New Parametric Model to Correlate the Gibbs Excess Function and

406

Ind. Eng. Chem. Res. 2010, 49, 406–421

New Parametric Model to Correlate the Gibbs Excess Function and Other Thermodynamic Properties of Multicomponent Systems. Application to Binary Systems Juan Ortega* and Fernando Espiau Laboratorio de Termodina´mica y Fisicoquimica de Fluidos, Parque Cientı´fico-Tecnolo´gico, UniVersidad de Las Palmas de Gran Canaria, Canary Islands, Spain

Jaime Wisniak Department of Chemical Engineering, Ben-Gurion UniVersity of the NegeV, Beer-SheVa 84105, Israel

A new empirical mathematical model for the Gibbs excess function, gE ) ψ(p,T,x), is presented for a multicomponent system. Dependence on the composition is achieved through the so-called active fraction, zi, which, in turn, is related to the molar fraction xi of the components of a solution and a parameter kij, the determination of which is also indicated. The efficacy of the model in relation to its extension of application is discussed, considering various cases and three possible ways to calculate the parameter kij. This produces different versions of the model for data correlation the advantages of which are discussed. The model proposed for the Gibbs excess function adopts the following generic expression, gE(P,T,x) ) z(x)[1 - z(x)]∑i)0gi(P,T)zi where gi(P,T) ) gi1 + gi2P2 + gi3PT + gi4/T + gi5T2, which can be applied to a general case of vapor-liquid equilibrium with variation of the three main variables xi, p, and T, or by considering the experimental values for two important situations, isobaric and isothermal, which are also studied here. Other mixing properties are obtained via mathematical derivation, and a simultaneous correlation is carried out on several of them. The model has been applied to various binary systems for which experimental data are available in the literature and over a wide range of p and T. The results obtained can be considered acceptable. 1. Introduction Many attempts have been made to develop a general and satisfactory theory for estimating or correlating thermodynamic properties. In the absence of an acceptable answer, an empirical approach may be considered as a step toward achieving this goal. In the few last years, several publications have presented different empirical expressions that can be used to directly obtain the values of some thermodynamic mixing quantities mainly for binary systems, when the experimental measurements are available. In general, these empirical functions that have no known relationship with molecular physics or statistical mechanics. To contribute to modeling solutions, the two most interesting properties, from a theoretical and a practical perspective, are the excess quantities of Gibbs’ energy and enthalpy. Entropy and other properties can be calculated from these. Another complementary study would be to analyze the changes in mixing and/or excess volumes, which can be determined quite precisely with available equipments based on the direct measurement of density. In this work, our main focus is on Gibbs’ excess energy function, since other thermodynamic properties of solutions can be obtained from it, as shown by the following relationships, presented in any standard thermodynamic book:1 G ∂G ∂G H )H)G-T , S) , ∂T p T T ∂T p ∂G ∂2G ∂G Cp ) -T , V) , A)G-p (1) ∂p T ∂P T ∂T2 p

( )

[ ]

( )

( )

[ ]

These expressions can be directly applied to the corresponding excess quantities and illustrate the great interest in developing * To whom correspondence should be addressed. E-mail: jortega@ dip.ulpgc.es.

a single equation or model for Gibbs’ function of a homogeneous solution, valid for different conditions of pressure and temperature. The experimental determination, theoretical estimation, and mathematical treatment carried out with these data are essential, since inter/extrapolation are often required when insufficient data are available. 2. Representation of Equilibrium Data The phase rule, developed by J. Willard Gibbs in 1875, establishes the number of independent variables, or degrees of freedom, which must be employed arbitrarily to define the intensive state of a system. This rule is mathematically expressed as: L ) C + 2 - F, where L is the number of independent variables, C is the number of components, and F is the number of phases present in the system. For a binary systems of components A and B in vapor-liquid equilibrium (VLE), the phase rule gives the following: L ) X[p, T, xA, yA] - Y[µAL ) µAV, µLB ) µVB ] ) 4 - 2 ) 2 (2) Here, the independent variables are the potentials p, T, and µi, because the previous expression is based on the general Gibbs-Duhem equation. Therefore, the mathematical approach to a real problem can be solved using eq 2, the definition of chemical potential, and the condition of equilibrium considered by the Gibbs-Duhem equation. However, it is also important to obtain numerical answers to the proposed model. To do this, the chemical potential must be related with the other fundamental variables appearing in eq 2, in other words, with p, T, and xi. Thermodynamics alone cannot strictly resolve this problem even when using other less abstract auxiliary functions, such as fugacity and activity. Even graphically, it is sufficient

10.1021/ie900898t  2010 American Chemical Society Published on Web 11/23/2009

Ind. Eng. Chem. Res., Vol. 49, No. 1, 2010

( ∂G ∂T )

dG )

p,n

dT +

( ∂G ∂p )

T,n

dp +

∑ ( ∂G ∂n )

407

dni

i p,T,nj*i

i

(4) Since G is a first-order homogeneous equation in relation to the amount of components, then G ) gn. From here, a general form of Gibbs function can be obtained in differential form. For intensive properties, this can be expressed as follows:

∑ gj dx

dg ) -sdT + Vdp +

i

(5)

i

i

In eq 5,the “g” function is related to the pressure p, temperature T, and composition. When working with solutions, the two first summands are established at constant composition and can be expressed as a function of other measurable thermodynamic quantities using the Maxwell’s relations, while the latter gives the corresponding partial molar properties of “G”, with derivatives in relation to the number of moles, ni, which is not very practical. Hence, in order to get closer to our objective, we will express this term as a function of the composition or molar fraction of each component. To do this, a change of variable must be carried out in the term that contains gj i Figure 1. 3D scheme of standard vapor-liquid phases for a two component system.

