Ind. Eng. Chem. Res. 2001, 40, 1261-1270
1261
Prediction of Excess Enthalpies Using a Gex/EOS Model Gonzalo N. Escobedo-Alvarado and Stanley I. Sandler* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
The Peng-Robinson equation of state (as modified by Stryjek and Vera), combined with the Wong-Sandler mixing rule, was used for the prediction of excess enthalpies of highly nonideal mixtures at both low and high pressures, and with supercritical components included. In this work, the model parameters were fitted only to experimental vapor-liquid equilibrium data and were then used to obtain simultaneous representation of both vapor-liquid equilibria and excess enthalpy data. The results show that this equation of state model is able to yield not only qualitative, but also reasonably quantitative, predictions of the thermodynamic properties of nonideal systems of varying complexity. Introduction Cubic equations of state (EOSs) have been used for the correlation and prediction of phase equilibria in mixtures. Recently, mixing rules that incorporate an excess free energy model have been introduced, allowing the EOS to be used for the description of highly nonideal mixtures. The important feature of these models is that the excess free energy obtained from an EOS is equated to that from an activity coefficient model. These EOS/ mixing rule models have been widely and successfully used for the correlation and prediction of vapor-liquid and liquid-liquid equilibria (Voustas et al.,1 Orbey and Sandler,2 Escobedo and Sandler3). However, little work has been done on the performance of these EOS/mixing rule models for the prediction of excess properties of mixtures that are also important for process design. In particular, excess enthalpies of mixing (Hex) are thermodynamic properties used in the design of chemical processes. The simultaneous representation of phase equilibria and excess enthalpies with cubic EOSs has been an important goal. Adachie and Sugie4 used the van der Waals and Peng-Robinson5 EOS with a modified van der Waals mixing rule to correlate the excess volumes and enthalpies of several binary mixtures at atmospheric pressure. Their mixing rule added a composition dependence to the binary interaction parameter of the traditional van der Waals mixing rule. This introduced two adjustable parameters into the model that were made temperature-dependent. Although the correlation of excess properties gave reasonable results, the extension of the method to predict phase equilibrium was not tested. Chen et al.6 used the virial-like Martin-Hou EOS to correlate excess enthalpies of both liquid and gas mixtures of several hydrocarbons and the mixture of ethanol and water. The adjustable binary interaction parameters obtained were temperature-dependent, and reasonable predictions of vapor-liquid equilibria were obtained by this method. An interesting feature of this work is that predictions were made using parameters that were determined by correlating excess enthalpy data of liquid mixtures. Lichtenstein et al.7 used a modified Peng-Robinson EOS to simultaneously cor* Corresponding author: telephone, 302-831-2945; Fax, 302831-4466; e-mail,
[email protected] relate vapor-liquid equilibria and excess enthalpy data of several mixtures of methanol and hydrocarbons covering a wide range of temperatures and pressures. In their approach, the UNIQUAC model (Abrams and Prausnitz)8 was used to calculate the residual excess contribution of the free energy of mixing, while the cubic EOS model was used to calculate the pure-component contributions. Reasonable results were obtained for these systems, even though, for the methanol + nalkane mixtures, significant deviations were observed. Despite the relative success of the above-mentioned approaches, none of them used the recently developed EOS mixing rules that include an excess free energy model. Orbey and Sandler9 studied the performance of