diagrams, both at 330 K. The diagrams differ in the values of the k12 parameter. Figure 6 shows that, when considering the “pressure-mole fraction” plane, the vapor-liquid immiscibility region does not seem to be wider for the linear ab mixture. We also observe in Figure 6 that, e.g., at 40 bar, the composition range for the liquid state is wider for the linear mixture. At this point it is convenient to come back to the original van der Waals (VdW) EOS on which eq 1 is based. The VdW EOS is the following: P)

a RT - 2 υ-b υ

(VdW EOS)

(12)

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Figure 6. Vapor-liquid equilibrium at 330 K for the system methane + n-heptane. Model: mSRK EOS with l12 ) 0 and with two different values of k12. The k12 ) -0.972 case corresponds to the linear mixture (Table 1/case I: ab mixture). At k12 ) 0, the system is an aabgm mixture (Table 1/case IIIgm).

The partial derivatives of pressure with respect to parameters a and b, corresponding to eq 12 are the following: ∂P ∂b

|

) T,υ

∂P ∂a

|

RT (υ - b)2 )-

T,υ

1 υ2

(VdW EOS)

(VdW EOS)

(13)

(14)

It is clear that pressure increases with the increase in the repulsive parameter b (eq 13) and that pressure decreases with the increase in the attractive parameter a (eq 14). In the VdW EOS (eq 12), parameter b appears only in the first term of the right-hand side, while parameter a appears only in the second term. Naturally, the first term is named the “repulsive term”, while the second is named the “attractive term”. Following van der Waals, we assign, in this work, the character of “attractive” to parameters whose increase, at constant T and υ, reduces the pressure (e.g., parameter a) and the character of “repulsive” to parameters whose increase increases the pressure (e.g., parameter b). When a given parameter changes, at constant T and υ, so that pressure decreases, we say in this work that “affinity” has increased. If the change makes the pressure increase, we say that affinity has decreased. Although our working definitions of affinity, attractiveness, and repulsiveness will be useful in our further discussion, the reader should bear in mind that “there is not direct relation between the attractive part of the intermolecular potential and the a parameter in a cubic EOS.”14 The clear separation between repulsive and attractive effects is at first sight lost in the SRK-EOS (see eq 1). In this case, parameter b appears in both terms. The partial derivatives are the following for the SRK-EOS: ∂P ∂b

|

RT a + (υ - b)2 υ(υ + b)2

)

T,υ

(SRK-EOS)

(15) ∂P ∂a

|

T,υ

)-

1 υ(υ + b)

(SRK-EOS)

(16)

Since a > 0 and υ > 0, it is clear, from eq 15, that ∂P/∂b|T,υ > 0, i.e., pressure increases with the increase in parameter b, for the SRK-EOS, in spite of the fact that parameter b influences both terms of the right-hand side of eq 1. Therefore, according to our previous definitions, we still classify parameter b of the

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Figure 8. Vapor-liquid equilibrium at 183 K for the system methane + n-heptane. Model: mSRK EOS with l12 ) 0 and at varying k12 values. The k12 ) -0.972 case corresponds to the linear mixture (Table 1/case I: ab mixture). For other values of k12, the system is an aab mixture (Table 1/case III). More specifically, Table 1/case IIIgm corresponds to k12 ) 0 (aabgm mixture).

Figure 7. Schematic representation, for the VdW, SRK, and PR EoSs, of the effects of the k12 and l12 binary parameters on the attractive and repulsive interactions of mixtures.

SRK-EOS as a “repulsive” parameter. Additionally, from eq 16, we have that ∂P/∂a|T,υ < 0. Thus, parameter a keeps its “attractive” character in the SRK-EOS. The Peng-Robinson (PR) EoS15 is the following: P)

a RT υ-b υ(υ + b) + b(υ - b)

(PR-EOS)

(17) The second term of the right-hand side of eq 17 is more complex than the corresponding term of the SRK-EOS (eq 1). From eq 17, we obtain the partial derivatives with respect to the a and b parameters: ∂P ∂b

|

) T,υ

RT 2a(υ - b) + (υ - b)2 [υ(υ + b) + b(υ - b)]2

(PR-EOS)

(18) ∂P ∂a

|

)T,υ

1 υ(υ + b) + b(υ - b)

(PR-EOS)

(19)

If the PR-EOS is used within the proper volume range, i.e., υ > b, the derivative ∂P/∂b|T,υ (see eq 18) is positive and the derivative ∂P/∂a|T,υ (see eq 19) is negative. Therefore even when the dependence of the second term of the right-hand side of eq 17 with respect to parameter b is more complex than for the SRK-EOS (eq 1), parameter b keeps its repulsive character in the PR-EOS. Parameter a also meets our working definition of attractiveness for the PR-EOS. Notice that our conclusions regarding the signs of partial derivatives, for the VdW, SRK, and PR EOSs, are also valid for mixtures, provided that we also impose the restriction of constant composition when generating the expressions for such partial derivatives. From the previous analysis and from Figures 1 and 2, we can summarize, for models such as the VdW, SRK, and the PR EOSs, the behavior of interaction parameters in the way illustrated in Figure 7. It shows in the l12 vs k12 plane, the following: [a] if k12 increases, then the mixture attractive a parameter aMIX decreases and [b] if l12 increases, then the

