Special Algebraic Properties of Two-Parameter Equations of State

1598

Ind. Eng. Chem. Res. 1998, 37, 1598-1612

Special Algebraic Properties of Two-Parameter Equations of State: Homogeneous Azeotropy Marcelo S. Zabaloy* and Esteban A. Brignole PLAPIQUI/UNS/CONICET, cc 717, 8000 Bahıá Blanca, Argentina

Two-parameter equations of state (EOSs), such as the Peng-Robinson and Redlich-Kwong EOSs, or their variants, are widely used for the simulation of industrial processes. In this work, some algebraic properties of two-parameter cubic and noncubic equations of state and of their translated forms are investigated. For saturated pure compounds, translated and untranslated forms describe the same universal curves for pairs of certain dimensionless variables. In the present work we provide general parameters, not previously given in the literature, representing the universal relations set by the van der Waals (VdW), Redlich-Kwong (RK), and modified Carnahan-Starling-van der Waals (MCSV) equations of state. Direct calculation procedures, which make use of the pure compound universal relations, are proposed for the calculation of properties at infinite dilution, leading to the determination of regions of azeotropy. For mixing rules which do not depend on density, it has been found that the universal relations which are valid for saturated pure compounds are also valid for homogeneous azeotropes of any number of components. This finding can be exploited within any existing algorithm designed to find azeotropic points, in order to get faster and more reliable outputs. On the basis of the identity between pure compound and azeotropic universal relations, noniterative methods of calculation are proposed to establish the existence of simple or multiple azeotropy and of pseudocritical points. Simplified calculation procedures are also proposed to obtain values of binary parameters from experimental azeotropic information. All numerical examples have been done using the Peng-Robinson EOS (PR EOS). Introduction

Universal Relations for Pure Compounds

Distillation is the most widely used separation process in the chemical and petrochemical industry, and it will remain the method of choice for the large-scale separation of liquid mixtures (Widagdo and Seider, 1996). The design and simulation of distillation processes is often performed with the aid of equation of state models capable of describing the equilibrium relations between temperature, pressure, and composition for the mixtures under consideration. In industrial practice simple equations of state coupled to flexible mixing rules are normally preferred. The occurrence of azeotropes governs the distillation schemes to be applied. Therefore, the computation of conditions of homogeneous azeotropy is of great interest. In the present work, we first study the restrictions imposed by simple equations of state on pure compounds at saturation and show how they can be used for the direct computation of properties at infinite dilution. Then, we study the relation between the pure compound constraints and the state of azeotropy, when using mixing rules which do not depend on density. This relation, besides its theoretical value, can be used for the actual calculation of binary azeotropes, including the case of polyazeotropy. Additionally, it can be used to upgrade existing algorithms for finding conditions of binary or multicomponent azeotropy, with regard to speed and reliability.

The family of van der Waals two-parameter cubic equations of state (EOSs) establishes universal relations among certain dimensionless variables for pure compounds at saturation (Soave, 1986; Zabaloy and Vera, 1996, 1997). In this section we show in a general way that similar restrictions are associated also with noncubic EOSs, including translated forms. Consider the following cubic equation of state:

* To whom correspondence should be addressed. Phone: 5491-861521/24. Fax: 54-91-88-3764. E-mail: przabalo@ criba.edu.ar.

P)

RT (v + c) - (b + d) a (v + c)2 + k1(b + d)(v + c) + k2(b + d)2 cubic EOS (1)

In the eq 1 P is the absolute pressure, R is the universal gas constant, T is the absolute temperature, and v is the molar volume. When eq 1 is applied to pure compounds, the parameters a, b, c, and d are specific of the compound considered and, in general, they are functions of temperature. Equation 1 is representative of a number of well-known EOSs. The parameters k1 and k2 are integer constants, which depend on the chosen EOS. The values of k1 , k2, c, and d for three of the most popular EOSs are given in Table 1. If c ) d ) 0, eq 1 reduces to a two-parameter cubic equation of state. If d ) 0, eq 1 represents a translation of the corresponding two-parameter EOS along the volume axis written in the format adopted by Soave (1984). In this case, only if c ) 0 is the parameter b the true covolume, i.e., the volume at infinite pressure. If d ) c, eq 1 represents also a translation along the

S0888-5885(97)00742-2 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/17/1998

Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1599 Table 1. Parameters for Cubic Equations of State EOS

k1

k2

c

d

van der Waals Redlich-Kwong (1949) Peng-Robinson (1976)

0 1 2

0 0 -1

0 0 0

0 0 0

volume axis but written in the format of Peńeloux and Rauzy (1982). In this case, the parameter b is the true covolume for any value of the parameter c (Zabaloy and Brignole, 1997). The translated MCSV (Aly and Ashour, 1994) EOS is the following:

P)

[

]

RT 1 + y + y2 - y3 a 3 v+c (1 - y) (v + c)2

where φ0i is the fugacity coefficient of the pure compound i. The fugacity coefficient can be expressed in terms of the dimensionless variables defined by eqs 7-9:

ln φ0i )

*

c ln φ0i B + B ) F(V*) - AG(V*) b+d

(3)

In eq 3 we have omitted a factor of 1/4 which is absorbed by the constants b and d (see the work by Aly and Ashour, 1994). In this way b becomes the value of the volume at infinite pressure when c ) d ) 0, as in the case of eq 1. Equations 1 and 2 can be respectively rewritten as follows:

B)

B)

A 1 V* - 1 V 2 + k V + k * 1 * 2

[

2

1 1+y+y -y V* (1 - y)3

]

3

-

A V*2

cubic EOS (4)

MCSV EOS (5)

where

y)

1 V*

(6)

In eqs 4-6 the dimensionless variables A, B, and V* are defined as follows:

P(b + d) RT

(7)

a RT(b + d)

(8)

v+c b+d

(9)

B) A)

V* )

Equations 4 and 5 can be expressed in the following general form:

B ) f(V*) - Ag(V*)

(10)

where the forms of the functions f and g depend on the chosen equation of state and may or may not correspond to a cubic EOS. The expression for the fugacity coefficient of the pure compound can be found in standard textbooks (Prausnitz et al., 1986)

ln φ0i )

P 1 Pv Pv - ) ∂v - ln + -1 ∫v∞(RT v RT RT

(11)

(13)

where

F(V*) )

b+d v+c

∂V* - ln BV* + BV* - 1 -

After we introduce eq 10 into eq 12, the following equation can be derived:

where

y)

)

c B (12) b+d

MCSV EOS (2)

(

∫V∞ B - V1*

[

]

∫V∞ f(V*) - V1* *

G(V*) )

∂V* + V*f(V*) - ln V* - 1

∫V∞g(V*) ∂V* + V*g(V*) *

(14) (15)

At conditions of vapor-liquid equilibrium for the pure compound, the temperature, the pressure, and the fugacities (or fugacity coefficients) of both pure phases are respectively equal. In addition, a, b, c, d, A, and B have respectively the same values for both phases. Therefore, we write

B ) f(V*V) - Ag(V*V)

(16)

B ) f(V*L) - Ag(V*L)

(17)

F(V*V) - AG(V*V) ) F(V*L) - AG(V*L)

(18)

Superscripts L and V correspond to the liquid and vapor phases, respectively. The system of nonlinear equations (16)-(18) sets a universal relation among the variables A, B, V*L, and V*V. This system has 1 degree of freedom usually chosen as A. The variable A depends only on temperature. The form of the universal relation depends on the particular equation of state chosen. Figures 1 and 2 show the universal relations associated with the PR, RK, VdW, and MCSV equations of state. These figures are valid for the original EOSs and for any of their translated forms. There is a minimum possible value for the variable A, i.e., the critical value Ac. For values of A less than Ac there is no van der Waals loop for the pressure as a function of the molar volume (or for B vs V*). For checking any computation scheme using the results of the present work, we provide in Appendix A some points which satisfy the pure compound equilibrium equations for the PR, RK, VdW, and MCSV EOSs and their translations. The universal relation between A and B has been studied by Soave (1986) for the case of two-parameter cubic equations of state. Note that the cubic nature of the chosen EOS is not a necessary condition for the existence of universal relations. Soave (1986) correlated the variable B as a function of A, using as many adjustable parameters as needed, to replace iterative calculations at saturation by a direct calculation procedure without reducing the quality of the calculated result. The parameters are general for the chosen

1600 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998

(1997) for the particular case of the (cubic and untranslated) Peng-Robinson EOS. If c ) 0, there is also a universal relation between A and φ0i at saturation (eq 13). In order to facilitate the next discussion, we will assume that for the chosen equation of state the expressions which explicitly relate the variables A, B, V*L, and V*V are available, so that we can write

Figure 1. Universal relationship between B and A for pure saturated compounds for several equations of state.