to represent the potential variables for each compound in each phase and to find the line of coexistence. The aim of this work is, therefore, to provide an initial contribution, which will correspond to the first step in the process to try and achieve relationships of this kind. We consider that an empirical approach could help to do this. The behavior of a system needs values for several variables mentioned above. The measurement of four variables, p, T and x, y, of a two component and two phase system can be fully described by a 3D diagram, as shown in Figure 1. Although this results in an overspecification in terms of the phase rule, it is the best procedure to accurately present the data. This representation of equilibrium data is not always simple, so partial sections are obtained in which one of the variables is maintained constant. For example, the compositions of a compound in two phases can be chosen as the variables, with the others remaining constant, which would imply that the molar fractions of both phases depend on p and T; in other words, they would be expressed as x(p,T) and y(p,T). These functions correspond to the two surfaces f1(p,T,x) ) 0 and f2(p,T,y) ) 0 in the space of Figure 1. The case being studied here could be more complex, because, in addition to the four variables mentioned above, for all measured experimentally in the between-phase equilibrium studies, there may be some degree of dependence, inter-related with other indirectly obtained quantities, such as the Gibbs excess function.

The Gibbs’ function for a material system composed of several species depends on temperature, pressure, and the number of moles of each of the species present. Mathematically, this functional dependence of G can be written as G ) G(p,T,ni), where ni is the number of moles of each of the species or substances involved in a multicomponent system. Hence, for a mixture with C components

Differentiating eq 3 in relation to each of the variables

( ) ∂G ∂ni

) p,T,nj*i

(3)

( ) ∂ng ∂ni

)n p,T,nj*i

g

( ) ∂n ∂ni

( ) ∂g ∂ni

+

p,T,nj*i

)n

p,T,nj*i

( ) ∂g ∂ni

+g

(6)

p,T,nj*i

given that n ) ∑ni, and thus (∂n/∂ni)p,T,nj*i ) 1. The intensive property g can be expressed as a function of the compositions or molar fractions xi, of the constituents of the solution, which satisfy the condition ∑xj ) 1. For a solution of C components at constant pressure and temperature, the property depends on (C - 1) molar fractions. Choosing one of them as the dependent variable, that is xi ) 1 - ∑xj, we get g ) g(x1, x2, ..., xi-1, xi+1, ..., xC)

(7)

and from here: ∂g ∑ ( ∂x ) C

dg )

dxk

(8)

k p,T,xj*k,i

k*i

To obtain the derivative of the property in relation to the number of moles, see eq 6, we must return to the quotient, xk ) nk/n, from which

( ) ∂xk ∂ni

)-

nk n

p,T,nj*i

2

)-

xk n

∀k ) 1, 2, ..., i - 1, i + 1, ..., C (9)

The derivative in eq 8 becomes

( ) ∂g ∂ni

3. Gibbs Function for a Multicomponent System

G ) G(p, T, n1, n2, ..., nC)

gji )

)

p,T,nj*i

( )

∂g ∑ ( ∂x )

k p,T,xj*k,i

k*i

) -

∂g ∑ ( ∂x )

∂xk ∂ni

()

k p,T,xj*k,i

k*i

p,T,nj*i

xk n

(10)

Substituting in eq 6 yields ji ) g G

∂g ∑ x ( ∂x ) C

k

k*i

and then replacing in eq 5:

k p,T,xj*k,i

(11)

408

Ind. Eng. Chem. Res., Vol. 49, No. 1, 2010 C

∑

dg ) -sdT + Vdp +

i

C

Vdp + g

[

∑ dx

i

]

∑ x ( ∂g ∂ )

g-

k

k

k

dxi ) -sdT +

∂g ∑ ∑ x ( ∂x )

-

C

C

i

k

dxi

k

i

(12)

k p,T,xj)k,i

Since ∑ dxi ) 0, eq 12 becomes C

C

dg ) -s dT + V dp -

∑∑ i)1

k)1 k*i

xk

dxi

ζ(p, T, x1, x2, ...) ) 0

(14)

This is a better way to obtain a simpler model for data treatment and for calculations programming than the initial one, eq 3. The variables contemplated in the possible empirical eq 21 shown below are intensive, regardless of the total amount of substance present; although the mole fractions xi are not all independent of each other, they are independent of T and p. An expression such as eq 14 is also obtained by applying one of the properties of the homogeneous functions to eq 4, by dividing it by the total number of moles n of the solution, in other words: G(p, T, n1, n2, ..., nC) ) g(p, T, x1, x2, ..., xC) (15) n The differential of eq 14 generates an equation mathematical analogous to eq 13. Even when this reaches the physical/ thermodynamic significance proposed, it can be used to obtain a suitable empirical equation to model the data treatment, as mentioned previously. For this situation, the total differential of eq 15 becomes the following:

( )

p,xj

dT +

∂g ∂p

( )

T,xj

∂g ∑ ( ∂x )

C-1

dp +

k)1

dxi

k p,T,xj*k,C

(16) The above arguments will be expanded in the following sections. Clearly, this approach is applicable to the corresponding excess magnitude, gE, the analysis of which is treated in phase equilibrium studies. Therefore, from hereon the present paper focuses on developing this quantity. 4. Representation of Gibbs Excess Function with Composition To implement now a model for solutions that is widely applicable, the relationship between excess Gibbs energy function and the variables pressure, temperature, and composition must be defined first. The most generic expression of the model will be the following: Q

E ) Mn,Q

∑ p)2

[

∑

i1i2...ip∈CR*(n,p)

ai1i2....ipzi1zi2...zip

]

(18)

∑k

i1xi

i)2

This equation clearly contains the derivatives corresponding to the variables pressure, temperature, and the molar fractions of the components of the system. Therefore, from a mathematical perspective, this equation resembles the total differential of a multivariable function of the type