mixture repulsive b parameter bMIX decreases. An increase of l12 (decrease in repulsion) coupled to a decrease of k12 (increase in attraction) leads to an increase in affinity (lower system pressure). If we change the l12 and k12 values in the opposite directions, we obtain a decrease in affinity (higher system pressure). Notice that there is a maximum possible value for k12, i.e., the value at which a line such as line “e” in Figure 1 becomes tangent to the horizontal line at a ) 0. On the other hand, there is not a minimum value for k12, i.e., there is not a maximum value for the attractive parameter a. In other words, there is no limit for affinity increases through changes in k12, but there is a limit for the decrease in affinity by changing k12. Analogously, affinity can be decreased without limit by increasing parameter b (i.e., by decreasing parameter l12, Figure 2), but affinity cannot be increased without limit by decreasing parameter b (i.e., by increasing l12). Figure 7 is useful to facilitate our understanding of the results of the present work. In Figure 4, we observed that a low temperature heterogeneous zone does not exist for the linear mixture (Table 1/case I: ab mixture). This behavior is analogous to that of ideal solutions, which cannot access situations of liquid-liquid immiscibility. The low k12 value (k12 ) -2.7) for the linear mixture (Figure 3) implies a high attractive parameter (Figure 7), i.e., a high affinity, consistent with the absence of liquid-liquid immiscibility. The wider composition range for the liquid state, at 40 bar, in Figure 6, for the linear mixture (k12 ) -0.97), is consistent with the fact that this linear mixture has at any composition a higher value for the attractive mixture parameter than the system with k12 ) 0 (see Figure 7). This also explains the lower n-heptane concentration required at 40 bar for producing a saturated vapor phase (higher tendency to the liquid state for the linear mixture). We could have also reached these conclusions by considering, e.g., bubble pressures at methane mole fraction values such as 0.5: the bubble pressure of the linear mixture (k12 ) -0.97) is less than that of the aabgm mixture (k12 ) 0), which corresponds to the more “attractive” nature of the former. In other words, LMs (which have l12 ) 0; see Table 1) show a stronger preference for the liquid state than other lin of eq mixtures, also with l12 ) 0 and with k12 greater than k12 11. Figure 8 is analogous to Figure 6 but corresponds to a much lower temperature. Under such conditions the vapor-liquid immiscibility area is much narrower for the linear mixture, when seen on the pressure-composition plane, i.e., the area where


Figure 9. Vapor-liquid equilibrium at 530 K for the system methane + n-heptane. Model: mSRK EOS with l12 ) 0 and at varying k12 values. The k12 ) -0.972 case corresponds to the linear mixture (Table 1/case I: ab mixture). For other values of k12, the system is an aab mixture (Table 1/case III). More specifically, Table 1/case IIIgm corresponds to k12 ) 0 (aabgm mixture).

the single-phase liquid state exists is narrower for the less attractive aab mixtures of Figure 8. The situation seems different at 530 K (Figure 9) where the vapor-liquid immiscibility region has maximum size for the linear mixture. At this temperature, the size of such a region does not have a monotonical behavior, i.e., it has a minimum in the range from kij ) -0.97 to 0. Notice, however, that within the pressure range where all three systems give vapor-liquid immiscibility (i.e., up to about 31.5 bar), the composition range at set pressure within which a single liquid phase exists increases monotonically with the decrease in kij, being maximum for the linear mixture, for which the attractive a parameter is also maximum (Figure 7). It is interesting to wonder whether at 100 bar and 0.90 methane mole fraction, in Figure 6, the linear system (k12 ) -0.97), which has a vapor-liquid state, has an overall density higher than that of the homogeneous aabgm system. To answer the question in a more comprehensive way, we generated isobars which give the ratio FLM/FQM (density of the linear system/ density of the aabgm system) as a function of the overall mole fraction of methane, at 330 K. For generating such isobars, we carried out multiphase flash calculations, using SPECS16 for all phase conditions, i.e., when one, both, or none of the two systems were heterogeneous. Figure 10 presents the results. It is clear that at 100 bar and 0.90 overall methane mole fraction the biphasic linear system is denser than the homogeneous aabgm system. More generally, and irrespective of the number of equilibrium phases for either of the systems, the linear system is always denser than the aabgm system according to Figure 10, i.e., throughout the methane mole fraction range in Figure 10, the overall density of the linear system (LS) is always higher than that of the aabgm system. This is what we observe for all isobars in Figure 10. This is consistent with the higher attractive parameter for the linear system (k12 ) -0.97) with respect to the aabgm system, which implies a preference for states of higher density for the linear system (see Figure 7). Notice that for a given point in Figure 10 the possibilities are the following: LS and aabgm system both homogeneous, LS homogeneous and aabgm system heterogeneous, LS and aabgm system both heterogeneous, and, finally, LS heterogeneous and aabgm system homogeneous (see Figure 6). When we decrease the k12 parameter from positive to negative values, we obtain increasingly attractive systems (Figure 7). A natural question is whether the evolution we observe for a set of increasing-attraction systems is roughly the same than that of a set of decreasing-repulsion systems. Figure 11 provides an

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Figure 10. Density ratio FLM/FQM vs overall mole fraction of methane at different pressures for the system methane + n-heptane at equilibrium at 330 K. Model: mSRK EOS with l12 ) 0. The symbol FLM represents the overall density of the linear mixture (Table 1/case I: ab mixture, k12 ) -0.972), while FQM represents the overall density of the aabgm mixture (Table 1/case IIIgm). Notice that, depending on the overall composition, the system may be a homogeneous liquid, homogeneous vapor, or vapor-liquid heterogeneous system. The phase condition at set pressure and set overall methane mole fraction can be obtained from Figure 6.