the Wong-Sandler mixing rule (Wong and Sandler),10 the HVOS mixing rule (Orbey and Sandler),11 the LCVM model (Boukouvalas et al.),12 and the MHV1 (Michelsen)13 and MHV2 (Dahl and Michelsen)14 mixing rules for the simultaneous representation of vapor-liquid equilibrium (VLE) and excess enthalpy of four binary mixtures using temperature-dependent parameters in the models. These authors found that, although the models could give good correlations of VLE and Hex separately, attempting to predict the values of one property from the other did not give satisfactory results. Moreover, although the simultaneous correlation of VLE and Hex at a single temperature was possible, when the parameters determined in this way were used for extrapolations, the predicted results of either property were not very accurate. It is worth mentioning that most of the mixtures examined by these authors were at room temperature and moderate pressures. Ohta15 examined the performance of the PengRobinson EOS as modified by Stryjek and Vera16 (PRSV) in combination with the Wong-Sandler and MHV1 mixing rules for the correlation of Hex data for several binary mixtures. In that study, the parameters in the excess free energy model were assumed to be temperature-dependent, and this dependence was found using the available parameters reported in the literature obtained from experimental VLE data. Most of the systems studied were at atmospheric pressure. Good predictions of Hex were obtained for the temperature and pressure ranges covered by the experimentally obtained parameters; however, as the author pointed out, it was not possible to obtain good VLE predictions outside of
10.1021/ie000682z CCC: $20.00 © 2001 American Chemical Society Published on Web 01/26/2001
1262
Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001
this range. Djordjevic et al.17 used the PRSV EOS in combination with the van der Waals and HVOS mixing rules for the simultaneous description of VLE, Hex, and excess heat capacity data. These authors used the NRTL model (Renon and Prausnitz)18 with temperature-dependent parameters in the HVOS mixing rule and found that the results became worse when two or three properties were simultaneously correlated. Furthermore, it was necessary to assign a strong temperature dependence to the parameters in order to obtain acceptable correlations of the experimental data. Most of the cited works cover only a limited range of temperatures and pressures. Dai et al.19 did a comprehensive review of Hex data of systems at supercritical conditions. Furthermore, they correlated these data using a modified Peng-Robinson EOS, the NRTL activity coefficient model, and the original Peng-Robinson EOS with MHV1 mixing rules. Their conclusion was that the ad hoc modified Peng-Robinson EOS was more suitable for correlating these data. Among all of the models tested by these authors, the MHV1 proved to give the poorest results. However, the models were not tested for their ability to predict VLE using the Hex correlated parameters. In this contribution, we will use the PRSV EOS in combination with the Wong-Sandler mixing rule for the prediction of excess enthalpies of highly nonideal mixtures covering a wide range of temperatures and pressures with parameters obtained by correlation of experimental VLE data. Thermodynamic Framework The Peng-Robinson EOS as modified by Stryjek and Vera16 (PRSV) was used
a(T) RT P) - 2 (V - b) V + 2bV - b2
a(T) ) 0.45724
Pc
b ) 0.07780
1/2
2
RTc Pc
[ ()( T Tc
0.5
0.7 -
T Tc
)]
(7)
where
(
)
a
Q)
∑i ∑j xixj b - RT ij
(8)
D)
aii Gex γ xi + biiRT C*RT
(9)
∑i
In eq 8, the cross parameter is given by a geometric combining rule
(b - RTa )
xaiiajj(1 - k ) 1 ) (bii + b) ij ij 2 RT
1 C* ) - ln(1 + x2) x2 In eq 9, Gex γ is the excess Gibbs free energy from an activity coefficient model. In this study, two Gex γ models were used. The first is the nonrandom two fluid (NRTL) model of Renon and Prausnitz18
