A comparison of various cubic equation of state mixing rules for the simultaneous description of excess enthalpies and vapor-liquid equilibria

HglDPBF EOIIIUDRIA ELSEVIER

Fluid Phase Equilibria 121 (1996) 67-83

A comparison of various cubic equation of state mixing rules for the simultaneous description of excess enthalpies and vapor-liquid equilibria Hasan Orbey *, Stanley I. Sandier Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware Newark DE 19716, USA Received 11 August 1995; accepted 30 January 1996

Abstract

Recent new mixing and combining rules for cubic equations of state (EOS) have extended the range of such equations to the accurate description of the vapor-liquid equilibria (VLE) of highly nonideal mixtures. However, the simultaneous correlation a n d / o r prediction of vapor-liquid equilibrium (VLE) and liquid mixture excess enthalpies ( H ex) by either activity coefficient models or equations of state has been a difficult problem in applied thermodynamics. In this communication, we re-examine this problem using a modified version of the Peng-Robinson equation of state and the two-parameter van der Waals one-fluid, Wong-Sandler and modified Huron-Vidal mixing rules. For comparison, the direct use of activity coefficient models is also considered. In each case a temperature dependence of the model parameters is introduced in an attempt to represent simultaneously VLE and H ex behavior. Four highly nonideal binary mixtures (2-propanol + water, methanol + benzene, benzene + cyclohexane, and acetone + water) are considered. The results indicate that while all the models can accurately correlate VLE and H ex data separately, attempting to predict the values of one property with parameters obtained from the other does not give satisfactory results with any model. Also, we find that the simultaneous correlation of both VLE and H ex with the EOS models at one temperature is possible, but extrapolations to other temperatures with parameters obtained in this way did not result in accurate predictions of either VLE or H ~x. The main problem appears to be that the excess free energy (activity coefficient) models used are not capable of representing both VLE and H ~x over a range of temperatures, and so equations of state that incorporate these free energy models have the same shortcoming. Keywords: Excess enthalpy; Correlation; Equation of state

* Corresponding author. 0378-3812/96/$15.00 © 1996Elsevier Science B.V. All fights reserved. PII S0378-3812(96)03030-0

68

H. Orbey, S.I. Sandier/Fluid Phase Equilibria 121 (1996)67-83

1. Introduction

Equations of state (EOS) with the van der Waals one-fluid mixing rules have long been used for phase equilibrium calculations in the hydrocarbon industries. With the advent of multiparameter mixing rules, especially those that incorporate the excess free energy, equations of state are now being used for the phase equilibrium calculations of complicated mixtures that were traditionally described with activity coefficient models. However, these equations of state/mixing rule combinations have not been tested for the correlation and prediction of the excess properties of liquid mixtures which are also important for process design. The simultaneous representation of excess enthalpy ( H ex) and VLE data with activity coefficient models has been an important goal, and there are excellent reviews of the subject by various investigators [1,2]. The subject has been dealt with more recently by several others (for example, [3,4]). These investigators reached several conclusions about the cross-predictability of excess properties and VLE with activity coefficient models. Basically they found that at a given temperature only a crude estimation of excess enthalpy prediction can be achieved with parameters obtained from VLE, and that the reverse prediction is even less accurate. They have observed, however, that the simultaneous correlation of VLE and excess enthalpy data with at least four parameters has been successful for some systems [1], [3]. Whether these conclusions can be extended to EOS models that contain these activity coefficient models in their mixing rules is the main concem of this work. Here we investigate the correlation of the molar excess enthalpy ( H ex) of liquid mixtures with cubic equations of state and various types of mixing rules. In particular, we consider a two-parameter version of the van der Waals one-fluid mixing rule [5], the Wong-Sandler [6] mixing rule, a recent modification of the Huron-Vidal mixing rule [7], and other excess free energy-EOS models (the LCVM (linear combination of the Vidal and Michelsen methods) model by Boukouvalas et al. [8]; the MHV2 (modified Huron-Vidal second-order) model by Dahl and Michelsen [9]; and the MHV1 (modified Huron-Vidal first-order) model by Michelsen [10]), as well as the direct use of activity coefficient models. We investigate first the simultaneous correlation, and then the prediction, of vapor-liquid equilibrium (VLE) and liquid excess enthalpy ( H ex) by these models. Note that VLE involves correlation of the excess Gibbs free energy (Gex), whereas correlating H ex data is equivalent to considering the temperature derivative of G ~x. Since phase equilibrium and enthalpy calculations are frequently done together, it is useful to consider the applicability of a single model and a single set of parameters to both these properties. There are a limited number of investigations of the correlation of liquid excess enthalpies with equations of state. Adachi and Sugie [5] used their two-parameter version of the van der Waals one-fluid model to calculate the excess enthalpies of binary liquid mixtures. Chen et al. [ 11 ] correlated excess enthalpies of binary liquid mixtures with the Martin-Hou [12] equation of state. Their method contained one interaction parameter per pair of components, and they reported only moderate success for liquid enthalpy calculations which they attributed to the use of only one binary interaction parameter. Lichtenstein et al. [13] used a van der Waals-type equation of state combined with an excess free energy model following the procedure of Peneloux et al. [14] for the simultaneous representation of vapor-liquid equilibria and excess enthalpies. They limited their analysis to methanol + hydrocarbon mixtures, and concluded that the mixing rules that combine equations of state with excess energy models could be useful.