Figure 11. Effect of the l12 interaction parameter on the critical loci: pressure-temperature projection of phase equilibrium univariant lines for the system methane + n-octane. Model: mSRK EOS with k12 ) -1.15 (linear a parameter) and different values of l12. The l12 ) 0 curve corresponds to the linear mixture (Table 1/case I: ab mixture). All other curves are for abb mixtures (Table 1/case II).

answer, at least for a particular case. We obtain a set of decreasing-repulsion systems for instance when we increase parameter l12 from negative values to zero (Figure 7). Notice that a set of increasing-attraction systems and a set of decreasingrepulsion systems are both sets of increasing affinity systems. In Figure 11, we observe the behavior of a linear mixture (Table 1/case I: ab mixture) and of a set of abb mixtures (Table 1/case II). The l12 parameter ranges from zero, that represents the linear mixture, to l12 ) -0.5. As l12 increases from l12 ) -0.5 to l12 ) -0.3, i.e., as repulsion decreases (Figure 7) (i.e., as affinity increases), the liquid-liquid critical line is shifted to lower temperatures, and thus, on the PT plane, the size of the low temperature liquid-liquid region decreases with decreasing repulsion. The mixture with minimum repulsion (maximum affinity) in Figure 11, i.e., the linear mixture, does not present a liquid-liquid critical line and therefore does not show low temperature liquid-liquid immiscibility. Also, as l12 increases from l12 ) -0.5 to l12 ) 0, the liquid-vapor critical locus confines an increasingly larger vapor-liquid immiscibility region in the P-T plane. In Figure 11, the linear mixture is of type V, while the mixtures with l12 ) -0.5 to l12 ) -0.3 are of type II. From comparing Figure 11 with, e.g., Figures 3 and 4, we conclude that we obtain similar effects by decreasing k12 (i.e., by increasing attraction) than by increasing l12 (i.e., by

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of the overall behavior of the system under study. The variables involved in Figures 5 and 6 are temperature, pressure, and composition. For the sake of an even deeper understanding of the behavior of the models, it is possible to generate sets of even more abstract diagrams in a space of variables defined in terms of the model parameters. One such variable is the asymmetry parameter for a binary system, which we define in this work as follows: AP12 )

Figure 12. Pressure-temperature projection of phase equilibrium univariant lines for the systems methane + n-octane and methane + n-octadecane. Model: mSRK EOS with l12 ) 0 and different values of k12. The curves labeled kC1C8 ) -1.15 and kC1C18 ) -2.7 correspond to linear mixtures (Table 1/case I: ab mixture). The other curves are for aabgm mixtures (Table 1/case IIIgm).

decreasing repulsion). Both cases correspond to situations where affinity is increased. Figure 11 does not present results for l12 > 0. This is done in Figures 17-20 and in Figures 22 and 23 of the next subsection. All such figures correspond to ranges of variation for parameter l12 including both, negative and positive, l12 values. We compare two different asymmetric systems in Figure 12. It shows the PTUL diagrams at lij ) 0 of systems methane + n-octane (at kC1C8 ) 0 and at kC1C8 ) -1.15) and methane + n-octadecane (at kC1C18 ) 0 and at kC1C18 ) -2.7). The linear mixtures correspond to kC1C8 ) -1.15 and kC1C18 ) -2.7. The mixtures with zero attractive interaction parameters are aabgm mixtures (Table 1/case IIIgm). We observe that the effect of increasing the molecular weight of the heaviest component on the vapor-liquid critical locus is much more dramatic for the linear mixtures (Table 1/case I: ab mixture) than for the aabgm mixtures, i.e., the growth in size for the vapor-liquid segregation region (in the PT plane) is larger when going from the more symmetric (C1C8) LS to the less symmetric (C1C18) LS than for the case of the aabgm systems. We also observe that none of the two linear systems shows low temperature liquid-liquid heterogeneity. In the aabgm mixtures, we see that there is a shift to higher temperatures of the liquid-liquid critical locus, as we increase the asymmetry of the system. For the linear methane + n-octadecane system, Figure 12 shows incipient gas-gas equilibria at temperature values slightly higher than the pure n-octadecane critical temperature. Diagrams Similar in Nature to the Master Diagrams of van Konynenburg and Scott.4 In Figure 6 (which, as stated above, shows a couple of familiar isothermal cuts of the pressure-temperature-composition equilibrium surfaces for the system methane + n-heptane), the constancy of a condition is defined by a single number, i.e., T ) 330 K. In contrast, for a given line in Figure 5, the constancy of a condition is not defined by a single number. This is for instance the case for the line labeled k12 ) -0.97, where the condition of criticality is the one that remains constant. Thus, Figure 5 has a level of abstraction higher than Figure 6. For a given value of k12 in Figure 6, only two out of all points in Figure 6 are shown in Figure 5, i.e., the critical point at T ) 330 K and the pure n-heptane vapor-liquid saturation point at T ) 330 K. Therefore, there is some information loss when going from particular figures such as Figure 6, to figures showing univariant phase equilibrium lines such as Figure 5. Such information loss is the price we pay to attain a more systematic understanding

a1b1 a2b2

(20)