Gex γ
)
∑i xi
∑j τjigjixj
(11)
∑k gkixk
where
∆Gij RT
(12)
ln gij ) -Rijτij
(13)
Here, ∆Gij is a characteristic energy, and Rij is a nonrandomness parameter. The other free energy model that was used is the UNIQUAC model of Abrams and Prausnitz8
Gex γ RT
)
∑i
xi ln
() Φi
+
xi
z 2
∑i
qixi ln
() θi
Φi
-
∑i qixi ln(∑j θjτji)
(14)
(4)
where:
κ0 ) 0.378893 + 1.4891753ω - 0.17131848ω2 + 0.0196554ω3 (5) Here, ω is the acentric factor, and κ1 is a substancedependent parameter that is found by fitting vapor pressure data to the EOS. Equation 2 was extended to mixtures using the Wong-Sandler (WS) mixing rule as
(10)
In eq 9, C* is a constant that depends on the EOS, and for the PR EOS, it is equal to
(2) (3)
(6)
Q 1-D
τij )
where Tc is the critical temperature and Pc is the critical pressure. In eq 2, κ is a parameter that ensures that the EOS reproduces the vapor pressures of the pure components and is given by
κ ) κ0 + κ1 1 +
b)
RT
{ [ ( ) ]} T 1+κ 1Tc
QD a ) RT 1-D
(1)
where P is the pressure, R is the universal gas constant, T is the temperature, and V is the molar volume. The pure-component parameters a, which is a function of temperature, and the constant b are obtained from
RT2c
modified by Orbey and Sandler.19 Using this mixing rule, the EOS mixture parameters are given by
Φi )
rixi
∑j
and θi )
rjxj ln τij )
uij RT
qixi
∑j
(15)
qjxj (16)
where u is an energy parameter characterizing the interaction of molecule i with molecule j; r and q are the pure-component volume and area parameters, re-
Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001 1263 Table 1. Parameters for the Wong-Sandler Mixing Rule Including the NRTL and UNIQUAC Models WS-NRTL mixture
ref
carbon dioxide (1) ethanol (2) methanol (1) carbon dioxide (2) ethane (1) methanol (2) propane (1) methanol (2) tetrahydrofuran (1) n,n-dimethylformamide (2) cyclohexane (1) dimethylformamide (2) benzene (1) cyclohexane (2) cyclopentane (1) tetrachloroethylene (2) tetrahydrofuran (1) cyclohexane (2)
a12
∆G12 ∆G21 (cal mol-1) (cal mol-1)
k12
1
k12
u12 u21 (cal mol-1) (cal mol-1)
F (eq 19)
0.35
0.128
679.669
232.919
7.12 × 10-5
0.115
158.082
536.605
8.64 × 10-5
26
0.40
0.164
143.284
432.887
1.78 × 10-3
0.359
129.358
295.907
1.98 × 10-3
27
0.65
0.473
-167.257
1228.077
1.66 × 10-3
0.451
-327.080
1067.400
2.14 × 10-3
28
0.50
0.259
2800.285
1154.637
5.16 × 10-4
0.459
-83.541
619.823
1.77 × 10-3
29
0.30
0.095
461.171
-38.780
3.85 × 10-4
0.084
233.513
-69.519
3.60 × 10-4
29
0.13 -0.227
1765.452
764.721
2.78 × 10-3 -0.269
698.852
247.959
6.80 × 10-3
29
0.40 -0.174
477.883
365.155
1.25 × 10-5 -0.168
63.340
200.504
1.10 × 10-5
30
0.20 -0.043
123.721
101.836
1.38 × 10-5 -0.065
-53.079
133.278
1.25 × 10-5
29,31 0.40 -0.092
293.139
436.740
1.06 × 10-4 -0.127
-11.609
237.297
1.31 × 10-4
( )
Ndata
∑
Ndata i)1
Pe - Pc
2
(17)
Pe
where Ndata is the number of data points and Pe and Pc are the experimental and calculated bubble point pressures, respectively. Equation 17 was minimized using the Levenberg-Marquardt21 method. The excess enthalpy is calculated from the equation
Hex ) ∆Hrmix -
∑xi∆Hri
(18)
The enthalpy departure functions for pure components (∆Hri ) and mixtures (∆Hrmix) are related to a cubic EOS by
∂P ∫∞V[P - T(∂T )V] dV - RT + PV
∆Hr ) -
(19)
For the PRSV EOS, these functions are given by
[( ]
( )]
1 a ∂b ∂a T -a- T × ∆Hr ) PV - RT + ∂T b ∂T 2x2b V + (1 + x2)b RT2 ∂b ln + V - b ∂T V + (1 - x2)b
[
F (eq 19)
25
spectively; and the coordination number, z, has been set equal to 10 (Reid et al.).20 The adjustable parameters in the model are kij in eq 10, ∆Gij in eq 12, and uij in eq 16. These parameters were determined by fitting VLE data to the EOS/mixing rule model. To do so, the following objective function was used:
F)
WS-UNIQUAC
)
()
()