H. Orbey, S.I. Sandier/Fluid Phase Equilibria 121 (1996)67-83

69

2. Models

2.1. Excessfree energymodels In this work excess free energy or activity coefficient models are used in two ways. First they are used directly, in the conventional manner, for the correlation of VLE and excess enthalpies. Second, excess free energy models are used in an equation of state mixing rule for the same purpose. Initially we selected the non-random two-fluid (NRTL) model of Renon and Prausnitz [15], and the van Laar model for our study. A temperature dependence of the parameters in each of these models was introduced, as described below, to improve the calculated excess enthalpies. The NRTL model is

I Y'j ~xjgJirji]

Gex

(1)

with

gji = exp(

-

oljiT"ji )

(2)

xi

where G ex is the molar excess Gibbs free energy, the mole fraction of species i, R the gas constant and T the temperature. Also, "r/j are dimensionless binary interaction parameters expressed as in the original NRTL model, with A~j being a temperature-independent binary interaction parameter with units of energy per mole. To be better able to correlate excess enthalpy data, we have made A;j temperature dependent and used

AiJRT

(3)

aij = mij + nijT

Renon et al. [16] had suggested a similar modification for enthalpy calculations with the NRTL model. This model has four parameters per binary; in order to evaluate all four parameters one needs either VLE data at two or more different temperatures, excess enthalpy data, or a combination of VLE and excess enthalpy data. We will come back to this point later. The van Laar activity coefficient model (VLACM) was chosen for its simplicity. It was originally developed for binary mixtures, and extended to multicomponent systems by various investigators [ 17] [18]. For the binary mixtures studied in this review the following form was used G ex

1

R----T--- ( 1 _ _ + _ _

1

)

(4)

Al2xl A21x2 In Eq. (4), A/2 and Ael are binary interaction parameters which, after preliminary trials, were taken to be the following functions of temperature A i j = m i j '1- n i j T

(5)

70

H. Orbey, S.I. Sundler / Fluid Phase Equilibria 121 (1996)67-83

2.2. Equation of state models Here we use the Peng-Robinson equation of state [19]

RT P= V-b

a V 2 + 2bV-b 2

(6)

where a and b are parameters specific for each substance, V is molar volume and P is pressure. This choice is not critical for excess enthalpy calculations since, as shown by Adachi and Sugie [20], calculated excess thermodynamic properties are not very sensitive to the type of cubic EOS used. It is, however, necessary to use an expression for the a term that produces the correct vapor pressure for phase equilibrium calculations [21]. Therefore, we use the form suggested by Stryjek and Vera [22] for the temperature-dependent a term. Three different mixing rules were tested in detail with this equation of state. These are the two-parameter modification of the van der Waals one-fluid mixing rule [VDWEOS] [23], [24], [25], the Wong-Sandler mixing rule [WSEOS] [6], which combines excess free energy models with an equation of state at infinite pressure, and a modified Huron-Vidal mixing rule proposed by Orbey and Sandier [HVOS] [7]. The reason for these choices is given below. We will not make a detailed comparison of the performance of these models for phase equilibrium calculations since this has already been documented [6,7],[21].