Notice that we have defined AP12 in terms of pure compound parameters only. Better known binary parameter definitions are the following:4 ξ12 )

b2 - b1 b2 + b1

(21)

ζ12 )

u2 - u1 u2 + u1

(22)

u2 + u1 - U12 u2 + u1

(23)

λ12 ) where

ui ) ai /bi2

(24)

and U12 ) 2

√a1a2(1 - k12) b1b2

(25)

Notice that if b1 ) b2 and λ12 ) 0, then, from eq 23, k12 becomes equal to the value given by eq 11. Notice also that parameters ξ12 and ζ12 depend on pure compound parameters only and that parameter λ12 depends both, on the pure compound parameters and on the interaction parameter k12. Parameters AP12, ξ12, ζ12, and λ12 are dimensionless. Table 2 shows the values of the mSRK EOS pure compound a and b parameters for the n-alkane homologous series up to n-octadecane and for CO2. Table 2 also presents calculated values of the parameters AP12, ξ12, and ζ12, for the methane (1) + n-alkane (2) series and for the CO2 (1) + n-alkane (2) series. From Table 2, we obtained Figure 13 which shows the covolume ratio bn-alkane/bmethane versus the attractive energy parameter ratio an-alkane/amethane given by the mSRK equation of state, for pure n-alkanes from methane to n-octadecane. Figure 13 clearly shows that, for the methane + n-alkane series of binary systems, the basic asymmetry related to the covolume parameter is not independent from the basic asymmetry related to the attractive energy parameter. In this case, the covolume parameter ratio increases with the energy parameter ratio. Therefore, the basic asymmetry for a given methane + n-alkane system can be characterized by a single parameter such as the AP12 parameter of eq 20. By “basic asymmetry”, we mean the one that can be estimated from just pure compound parameters, regardless the values of the interaction parameters. Table 2 shows that the asymmetry parameter AP12, because of the way in which we have defined it (eq 20), and since we have assigned the label “1” to methane, decreases with the increase in the asymmetry of the binary methane (1) + n-alkane (2) system.


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a

Table 2. Pure Compound a and b Parameters for the mSRK EOS and Characteristic mSRK-Based Binary System Dimensionless Parameters for the Methane + n-Alkane and CO2 + n-Alkane Series methane (1) + n-alkane (2) a

6

2 b

b

CO2 (1) + n-alkane (2)

compound

a (bar m /kmol )

b (L/mol)

ξ12

ζ12

AP12

ξ12

ζ12

AP12

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 CO2

2.33 5.65 9.51 14.07 19.35 25.17 31.47 38.38 45.62 53.44 61.88 70.30 80.15 90.40 100.09 110.34 119.46 129.84 3.70

0.030 0.045 0.063 0.081 0.100 0.121 0.142 0.165 0.187 0.211 0.236 0.260 0.289 0.318 0.345 0.372 0.396 0.424 0.030

0 0.20 0.36 0.46 0.54 0.60 0.65 0.69 0.72 0.75 0.78 0.79 0.81 0.83 0.84 0.85 0.86 0.87

0 0.029 -0.040 -0.096 -0.154 -0.206 -0.253 -0.297 -0.335 -0.371 -0.404 -0.433 -0.465 -0.491 -0.513 -0.533 -0.549 -0.567

1 0.2729 0.1167 0.0614 0.0359 0.0229 0.0156 0.0110 0.0082 0.0062 0.0048 0.0038 0.0030 0.0024 0.0020 0.0017 0.0015 0.0013

0.003 0.21 0.36 0.46 0.54 0.61 0.65 0.69 0.73 0.75 0.78 0.80 0.81 0.83 0.84 0.85 0.86 0.87 0

-0.23 -0.20 -0.27 -0.32 -0.37 -0.42 -0.46 -0.50 -0.53 -0.56 -0.58 -0.60 -0.63 -0.65 -0.67 -0.68 -0.69 -0.71 0

1.5786 0.4307 0.1843 0.0969 0.0566 0.0361 0.0246 0.0174 0.0129 0.0098 0.0075 0.0060 0.0047 0.0038 0.0032 0.0027 0.0023 0.0020 1

a C1-C18 represent the pure n-alkanes from methane to n-octadecane. b Calculated5 from the pure compound critical temperature and pressure.7 See eqs 20-25.

Figure 13. Covolume ratio bn-alkane/bmethane versus attractive energy parameter ratio an-alkane/amethane given by the mSRK equation of state, for pure n-alkanes from methane to n-octadecane (see Table 2).