V a ∂b T (20) b ∂T V2 + 2bV - b2 For the mixture, the partial derivatives are at constant composition and are obtained from eq 7; for the pure components, (∂a/∂T) ) da/dT and ∂b/∂T ) db/dT ) 0. In all of the Hex calculations a rigorous phase stability analysis was done. When the stability analysis showed that more than one phase was present in the system, the equilibrium compositions and the mass fraction of each phase in the system were computed using a flash calculation. Both the stability analysis and the flash
calculation were done using an algorithm that follows the procedure suggested by Michelsen.22,23 Systems Investigated Nine binary systems were selected for a more detailed analysis of the models considered here. These systems are: carbon dioxide + ethanol, ethane + methanol, methanol + carbon dioxide, propane + methanol, tetrahydrofuran + n,n-dimethylformamide, cyclohexane + dimethylformamide, benzene + cyclohexane, cyclopentane + tetrachloroethylene, and tetrahydrofuran + cyclohexane. There are considerable VLE and Hex data covering a wide range of temperatures and pressures for these systems, all of which show strong deviations from ideal solution behavior in both VLE and excess enthalpy. Consequently, these systems are a stringent test of the model considered here. The sources of experimental data for these systems are provided in Table 1 for VLE and in Table 2 for Hex. The pure-component critical constants required to evaluate eqs 2-5 were obtained from Reid et al.,20 and the pure-component vapor pressure data necessary to calculate the parameter κ1 in eq 4 were obtained from Boublik et al.24 Results and Discussion The binary mixture VLE data were correlated using the WS mixing rule. The correlated parameters and the final values of the objective function defined by eq 17 are shown in Table 1. In all cases, it was possible to use temperature-independent parameters. In this contribution, we present the results for four systems at high pressures and five systems for which Hex was measured at atmospheric pressure. In Table 2, the results obtained for the prediction of Hex with the WS mixing rule including the NRTL and UNIQUAC models are compared. In this analysis, the results have been compared in terms of the absolute average deviation (AAD) ex |, the maximum absolute value of the relative to |Hmax excess enthalpy for each mixture. a. Low-Pressure Systems. The mixture of tetrahydrofuran with n,n-dimethylformamide was studied using the UNIQUAC and NRTL models in the WS mixing rule. For this system, excellent results were obtained for the correlation of VLE data with the WS-
1264
Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001
Table 2. Comparison of the WS-NRTL and WS-UNIQUAC Models for the Prediction of Hex system carbon dioxide (1) ethanol (2) methanol (1) carbon dioxide (2) ethane (1) methanol (2) propane (1) methanol (2) tetrahydrofuran (1) n,n-dimethylformamide (2) cyclohexane (1) dimethylformamide (2) benzene (1) cyclohexane (2) cyclopentane (1) tetrachloroethylene (2) tetrahydrofuran (1) cyclohexane (2) a
ex |Hmax | (cal/mol)
ex AAD/|Hmax | × 100 NRTL
ex AAD/|Hmax | × 100 UNIQUAC
AADa-NRTL (cal/mol)
AADa-UNIQUAC (cal/mol)
32
23.179
30.344
420.650
5.51
7.21
33
81.021
71.456
2844.168
2.85
2.51
34
16.295
10.176
215.368
7.57
4.73
35
23.429
175.838
361.750
6.48
48.61
36
0.961
0.488
81.644
1.18
0.60
37
2.344
1.798
524.696
0.45
0.34
38
17.107
15.925
191.044
8.95
8.34
30
0.365
0.584
56.405
0.65
1.03
38,39
2.032
2.728
171.716
1.18
1.59
ref
AAD ) absolute average deviation defined as:
AAD )
1 Ndata
Ndata
∑ |H
ex i (experimental)
- Hex i (predicted)|
i)1
Figure 1. VLE correlation of the system tetrahydrofuran (THF) + n,n-dimethylformamide. The symbols represent the experimental data, and the lines represent the results using the PRSV EOS with the WS mixing rule including the UNIQUAC model.
Figure 2. Excess enthalpy for the mixture system tetrahydrofuran (THF) + n,n-dimethylformamide at 298.15 K. The symbols represent the experimental data, and the lines represent the prediction using the PRSV EOS with the WS mixing rule including the UNIQUAC model.