∂g ∂T

(i ) 2, ..., n)

n

x1 +

p,T,xi*k,i

(13)

dg )

ki1xi

zi )

( ) ∂g ∂xk

etc. Q represents the maximum order of molecular interactions considered, while ai1i2...ip are the influence particular coefficients of the p-aria combination in the excess quantity considered. In some cases, one or more of these coefficients can even be zero. For the active fraction zi, the following generic relationship with the molar fraction is considered:

(17)

Equation 17 shows that any excess quantity can be defined as the sum of the contributions of the effects of all the possible p-arias interactions of n effective fractions, where p ) 2, 3, 4,

Where ki1 is now a coefficient that must be determined. This possibility will be discussed separately (Appendices A.1 and A.2). When eq 17 is developed, a recurrent expression for n substances can be obtained, which can be written generically as E ) Mn,Q

∑

i1i2...in-1∈C(n,n-1)

E(i1-i2-i3...-in-1) Mn-1,Q + φnχQ

(19)

E(i1-i2-...-in-1) represents each fraction or partial where Mn-1,Q contribution of the excess quantity due to the interaction of (n - 1) components, to a maximum of Q molecules chosen from among the species i1, i2, ..., in-1. Each of the partial contributions to the total excess function can, in turn, be E reduced to the corresponding Mn-1,Q , and so on, until the binary interactions are reached. The term φnχQ essentially represents the product between z1z2 ... zn and a polynomial, the degree of which depends on Q. For ternary systems in which interactions up to the fourth-order are considered, eq 19 becomes

ME3,4 ) ME(1-2) + ME(1-3) + ME(2-3) + 2,4 2,4 2,4 z1z2z3(C0 + C1z1 + C2z2)

(20)

E(i-j) refer to the partial contribution of the Here, the terms M2,4 excess quantity considered in the binary system i-j, to the global calculation of mixing energy for the ternary system with a maximum of fourth-order interactions. Equation 17 becomes analogous to the empirical model proposed by Wohl,2 but with an approach based on statistical mathematics. Since the set of eqs 17-20 can be applied to any excess quantities, owing to the nature of this work, we will apply them to Gibbs’ excess function gE. To achieve our objective, instead of using one of the generalized equations, we use that corresponding to a simple binary system, where x1 + x2 ) 1 and also z1 + z2 ) 1. In this way, the dependence of the excess Gibbs function on composition, starting from eq 17, after arranging and simplifying terms, reduces to

gE(x) ) z1(1 - z1)[A0 + A1z1 + A2z12] (21) RT In eq 21, the coefficients Ai are a linear combination of the influence particular coefficients, ai1i2...ip and, at least here, are independent of temperature. Dependence on the composition xi can be clearly observed through zi [eq 18]. 5. Analytical Representation of the Gibbs Excess Function as a Function of Composition, Temperature, and Pressure The main question planned in this work is to generate a single model for the excess Gibbs energy function for a multicomponent system. In other words, to obtain an expression that contains the three main variables, within an explicit form of the gE ) gE(p,T,xi) type. Equation 21 reflects the dependence on composi-


tion; the dependence on composition and temperature was developed in a previous work3 as gE(x, T) ) z(1 - z) Aizi where: Ai ) RT i)0

∑

3

Here, the most rigorous expression for zi, for the case where the active fraction depends on the temperature and pressure is

∑A T

j-1

ij

(22)

2

gE(x, p) ) z(1 - z) Bizi where: Bi ) RT i)0

∑

2

∑b p

j

ij

j)0

(23) The function we are looking for will be an extended expression as a funtion of composition, similar to the one proposed for the two cases studied. In other words, not only a polynomial function of the active fraction z, but one in which the coefficients must include the intensive variables of temperature and pressure. Applying an extension of this for the Gibbs excess function, and unifying the coefficients to avoid confusion, we have: gE(p, T, x) ) z1z2[g0(p, T) + g1(p, T)z1(x) + g2(p, T)z12(x)] (24) since gi(p,T) can be replaced by a combination of the coefficients Ai of eq 22 and Bi of eq 23, as shown by the following expression that contains the partial expressions proposed for these equations gi4 + gi5T2 (25) T It is important to clarify that eqs 22 and 23 have been specially proposed for the adimensional Gibbs excess function, gE/RT, for the isobaric and isothermic cases, respectively. However, eq 24 represents an expression, including the three variables, which has been defined for Gibbs excess function in its intensive and dimensional form. When thermodynamic quantities are calculated using liquid-vapor equilibria (VLE), it is common to use data from an adimensional Gibbs excess function because of their direct relationship to the activity coefficients. However, it seemed more appropriate to establish a general model for the extensive excess Gibbs function since the adimensional conversion is immediate. The derivatives of eq 25 in relation to each of the variables p and T are

ki1(p, T)xi

zi )

j)0

The dependence on composition and pressure can be expressed as

[

]

p,x

[

gi4

]

dgi(p, T) ) gi3p - 2 + 2gi5T and ) dp T,x T 2pgi2 + gi3T (26)

(i ) 2, ..., n)

n

x1 +

∑k

i1(p, T)xi

Although eq 27 initially seems to have a large number of parameters, in practice, as we shall see below, the actual number is small. Let us now check if this expression is consistent with the findings of previous papers, following the mathematicalthermodynamic formalism relating eqs 1. To derive expressions for other thermodynamics properties, it is necessary to determine the derivatives of the excess Gibbs function in relation to temperature and pressure. In other words h E ) gE - T

( ) ∂gE ∂T

) gE - Tz1z2

p,x

2

[( (

∑gz i

(

g21

) )

)]

i-1

]

(29)

i)0

2

Y ) (1 - 2z)