Figure 14 presents the type of phase behavior,4 at l12 ) 0, as a function of the asymmetry parameter AP12 and of the interaction parameter k12, for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2). A given point in Figure 14 represents a specific binary system, e.g., C1 + C4, at a specific k12 value, e.g., k12 ) -1. At AP12 ) 0.2729 (see Table 2), Figure 14 shows a set of points corresponding to the system methane (1) + ethane (2), for which we observe a II f I f V evolution of phase behavior type when k12 decreases, i.e., when affinity increases. We observe the same evolution for the systems methane (1) + propane (2) [AP12 ) 0.1167] and methane (1) + n-butane (2) [AP12 ) 0.0614]. In general, we see that the evolution is III f II f I f V when k12 decreases. We stress that in Figure 14 a set of points falling on the same horizontal line corresponds to a binary system, e.g., C1 + C2, with a varying degree of affinity (depending on the k12 value). On the other hand, a set of points falling on the same vertical line corresponds to a series of 17 different binary systems (from C1 + C2 to C1 + C18) all at the same k12 value. Since on such vertical line asymmetry increases from the top to the bottom of Figure 14, we might expect a III f II f I f V evolution from bottom to top. The only exception to this expectation, in Figure 14, is the vertical line at k12 ) -1. The increase of asymmetry, i.e., the decrease of AP12, at constant k12, may or may not imply a decrease in

affinity. Affinity increases with the increase in the mixture attractive energy parameter “aMIX” and decreases with the increase in the mixture covolume parameter “bMIX”. When going, e.g., from C1 + C3 to C1 + C4, at, e.g., k12 ) -1, aMIX increases because aC4 is greater than aC3 (Table 2) and bMIX also increases since bC4 is greater than bC3 (Table 2). Therefore, it is not clear whether affinity increases or decreases for the transition C1 + C3 to C1 + C4. The solid line in Figure 14 corresponds to linear mixtures (Table 1/case I: ab mixture), which correspond either to type V (as we have previously found out) or to type I. Both types imply the absence of low temperature liquid-liquid immiscibility. On the other hand, type I and type II behaviors are shown in Figure 14 for aabgm mixtures (dashed line). We can represent, basically, the same information that Figure 14 provides, in terms of the more conventional parameters4 ξ12 and ζ12 rather than in terms of parameter AP12. We do that in Figure 15. It shows the type of phase behavior4 as a function of parameters ξ12, ζ12, and k12 for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2). Table 2 shows that, for this series, ξ12 and ζ12 are not independent: ζ12 decreases with the increase in ξ12, and also the absolute values of ξ12 and ζ12 both increase with the asymmetry of the methane (1) + n-alkane (2) system. In Figure 15, the system, e.g., C1 + C2 (ξ12 ) 0.20 and ζ12 ) 0.029, Table 2) corresponds to a set of points located on a straight line parallel to the k12 axis in Figure 15. Notice that the ξ12 and ζ12 coordinates of the points are the same for different constant k12 cuts. When we change the value of k12, then we produce changes on the labels on the points, i.e., on the mixture phase behavior type, but not on the ξ12 and ζ12 values, which depend only on pure compound parameters (eqs 21 and 22). Since Figures 14 and 15 basically provide the same information, the conclusions on the evolution of the phase behavior we have obtained from Figure 14 are the same than those that we could draw from Figure 15. To fix ideas, the information with regard to evolution in system type (I, II, III, IV, or V) that can be obtained from looking at Figures 3 and 4 is provided by a single horizontal line in the 2D chart of Figure 14 (bottom line) or by another single horizontal line, parallel to the k12 axis, in the 3D plot of Figure 15 (top). Similarly, a vertical line at k12 ) 0 in

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Figure 14. Type of phase behavior4 as a function of the asymmetry parameter AP12 and of the interaction parameter k12 for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2). Model: mSRK EOS with l12 ) 0. See Table 2. The solid line corresponds to linear mixtures (Table 1/case I: ab mixture) and the dashed line to aabgm mixtures (Table 1/case IIIgm).

Figure 15. Type of phase behavior4 as a function of parameters ξ12, ζ12, and k12 for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2). Model: mSRK EOS with l12 ) 0. See Table 2. The solid line corresponds to linear mixtures (Table 1/case I: ab mixture), and the dashed line corresponds to aabgm mixtures (Table 1/case IIIgm). Notice that ξ12 is greater than zero for every point in this figure (see Table 2).

Figure 14 (or the k12 ) 0 vertical plane in Figure 15) tells that type II behavior corresponds to the aabgm case for methane + heptane, methane + octane, and methane + octadecane, as we concluded previously from looking at Figures 3-5 and 12. However, information such as the temperature of the upper critical end point, or the slope of the liquid-liquid critical line, is lost in Figures 14 and 15. In these figures, in general, the aabgm mixtures correspond either to type I or to type II, and the transitions from the ab mixture (linear mixture) to the aabgm mixture which we see in Figures 14 and 15 are I(ab) f I(aabgm) or V(ab) f II(aabgm). J. R. Elliott17 has presented a 2D chart in terms of parameters ξ12 and ζ12 which shows the evolution of the system type (I, II, III, IV, or V), for a number of asymmetric systems. Elliott’s chart corresponds to the full SRK model (i.e., with temperature dependent pure compound parameters), and it is restricted to the case of k12 ) 0 (gm). Figure 16 presents the type of phase behavior4 for the systems methane (1) + n-heptane (2) and methane (1) + n-octadecane

Figure 16. Type of phase behavior4 as a function of parameters ζ12 and λ12 for the systems methane (1) + n-heptane (2) and methane (1) + n-octadecane (2). Model: mSRK EOS with l12 ) 0. See Table 2.