2.2.1. Multiparameter combining rules of the van der Waals one-fluid type In the van der Waals one-fluid model, the mixture EOS parameters are evaluated from a= E Exixjaij i

b=

(7)

j

xibi

(8)

i

Ordinarily, a geometric mean average for the a parameter, corrected with a single binary interaction parameter kij, is used for the cross coefficient in Eq. (7) aij = a,~.a~-(1-kij )

(9)

This form, however, is known to be inadequate for describing very nonideal phase and excess enthalpy behavior [5]. An empirical approach to overcome this has been to add an additional composition dependence and parameters to the combining rule of the a parameter, generally leaving the b parameter rule unchanged. Some examples are the combining rules of Panagiotopoulos and Reid [23], Adachi and Sugie [24], Stryjek and Vera [22], Sandoval et al. [25], and Schwartzentruber and Renon [26,27]. These combining rules can be reduced to one another and to the original van der Waals one-fluid combining rule with an appropriate selection of their parameter values. In spite of their several shortcomings [21], these mixing rules have been shown to provide a good correlation of the VLE of complex binary mixtures, including nonideal systems that previously could only be correlated with activity coefficient models. For all these models for the binary systems, the k o term in Eq. (9) reduces to the form first suggested by Stryjek and Vera [22]

kij =

g i j x i -[- K j i x j

(10)

H. Orbey, S.I. Sandler / Fluid Phase Equilibria 121 (1996)67-83

Here for K i j

we

71

use (tl)

K i j = m i j nt- n i j T

with Kji being obtained by index rotation.

2.2.2. The W o n g - Sandler model Wong and Sandier [6] developed a mixing rule that produces the desired EOS behavior at both low and high densities without being density dependent, uses existing activity coefficient parameter tables, allows extrapolation over wide ranges of temperature and pressure, and provides a simple method of accurately extending the UNIFAC or other low-pressure prediction methods to high temperature and pressure [28,29]. In this mixing rule, the EOS parameters a and b are obtained from the relations

a

b

(

a)

,(T- E Exixj b i

RT ij

j

and ex

Az, _ __a CRT bRT

Y'~ x i - a i i biRT

(13)

where C is a constant specific to the EOS chosen. For example, for the Peng-Robinson equation C = [ln(v/2 - 1)]/v/2 = - 0.62323]. The cross term in Eq. (12) is obtained from ( a ) 1[(ai) --= b i "-~ b R T ij -

+

(

bj-aj]](1-kij, RT]]"

]

(14)

which introduces a second virial coefficient binary interaction parameter kij. Any conventional excess eX Gibbs free energy model (such as those described in Section 2.1) may be used for the A.¢ term in Eq. (13).

2.2.3. Approximate combination o f free energy models and equations o f state Recently several investigators have combined EOS and activity coefficient models at zero pressure in various approximate ways. The most successful of these are the LCVM (Linear Combination of Vidal and Michelsen) model by Boukouvalas et al. [8]; the MHV2 (Modified Huron- Vidal second-order) model by Dahl and Michelsen [9] and the MHV1 (Modified Huron-Vidal first-order) model by Michelsen [10]. A somewhat more rigorous and simpler modification of the Huron-Vidal approach, proposed recently by Orbey and Sandler (HVOS) [7], is algebraically similar to these zero-pressure models and as good as the above-mentioned models for the correlation of VLE. Preliminary calculations in this work indicated that the same conclusion is true for the correlation of excess enthalpy. Consequently it is the only model of this class tested in detail here. In the HVOS model, the approximate relation that connects the excess Helmholtz free energy from an EOS and from an activity coefficient model is

Aex

~ RT

(a

~.,xiln

~

+ C

bRT

a/)

xi . biRT

(15)

72

H. Orbey, S.I. Sandier/Fluid Phase Equilibria 121 (1996)67-83

Eq. (8) and Eq. (15) are the Orbey-Sandler modification of the Huron-Vidal mixing rule. Any activity coefficient model can be used for the excess energy term in this equation.

3. Systems investigated We selected four binary systems for a more detailed analysis of the mixing rules considered here. These systems are 2-propanol + water, methanol + benzene, benzene + cylohexane, and acetone + water. There are considerable VLE data over a range of temperatures and excess enthalpy data (albeit mostly at or near 25°C) for these systems, all of which show strong deviations from ideal solution behavior in both VLE and excess enthalpy. Consequently this collection of systems will be used for a stringent test of the models considered here. The sources of data for these systems are provided in the figure captions.