B ) fB(A)

(19)

V*L ) fL(A)

(20)

V*V ) fV(A)

(21)

where the functions fB, fL, and fV are known explicit functions corresponding to the pure state at saturation, for the chosen equations of state. Soave (1986) has provided the function fB of eq 19 for the VdW, RK, and PR EOSs, while Zabaloy and Vera (1997) have given the functions fL and fV of eqs 20 and 21 for the PR EOS. Soave (1986) and Zabaloy and Vera (1997) have also provided functional forms for choices of independent variables different from A. We give the explicit functions fL and fV in appendices B-D for the VdW, RK, and MCSV EOSs, respectively. In appendix D we also provide the function fB for the MCSV EOSs together with the explicit relations useful for calculating A as a function of B. This last calculation is useful for constantpressure calculations or for regressing pure compound parameters (Soave, 1986). The conclusions of the present section are of practical interest for any calculation involving saturated pure compounds. We show in the following sections that the pure compound universal relations can be used to determine the existence of azeotropes, without the need of iterative calculations. Homogeneous Azeotropes A homogeneous azeotrope is a multicomponent liquid which boils without change in composition; i.e., the only liquid phase has the same composition as the vapor phase. For a general multicomponent system the isofugacity condition, i.e., the necessary condition of vapor-liquid equilibrium, is written as follows:

yiφVi ) xiφLi

Figure 2. Universal relationship between V* and A for pure saturated compounds for several equations of state.

equation of state and need to be obtained only once. Soave (1986) also provided parameters for calculating A as a function of B. Analogous correlations can be obtained for, e.g., the relations A - V*V and A - V*L, to make possible the explicit calculation of V*V and V*L from the value of A, i.e., from the value of the temperature. This has been done by Zabaloy and Vera

i ) 1, N

(22)

where N is the number of components, yi and xi are the vapor- and liquid-phase mole fractions of component i, respectively, and φVi and φLi are respectively the vaporand liquid-phase fugacity coefficients of component i, in the corresponding vapor and liquid mixtures. The fugacity coefficient is a function of composition, temperature, and mixture molar volume. The relative volatility of component i with respect to component j is written as

Rij )

φLi φVj φVi φLj

(23)

At the azeotropic point, we write

yi ) x i

i ) 1, N (azeotrope)

(24)

Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1601

From eqs 22 and 24, we write

φVi ) φLi

i ) 1, N (azeotrope)

(25)

The fugacity of a mixture is given by the equation (Smith and Van Ness, 1975) N

ln φmix )

zi ln φi ∑ i)1

(26)

where zi is the mole fraction of component i, φi is the fugacity coefficient of component i in the mixture, and φmix is the fugacity of the mixture. From eqs 24-26 it can be shown that at the azeotropic point the mixture fugacities of the vapor and liquid phases are equal:

φVmix ) φLmix ) φaz (azeotrope)

(27)

Equation 27 is a general result valid for any multicomponent homogeneous azeotrope, and it can replace one of the N equations (eq 25). Equation 27 is equivalent to equation “(1)” of the work of Wang and Whiting (1986), which was written in terms of the Helmholtz energies of the azeotropic phases. We use eq 27 in further sections. At this point it is convenient to set some definitions. Consider a mixture of composition z. The ith element of vector z is zi, i.e., the mole fraction of component i. We define the pseudo pressure, pseudo liquid-phase molar volume, and pseudo vapor-phase molar volume, at a given multicomponent composition z and temperature T, as respectively the pressure, liquid-phase molar volume, and vapor-phase molar volume which satisfy eq 27 and the equation of state written for the mixture (e.g., eq 1) but do not necessarily satisfy the remaining (N - 1) azeotropic equilibrium equations (25). In terms of the number of iteration variables the calculation of the pseudo pressure has the same cost as the calculation of a pure compound vapor pressure. To clarify the concept, we give a calculation procedure for the pseudo pressure in appendix E (procedure E0). A definition analogous to that of the pseudo pressure can be written for the pseudo temperature if T is changed by pressure as the independent variable. The pseudo vapor (liquid)-phase fugacity coefficient of component i in the mixture is obtained from the expression of φi evaluated at temperature T, at composition z, and at the pseudo vapor (liquid)-phase molar volume. The pseudo relative volatility of component i with respect to component j is calculated from the pseudo fugacity coefficients through eq 23. These definitions will prove useful in the following sections. Existence of Homogeneous Azeotropes for Binary Mixtures Consider the product Iaz ) (R∞1,2 - 1)(R01,2 - 1) for a binary mixture of components 1 and 2 at a temperature T. The variable R∞1,2 is the relative volatility of component 1 with respect to component 2 when component 1 is infinitely diluted in component 2, and R01,2 is the relative volatility of component 1 with respect to component 2 when component 2 is infinitely diluted in component 1. If the product Iaz ) (R∞1,2 - 1)(R01,2 - 1) is negative, then the model predicts, in general, the

existence of an uneven number of azeotropes at temperature T for the binary system. If the product Iaz ) (R∞1,2 - 1)(R01,2 - 1) is positive, then the model predicts either the absence of an azeotrope or the existence of an even number of azeotropes at temperature T for the binary system. If the product Iaz ) (R∞1,2 - 1)(R01,2 - 1) is equal to zero, then there is azeotropy at infinite dilution at one or both ends of the composition range, with the number of azeotropes at intermediate concentration being even, uneven, or zero. In fact, the case of multiple azeotropy at a given temperature is rare. A review on this subject has been provided by Christensen and Olson (1992). Therefore, for the majority of cases, a negative value for the product Iaz ) (R∞1,2 - 1)(R01,2 1) implies the existence of an azeotrope, a positive value implies the absence of an azeotrope, and a zero value implies incipient azeotropy at one end of the composition range. At infinite dilution, for a binary system at temperature T, the vapor-liquid equilibrium pressure and phase densities are respectively equal to the vapor pressure and phase densities of the concentrated component in a saturated pure state at temperature T. Hence, the infinite-dilution relative volatilities are given by the expressions V,∞ R∞1,2 ) φL,∞ 1 /φ1

(28)

L,0 R01,2 ) φV,0 2 /φ2

(29)

and

where φL,∞ and φV,∞ are the fugacity coefficients of 1 1 component 1 in the liquid and vapor phase, respectively, when component 1 is infinitely diluted in component 2; V,0 and φL,0 2 and φ2 are the fugacity coefficients of component 2 in the liquid and vapor phase, respectively, when component 2 is infinitely diluted in component 1. For the derivation of eq 28, it was considered that the fugacity coefficients in the vapor and liquid phases, for the concentrated component, i.e., component 2, are equal. Similarly, the same consideration was used for component 1 when deriving eq 29. The fugacity coefficient of a component in a mixture depends on the following variables: mixture molar volume, temperature, composition, parameters of pure compounds, and interaction coefficients. For a binary mixture in vapor-liquid equilibrium at infinite dilution, the composition dependence disappears and the molar volume is equal to the molar volume of the concentrated component in a pure saturated state. Then, if explicit functions relating the variables A, B, V*L, and V*V for pure saturated compounds (e.g., eqs 19-21) are available, they can be used to calculate the product Iaz ) (R∞1,2 - 1)(R01,2 - 1) in a direct way. Thus, it can be established whether the model predicts the existence of azeotropes, in a range of temperatures, without having to perform any iterative calculation. Existence of Homogeneous Azeotropes Using the PR EOS The original Peng-Robinson (1976) equation of state (PR EOS) is eq 1 with the parameters of Table 1. The


classical quadratic mixing rules (QMR) which are normally coupled to this EOS to describe the behavior of mixtures are N

xibi ∑ i)1

(30)