2

∑

gizi + z(1 - z)

i)0

∑ ig z

i-1

(30)

i

i)0

The last expression may be written in the general form Y ) 3 (i + 1)(gi - gi-1)zi, which in our case gives g-1 ) g3 ) 0. ∑i)0 Introducing eq 30 in eq 29 yields hE ) gE - Tz1z2

∑ ( ∂T )z ∂gi

z(1 - z)

-T

i

∑ [g

i

( dTdz )Y )

-T

∑ ( ∂T )]z ∂gi

i

-T

( dTdz )Y

(31)

Other quantities are

( )

dz ( )[(1 - 2z) ∑ ( ∂T )z + dT ∂T ∂g dY dz (32) z(1 - z) ∑ ( )iz +( )] - TY( ∂T dT dT )

cEp ) -z(1 - z)T

∑

∂2gi 2

∂gi

zi - T i

i

2

i-1

2

The term in brackets in eq 32 can be expressed in a more generic form as follows: cEp ) -z(1 - z)T

∑

( ) ∂2gi ∂T2

zi -

( dTdz )[2 ∑ (i + 1)(

T

( dTdz ) ∑ (

)

)

∂(gi - gi-1) i z + ∂T

] ( )

∂gi d2 z i(i + 1)(gi - gi-1)zi-1 - TY ∂T dT2

-sE ) z(1 - z)

∑ ( dT )z dgi

i)0

i

+

( dTdz )Y

(33)

(34)

Similarly, for the variation with p

g04 + g05T2 + T g14 + g15T2 z + g11 + g12p2 + g13pT + T g24 + g25T2 z2 + g22p2 + g23pT + T

z1z2 g01 + g02p + g03pT +

2

which simplifies to

) z1z2[g0 + g1z + g2z ] )

i)0

-

i

i)0

2

2

i

i

2

g (p, T, x) ) z(1 - z)

∂gi

i

2

i

∑ ( ∂T )z

( ∂k∂z )( dk(p,dT T) )[(1 - 2z) ∑ g z + z(1 - z) ∑ ig z

T

Both equations are of the same type as eqs 22 and 23. The following is one of the possible expanded forms of the equation we are aiming for: E

(28)

i)2

gi(p, T) ) gi1 + gi2p2 + gi3pT +

dgi(p, T) dT

409

VE )

( ) ∂gE ∂p

T,x

) z1z2

∑ ( ∂p ) ∂gi

T,x

zi +

( dkdz )( dk(p,dp T) )[(1 - 2z) ∑ g z + z(1 - z) ∑ ig z 2

2

i

i

i)0

(27)

Using eq 30, transforms eq 35 into

i

i)0

]

i-1

(35)

410


∑ ( ∂p ) ∂gi

np

dz Y T,x dp A similar treatment yields for the Helmoltz function VE ) z1z2

∑ [g 2

aE ) z1z2

i

-p

i)0

( )( )[

p

zi +

( )

( )]

dgi i z dp 2

∑

(36)

∑

(37)

or, after simplifying

∑ [g 2

i

i)0

-p

(

Pi,exp - Pi,cal Pi,exp

)

2

being

P ) γ(iso - T), γ(iso - p), hE, VE, cEp (39)

dz dk igizi-1 (1 - 2z) gizi + z(1 - z) dk dp i)0 i)0

aE ) z1z2

N

∑∑

np)1 i)1

]

2

OF )

( )] ( )

dgi i dz z -p Y dp dp

(38)

The latter relations generate polynomial eqs 31 and 35 similar to the original ones to be used in the data treatment of excess volumes and enthalpies. Therefore, from hereon we will use the basic model [eq 27] for application to binary systems for which for several properties have been published in the literature, such as VLE (isobaric and isothermic), hE, and VE, at different T and p.

where N is the number of points and np the number of properties considered in the correlation. Figure 2 shows a diagram of the entire process followed when the model is applied and is explained in detail below. For the practical application of this model, two main cases can be distinguished. The first case corresponds to application of the model to either an isobaric or an isothermal process, considering the different versions of the model shown in Table 1. We have found that for parameter kij, at least initially, that free kij is the one that produces the best correlation (for the correlation of VE we refer to kV, kh for the correlation of hE, and for the correlation of gE, we refer to kg).

6. Application of the Proposed Model: Description of the Procedure The model established by eq 27 can be applied to systems for which the behavior of the phases is known over a wide range of temperatures and pressures and can overlap, or at least present a degree of approximation in p and/or in T. In spite of the numerous researchers working on phase equilibria, there are very few systems for which experimental data are known for several of the properties used in the model proposed here. An exhaustive search of the literature has shown that these data are available for some sets/families of mixtures of alkanol + alkane, dialkylcarbonate + alkane, ester + alkanol, and aromatic + alkane. Table 1 gives a summary of several versions of the model using the one shown in eq 27 as its standard form. Different cases are obtained depending on whether the degree of the polynomial equation is modified in relation to the variable composition, or active fraction, or with respect to the order of the polynomial equation for each coefficient of “z” as a function of the variables temperature and pressure. Hence, three different degrees are considered for the polynomial, each with four or five coefficients, which originate the six cases called “A” to “F” in Table 1. The possibility that the parameter kij be dependent on p and T is also considered. The fitting has been carried on using the following objective funtion (OF) including the different quantities or thermodynamic properties employed in the data correlation.

Figure 2. Flow chart corresponding to the vias/procedure planned in the correlation of experimental values using the model of eq 27 and its derivatives.