4. Modeling 4.1. Direct correlation of excess enthalpy with activity coefficient models It is well known that activity coefficient models can accurately correlate VLE behavior at low and moderate pressures, and this was not tested here again. We only investigated the correlation of excess enthalpy data using the NRTL and van Laar models. With the NRTL model, the four parameters

300 200 1 00 O

0

E v

-1 00

Q. ¢g ,.c

-200

"E

-300

o

X",

r/

-400 -500 -600

-700 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

.0

mole fraction 2-propanol

Fig. 1. Correlation of the excess enthalpy of the 2-propanol + water system at 25°C by various activity coefficient models. Data (filled circles) are from [30] (Vol. 3, Part 1, p. 583). Solid line is the correlation of the van Laar model, and the dashed lines is the correlation of the NRTL model. The long, medium and short-dashed lines are with ot = 0.45, 0.25 and 0.65, respectively.

H. Orbey, S.I. Sandier/Fluid Phase Equilibria 121 (1996)67-83

73

200 1 O0

o! o

E

-1 00

v

-200 o0 J~

-300 o

-400 -500 -600 -700

o oolt

02013 014015 06 017 06 o, mole fraction 2-propanol

Fig. 2. Correlation of the excess enthalpy of 2-propanol + water system at 25°C. Data (filled circles) are from [30] (Vol. 3, Part 1, p. 583). The dotted lines are the correlation using the conventional van Laar activity coefficient (VLACM) model, the solid line is the correlation by the two-parameter van der Waals EOS (VDWEOS) model, the long-dashed line results from correlation using the Wong-Sandler [WSEOS] model, and the short dashed line from correlation using Orbey-Sandler [HVOS] model.

given in Eq. (3) were fitted to excess enthalpy data for various values of ct, while for the van Laar model the four parameters of Eq. (5) were fitted. In all cases, a simplex optimization routine was used for correlation with the objective function of F = He'all2. The results are shown for the 2-propanol + water binary system in Fig. 1. Similar results were obtained for the other systems investigated here. The van Laar model (solid lines) can correlate the excess enthalpy behavior, though the best results are obtained with the NRTL model (with a different optimum value of c~ for each system). Consequently, the optimum NRTL model has five parameters, and the accuracy of the excess enthalpy correlation is strongly dependent on the value of the ct parameter; this is especially true for the 2-propanol + water system which has a transition from exothermic to endothermic behavior over the concentration range. In contrast, the correlation of VLE data has been found to be much less sensitive to the value of this parameter. Consequently, we decided to use only the van Laar model in the remainder of this review in order to limit the number of parameters.

EilHg~xXp-

4.2. Excess enthalpy correlations with an equation of state To examine the use of the equation of state models, we first correlated data for the four test systems near room temperature. The results for the 2-propanol + water binary system are presented in Fig. 2; similar results were obtained with the other systems. Enthalpy correlation by the direct use of the van Laar activity coefficient model [VLACM] is also shown in Fig. 2; this activity coefficient model was also used in the two EOS models (WSEOS and HVOS). For the WSEOS model, the binary interaction parameter of Eq. (14) was set to zero for enthalpy correlations, so that four

74

H. Orbey, S.I. Sandier/Fluid Phase Equilibria 121 (1996)67-83

parameters were fitted in each model. For the VLACM, WSEOS and HVOS models, these are the four parameters in Eq. (5), while for the VDWEOS model, it is the four parameters in Eq. (11) that were fitted. The experimental excess enthalpy data for each of the mixtures could be correlated accurately and comparably by the different equation of state models (except that the correlation with the VDWEOS model for the methanol + benzene system, not shown here for brevity, is slightly inferior to that of the other models). The direct use of the activity coefficient model (dotted lines in Fig. 2) did not lead to better results than the other EOS models. One observation we made in these calculations is that when the optimized parameters based on H e× data were used in Eq. (5) (or Eq. (11) for the VDWEOS model) to predict VLE behavior, very poor results were obtained, as has previously been found by others [5]. For brevity we do not present those results here. Therefore, we conclude that equation of state parameters fitted only to excess enthalpy data do not result in accurate VLE predictions. We also observed similar behavior when the activity coefficient models considered in this work were used in the same manner. Nicolaides and Eckert [1] have reported the same conclusion.