∑ ∑xixj(aiaj)0.5(1 - kij) i)1 j)1

(31)

bmix ) and N N

amix )

N is the number of components, xi is the mole fraction of component i, ai, and bi are the energetic and covolume parameters of pure component i, respectively, amix and bmix are the energetic and covolume parameters of the mixture, respectively, and kij is a symmetric binary parameter, which has a value of 0 when i ) j. Since the molar volume of the mixture does not appear in the expressions of amix and bmix, the quadratic rules are mixing rules which do not depend on density. The fugacity coefficient of component i in the mixture is φi, and it is given by the expression

ln φi )

Figure 3. Product Iaz ) (R∞1,2 - 1)(R01,2 - 1) as a function of temperature for the system carbon dioxide (1)-ethane (2), using the QMR-PR-EOS.

bi

(BV* - 1) - ln B(V* - 1) bmix

[

N

2

A 2x2

∑ xkaik k)1

-

amix

bi

][

bmix

ln

]

V* + 1 + x2

V* + 1 - x2

(32)

where B, A, and V* are given by eqs 7-9 with c ) 0, d ) 0, a ) amix, and b ) bmix. The expressions for the pure compound parameters a and b used in this work are those of the original Peng-Robinson (1976) model, for which a is a function of temperature, acentric factor, and critical temperature and pressure, while b depends only on the critical coordinates. From eq 32, it can be shown that the fugacity coefficient of component 1 when component 1 is infinitely diluted in component 2 has the form

ln φ∞1 ) A2

b1 (B V - 1) - ln B2(V*,2 - 1) b2 2 *,2

[( )

2x2

2

a1 a2

1/2

(1 - k12) -

][

]

b1 V*,2 + 1 + x2 ln (33) b2 V*,2 + 1 - x2

where k12 is a symmetric binary interaction parameter. Similarly, the fugacity coefficient of component 2 when component 2 is infinitely diluted in component 1 has the form

ln φ02 ) A1

b2 (B V - 1) - ln B1(V*,1 - 1) b1 1 *,1

[( )

2x2

2

a2 a1

1/2

(1 - k12) -

][

]

b2 V*,1 + 1 + x2 ln (34) b1 V*,1 + 1 - x2

From eqs 33 and 34, it is evident that the fugacity coefficients at infinite dilution in binary mixtures can be calculated explicitly from the pure compound properties and from the binary parameter k12. Therefore, a

noniterative calculation procedure can be used to compute the product Iaz ) (R∞1,2 - 1)(R01,2 - 1) at a given temperature T, using the QMR-PR-EOS. Such a calculation procedure is identified as procedure E1 and given in detail in appendix E. Figure 3 shows a plot of the product Iaz ) (R∞1,2 1)(R01,2 - 1) as a function of temperature for the system carbon dioxide (1)-ethane (2), generated by the PR EOS with quadratic mixing rules. The values of the pure component and interaction parameters are given in appendix F. It can be seen in Figure 3 that since Iaz is negative, the model predicts the existence of azeotropes throughout the temperature range of Figure 3 for this system. We stress that Figure 3 has been generated without performing any iterative calculation. Threshold of Homogeneous Azeotropy for Binary Mixtures The search for the conditions at which incipient azeotropy takes place is important to define suitable operating conditions for separation units. In the present section we show that with quadratic mixing rules, coupled either to cubic or to noncubic EOSs, it is possible to compute the regions of binary azeotropy, from only the pure compound data, without performing iterative calculations. We illustrate this for the case of the QMR-PR-EOS. For a given binary mixture we assume that, at a given temperature T, the model predicts at most one azeotrope. In other words, we assume that the relative volatility R12 is a monotonically increasing (or decreasing) function of composition at constant temperature. Then, for a binary mixture, the threshold of azeotropy takes place when at infinite dilution the relative volatility between the two components is unity. Thus, the conditions for incipient azeotropy can be written as


ln φL,∞ ) ln φV,∞ 1 1

(35)

for the infinite dilution of component 1 in component 2 and V,0 ln φL,0 2 ) ln φ2

(36)

for the infinite dilution of component 2 in component 1. From eqs 33 and 35, we obtain the value k∞12, i.e., the value of k12 which gives incipient azeotropy when component 1 is infinitely diluted in component 2.

k∞12 ) 1 -

()

1 a2 2 a1

[(

1/2

) ]

b1 SA,2 - SB,2 b1 2x2 b2 + A2 SC,2 b2

(37)

The functions SA, SB, and SC are given by the expressions

SA ) B(V*V - V*L)

(38)

( )

SB ) ln and

V*V - 1

[

SC ) ln

(39)

V*L - 1

Figure 4. Regions of azeotropy for the system carbon dioxide (1)ethane (2), using the QMR-PR-EOS.

]

(V*V + 1 + x2)(V*L + 1 - x2)

(V*V + 1 - x2)(V*L + 1 + x2)

(40)

where B and V* are given by eqs 7 and 9 with c ) 0 and d ) 0. The functions SA,2, SB,2, and SC,2 are the functions SA, SB, and SC evaluated for the saturated pure compound 2. Similarly, we obtain the value k012, i.e., the value of k12 which gives incipient azeotropy when component 2 is infinitely diluted in component 1, from the combination of eqs 34 and 36.

k012 ) 1 -

()

1 a1 2 a2

[(

1/2

) ]

b2 SA,1 - SB,1 b2 2x2 b1 + A1 SC,1 b1

(41)

The functions SA,1, SB,1, and SC,1 are the functions SA, SB, and SC evaluated for the saturated pure compound 1. The noniterative calculation procedure to obtain the values of k∞12 and k012, at a given temperature T, is given in appendix E as procedure E2. Figure 4 shows the QMR-PR-EOS regions of azeotropy for the system carbon dioxide (1)-ethane (2) and the vapor pressure curves for the two components in the pure state as calculated from the PR-EOS, in a range of temperatures. There is no azeotrope formation for values of the interaction coefficient k12 located between the lines of k∞12 and k012. At the line labeled k∞12 the system pressure is equal to the vapor pressure of the pure component 2, i.e., ethane. Since, as is shown in Figure 4, the vapor pressure of pure ethane (2) is less than the vapor pressure of pure carbon dioxide (1), the azeotropes located below the line of k∞12 are minimum pressure azeotropes. At the k012 line the system pressure is equal to the vapor pressure of pure carbon dioxide (1). Because pure carbon dioxide (1) has always a vapor pressure higher than that of pure ethane (2),

all azeotropes above the k012 line are azeotropes of maximum pressure. The line labeled k12 corresponds to the temperature-dependent interaction coefficient of eq F-1 of appendix F, which gives a good reproduction of the experimental azeotropic compositions of Fredenslund and Mollerup (1974; Knapp et al., 1982, p 527). Since the k12 line is located above the k012 curve, the temperature dependence of eq F-1 of appendix F for the system carbon dioxide (1)-ethane (2) gives azeotropes of maximum pressure. Figure 4 shows that a value of zero for the interaction parameter gives azeotropic behavior at lower temperatures and nonazeotropic behavior at higher temperatures. The k∞12 and k012 lines of Figure 4 have been obtained following the calculation procedure E2, which is noniterative. The pure compound vapor pressures have been obtained directly [Soave (1986), eq 19]. Figure 4 is useful to classify ranges of k12 from the point of view of azeotrope formation. For a system of known k12 expression, the temperature of incipient azeotropy can be obtained by finding the values of temperature at which the product Iaz of the previous section is zero. Though the model used in previous examples has mixing rules which do not depend on density, noniterative computations at infinite dilution are also possible for mixing rules which do depend on density. The proposed direct procedures at infinite dilution are based on the fact that the system pressure and the molar volumes of the phases, needed to calculate the fugacity coefficients, are identical with those of the concentrated compound in a saturated pure state. Such a boundary condition is met by any well-behaved mixing rule, regardless of its dependence from density. Universal Relations at the Multicomponent Azeotropic Point In the previous sections we have developed a technique to determine if the chosen model predicts the


existence of azeotropes and a method to find the regions of azeotropy. These techniques do not require iterative calculations and are based on the existence of universal relations between dimensionless variables for pure compounds at saturation. In this section, we demonstrate that cubic and noncubic translated or untranslated EOSs such as eqs 1 and 2 set also universal relations for multicomponent homogeneous azeotropes, which are identical with those corresponding to the pure compound state. In practical calculations it is very often assumed that the form of the equation of state for the pure fluid is valid also for the mixture, but the parameters a, b, c, and d, are now dependent on composition, on temperature, on interaction parameters, and on the temperature-dependent pure compound parameters of the components present in the mixture but not on the mixture density. An example of mixing rules which do not depend on density is the QMR-PR-EOS. The working assumption of this approach implies that eq 10 is also valid for the mixture. Thus, we write