(2) at l12 ) 0 as a function of parameters ζ12 and λ12, which are also more conventional than those of Figure 14. Parameter k12 has no influence on parameter ζ12. On the other hand, λ12 increases with the increase in k12 (eqs 23 and 25). Notice that


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Figure 17. Type of phase behavior4 as a function of the asymmetry parameter AP12 and of the interaction parameter l12 for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2). Model: mSRK EOS with k12 ) 0.1. See Table 2.

Figure 19. Type of phase behavior4 as a function of the asymmetry parameter AP12 and of the interaction parameter l12 for CO2 (1) + n-alkane (2) systems from CO2 (1) + methane (2) to CO2 (1) + n-octadecane (2). Model: mSRK EOS with k12 ) 0.1. See Table 2.

the linear system is not located at λ12 ) 0. We observe that for both systems the evolution sequence is III f II f I f V when λ12 decreases, i.e., when affinity increases. This conclusion can also be obtained from Figures 14 and 15. In other words, basically, a more conventional plot such as Figure 16 does not provide more information than a plot such as Figure 14. Diagrams such as Figures 14 and 15 are useful to see at a glance the influence of the basic system asymmetry and of the interaction parameters on the qualitative phase behavior of the model. However they do not show relevant information such as the composition range of existence of the liquid state or the size of the region lying below the vapor-liquid critical line. In practical correlation of phase equilibria it is often needed to assign positive values to k12, while the required values for l12 may be negative or positive. In view of that, we generated Figure 17, which presents the type of phase behavior4 at k12 ) 0.1, as a function of the asymmetry parameter AP12 and of the interaction parameter l12 for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2). The l12 range in Figure 17 includes negative and positive values. In general, we see that, at constant AP12, the evolution

is III f II f I f V when l12 increases, i.e., when affinity increases. At l12 ) 0.5, we observe a I f V transition, that may seem unexpected, when going from the less asymmetric system C1 + C2 (AP12 ) 0.2729) to the more asymmetric system C1 + C3 (AP12 ) 0.1167). Such transition may be explained by the fact that when going from C1 + C2 to C1 + C3, at constant l12 and constant k12, both mixture parameters increase, i.e., aMIX and bMIX increase simultaneously, since aC3 > aC2 and also bC3 > bC2 (see Table 2). Figure 18 shows the type of phase behavior4 as a function of parameters ξ12, ζ12, and l12 for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2) at k12 ) 0.1. The information in Figure 18 is basically the same than that of Figure 17. Figures 17 and 18 relate to each other in a way analogous to that of Figures 14 and 15. Figure 19 shows the type of phase behavior4 as a function of the asymmetry parameter AP12 and of the interaction parameter l12 for CO2 (1) + n-alkane (2) systems, from CO2 (1) + methane (2) to CO2 (1) + n-octadecane (2). Figure 19 also corresponds to k12 ) 0.1. At constant AP12, we observe the evolution III f

Figure 18. Type of phase behavior4 as a function of parameters ξ12, ζ12, and l12 for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2). Model: mSRK EOS with k12 ) 0.1. See Table 2. Notice that ξ12 is greater than zero for every point in this figure (see Table 2).

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Figure 20. Type of phase behavior4 as a function of parameters ξ12, ζ12, and l12 for CO2 (1) + n-alkane (2) systems from CO2 (1) + methane (2) to CO2 (1) + n-octadecane (2). Model: mSRK EOS with k12 ) 0.1. See Table 2. Notice that ξ12 is greater than zero for every point in this figure (see Table 2).

II f I when l12 increases, i.e., when affinity increases. At l12 ) -0.5, we observe that when going from CO2 (1) + methane (2) (AP12 ) 1.5786, Table 2) to the more asymmetric system CO2 (1) + ethane (2) (AP12 ) 0.4307, Table 2) the transition is III f II which, while unexpected at first sight, could be regarded as acceptable because, again, both aMIX and bMIX increase simultaneously in the transition (because aC2 > aC1 and bC2 > bC1, Table 2). The 3D plot associated to Figure 19 is Figure 20. Notice that van Konynenburg and Scott4 restricted their work on the van der Waals equation of state to the case of a linear mixing rule for the covolume parameter (i.e., l12 ) 0). In the previous paragraphs, we identified three transitions in system type that seemed unexpected at first sight. For such transitions, it was not clear whether affinity increased or decreased with the increase in the degree of asymmetry, at constant values of the interaction parameters. This prompted us to consider, for representing changes in phase behavior type, parameters which, while being independent from the binary system composition, would be related to the mixture attractiveness or repulsiveness. The most natural choices are probably the well-known crossed attractive and repulsive parameters, i.e., a12 and b12, which are defined as follows: a12 ) √a1a2(1 - k12)