4.3. Prediction of excess enthalpy with parameters obtained from VLE data Next we investigated fitting the EOS parameters in Eq. (5) or Eq. (11) to VLE data, and then calculating excess enthalpies. To do this, VLE data are needed at a minimum of two different temperatures. Our preliminary trials have shown that for best results in predicting H ex from VLE data, these two temperatures should not be too close to each other, and preferably should be in the same temperature range as the excess enthalpy data. Since most H ex data are available at 298.2 K, we have selected VLE data in the 293-353 K range for the evaluation of parameters. All of the models

b

•

i

i

i

i

i

i

'

i

400 3O0 20O -$ -6 E

1 00

0

v

-1 00 -200 -300 o x

-400 -500 -600 -700 -800

0.0

i

'o.t

i

i

0.2 013 0.,

,

:

i

015 016 0.7 0.8 0.9

.0

mole fraction acetone

Fig. 3. Prediction of the excess enthalpy of the acetone + water system at 20°C with parameters obtained from the correlation of VLE data at 25 and 35°C [30] (Vol. 1, Part lb, p. 148 and p. 149). The excess enthalpy data (filled circles) are from [30] (Vol.3, Part 1, p. 518), and the legend is as in Fig. 2.

75

H. Orbey, S.I. Sandier~Fluid Phase Equilibria 121 (1996)67-83 1 400 1 200 1 000 0

I=

800

o o°

°Oo

/

•

600 J~

•

400 o

--J

..'"

200

\\

/

?' o / . . . . . . .

/ /

,/

•

...-

\1

"'., \ / •

!, /

0 -200 -400

0.0 0',1 01,2 0.3 0.4 0.5 0.6 0,7 0'.8 0.9

,0

mole fraction methanol

Fig. 4. Prediction of the excess enthalpy of the methanol+benzene system at 25°C with parameters obtained from the correlation of VLE data at 20°C [30] (Vol. 1, Part 2a, p. 220) and 60°C ([30] p. 223). The excess enthalpy data (filled circles) are from [30] (Vol.3, Part 1, p. 280), and the legend is as in Fig. 2.

tested were capable of fitting VLE to within the experimental accuracy of the data; for brevity, we do not present those results here. The excess enthalpy predictions with parameters obtained in this way are presented in Figs. 3 and 4 for the acetone + water and methanol + benzene binary systems respectively. (The isotherms used to obtain parameters are indicated in the figure captions.) In general, the predictions of all of the models were only of semi-quantitative accuracy for the excess enthalpies, as shown in the figure for the acetone + water binary system, and there was no model that was more successful than the others in all cases. The activity coefficient and EOS models performed comparably and yielded acceptable results. One notable exception was the methanol + benzene system for which excess enthalpy was predicted poorly by all models as is shown in Fig. 4. For this system, the results presented in Fig. 4 were based on parameters obtained by fitting VLE data at 293 K and 333 K. To investigate whether better results could be obtained with VLE information closer to the temperature of the available excess enthalpy data, we reexamined this system with parameters obtained from VLE data at 293 K and 303 K. The shape of the curves and the numerical results change significantly, but generally for the worse for all the models considered. The results obtained in this section suggest that for all the models tested, only semi-quantitative estimates of the excess enthalpy of binary mixtures can be obtained by using only VLE data; in general the accuracy is not very good, and for highly nonideal systems, such as the methanol + benzene system, very poor results may be obtained. Again this result was expected for activity coefficient models [1], and the EOS models exhibit the same behavior. 4.4. The simultaneous correlation of VLE and H ~x data

Having shown that for the highly nonideal mixtures considered here prediction of excess enthalpy from VLE, and vice versa, is not very successful, but that each type of data can be separately

76

600

H. Orbey, S.I. Sandier~Fluid Phase Equilibria 121 (1996)67-83 800

_~

400

O

E

i

200

•

0

¢/t

==

-2oo

o x

•

-400 -600 I

80000 01 0 2 03 014

I

,

,

,

,

'