Bmix ) f(V*,mix) - Amixg(V*,mix)

(42)

where f and g are the same functions of eq 10. From classical thermodynamics the expression for the fugacity coefficient of component i in the mixture, φi, becomes defined once the functions f and g are defined (e.g., eq 32):

ln φi ) fφ[T,vmix,(xk,ak,bk,ck,dk, k ) 1, N), interaction parameters] (43) On the other hand, from exact thermodynamics, the relation between the mixture fugacity coefficient and the mixture equation of state is identical with that corresponding to a pure compound, i.e., eq 11 (Smith and Van Ness, 1975). Thus, for a mixture, we write

ln φmix )

∫v∞

mix

(

)

Pvmix P 1 + ∂vmix - ln RT vmix RT Pvmix - 1 (44) RT

The integration involved in eq 44 is performed not only at constant temperature but also at constant composition. Starting from eqs 42 and 44 and following steps analogous to those which led to eq 13, it can be shown that eq 13 is also valid for a mixture, as long as the mixture parameters a, b, c, and d do not depend on the volume of the mixture. Hence, we write

cmix ) F(V*,mix) ln φmixBmix + Bmix bmix + dmix AmixG(V*,mix) (45) where F and G are the functions defined by eqs 14 and 15. For a multicomponent vapor-liquid equilibrium, φmix, Amix, and Bmix are in general different for the liquid and vapor phases. At the azeotropic point the composition and the temperature are the same for the liquid and vapor phases. Hence, the liquid phase a, b, c, and d parameters are respectively equal to the corresponding parameters of the vapor phase. Thus, we write

aVaz ) aLaz

(46)

bVaz ) bLaz

(47)

cVaz ) cLaz

(48)

dVaz ) dLaz

(49)

Note that eqs 46-49 would not be valid for densitydependent mixing rules. From definitions (7) and (8), from eqs 46-49, and because the temperature and pressure are the same for the azeotropic phases, we write

AVaz ) ALaz ) Aaz

(50)

BVaz ) BLaz ) Baz

(51)

From eqs 27, 42, and 46-51, i.e., from the conditions of vapor-liquid equilibrium at the azeotropic point, we write

Baz ) f(VV*,az) - Aazg(VV*,az)

(52)

Baz ) f(VL*,az) - Aazg(VL*,az)

(53)

F(VV*,az) - AazG(VV*,az) ) F(VV*,az) - AazG(VL*,az)

(54)

The system of equations (52)-(54) is a subsystem to which it is necessary to add N - 1 out of the N equations (25) to have a number of equations equal to the number of unknowns, i.e., a determined system of equations, corresponding to the necessary condition of equilibrium of the multicomponent homogeneous azeotrope. As for the case of pure compounds we see that eqs 52-54 establish a universal relation between the variables Aaz, Baz, VL*,az, and VV*,az. Note also that if the variables of eqs 52-54 are properly renamed the subsystem becomes identical with the system of eqs 16-18. Therefore, the universal relationship among the variables Aaz, Baz, VL*,az, and VV*,az is identical with the universal relationship among the variables A, B, V*L, and V*V set by the chosen equation of state for the saturated pure compounds. Thus, if explicit functions relating the variables A, B, V*L, and V*V for pure compounds, e.g., eqs 19-21, are available, then they can be coupled to any algorithm specially designed for the calculation of multicomponent azeotropic conditions, in order to make the algorithm more efficient and reliable. Examples of proposed algorithms for the calculation of azeotropes are those of Wang and Whiting (1986) and of Fidkowski et al. (1993). The later researchers describe a homotopy method to compute the temperatures and compositions of all multicomponent azeotropes predicted by thermodynamic models. A recent review on the computation of azeotropes within the framework of azeotropic distillation has been given by Widagdo and Seider (1996). A possible calculation procedure to find the homogeneous (multicomponent) azeotrope composition at a given temperature, which exploits the identity between pure compound and azeotropic universal relations, is given in appendix E (procedure E3). The variables of iteration in procedure E3 are only the compositions (which are the same for both phases). No internal iterative loops are necessary for the calculation of the


pressure and phase densities of the azeotrope, provided that the pure compound universal relations for molar volumes and vapor pressure are available. The lack of convergence for the above procedure indicates either the failure of the numerical method to solve the nonlinear system of equations or that the model does not predict the existence of a multicomponent homogeneous azeotrope at temperature T. If a solution is found, then the azeotropic pressure can be calculated explicitly using eq 19. Mono- and Polyazeotropy for Binary Mixtures For a binary mixture, the validity of the pure compound universal relations at the azeotropic point, for mixing rules independent from density, makes it possible to determine the number and compositions of all the azeotropes predicted by a model at a given temperature T, through a plot generated doing noniterative calculations. Therefore, the procedure that we describe next is free from the assumption of the procedures which led to Figures 3 and 4, i.e., the existence of at most only one azeotrope at a given temperature. The following calculation procedure needs to be done for z1 varying between 0 and 1. The symbol z1 is the tentative composition of the binary azeotrope expressed as the mole fraction of component 1 in any of the phases. At temperature T and at the current value of z1, follow the calculation procedure E3 of appendix E but replacing the last step by the evaluation of the pseudo relative volatility of component 1 with respect to component 2, as follows:

Rps 12 )

φL1,az φV2,az φV1,az φL2,az

Figure 5. Pseudo relative volatility Rps 12 as a function of the mole fraction of CO2 for the system carbon dioxide (1)-ethane (2), using the QMR-PR-EOS, at 223.15, 263.15, and 288.15 K.

(55)

At values of z1 strictly different from 0 or unity, eq 55 gives a true value of the relative volatility only if Rps 12 is equal to unity. At values of z1 equal to 0 or unity, eq 55 also gives true values of R12, i.e., the values of the infinite-dilution relative volatilities R∞12 and R012. This is because the pure compound universal relations are also valid for binary mixtures for which one component is at infinite dilution in the other. Thus, if at a composition value of z1 the pseudo relative volatility Rps 12 is equal to unity, then the model predicts the existence of an homogeneous binary azeotrope of composition z1 at temperature T. If this happens for two or more values of z1, then the model predicts multiple azeotropy at temperature T. If the condition Rps 12 ) 1 is not met for any value of z1, then the model predicts that no homogeneous binary azeotrope is formed at temperature T. Figure 5 shows a plot of Rps 12 as a function of z1, at three temperatures for the system carbon dioxide (1)ethane (2), modeled with the QMR-PR-EOS. All calculations were noniterative. For all temperature values, the curves intersect the unity line. Then, the model predicts the existence of azeotropes at all three temperatures. The intersection values of z1 are close to the experimental azeotropic values reported by Fredenslund and Mollerup (1974; Knapp et al., 1982, p 527). Note that for a given isotherm of Figure 5 all points are nonequilibrium points except that of the azeotrope and the two points at infinite dilution, i.e., the points at z1 ) 0 and z1 ) 1. Figure 6 shows again the curve of Rps 12

Figure 6. Pseudopressure and pseudo relative volatility Rps 12 as a function of the mole fraction of CO2 for the system carbon dioxide (1)-ethane (2), using the QMR-PR-EOS, at 263.15 K. 9: Azeotropic experimental datum of Fredenslund and Mollerup (1974; Knapp et al., 1982, p 527). s: QMR-PR-EOS.

at T ) 263.15 K for the system carbon dioxide (1)ethane (2), together with the calculated pseudopressure (eq 19) as a function of the carbon dioxide mole fraction. The squares are experimental data of Fredenslund and Mollerup (1974; Knapp et al., 1982, p 527). At the composition for which the pseudo relative volatility is unity, the pseudopressure has a maximum value. This is the predicted value for the azeotropic pressure. The azeotrope is evidently a maximum pressure azeotrope. The only values of pressure which are solutions of the


Figure 7. Pseudo relative volatility Rps 12 as a function of the mole fraction of CO2 for the system carbon dioxide (1)-ethane (2), using the QMR-PR-EOS, at 291.15, 298.15, and 304.5 K.