Table 1. Different Versions of the Model, Noted as A-F, as a Function of the Polynomial Degree, in z, and of the Extension of the Coefficients gi(p,T) model, degree z

equation

coefficients gi(p,T)

A, n ) 1

gE ) z1z2[g0 + g1z1]

gi(p, T) ) gi1 + gi2p2 + gi3pT + gi4 /T

B, n ) 1

gE ) z1z2[g0 + g1z1]

C, n ) 2

gE ) z1z2[g0 + g1z1 + g2z12]


D, n ) 2

gE ) z1z2[g0 + g1z1 + g2z12]

gi(p, T) ) gi1 + gi2p2 + gi3pT + gi4 /T + gi5 /T2

E, n ) 3

gE ) z1z2[g0 + g1z1 + g2z12 + g3z13]


F, n ) 3

gE ) z1z2[g0 + g1z1 + g2z12 + g3z13]




411 E

Table 2. Standard Error s, Obtained in the Simulataneous Fitting of VLE Data at Isothermal and Isobaric Conditions, Excess Enthalpies, h (J · mol-1), Excess Volumes, 109WE (m3 · mol-1), and Excess Thermal Capacities, cpE (J · mol-1 · K-1) (When Available), Using Different Versions of Equation 27, Presented in Table 1a

412


Table 2. Continued

a N (Pi,exp - Pi,cal)2/ Parameters kg, kh, kV, and kc are obtained in the same fitting procedure. * Isothermal VLE data. ** Isobaric VLE data s(P) ) (∑i)1 N)1/2.

The data treatment carried out in the first step of the process, maintaining each of these values free in the fitting procedure, yields the results shown in Table 2 for the group of systems shown. Figure 3 shows the results and standard deviations for the case of the free kij parameter for the six models proposed in Table 1 (A-F) for each the properties. The columns corresponding to the model that gives the minimum deviation of the properties considered together are shaded “gray”, to give a global qualitative evaluation of this application. It is seen that the D and F versions of the model give the smallest standard deviations in almost all cases (in some cases C and B can also be included). Version D is preferred to F for practical reasons, since the former has a lower degree of polynomial in z, thus avoiding overparametrization. It is interesting that the deviations found for cpE were not very significant in any of the versions. Figure 3 shows that the results for all the models for each of the systems are quite uniform. The largest deviations shown in Figure 3 for hE, VE, and cpE of the systems 8 and 9 are not significant. The values of s(gE) and s(γi) presented in Figure 3 correspond to the mean values of those obtained for the isobaric and isothermic cases; these are shown separately in Table 2 However, for hE and VE, the mean values of the statistical coefficient s obtained at different temperatures are shown when available. In the second case studied, parameter kij is considered to depend on pressure and temperature, according to considerations

described in Appendix A.1. The model has, therefore, acquired stronger physical reasons by establishing as part of the correlation procedure the corresponding functions of dependence for kV, kh, and kg, for eqs A.5, A.7, and A.10, respectively. To facilitate evaluation of the results, these are presented in the same way in Table 3 and Figure 4. For this case, the exception for correlations of cpE is noteworthy, owing to insufficient data/ functions for cpE(T). Here, parameter kij is considered to have a constant value, calculated from the quotient of the thermal o o /cp,1 ; a capacities of the compounds in the mixture, kc ) cp,2 similar definition to that used for kV for the volumes. The most immediate conclusion is that, as expected, data correlation with the parameter kij(p,T) produces quantitatively inferior results than when free kij is used. This can, also, be observed by comparing the corresponding graphs (Figures 3 and 4) of the parameters of goodness of fit s, although these differences are not clearly translated into representations of the thermodynamic properties studied. The quantitative differences in Tables 2 and 3 are not very significant. Summarizing, comparison of the results of the different versions used for all cases, indicates that from a practical perspective, model D in Table 1 can be considered to be acceptable. To facilitate the understanding of the results presented in Tables 2 and 3, a detailed numerical application of the calculation procedure is given in Appendix A.2 for the binary system benzene + hexane. This binary mixture has been repeatedly used as a standard systems to compare experimental


413

Figure 3. Standard deviations, s(P), obtained for several properties using the different versions of the model shown in Table 1, applied to the correlation of the systems from literature. Labels indicate the asignation of system according with Table 1 and considering the parameter kij-free, obtained in the fitting procedure (a) for gE/RT, (b) for γi, (c) for hE, (d) for VE, and (e) for cEp .

measurements of the different properties made by the different authors. Moreover, the system benzene + hexane corresponds to one of the few systems in the literature that offers a large amount of experimental data for properties, which can be used to test the capacity of the polynomial model proposed. 7. Conclusions

applied to the simulataneous correlation of experimental data of several thermodynamic properties of binary systems. (2) The generic expression of the model for a binary system is gE(p, T, x) ) z1(x)[1 - z1(x)]

∑ g (p, T)z i

i 1

where

i)0

The following conclusions can be drawn: (1) A flexible mathematical model is presented, which can be used to correlate any excess thermodynamic property, particularly the excess Gibbs funtions of a multicomponent systems, gE ) gE(p,T,x1,x2, ...). The model has been

z1 )

x1 x1 + kij(p, T)x2

(40)

Application of this model to the systems reported in the literature allows selecting the most appropriate polyno-

414


Table 3. Standard Error s, Obtained in the Simulataneous Fitting of VLE Data at Isothermal and Isobaric Conditions, of Excess Enthalpies, hE (J · mol-1), Excess Volumes, 109WE (m3 · mol-1), and Excess Thermal Capacities, cpE (J · mol-1 · K-1) (When Available), Using Different Versions of Equation 27, According to Table 1a


415

Table 3. Continued

a In all cases, the parameter kij was considered to be dependent on the pressure and temperature, eqs A.5, A.7, and A.11, to calculate, respectively, the values corresponding to kV, kh, γ, and kg. For correlations of cEp , a fixed value for kij ) cp,2/cp,1 was considered. * Isotherml VLE data. ** Isobaric N VLE data s(P) ) (∑i)1 (Pi,exp - Pi,cal)2/N)1/2.