,

06 017 016 019 10

mole fraction 2-propanol Fig. 5. Excess enthalpy of the 2-propanol + water system obtained from simultaneous correlation with VLE data at 25°C, and by prediction at 75°C with the same parameters. Data are from [30] (filled circles are for 25°C from Vol. 3, Part 1, p. 583 and the triangles are at 75°C from Vol. 3, Part 1, p. 601), and the legend is as in Fig. 2.

correlated, we next considered the simultaneous fitting of both VLE and H ex data to determine if a single set of parameters could be used to represent both. Demirel and Gecegormez [3] have already shown that this is possible with NRTL and UNIQUAC excess free energy models. Here we used both excess enthalpy and VLE data at or near room temperature to obtain the four parameters in each model. The procedure we followed was to first use VLE data at room temperature to obtain the best fit parameters Ai) in Eq. (5) or Kij in Eq. (11) to ensure accurate representation of the VLE data at that temperature (in the case of the WSEOS model, we also allowed non-zero kij values in Eq. (14) when fitting VLE). Next, excess enthalpy data at the same (or at a close) temperature were used to obtain the mij and n i j terms in Eq. (5) or Eq. (11) with the boundary condition that the mij and nij pairs selected give the previously optimized values of Aij (or Kij) in Eq. (5) or Eq. (11). Then, with the parameters thus obtained at room temperature, we attempted to predict both the excess enthalpy and VLE behavior at other temperatures. For brevity, only the results for the 2-propanol + water (Figs. 5 - 7 ) and benzene + cyclohexane (Figs. 8 and 9) systems are presented. Similar results were obtained with the methanol + benzene and acetone + water binary systems. Again the correlation of the VLE data at 25°C is excellent with all models, and is not shown. In Fig. 5, the filled circles are the excess enthalpy data at 25°C that were used to obtain the model parameters. All models could be fitted acceptably to the data at this temperature, although the behavior of the VDWEOS model is slightly better than the other models which produced results that almost coincide. In the same figure, we compare the measured excess enthalpies at 75°C (triangles) with the predictions of various models. The VDWEOS model (upper solid line) leads to an acceptable extrapolation of the excess enthalpy for this system, whereas the other models are less accurate.

H. Orbey, S.I. Sandier/Fluid Phase Equilibria 121 (1996)67-83

1.1

'

'

'

'

77

l ]

1.0

/yy-~

0.9 ~

0.8

~

0.7

ff

[1.6

0.5 0.4

'

0,0

0.1

'

'

'

0 . . 2 03

0.4

'

0.5

0.6

'

'

0.7

'

0.8

0.9 1.0

mole fraction 2-propanol

Fig. 6. Prediction of the VLE for the 2-propanol+water system at 80°C with parameters obtained from simultaneous correlation of excess enthalpy and VLE data at room temperature. The data (filled circles) are from [31 ], and the legend is as in Fig. 2.

The prediction of the VLE behavior of the 2-propanol + water binary system at higher temperatures with parameters obtained from correlating VLE and H ex data at 25°C are shown in Figs. 6 and 7. At 80°C (Fig. 6), the activity coefficient model performs best. The VLE predictions at 250°C in

70

/-

/ f

°0V/

~

....

40

3O 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

mole fraction 2-propanol

Fig. 7. Prediction of the VLE for the 2-propanol+water system at 250°C with parameters obtained from simultaneous correlation of excess enthalpy and VLE at room temperature. The data (filled circles) are from [32], and the legend is as in Fig. 2.

78

H. Orbey, S.I. Sandier/Fluid Phase Equilibria 121 (1996) 67-83 1 400 1 200 J

1 000

/'/

v

>,

800

\',

600

== o

200

~

0 I

o.o'olt

o.,

i

i

o'.9

,

mole fraction benzene

Fig. 8. The excess enthalpy of the benzene +cyclohexane system obtained by simultaneous correlation with VLE data at 25°C, and by prediction at 120°C with the same parameters. Data are from [30] (filled circles are at 25°C from Vol. 3, Part 2, p. 992 and the triangles are at 120°C from Vol. 3, Part 1, p. 995), and the legend is as in Fig. 2.