Figure 8. Pseudopressure and pseudo relative volatility Rps 12 as a function of the mole fraction of CO2 for the system carbon dioxide (1)-ethane (2), using the QMR-PR-EOS, at 291.15 K.

equilibrium equations are the maximum value and the values at 0 and 1 mole fractions for carbon dioxide (vapor pressure of the pure compounds). For a binary mixture, the problem of finding the azeotropic composition consists of finding the roots of Rps 12 - 1, or for the examples studied here, it consists of finding the extrema of the pseudopressure. In the first case the functions of eqs 19-21 and the expressions for the fugacity coefficients need to be known, while in the second case only the function fB of eq 19 needs to be known. Therefore, it is simpler to plot just the pseudopressure as a function of composition instead of plotting the pseudo relative volatility, in order to evaluate the occurrence of azeotropes. Figure 7 shows a plot of Rps 12 as a function of z1, at three temperatures for the system carbon dioxide (1)ethane (2), modeled with the QMR-PR-EOS. The difference with respect to Figure 5 is that the values of temperature are higher. For any of the temperatures there is a range of carbon dioxide compositions for which no values of the pseudo relative volatility are reported. Within this range the value of the dimensionless variable A is less than Ac and hence no van der Waals loop exists. The existence of vaporlike and liquidlike molar volume values at fixed composition, in a range of pressures, is a necessary condition for the existence of an azeotrope. The composition value at which A ) Ac at a given temperature is the pseudo critical composition, for which the pseudo relative volatility is unity because the molar volumes of the liquid and vapor phases are the same. A point at which the first and second derivatives of the pressure with respect to the molar volume, at constant temperature and composition, are both equal to zero is a pseudo critical point (Sandler and Dodd, 1986). A pseudo critical point does not necessarily coincide with the true critical point of the mixture (Sandler and Dodd, 1986). Figure 7 shows that there are two pseudo critical points at 291.15 K. The same holds at 298.15 K. However, there is only one pseudo critical point at 304.5 K. The existence of two

pseudo critical points at a given temperature indicates the existence of two unconnected vapor-liquid equilibrium regions at that temperature. This has been experimentally shown by Ohgaki and Katayama (1977) for the carbon dioxide-ethane system. Since the right-hand branch of the 291.15 K curve crosses the unity line, the QMR-PR-EOS predicts the existence of an azeotrope at 291.15 K. Figure 8 shows this curve together with the pseudopressure curve. At the composition at which the unity line is crossed by the pseudo relative volatility curve, the pseudopressure reaches a maximum value, i.e., the azeotropic pressure value. The two pseudo critical pressures are less than the azeotropic pressure. In summary, Figure 8 indicates that there are an azeotropic point and two potential critical points at 291.15 K. The existence of two critical points and of an azeotropic point at 291.15 K for the carbon dioxide (1)-ethane (2) system has been experimentally found by Ohgaki and Katayama (1977). Figures 5-8 have been generated without iterative calculations. At a fixed temperature, if the interaction parameter is known, the pseudo critical compositions can be calculated by setting A ) Ac. For quadratic mixing rules this equation leads to a second-order polynomial equated to zero, for which the unknown is the molar fraction of one of the components of the binary mixture. The solution is explicit. Once the pseudo critical composition is known the pseudo critical pressure is calculated explicitly from the critical value of B (Bc). Locating the pseudo critical points is useful to identify the region of search for the azeotropic composition (see Figure 8). Additionally, they are of practical interest in phase identification techniques for mixtures (Sandler and Dodd, 1986). For the technique developed above no restrictions have been placed on the number of binary homogeneous azeotropes predicted at a given temperature. Hence, a plot such as that of Figure 6 is useful to locate all the binary azeotropes at a given tempera-


iteration variables, than calculating all bubble points in the chosen composition region. In fact, the pseudopressure can be calculated using any equation of state, even those EOSs not matching eqs 1 or 2. In all the examples studied in the present work the pseudopressure was a well-behaved variable. This result suggests that this variable could be used for the automatic initialization of normal vapor-liquid equilibrium calculations. Estimation of Binary Parameters

Figure 9. Pseudo relative volatility Rps 12 as a function of the mole fraction of propylene (1) for the system propylene (1)-1-butene (2), using the QMR-PR-EOS, in a range of temperatures.

ture in the event of polyazeotropy, without having to perform iterative calculations. Figure 9 shows five pseudo relative volatility isotherms for the system propylene (1)-1-butene (2), modeled with the QMR-PR-EOS using a value of k12 equal to 0.0004. This value was taken from the Dechema Data Base (Knapp et al., 1982, p 661). Because the unity line is never crossed, no azeotropes are predicted for this system in the temperature range of Figure 9. At, for example, 310.93 K none of the points are equilibrium points, except those at infinite dilution. At 377.59 K there is a pseudo critical point. The same happens at 410.93 K. From Figure 6 it is evident that, for the case of single azeotropy without pseudo critical points, the problem of finding the azeotropic composition at a given temperature for a binary system can be reduced to a problem in only one iteration variable. A robust procedure can be applied, such as the simple bisection method, to find the single root of (Rps 12 - 1). Once the azeotropic composition is found, the pressure of the azeotrope can be calculated explicitly from eqs 19 and 7. For studying the likelihood of ternary azeotrope formation the pseudopressure can be calculated in a region of the composition plane. For mixing rules not depending on density, coupled to EOSs which give universal relations, the multicomponent pseudopressure can be explicitly calculated, at a given temperature and composition, from eqs 7, 8, and 19. Preliminary calculations indicate that extrema in the values of the ternary pseudopressure correspond to ternary azeotropes, while the absence of such extrema corresponds to the absence of ternary azeotropes. Hence, the evaluation of the pseudopressure, which is noniterative, could be used as a tool to study multicomponent azeotropes. For equations of state not giving universal relations and/or for EOSs coupled to density-dependent mixing rules the pseudopressure can be calculated as stated by procedure E0 of appendix E. This is less expensive, in terms of

The estimation of the value of a binary parameter from the experimental temperature and composition of the binary azeotrope is at first sight a problem for which it is necessary to solve a system of four nonlinear equations. The unknowns are the azeotropic pressure, vapor and liquid molar volumes, and the interaction parameter k12. Due to the validity of the pure compound universal relations at the azeotropic point and to the algebraic form of the QMR-PR-EOS, it is possible to reduce the problem of estimating the value k12 from the experimental temperature and composition of the binary azeotrope to a problem in only one unknown, i.e., the parameter k12, which then becomes the only iteration variable in a direct substitution scheme. In a binary mixture the conditions of azeotropy for components 1 and 2 are

φV1,az ) φL1,az

(56)

φV2,az ) φL2,az

(57)

and

If we take advantage of the universal relations, which are valid at the azeotropic point, one of the equations 56 and 57 becomes redundant. Thus, at finite values of the azeotropic composition, either of the equations (56) or (57) can be used to find the value of kaz 12. The result is the same in spite of the equation chosen. For the case of the QMR-PR-EOS it can be shown that the equation which results from the combination of eqs 32 and 56 can be rearranged to give the expression

kaz 12 where

)1-

F1(x12a1 + x22a2) - x1a1 (x2 - 2F1x1x2)(a1a2)1/2

[(

) ]

b1 SA,az - SB,az b1 b 1 2x2 az + F1 ) 2 Aaz SC,az baz

(58)