(26)

b1 + b2 (1 - l12) (27) 2 From eq 26, we see that a12 increases if the attractiveness of the system increases, regardless the reason for such increase, i.e., the increase of a1 and/or the increase of a2 and/or the increase of the factor (1 - k12). Analogously, eq 27 tells that b12 increases if the repulsiveness of the system increases, which happens if b1 increases and/or if b2 increases and/or if the factor (1 - l12) increases. Notice that a12 appears in eq 3 and that b12 appears in eq 4. Figure 21 shows the type of phase behavior4 as a function of the crossed covolume parameter b12 and of the crossed attractive parameter a12 for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2), at l12 ) 0. The calculation results in Figure 21 are the same b12 )

than those in Figure 14, i.e., the way for presenting the results is the only difference between Figures 14 and 21. For a given value of a12 (i.e., for a given vertical line in Figure 21), the increase of b12 implies an evolution V f I f II f III, i.e., at constant attractiveness the system follows the evolution V f I f II f III when the repulsiveness increase, i.e., when affinity decreases. We can also see that for a given value of b12 (i.e., for a given horizontal line in Figure 21) the decrease of a12 implies also an evolution V f I f II f III, i.e., at constant repulsiveness the systems follow the evolution V f I f II f III when the attractiveness decreases, i.e., when affinity decreases. The pattern is the same anywhere in Figure 21, independently on the constant value set for b12 or on the constant value set for a12. More synthetically, we see for Figure 21 that a decrease in affinity is associated to the evolution V f I f II f III, regardless the way in which affinity is decreased. Figure 22 presents the type of phase behavior4 as a function of the crossed covolume parameter b12 and of the crossed attractive parameter a12 for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2) at k12 ) 0.1. Figures 22 and 17 differ only in the presentation of the same calculation results. An analysis of Figure 22 analogous to the one we carried out for Figure 21 leads to the same conclusions stemming from Figure 21. Figure 23 shows the type of phase behavior4 as a function of the crossed covolume parameter b12 and of the crossed attractive parameter a12 for CO2 (1) + n-alkane (2) systems from CO2 (1) + methane (2) to CO2 (1) + n-octadecane (2) at k12 ) 0.1. Figure 23 corresponds to an alternative way of presenting the same calculation results of Figure 19. An analysis of Figure 23 performed as the one for Figure 21 leads us to conclude that a decrease in affinity is associated to the evolution I f II f III, regardless of the way in which affinity is decreased, i.e., the evolution we observe, for binary CO2 (1) + n-alkane (2) systems, is I f II f III if attractiveness decreases or if repulsiveness increases. In summary, we see, on one hand, that the pattern observed is the same for Figures 21-23 and, on the other hand, that the qualitative effect of increasing the system attractiveness is the


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Figure 21. Type of phase behavior4 as a function of the crossed covolume parameter and of the crossed attractive parameter for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2). Model: mSRK EOS with l12 ) 0. See Table 2. The solid line corresponds to linear mixtures (Table 1/case I: ab mixture) and the dashed line to aabgm mixtures (Table 1/case IIIgm). The systems, pure compound parameter values, and range of variation of parameter k12 in this figure are the same as those in Figure 14.

of EOS type models, when applied to sets of binary systems, can complement the use of the well-known4 parameters ξ12, ζ12, and λ12. It is important to stress that for the systems and parameter values here studied we have never observed type IV behavior. We have not intended to find it either. However, a finer division of the k12 domain, or of the l12 domain, might show the appearance of such a type, for some binary systems. This appearance would add an additional intermediate type to the evolution of the phase behavior of a given system as a function of a given interaction parameter. Remarks and Conclusions Figure 22. Type of phase behavior4 as a function of the crossed covolume parameter and of the crossed attractive parameter for methane (1) + n-alkane (2) systems from methane (1) + ethane (2) to methane (1) + n-octadecane (2). Model: mSRK EOS with k12 ) 0.1. See Table 2. The systems, pure compound parameter values, and range of variation of parameter l12 are in this figure the same than those in Figure 17.

Figure 23. Type of phase behavior4 as a function of the crossed covolume parameter and of the crossed attractive parameter for CO2 (1) + n-alkane (2) systems from CO2 (1) + methane (2) to CO2 (1) + n-octadecane (2). Model: mSRK EOS with k12 ) 0.1. See Table 2. The systems, parameter values, and range of variation of parameter l12 are in this figure the same than those in Figure 19.

same than that of decreasing the system repulsiveness, at least for the cases studied here. It is clear from the previous analysis that the use of the crossed parameters a12 and b12 for systematically studying the behavior

In this work, we studied the fluid phase equilibrium properties of a model system, named a linear mixture (LM) or linear system (LS). It corresponds to linear mixing rules used in EOS type models. To better understand the behavior of LSs, we also considered in this work quadratic mixing rules (QMRs), i.e., nonlinear systems (NLSs). This made possible to visualize the transition between NLSs and LSs. We studied LSs and NLSs of varying asymmetry using the mSRK model. We did not account in this work for the possible appearance of solid phases. Our main findings are that, in comparison with typical quadratic systems (TQSs), linear systems [a] do not have low temperature liquid-liquid (LL) immiscibility, [b] have vaporliquid critical lines that extend to higher pressures, and [c] have access to the liquid state in a wider composition range. Feature a makes linear systems analogous to ideal solutions. The absence of low temperature LL immiscibility for LSs is due to the high value of the LS attractive energy parameter. By TQSs we mean the use of quadratic mixing rules with positive kij and zero lij interaction parameters. We observed that, under the conditions of the present study, LSs correspond to types V or I in the van Konynenburg and Scott4 classification of phase equilibrium behavior. From observing different kinds of phase equilibrium diagrams, we concluded that roughly, at least qualitatively, the effect of increasing the mixture attractive parameter is the same as the effect of decreasing the mixture repulsive parameters. In both cases, affinity is increased. Thus, roughly, a desired modification of the fluid phase equlibria corresponding to the adopted EOS model could be, in principle, obtained by changing the affinity level of the system, regardless the way in which