mial degree for the coefficients gi(p,T), corresponding to the more following gi4 + gi5T2 (41) T (3) From formal thermodynamic relationships, see eqs 1, specific expressions can be obtained for each of the properties of the solution. The advantage of the polynomial form of the model given by eq 40 is that its degree can be selected depending on the number of thermodynamic quantities considered in the mathematical treatment. (4) The model is applied by simultaneously correlating several properties using a least-squares procedure with a genetic algorithm (GA), minimizing the deviations of each property considered. Optimum values are thus determined for the gij coefficients of eq 41. Two possibilities are discussed for parameter kij; it is first considered as an additional parameter in the fitting process (free-kij mode). In general, this alternative gives the best correlations. In the second case, the value of the parameter is given by the expressions A.5, A.7, and A.10, for correlationg the quantities VE, hE, and gE, respectively, taking into account variations in these quantitities pressure and temperature. When no data of excess heat capacities gi(p, T) ) gi1 + gi2p2 + gi3pT +

as a function of T are available, cpE(T), the parameter is o o established as a known value of kc ) cp,2 /cp,1 . (5) The results obtained indicate that the model gives acceptable estimations of the excess properties for the set of binary systems studied here. Although its polynomial form includes a large number of parameters (a minimum of 15 with version D, with an even higher number if the kij are free in the fitting process), it is, actually, the most appropriate expression. This is because successive derivations can result in a reduction in the parameters and a stepwise correlation procedure can even be applied in the opposite order to the derivation. One example of this stepwise procedure is (x,cpE) f (x,hE) f (x,gE), provided that data are available for each of the properties. It also permits the approximate estimation of other thermodynamic quantities not obtained experimentally, such as the entropy, contributing in the interpretation of the behavior of the solution studied. (6) Usually, in practical application of the model, the terms corresponding to the variation in the active fraction “z” with pressure and temperature can be ignored, since they only contribute to a very small percentage of the total. This can be observed for the case shown in the appendix

416


Figure 4. Standard deviations, s(P), obtained for several properties using the different versions of the model shown in Table 1, applied to the correlation of the systems from the literature. Labels indicate the asignation of system according with Table 1 and considering the parameter kij(p,T) as a fixed value, obtained from eqs A.5, A.7, A.10 (a) for gE/RT, (b) for γi, (c) for hE, (d) for VE, and (e) for cEp .

for the standard binary system of benzene + hexane, which gives acceptable results for the correlation of all properties. (7) Finally, we must remember that the model proposed in this work is only the first step in the process to find a more general solution, since the absence of a more extensive database means that it cannot be verified for a broad interval of data. Therefore, an intense research labor is required in the medium- and long-term since, as well as expanding on the available database, further applications of the model must be developed, such as its

incorporation into the general Gibbs-Duhem equation, its validation in the reproduction of different thermophysical quantities, such as those included in eqs 1, and other applications.

Acknowledgment The authors (J.O. and F.E.) acknowledge the financial support received from the Ministerio de Educacioń y Ciencia (Spain) for the Project CTQ2006-12027.

Ind. Eng. Chem. Res., Vol. 49, No. 1, 2010 [(βi-βj)(p-p0)-(Ri-Rj)(T-T0)]

Appendix

kg(p, T) ) kg(p0, T0)e

A.1. Calculation of kij Parameters To complement the study carried out, a general equation must be developed to describe the dependence of parameter kij on temperature and pressure. For a pure substance, assuming V(p,T), we have ∂V ∂V dT + dV ) ∂T p ∂p Dividing eq A.1 by V, we get

( )

( )

T

dp

d ln V ) R dT - β dp

(A.1)

(A.2)

where R and β are the isobaric expansion coefficient and the isothermal compressibility, respectively. Integrating between an initial reference state and any other state (p,T) leads to F(p, T) ) F0(p0, T0)e[β(p-p0)-R(T-T0)]

(A.3)

where F is the density. Equation A.3 corrsponds to an expression of the type φ(p,V,T) ) 0, in other words, with an equation of state as a function of the characteristic coefficients of a pure compound. So, the more general equation for kV(p,T), used to correlate excess volume and established as a quotient of molar volumes, has the following form Voj (p, T)

kV(p, T) ) kij(p, T) )

Voi (p, T)

)

( )( )

Mj Fi(p, T) ) Mi Fj(p, T)

Mj Fi,0 [(βi-βj)(p-p0)-(Ri-Rj)(T-T0)] e Mi Fj,0

(A.4)

(A.5)

These expressions allow calculating the coefficient kh, using the following equation developed in a previous work:2 kh(p, T) )

( )( ) qi rj qj ri

2/3

(A.8)

from a practical perspective, by referring to the value of kg(p0,T0) as a constant in reference conditions. It is important to note that these conditions will not always coincide with those for which the thermal coefficients were obtained, so eq A.8 can be generalized even further as follows kg(p, T) ) Ae[(βi-βj)(p-p0)-(Ri-Rj)(T-T0)]

(A.9)

kg(p, T) ) Ae[B(p-p0)-C(T-T0)]

(A.10)

or

in other words, by replacing the thermal coefficients R and β and the value of kg(p0,T0) by other parameters that are determined in the fitting procedure. This produces a clear increase in the number of parameters and, possibly, a slight reduction in the standard deviations. In most cases, it results in an unnecessary overparametrization. Finally, for correlations of excess specific thermal capacities, cEp , values of this quantity have not been reported in the literature at different temperatures and pressures for the pure compounds and binary systems studied here. Therefore, a constant value o o /cp,1 at a specific pressure and has been taken for kij ) cp,2 temperature. A.2. Application of the Proposed Model: Analysis of the Function kij(p,T)

in other words kV(p, T) ) kV(p0, T0)e[(βi-βj)(p-p0)-(Ri-Rj)(T-T0)]