(59)

where the functions SA,az, SB,az, and SC,az are the functions SA, SB, and SC, given by eqs 38-40, respectively, evaluated from the values of Baz, VV*,az, and VL*,az setting c ) 0 and d ) 0. The calculation procedure to obtain the value of k12 (kaz 12) which reproduces exactly the experimental azeotropic composition at a given temperature T, through direct substitution, when using the QMR-PR-EOS, is given in appendix E (procedure E4). At the infinite-dilution limits, relations (37) and (41) should be used, instead of procedure E4. Procedure E4 can be followed by a stability test in order to ensure that the azeotropic liquid phase is stable.


ertheless, they can provide excellent initial estimates for the phase densities of the pure compounds, i.e., the molar volumes at saturation. These estimates are obtained explicitly using constants such as those of appendices B-D. Conclusions

Figure 10. Binary interaction parameter as a function of temperature for the system carbon dioxide (1)-ethane (2), using the QMR-PR-EOS. 9: Values obtained from the iterative application of eq 58. s: linear regression.

For the obtained value of kaz 12, at temperature T, the model will give an azeotropic behavior at a composition equal to the experimental azeotropic composition. Due to the phase rule, the pressure of the azeotrope becomes a predicted azeotropic pressure which, in general, will be different to some extent from the experimental azeotropic pressure. This difference is small for the system carbon dioxide (1)-ethane (2), but, in general, it will depend on the chosen combination of the equation of state and mixing rule. We have applied the above calculation procedure to the system carbon dioxide (1)-ethane (2) using the azeotropic data of Fredenslund and Mollerup (1974; Knapp et al., 1982, p 527), at 223.15, 243.15, 263.15, and 283.15 K. In all cases the initial value of the interaction parameter was zero. Convergence was always achieved in no more than six steps. Figure 10 shows a plot of the calculated values of the binary parameter kaz 12 which exactly reproduce the azeotropic composition. The infinite-dilution activity coefficient of component 1 in component 2 is the ratio between the fugacity coefficient of component 1 infinitely diluted in component 2 and the fugacity coefficient of the pure compound 1, both at the temperature and pressure of the system. For the QMR-PR-EOS, it can be shown that the value of the interaction parameter which exactly reproduces an experimental value of the infinite-dilution activity coefficient of component 1 in component 2 can be obtained explicitly from the pure compound densities of components 1 and 2 at the temperature and pressure of the system (see eq 33) and from the experimental value of the infinite-dilution activity coefficient. Since, in general, the temperature and pressure of the system do not correspond to the saturation coordinates of components 1 and 2 as pure compounds, the universal relations given by the PR EOS cannot be directly used for the computation of interaction parameters from experimental infinite-dilution activity coefficients. Nev-

Two-parameter, cubic or noncubic, equations of state and their translated forms establish universal relations between dimensionless variables, for pure compounds at saturation. These facts make possible the calculation of saturated pure compound properties and infinitedilution properties of binary mixtures through noniterative calculations. For instance, the values of the incipient azeotropy binary parameters can be obtained, or the existence of an azeotrope can be studied in a temperature range by performing direct calculations. The pure compound universal relations are also valid for multicomponent homogeneous azeotropes, provided that the mixing rules do not depend on density. This conclusion is valid for any degree of complexity in the temperature and composition dependence of the mixing rule used. This finding makes it possible, in the case of binary polyazeotropy, to determine the number and compositions of all azeotropes at a given temperature through a graph generated without using iterative calculations. Another consequence of the validity of the pure compound restrictions at the azeotropic point is the possibility of reducing the calculation of a binary azeotropic composition, or the calculation of binary parameters from the binary azeotropic data, to problems in only one iteration variable. Through examples, we have shown how the identity between the pure compound and the azeotropic universal relations can be used in practical calculations. All the examples have been limited to constant-temperature problems. For the case of constant-pressure problems it is still possible to take advantage of the pure compound universal expressions in order to produce faster, simpler, and reliable calculation procedures for homogeneous azeotropes. In the case of multicomponent homogeneous azeotropy, any existing algorithm for the computation of azeotropes can take advantage of available pure compound universal relations, thus reducing the size of the problem. In this regard the present work should be seen as complementary of previous works dealing with computation methods for azeotropes. In summary, we have shown that the special algebraic properties of van der Waals type equations of state make it possible, depending on the problem, either to replace iterative calculations by direct procedures or at least to reduce the number of iteration variables, not only for saturated pure compounds but also for mixtures, either at the limit of infinite dilution or at a state of multicomponent azeotropy. List of Symbols a ) energetic parameter b ) covolume parameter c ) translation parameter d ) translation parameter kij ) binary interaction parameter N ) number of components P ) absolute pressure R ) universal gas constant

Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1609 Table 2. Points of the Universal Relations for Pure Saturated Compounds equation of state van der Waals

Redlich-Kwong

Peng-Robinson

MCSV

a

V*L

V*V

1.027 59 1.125 59 1.223 59 1.321 59 1.426 59 1.524 59 1.825 59 2.826 59 2.994 59 1.057 299 1.157 299 1.257 299 1.357 299 1.457 299 1.557 299 1.857 299 2.857 299 3.837 299 1.06 1.16 1.26 1.36 1.46 1.56 1.86 2.86 3.76 3.95 1.757 000 1.855 000 1.953 000 2.051 000 2.254 000 2.352 000 2.856 000 2.954 000 7.609 000

1.122 926 × 2.643 771 × 103 1.302 355 × 102 4.047 561 × 101 2.054 193 × 101 1.370 437 × 101 6.878 364 3.193 546 3.005 428 1.469 663 × 1010 8.358 619 × 103 4.212 522 × 102 1.159 680 × 102 5.549 371 × 101 3.409 723 × 101 1.484 223 × 101 5.538 840 3.857 387 7.474 714 × 108 4.402 889 × 103 3.251 773 × 102 1.027 449 × 102 5.256 989 × 101 3.355 621 × 101 1.536 301 × 101 5.874 262 4.158 482 3.952 747 8.034 828 × 106 3.140 391 × 105 3.573 981 × 104 7.799 449 × 103 1.066 119 × 103 5.663 281 × 102 9.251 992 × 101 7.526 357 × 101 7.723 956 1015

a/RT(b + d)

P(b + d)/RT

3.827 261 × 1.008 753 × 101 6.685 067 5.392 644 4.689 234 4.301 113 3.732 840 3.378 142 3.375 003 3.796 192 × 101 1.587 164 × 101 1.102 378 × 101 8.929 128 7.773 971 7.049 169 5.940 909 5.039 441 4.933 969 3.739 336 × 101 1.665 940 × 101 1.194 307 × 101 9.883 751 8.741 612 8.022 203 6.916 163 5.998 648 5.879 734 5.877 360 3.753 789 × 101 3.169 430 × 101 2.756 536 × 101 2.453 039 × 101 2.032 541 × 101 1.892 482 × 101 1.480 116 × 101 1.433 483 × 101 1.060 132 × 101

8.881 784 × 10-16 3.769 474 × 10-4 7.343 678 × 10-3 2.204 044 × 10-2 4.005 932 × 10-2 5.581 163 × 10-2 9.121 681 × 10-2 1.246 514 × 10-1 1.249 997 × 10-1 6.804 246 × 10-11 1.194 242 × 10-4 2.317 549 × 10-3 8.039 802 × 10-3 1.587 104 × 10-2 2.432 360 × 10-2 4.697 664 × 10-2 8.117 713 × 10-2 8.664 000 × 10-2 1.337 843 × 10-9 2.263 162 × 10-4 2.972 474 × 10-3 8.910 026 × 10-3 1.634 290 × 10-2 2.398 681 × 10-2 4.359 803 × 10-2 7.260 850 × 10-2 7.769 073 × 10-2 7.779 607 × 10-2 1.244 576 × 10-7 3.184 036 × 10-6 2.796 155 × 10-5 1.278 767 × 10-4 9.236 267 × 10-4 1.719 282 × 10-3 9.559 533 × 10-3 1.148 621 × 10-2 4.682 198 × 10-2

101

For the Peng-Robinson EOS Ac is equal to 5.877 359 and Bc is equal to 0.077 796 07.