Fig. 7 obtained using parameters derived from correlating VLE and H eX data at room temperature are more surprising. The activity coefficient model cannot be used at this temperature because 2-propanol is supercritical, and the HVOS model fails to converge. The WSEOS model converges, but gives poor results; only the VDWEOS model represents the VLE behavior with acceptable accuracy. 1,1 0

i

•

i

i

=

1.09

1.07 1.06

o/

i

,./'"'"""

1.08

..Q

i

eo e °

1.05

1.04

B.

1.03 1.02

1.01 1.00 0.99 i

0.0

o.1

i

0.2 0.3 o14 0.5 o16 o.7 0.8 0.9 1.o mole fraction benzene

Fig. 9. Prediction of the VLE for the benzene +cyclohexane system at 80°C with parameters obtained from simultaneous correlation of excess enthalpy and VLE at 25°C. The data (filled circles) are from [30] (Vol. 1, Part 6, p. 218), and the legend is as in Fig. 2.

H. Orbey, S.I. Sandier / Fluid Phase Equilibria 121 (1996) 67-83

79

For the benzene + cyclohexane system, results of enthalpy correlation are shown in Fig. 8. All models fit the excess enthalpy data at 25°C very accurately (filled circles are the experimental data), and are almost indistinguishable. However, when the same parameters are used to predict the excess enthalpy at 120°C (triangles), only the VDWEOS model predicts a qualitatively correct reduction of the excess enthalpy as the temperature increases, and gives quantitatively acceptable results. All other models predict that excess enthalpy will increase with increasing temperature. Interestingly, the worst predictions result from the direct application of the activity coefficient model. The predictive capabilities of the models for the VLE of the benzene + cyclohexane mixture at 80°C are shown in Fig. 9. The results indicate that the best option in this case is again the VDWEOS model.

5. Conclusions We have used recent multiparameter EOS mixing and combining rules, including those that incorporate excess free energy models, for the simultaneous correlation and prediction of VLE and H ex properties, and compared the performance of these models among themselves and with the direct use of activity coefficient models. Several conclusions can be made from this work. 1. The excess enthalpy of liquid mixtures can be correlated very accurately using equation of state models with temperature-dependent parameters. All of the EOS models used here led to correlations that are as good as or better than activity-coefficient-based models. This point is important because it suggests that it is possible to use the same EOS model to represent both phase equilibrium and excess properties of solutions. 2. With the multiparameter (here four) approach used here, fitting parameters only to excess enthalpy data or only to VLE data and then attempting to predict another property did not give satisfactory results. This observation was made earlier for the direct use of excess free energy models [ 1,2], and what we see here is also seen with the multiparameter EOS models. Consequently, one must consider the simultaneous correlation of VLE and H ex data. 3. For many systems at room temperature both excess enthalpy and VLE data are available. We have found that EOS models with temperature-dependent parameters can be used to fit these data simultaneously. While the resulting VLE correlations are excellent, the H ~x correlations are less accurate than fitting this property alone (see conclusion 1) but still acceptable for design purposes. This suggests that at or around room temperature EOS models can be used with the same parameters to represent both the VLE and excess enthalpy of mixtures. 4. In most cases the availability of H ex data and in some cases VLE data is limited to room temperature. It would be desirable to be able to use EOS model parameters obtained by fitting these data to predict these properties at other temperatures. Unfortunately, fitting model parameters to excess enthalpy and VLE data at a single temperature did not generally result in accurate predictions of the excess enthalpy at other temperatures for either the EOS or activity coefficient models tested here. Interestingly, only the modified van der Waals mixing rule, the VDWEOS model, led to extrapolations of reasonable accuracy for the excess enthalpy and vapor-liquid equilibrium at other temperatures. The other EOS mixing rules that make use of the excess free energy inherit the problems of those models.

H. Orbey, S.I. Sandier/Fluid Phase Equilibria 121 (1996)67-83

80

6. List of symbols AeX

Aij

a b C G ex

gij

nex

Kij kij mij ~ij

P R T V xi

molar excess Helmholtz free energy binary interaction parameter in excess free energy models equation of state parameter equation of state parameter an equation of state dependent mixing-rule parameter molar excess Gibbs free energy NRTL model parameter in Eq. (2) molar excess enthalpy binary interaction parameter in equations of state binary interaction parameter in equations of state binary interaction parameter in excess free energy models binary interaction parameter in excess free energy models absolute pressure gas constant absolute temperature molar volume mole fraction

6.1. Greek letters O/.

Tij

NRTL excess free energy model parameter NRTL excess free energy model binary interaction parameter

Acknowledgements The preparation of this manuscript was supported by Grant. No. DE-FG02-85ER13436 from the U.S. Department of Energy, and Grant No. CTS-9123434 from the U.S. National Science Foundation, both to the University of Delaware.