417

kV2/3(p, T) ) kh(p0, T0)[kV(p, T)]2/3 (A.6)

or kh(p, T) ) kh(p0, T0)[kV(p0, T0)e[(βi-βj)(p-p0)-(Ri-Rj)(T-T0)]]2/3 (A.7) since the area parameters qk and rk do not depend on p and T. To use this equation, the mean values of R and β must be known for the pure compounds in suitable intervals of temperature and pressure. In a previous article,2 variation in the parameter kV with temperature is represented graphically (isobaric case). We consider it appropriate to include a 3D representation with both intensive variables for a group of pure substances and their mixtures (ethanol + alkane) for which data are available in the literature studied.62 The results are shown in Figure A1, in which kV is only slightly dependent on pressure and temperature (for the mixtures considered here). This possibly reflects what would happen in many cases, if data were available for a range of conditions. In accordance with this, for the Gibbs excess function, the parameter kg can adopt an analogous form to that used for the other thermodynamic quantities studied. In other words, we could write an expression of the following type

Here, we present a detailed application of version D of Table 1 as an example of the utility of the model proposed, since this version was found to be reasonably suitable for most of the cases shown in Tables 2 and 3. The numerical and graphical data corresponding to the binary system benzene + hexane are shown later, since this is the system with the most data recorded (isothermic and isobaric) for several properties used in the model. The complete equation for version D of the model for the Gibbs excess function is presented in eq 27. As mentioned in section 5, it is common practice to describe phase equilibrium using the excess Gibbs function in adimensional form gE/RT. However, for the first derivative of this function in relation to temperature, it is possible to use the dimensional form expressed by eq 27. Hence, gE/RT can be related to enthalpy hE, by the following expression:

[

hE ∂(gE /RT) ) ∂T RT2

]

)p,x

[ ( ) 1 ∂gE RT ∂T

]

gE gE ) p,x RT2 RT2 T ∂gE (A.11) RT2 ∂T p,x -

( )

which is an equation identical to eqs 29 and 31 that describe hE as a potential function of gE. The objective is to define both of the summands of eq A.11, hence using eq A.7 to determine (dz/dT) and considering eq 30 for the term Y and its derivative. Using the coefficients gi(p,T) of eq 27 yields the following for hE :

∑ (g 2

hE ) z(1 - z)

i1

i)0

+ gi2p2 +

)

2gi4 dz - gi5T2 zi - T Y T dT

( )

(A.12) The expressions for the other excess thermodynamic quantities, such as cpE and sE, can be easily obtained by expanding the

418


Figure A1. (a) Variation of the molar volumes Vio of pure compounds (1 ethanol, 2 hexane, 3 octane, 4 decane) as a function of pressure and temperature. (b) Variation of kV parameter of binary mixtures formed by previous compounds as a function of pressure and temperature.

Figure A2. Binary system x1 benzene + x2 hexane: plot of the different contributions of the addends of eq 34 separately, (upper part) with (dz/dT) ) 0 and (lower part) contribution of term with (dz/dT), to the total excess property (a) on hE according to eq A.12 and (b) on VE according to eq A.15.

respective equations. For example, eq 33 for cpE can now be written cEp ) -z(1 - z)2T

∑

(

gi4 T3

(

( )

)

d(gi - gi-1) i d2z (i + 1) z - TY dT dT2 i)0 3

( )∑

dz T dT

)

+ gi5 zi (A.13)

Similarly, using eq 34, yields for the excess entropy 2

-sE ) z(1 - z)

∑ i)0

(

gi3p -

gi4 2

T

)

+ 2gi5T zi - T

( dTdz )Y

(A.14)

Using isothermic VLE data and eqs 36 and 37 allows calculating the excess volumes VE, and the Helmoltz function aE, as follows: VE ) z(1 - z)

∑ (2g

i2

+ gi3T)zi +

i)0

∑ (g 2

aE ) z(1 - z)

i1

i)0

- gi2p2 +

( dpdz )Y

(A.15)

) ( )

gi4 dz + gi5T2 zi - p Y T dp (A.16)

Now we must study the application of the corresponding kij parameters and their derivatives in relation to p and T, since they are clearly involved in all the previous equations. Hence, for the change with temperature: ∂z dk dk dz(k, T) x(1 - x) ) ) (A.17) dT ∂k dT [x + k(1 - x)]2 dT The variation in parameter k with T can be observed from one of the eqs A.5, A.7, A.8, or A.10. If the variation of k with T is neglected, the nullity of the term A.17 will mean that these terms are eliminated from eqs A.12-A.14, giving a more simplified expression. The variation of z with p, eqs A.15 and A.16, is studied with an analogous procedure, so it is not explained here to avoid repetition. A.2.1. Correlation of Experimental Data of the Binary (Benzene + Hexane) with the Proposed Model. In this section, we present the results of applying the model to the system (benzene + hexane). This is then used to achieve an exhaustive analysis of the results, quantitative and qualitative validations, and, if necessary, verification by other authors. As mentioned before, one of the reasons for choosing this system is the large amount of data available in the literature for several of the

( )( )

( )