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such affinity change is imposed. We reached this conclusion by using the mSRK EOS, for which the partial derivatives of pressure with respect to the mixture energy (a) and covolume (b) parameters (at constant volume, temperature, and composition) are well-behaved. All these conclusions stem from our observation of the behavioral patterns of a large but still limited number of systems. Our conclusions are restricted to the model (mSRK), the systems and the ranges of conditions here studied. For assessing more thoroughly the behavior of equations of state based on the one-fluid approach, or for defining new models, besides studying the departure of a given mixing rule with respect to any reference that the user could regard as meaningful, he/she could also consider the departure from the linear mixture reference, since this last reference seems to always imply the absence of low temperature liquid-liquid immiscibility. To understand the effect of interaction parameters on the vapor-liquid equilibrium, it is important not only to consider how the changes in such parameters influence the size of vapor-liquid coexistence regions, but also to look at how the composition range of liquid homogeneity, or of existence of the liquid state, changes. Finally, our results could be useful when intending to fit complex phase behavior, such as LLE, when using models of the equation of state type. Acknowledgment We are grateful to Diego Nun˜ez for his continued support in software development, and for their financial support, to Fundaçaõ Arauca´ria (Brazil), Consejo Nacional de Investigaciones Cientı´ficas y Tećnicas de la Repu´blica Argentina (CONICET), Universidad Nacional del Sur (U.N.S., Arg.), Universidad Nacional de Co´rdoba (U.N.C., Arg.), and Agencia Nacional de Promocioń Cientı´fica y Tecnolo´gica (ANPCyT, Arg.). We are also grateful to one of the reviewers of the present work, whose input led us to include in this manuscript a number of additional figures (set in terms of van Konynenburg and Scott4 parameters) useful for improving the understanding of the behavior of EOS type models. Literature Cited (1) Valderrama, J. O. The State of the Cubic Equations of State. Ind. Eng. Chem. Res. 2003, 42, 1603–1618.

(2) Zabaloy, M. S.; Brignole, E. A.; Vera, J. H. A Conceptually New Mixing Rule For Cubic And Non-Cubic Equations Of State. Fluid Phase Equilib. 1999, 158-160, 245–257. (3) Zabaloy, M. S.; Brignole, E. A.; Vera, J. H. A Flexible Mixing Rule Satisfying the Ideal-Solution Limit for Equations of State. Ind. Eng. Chem. Res. 2002, 41, 922–930. (4) van Konynenburg, P. H.; Scott, R. L. Critical Lines and Phase Equilibria in Binary Van Der Waals Mixtures. Philos. Trans. R. Soc. London 1980, 298, 495–540. (5) Soave, G. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 1972, 27, 1197–1203. (6) Smith, J. M.; Van Ness, H. C.; Abbot, M. M. Introduction to Chemical Engineering Thermodynamics, 5th int. ed.; McGraw Hill: New York 1996, p 482. (7) DIPPR 801; Evaluated Process Design Data, Public Release (2003); American Institute of Chemical Engineers, Design Institute for Physical Property Data, BYU-DIPPR, Thermophysical Properties Laboratory: Provo, UT, 2003. (8) Cismondi, M.; Michelsen, M. L. Global phase equilibrium calculations: Critical lines, critical end points and liquid-liquid-vapour equilibrium in binary mixtures. J. Supercrit. Fluids 2007, 39, 287–295. (9) Cismondi, M.; Michelsen, M. L. Automated Calculation Of Complete Pxy And Txy Diagrams For Binary Systems. Fluid Phase Equilib. 2007, 259, 228–234. (10) Cismondi, M.; Michelsen, M. L.; Zabaloy, M. S. Automated Generation of Phase Diagrams for Binary Systems with Azeotropic Behavior. Ind. Eng. Chem. Res. 2008, 47, 9728–9743. (11) Allgower, E. L.; Georg, K. Numerical path following. In Handbook of Numerical Analysis; Ciarlet, P. G., Lions, J. L., Eds.; North-Holland, 1997; Vol. 5. (12) Koningsveld, R.; Stockmayer, W. H.; Nies, E. Polymer Phase Diagrams; Oxford University Press: Oxford, NY, 2001, p 35. (13) Elliott, J. R.; Lira, C. T. Introductory Chemical Engineering Thermodynamics; Prentice-Hall PTR: Upper Saddle River, NJ, 1999, pp 445-458. (14) Sandler, S. I.; Orbey, H. Mixing and Combining Rules. In: Equations of State for Fluids and Fluid Mixtures; Sengers, J. V., Kayser, R. F., Peters, C. J., White, H. J., Jr., Eds.; Elsevier: Amsterdam, 2000, p 324. (15) Peng, D. Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59–64. (16) SPECS V5.24 - IVC-SEP; Institut for Kemiteknik. Danmarks Tekniske Universitet: Lyngby, Denmark, 2006; www.ivc-sep.kt.dtu.dk. (17) Lee, S., Ed. Encyclopedia of Chemical Processing; Taylor & Francis: London, 2005-2009.

ReceiVed for reView June 5, 2009 ReVised manuscript receiVed January 28, 2010 Accepted February 3, 2010 IE900927F

Phase Behavior of Linear Mixtures in the Context of Equation of State

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