Appendix A. Residual enthalpy from cubic equations of state with various mixing rules The residual thermodynamic property is a difference in value of the thermodynamic property between real fluid and an ideal gas at the same temperature and pressure. These changes may be calculated using an equation of state from the relations A M r = H/ig(T,p ) - H i ( T , P ) r ig Amix = 2 x i H i ( T , P ) - n m i x ( x i , T , P )

(A1)

In Eq. A1, superscripts ig and r represent ideal gas and residual properties respectively, the subscript mix indicates the property of a homogeneous real mixture, T is absolute temperature, P is pressure, and H is the molar enthalpy. In Eq. A 1, the first relation is for a pure component and the second is for a mixture. These residual terms can be calculated from a selected equation of state for constituents of a mixture and for a mixture separately.

H. Orbey, S.I. Sandier/ Fluid Phase Equilibria 121 (I 996) 67-83

81

A general expression for the residual enthalpy from an equation of state, valid for both pure components and for mixtures is A Hmixorpur

e --

RT-

PV +

s:[ (0 )1 P - T

v

(A2)

dV

The algebraic expression for the residual enthalpy depends on the selection of the equation of state, and for mixtures, on the mixing rules for its parameters, through the derivative Eq. A2. The residual enthalpy expressions are given below for the models used here. 1. Two-parameter van der Waals (VDWEOS) model

AHr=RT-PV

(A3)

[V + ]'-- ~ yb

-~-~

The temperature derivative of the a term for pure components from the PRSV equation is straightforward and can be found elsewhere [33]. For mixtures, in addition to the pure component derivative terms, the cross term aij ( i ~ j ) needs to be evaluated. Since Eq. (10) and Eq. (11) introduce an additional temperature dependence, this derivative becomes Oaij 1 [ Oay OT = ~ a ~ i a ~ ( 1 - K i j x i - K j i x i ) [ a i - ~ + a J - ~

Oa i q- a ~ i a j ( - - x i n i j - - x j n j i )

(A4)

2. The Wong-Sandler (WSEOS) model When the Wong-Sandler mixing rule is used for mixtures, the b parameter of the EOS is also a function of temperature. In this case the expression for the residual enthalpy from the Peng-Robinson EOS is V + (1 + V/2-)b

AH r = R T - P V -

~--Vr~/~

[

V+(l+v~-lb]

X In

V+(1-V~)b]

In

V+(1- q)b r[ Ob) a [-~

b(V2+2bV-b

+

+ V- b

V

2)

2v/2-b 2

(AS)

The expressions for the a and b parameters are

aRT

bmix ami x =

RT- DRT DRTbmi x

with Q R T = Y'. Y'.x, x j ( b R r - a ) , j i j ai A ex

DRT= Y'.x i - + "-"

i

C

(A6)

H. Orbey, S.I. Sandier~Fluid Phase Equilibria 121 (1996)67-83

82

and the temperature derivatives are Obmix aT

QRT ( ( R T - DRT) 2 R

1 a( QRT) R T - DRT OT

O( DRT) ) aT

and

Oamix O( DRT) aT - bmi~ OT + ( DRT)

0bmix

(A7)

o----T-

and U

~-- Z i

XiX j bR

-~

ij

j

3. Modified Huron-Vidal (HVOS) model In this case, Eq. A3 is applicable when the following substitution is made X i [Oai I ai T ,~.-b-7{,-~]-7}+

,0a, a [ 'Ob]

'~-)C-

CT

1

(19)

Finally, to calculate the excess enthalpy of mixing at constant temperature and pressure we use n ex =

Hmix(v,e,xi) -- 2 x i H i ( r , P )

r = E x i H i i g ( v , P ) - AHmix(T,P,xi) - ]~__~xi[H iig (T,P) __ A Hir(T,p)]

= 2xiAHir(T,P) - AHri,,(T,P,xi)

(AIO)

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