T ) absolute temperature v ) molar volume xi ) liquid mole fraction of component i yi ) vapor mole fraction of component i zi ) mole fraction of component i

Appendix A

Greek Letters

Appendix B: Expressions for the Explicit Calculation of Dimensionless Volumes, for Saturated Pure Compounds, Corresponding to the van der Waals EOS

Rij ) relative volatility of component i with respect to component j φ ) fugacity coefficient ω ) acentric factor Superscripts az ) azeotropic L ) liquid phase V ) vapor phase ps ) pseudo 0 ) pure compound or infinite dilution of component 2 in component 1 ∞ ) infinite dilution of component 1 in component 2 Subscripts az ) azeotropic c ) critical i ) ith component mix ) mixture VdW ) van der Waals MCSV ) modified Carnahan-Starling-van der Waals PR ) Peng-Robinson RK ) Redlich-Kwong

Table 2 shows some points of the universal relations set by several equations of state for pure saturated compounds.

11

VL ) 1 + (Vc,VdW - 1) exp[(-5)(

KVdW,L,mXEVdW ∑ m)1

VdW,m

)2]

when Ac,VdW e A e 38.272 61 (B-1) 10

VV ) Vc,VdW exp[40

KVdW,V,mXm/2 ∑ VdW] m)1

when Ac,VdW e A e 38.272 61 (B-2) where

Vc,VdW ) 3 XVdW )

A - Ac,VdW 40

Ac,VdW ) 3.375

(B-3) (B-4) (B-5)

The constants KVdW,L,m, EVdW,m, and KVdW,V,m are given


VL )

Table 3. van der Waals EOS m

KVdW,L,m

[EVdW,m]1/2

KVdW,V,m

1 2 3 4 5 6 7 8 9 10 11

0.919 052 9 2.975 368 4 -2.089 443 6 1.705 046 7 -2.646 011 6 2.984 662 4 -1.912 882 4 0.673 634 4 -0.223 674 7 -0.452 392 4 -0.995 064 6

0.453 497 7 0.680 651 9 0.803 612 0 0.473 419 8 0.616 334 4 0.688 493 5 0.721 811 8 1.141 433 7 1.357 708 9 0.675 264 0 0.428 619 7

0.170 070 0 0.536 108 6 -0.196 320 2 0.402 380 4 2.688 128 9 -4.531 846 6 0.605 523 2 1.496 473 2 0.821 590 5 -1.034 404 5

11

1 + (Vc,MCSV - 1) exp[(-3)(

KRK,L,m

[ERK,m]1/2

KRK,V,m

1 2 3 4 5 6 7 8 9 10 11

1.118 865 2.036 760 -2.657 123 0.874 775 -1.166 941 2.567 540 -2.491 671 1.833 267 -1.465 162 1.598 083 -1.232 013

0.497 803 5 0.713 057 8 0.807 305 7 0.557 725 0 0.624 477 8 0.752 961 0 0.863 084 9 0.944 239 6 0.592 997 1 0.572 502 8 0.636 119 5

0.256 060 0 0.543 672 4 -0.115 425 4 -0.095 517 8 2.584 959 8 -2.134 069 9 -2.109 948 2 2.308 519 0 0.636 545 3 -0.807 657 6

in Table 3. Equation B-1 reproduces the values of V*L of Figure 2, for the VdW EOS with a maximum absolute deviation of 0.02%. Equation B-2 reproduces the values of V*V of Figure 2, for the VdW EOS with a maximum absolute deviation of 0.27%. Appendix C: Expressions for the Explicit Calculation of Dimensionless Volumes, for Saturated Pure Compounds, Corresponding to the Redlich-Kwong EOS

10

VV ) Vc,MCSV exp[15

KMCSV,V,mXm/2 ∑ MCSV] m)1

when Ac,MCSV e A e 37.537 89 (D-2)

KRK,L,mXERK ∑ m)1

RK,m

)2]

when Ac,RK e A e 37.961 92 (C-1) 10

KRK,V,mXm/2 RK ]

m)1

when Ac,RK e A e 37.961 92 (C-2) where

Vc,RK ) 3.847 322 XRK )

A - Ac,RK 40

Ac,RK ) 4.933 962

(C-3) (C-4) (C-5)

The constants KRK,L,m, ERK,m, and KRK,V,m are given in Table 4. Equation C-1 reproduces the values of V*L of Figure 2, for the RK EOS with a maximum absolute deviation of 0.09%. Equation C-2 reproduces the values of V*V of Figure 2, for the RK EOS with a maximum absolute deviation of 0.1%. Appendix D: Expressions for the Explicit Calculation of Dimensionless Variables, for Saturated Pure Compounds, Corresponding to the MCSV EOS

KMCSV,B,mX(m+1)/2 ∑ MCSV + m)1

10

∑ KMCSV,B,mXm-2 MCSV]} m)5 when Ac,MCSV e A e 37.537 89 (D-3) A ) Ac,MCSV + 40[KMCSV,A,1YMCSV + KMCSV,A,2Y3/2 MCSV + 10

∑ KMCSV,A,mYm-1 MCSV] m)3 when 1.244 576 × 10-7 e B e Bc,MCSV (D-4) where

Vc,MCSV ) 7.666 132

(D-5)

Ac,MCSV ) 10.601 227

(D-6)

Bc,MCSV ) 0.046 823 64

(D-7)

XMCSV )

11

∑

)2]

4

m

VV ) Vc,RK exp[25

MCSV,m

when Ac,MCSV e A e 37.537 89 (D-1)

B ) Bc,MCSV exp{(-15)[

Table 4. Redlich-Kwong EOS

VL ) 1 + (Vc,RK - 1) exp[(-4)(

KMCSV,L,mXEMCSV ∑ m)1

YMCSV )

A - Ac,MCSV 40

( ) (

(D-8)

)

-1 B ln 15 Bc,MCSV

(D-9)

The constants KMCSV,L,m, EMCSV,m, KMCSV,V,m, KMCSV,B,m, and KMCSV,A,m are given in Table 5. Equation D-1 reproduces the values of V*L of Figure 2, for the MCSV EOS with a maximum absolute deviation of 0.08%. Equation D-2 reproduces the values of V*V of Figure 2, for the MCSV EOS with a maximum absolute deviation of 0.08%. For the case of eq D-3 the maximum absolute deviation in the values of B is 0.03%, while for eq D-4 the maximum absolute deviation in the values of A is 0.004%. Appendix E: Calculation Procedures Procedure E0. Computation of the pseudopressure for a given multicomponent composition z and temperature T: (1) Propose a value for the pseudo pressure Pps. (2) Obtain a liquidlike density and vaporlike density at T, z, and Pps, from the PvT relation written for the mixture (e.g., equation of state 1). (3) Calculate the fugacity coefficients of all components in the vapor-phase and liquid-phase mixture, and obtain the vapor- and liquid-phase fugacity coefficients using eq 26. If they are equal, then Pps is the pseudopressure at T and z, and the phase volumes and component fugacity coefficients are the pseudovolumes and pseudo fugacity coefficients, respectively.

Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1611 Table 5. MCSV EOS m

KMCSV,L,m

[EMCSV,m]1/2

KMCSV,V,m

KMCSV,B,m

KMCSV,A,m

1 2 3 4 5 6 7 8 9 10 11

1.171 847 1.642 443 -1.987 207 1.485 701 -1.484 509 1.301 619 -1.308 602 1.169 648 -1.165 114 1.162 961 -1.096 562

0.488 588 2 0.629 594 4 0.756 653 4 0.740 028 7 0.835 937 0 0.855 144 7 0.901 794 0 0.919 731 8 0.915 075 3 0.946 545 3 0.609 520 7

0.319 717 0 0.462 295 5 0.358 394 7 3.343 209 3 × 10-4 -3.879 961 9 × 10-2 2.185 875 9 -1.138 868 0 -3.014 203 2 2.678 840 8 -0.343 538 7

0.968 439 9 1.459 705 6 × 10-2 0.172 174 1 0.445 132 4 0.313 483 2 -0.883 644 9 -0.261 178 1 0.955 620 9 0.115 248 5 -0.527 619 4

1.027 543 1 5.822 417 4 × 10-2 -0.532 119 1 3.848 654 2 × 10-2 0.555 891 6 -0.255 725 8 -0.536 424 8 0.322 994 6 0.318 939 6 -0.230 973 9

Procedure E1. Computation of the product Iaz ) - 1)(R01,2 - 1) at a given temperature T: (1) Calculate the values of A for the pure compounds (A1 and A2) from eq 8 using d ) 0. (2) Obtain B1, VL*,1, and VV*,1 from the value of A1 using eqs 19-21, respectively. Calculate the value of L V,0 φL,0 2 from the values of B1 and V*,1 and the value of φ2 V from the values of B1 and V*,1 using in both cases eq 34. Obtain R01,2 from eq 29. (3) Obtain B2, VL*,2, and VV*,2 from the value of A2 using eq 19-21, respectively. Calculate the value of from the values of B2 and VL*,2 and the value of φV,∞ φL,∞ 1 1 from the values of B2 and VV*,2 using eq 33. Obtain R∞1,2 from eq 28. (4) Calculate the product Iaz ) (R∞1,2 - 1)(R01,2 - 1). Procedure E2. Computation of k∞12 and k012, at a given temperature T: (1) Calculate the value of A for the pure compound 2 (A2) using d ) 0, and obtain the values of B2, VL*,2, and VV*,2 from eqs 19-21, respectively. (2) Calculate SA,2, SB,2, and SC,2 using eqs 38-40, and obtain k∞12 from eq 37. (3) Calculate the value of A for the pure compound 1 (A1) using d ) 0, and obtain the values of B1, VL*,1, and VV*,1 from eqs 19-21, respectively. (4) Calculate SA,1, SB,1, and SC,1 using eqs 38-40, and obtain k012 from eq 41. Procedure E3. Multicomponent azeotrope calculation at temperature T: (1) Propose the composition of the azeotrope at the given temperature T. (2) Calculate aaz, baz, daz, and from them calculate Aaz using eq 8. (3) Calculate explicitly Baz, VL*az, and VV*,az from the value of Aaz using the pure compound universal relations, i.e., eqs 19-21, respectively. (4) Calculate the fugacity coefficients of N - 1 components in the vapor phase and in the liquid phase using eq 43 (e.g., eq 32, which is an example corresponding to the QMR-PR-EOS, for which c ) 0 and d ) 0]. (5) Check if equations (25) are satisfied for the N - 1 components chosen in the previous step. If not, change the composition and repeat the calculation procedure. It is only necessary to consider N - 1 equations of the N equations (25) in this step since the equality of the mixture fugacity coefficients (eq 27) of the azeotropic phases is implicit in the third step. Procedure E4. Direct substitution computation of the value of k12 which reproduces exactly the experimental azeotropic composition at a given temperature T, using the QMR-PR-EOS: (R∞1,2

Table 6. Pure Compound Parameters compound

carbon dioxidea

ethanea

propylenea

1-butenea

Tc/K Pc/atm ω

304.15 72.80 0.231

305.43 48.20 0.099

365.05 45.40 0.140

419.55 39.70 0.191

a

Knapp et al., 1982.

(1) Propose a value for kaz 12. This initial value can be placed in the proper range with the aid of eqs 37 and 41 and the known experimental behavior of the system at temperature T (maximum-pressure or minimumpressure azeotrope). (2) Calculate the values of aaz, baz, and Aaz, using eqs 30, 31, and 8, respectively, with the experimental value of temperature and azeotropic composition, and d ) 0. (3) Obtain the values of Baz, VL*,az, and VV*,az from eqs 19-21, respectively. (4) Compute SA,az, SB,az, and SC,az using eqs 38-40, respectively. (5) Calculate a new value of kaz 12 using eq 58 with the experimental azeotropic composition. Follow again the previous steps to achieve convergence. The predicted azeotropic pressure can be obtained from the final value of Baz, using the defining equation (7), with d ) 0. Appendix F: Parameters Used in the Present Work with the Peng-Robinson EOS Table 6 shows the pure compound values of critical temperature and pressure and the acentric factor used in the present work. Interaction Parameters. Carbon Dioxide (1)Ethane (2). The temperature-dependent k12 interaction parameter used in this work for the carbon dioxide (1)ethane (2) system is given by the following equations:

k12 ) 3.705 × 10-4T + 0.0351 for T e 283.15 K (T in K) (F-1) k12 ) -2.139 × 10-3T + 0.7457 for T g 283.15 K (T in K) (F-2) Literature Cited Aly, G.; Ashour, I. A Modified Perturbed Hard-Sphere Equation of State. Fluid Phase Equilib. 1994, 101, 137. Christensen, S. P.; Olson, J. D. Phase Equilibria and Multiple Azeotropy of the Acetic Acid-Isobutyl Acetate System. Fluid Phase Equilib. 1992, 79, 187. Fidkowski, Z. T.; Malone, M. F.; Doherty, M. F. Computing Azeotropes in Multicomponent Mixtures. Comput. Chem. Eng. 1993, 17, 1141.

1612 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 Knapp, H.; Do¨ring, R.; Oellrich, L.; Plo¨cker, U.; Prausnitz, J. M. Vapor-liquid equilibria for mixtures of low boiling substances. DECHEMA Chem. Data Ser. 1982, VI. Ohgaki, K.; Katayama, T. Isothermal Vapor-Liquid Equilibrium Data for the Ethane-Carbon Dioxide System at High Pressure. Fluid Phase Equilib. 1977, 1, 27. Peńeloux, A.; Rauzy, E. A Consistent Correction for RedlichKwong-Soave Volumes. Fluid Phase Equilib. 1982, 8, 7. Peng, D.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15 (1), 59. Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1986. Redlich, O.; Kwong, J. N. S. On the Thermodynamics of Solutions. V: An Equation of State. Fugacities of Gaseous Solutions. Chem. Rev. 1949, 44, 233. Sandler, S. I.; Dodd, L. R. On the problem of phase identification in a mixture. Fluid Phase Equilib. 1986, 31, 313. Smith, J. M.; Van Ness, H. C. Introduction to Chemical Engineering Thermodynamics; 3rd ed.; McGraw-Hill Book Co.: New York, 1975; Appendixes D-2 and D-4. Soave, G. Improvement of the van der Waals Equation of State. Chem. Eng. Sci. 1984, 39, 357.

Soave, G. Direct Calculation of Pure-Compound Vapour Pressures through Cubic Equations of State. Fluid Phase Equilib. 1986, 31, 203. Wang, S.-H; Whiting, W. B. New Algorithm for Calculation of Azeotropes from Equations of State. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 547. Widagdo, S. ; Seider, W. D. Azeotropic Distillation. AIChE J. 1996, 42, 96. Zabaloy, M. S.; Vera, J. H. Cubic Equation of State for Pure Compound Vapor Pressures from the Triple Point to the Critical Point. Ind. Eng. Chem. Res. 1996, 35, 829. Zabaloy, M. S.; Brignole E. A. On Volume Translations in Equations of State. Fluid Phase Equilib. 1997, 140, 87. Zabaloy, M. S.; Vera, J. H. Extension of EOS Non-Iterative Methods to Density Calculations: Correlation of Saturated Molar Volumes and Heats of Vaporization. Can. J. Chem. Eng. 1997, 75, 214.

Received for review October 22, 1997 Revised manuscript received February 24, 1998 Accepted February 28, 1998 IE970742+

Special Algebraic Properties of Two-Parameter Equations of State

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