419

Table A1. Coefficients gij for Different Versions of the Model Presented in Table 1, Obtained from Simultaneous Correlation of γi (Isothermal and Isobaric VLE Data) and the Properties hE, WE, and cpE, of the Binary System x1 Benzene + x2 Hexane (12; See Tables 2 and 3), and Standard Deviation for the Quantity Ma parameters kV, kh, kg, and kc obtained in the fitting procedure along to values for gij

parameters kV, kh, kg, and kc obtained as a function of p and T according to eqs A.5, A.7, and A.10

g0j

g1j

g2j

g0j

-21.0 2.27 × 10-8 -8.61 × 10-9 516069.1 -3.92 × 10-4

-889.2 8.35 × 10-9 -8.29 × 10-9 146170.7 4.46 × 10-4

-782.8 2.19 × 10-9 -1.60 × 10-9 209989.7 2.39 × 10-3 0.004b,c,d,e 0.013b,c,d,e 31f,g,h 22h,i,j 0.08k

g1j

g2j

Model D j)1 j)2 j)3 j)4 j)5 s(gE/RT) s(γi) s(hE) s(VE) s(cpE)

kg ) 1.124 kh ) 1.350 kV ) 1.328 kc ) 0.938

-1394.2 2.78 × 10-8 -1.14 × 10-8 914559.6 -2.08 × 10-3

833.4 -1.13 × 10-10 -4.39 × 10-9 -540801.0 7.4510-3

58.6 6.24 × 10-10 2.07 × 10-10 196343.6 -1.27 × 10-5 0.004b,c,d,e 0.022b,c,d,e 31f,g,h 22h,i,j 0.06k

a

For the parameter kij are considered two cases: kij-free and that where the parameter is a function of p and T according to eqs A.5, A.7, and A.10. Reference 42. c Reference 43. d Reference 44. e Reference 28. f Reference 31. g Reference 32. h Reference 50. i Reference 38. j Reference 39. k Reference 59. b

Figure A3. Plot of experimental values for different thermodynamic quantities and correlation curves obtained by version D of the model, with parameters kg, kh, kV, γ, kc calculated in the fitting process (red line), and the version F of the model with the parameters kg, kh, kV, γ, kc as a function of p and T according to expressions A.5, A.7, and A.10 (dashed line), against the composition of binary system x1 benzene + x2 hexane (12) in Tables 2 and 3. (a) Gibbs excess energy function of isothermic VLE at T ) 298.15 (9), 328.15 (2), and 333.15 K (•). (b) Gibbs excess energy function of isobaric VLE at p ) 101.32 kPa (•). (c) Excess enthalpy at T ) 293.15 (O), 298.15 (4), 313.95 (0), 318.15 (]), and 323.15 K (3). (d) Excess volume at T ) 288.15 (O), 298.15 (4), 303.15 (0), 308.15 (]), 318.15 (3), and 323.15 K (\). (e) Excess thermal capacity at T ) 298.15 K (O).

420


mixing properties considered here. Since they are part of a longterm research project, the experimental data and/or correlations of thermodynamic quantities of this system are used by numerous researchers for reference or comparative purposes, to verify the procedures. In fact, the literature consulted includes isobaric VLE28 and isothermic data,42,43,45 hE,29-31,48,49 and VE data31,37,38,49 over a range of temperatures and also cpE data.58 Version D of the model in Table 1 was used for the data correlation, using the procedure described in Figure 3. Then, simultaneous correlations were performed of the quantities for which the experimental data for this mixture are available in the literature, as specified in the previous paragraph and eqs A.12-A.17. Another important consideration, about the polynomial form of the model which could, a priori, make its application more difficult, concerns whether or not to take into account variation of the parameter kij with pressure and temperature. If it is taken into account, via the term (dz/dT), the equation would be more scientifically rigorous but it would result in an increased number of summands in each expression established for the thermodynamic quantities; see eqs A.12-A.16). If it is excluded, then these would be reduced to the first term only. This procedure was utilized to determine the quantitative contribution of these summands in the final calculation for all the thermodynamic functions derived from gE. As shown in Figure A2, the contribution for the enthalpy is about 1.5% of the total, while for the volumes it is almost negligible. Since these are very small contributions, in the study of the usefulness of the proposed model, to the system benzene + hexane, only the first summand of each of eqs A.12-A.16 was considered. Table A1 reports the final results. Two large columns can be distinguished for the version of the model that is chosen, depending on the treatment of parameter kij. The group on the left corresponds to the values of coefficients gi(p,T) when the parameter is considered to be free in the fitting procedure, free kij. The last three columns on the right show the results obtained for the coefficients and deviations when the parameters kij vary with the pressure and temperature, for the experimental data used; in other words, using values of k(p,T) obtained from the expressions given in Appendix A.1. It can be observed that the differences in the results obtained with the different considerations for parameter kij in the correlation process are not significant, so that employment in this case one way or another in the correlation process is indistinct. We emphasize, once again, the good correlation achieved, at least from a quantitative perspective, of the thermodynamic quantities of this binary system. A 3D and 2D representation of the different data sets used is also shown. The graphs in Figure A3 include the experimental points recorded by different authors over a range of conditions of temperature and pressure. They also show the curves obtained by correlating the different quantities using two versions of the model for comparison: version D, which was selected in this work to demonstrate the working procedure, with the parameters of Table A1 and free kij, and version F, in which the kij parameter varies with p and T. It is seen that the differences between them are negligible. A 3D graph has also been constructed with the model studied here, for the binary benzene + hexane, representing the dew and bubles surfaces that define version D (see Figure A4). Red lines on the surfaces correspond to three isothermic equilibria (T ) 298.15, 328.15, and 333.15 K), for which data are available in the literature. The red lines in the p-T planes correspond to estimates for the vapor pressure of the pure substances using

Figure A4. 3D representation of experimental data (O) and correlation curves (red line) using version D of the model for binary system x1 benzene + x2 hexane.

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ReceiVed for reView June 1, 2009 ReVised manuscript receiVed September 29, 2009 Accepted October 27, 2009 IE900898T

New Parametric Model to Correlate the Gibbs Excess Function and

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