UNIQUAC model, as is shown in Figure 1. Here, the two isotherms (313.15 and 333.15 K) were fitted using a single set of temperature-independent parameters. Figure 2 shows the predicted excess enthalpy values at 298 K and 1.0133 bar. The model gives a reasonable representation of Hex for this mixture. It is noteworthy that, in this case, Hex was predicted at conditions outside the temperature range used for the parameter estimation and that temperature-independent parameters were used. For this system, both models, the WS-NRTL and WS-UNIQUAC, yielded good results. Similar results were obtained for the system cyclohexane + n,n-dimethylformamide. Figure 3 shows a prediction of the excess enthalpy of this mixture at 298.15 K using the WS mixing rule in combination with the NRTL model. In this case, the parameters were obtained by correlating VLE data at 298.15, 313.15, and 333.15 K. For the system tetrahydrofuran + cyclohexane, four isotherms were available for the correlation of VLE
data. The excellent results of this correlation with temperature-independent parameters and the NRTL model are shown in Figure 4. Figure 5 shows the good agreement between the predicted excess enthalpy and the experimental data at 303.15 K. Comparable results were obtained using the WS-UNIQUAC model. Figure 6 shows the correlation of six isotherms for the system benzene + cyclohexane using the WS-NRTL model. It is observed that the model gives an excellent representation of this system over a wide range of temperature and pressure. The parameters then were used to predict the excess enthalpy of this mixture at 298.15, 308.15, 313.15, 318.15, 323.15, and 328.15 K. The results are summarized in Table 2 and show that the model is able to give a good prediction of this property for this mixture over this range of temperatures. In Figure 7, the predictions of Hex for 298.15 and 328.15 K are shown. These results indicate that these models are able to give good quantitative results for this mixture; however, as the temperature increases, the Hex
Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001 1265
Figure 3. Excess enthalpy for the mixture system cyclohexane + n,n-dimethylformamide at 298.15 K. The symbols represent the experimental data, and the lines represent the prediction using the PRSV EOS with the WS mixing rule including the NRTL model.
Figure 5. Excess enthalpy for the mixture system THF + cyclohexane at 298.15 and 303.15 K. The symbols represent the experimental data, and the lines represent the prediction using the PRSV EOS with the WS mixing rule including the NRTL model.
Figure 6. VLE correlation of the system benzene + cyclohexane. The symbols represent the experimental data, and the lines represent the results using the PRSV EOS with the WS mixing rule with the NRTL model. Figure 4. VLE correlation of the system tetrahydrofuran (THF) + cyclohexane. The symbols represent the experimental data, and the lines represent the results using the PRSV EOS with the WS mixing rule with the NRTL model.
predictions become less accurate. Similar results are observed for the system cyclopentane + tetracholorethylene. Figure 8 shows the correlation of low-pressure VLE data for this system at 298 K. Figure 9 shows the prediction of the excess enthalpy at the same temperature. The results show that the model gives an accurate prediction of this mixing property. b. High-Pressure Systems. Figure 10 shows the result of the VLE correlation for the 314.5, 325.2, and 337.2 K isotherms of the system CO2 + ethanol that cover a wide range of pressures using the WS-NRTL model. It is evident that this model gives an accurate representation of the system. Figure 11 shows the
predicted excess enthalpies for this mixture at 373.15 K at four different pressures. It is important to notice that this prediction is made at a temperature higher than the range used to correlate the VLE data and that these three isobars are also outside of this range. These results show that the model is able to give a good prediction of both the phase behavior and Hex for this system. In Figure 11, the Hex values at low x are negative because of the condensation of gaseous carbon dioxide into liquid ethanol during mixing and the effect of the enthalpy of vaporization of ethanol. At high x, liquid ethanol vaporizes into gaseous carbon dioxide during mixing, and Hex is positive and changing rapidly with composition because of the absorption of the enthalpy of vaporization of ethanol. This is an interesting result because, at these conditions, CO2 is a supercritical fluid. Furthermore, mixtures of CO2 + ethanol are frequently used in supercritical extraction, which
1266
Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001
Figure 7. Excess enthalpy for the mixture system benzene + cyclohexane. The symbols represent the experimental data, and the lines represent the prediction using the PRSV EOS with the WS mixing rule including the NRTL model.
Figure 8. VLE correlation of the system cyclopentane + tetrachloroethylene at 298.15 K. The symbols represent the experimental data, and the lines represent the results using the PRSV EOS with the WS mixing rule with the NRTL model.
shows that the WS mixing rule might be suitable for the modeling of different aspects of supercritical processes. Although the WS-UNIQUAC model was successful in correlating the VLE data, its performance for the prediction of Hex for this mixture was less accurate than that of the WS-NRTL model. Similar results were obtained for the system ethane + methanol, where the EOS model was able to provide a good representation of the limited experimental VLE data for three isotherms (Figure 12) and for Hex in the vapor-liquid equilibrium region (Figure 13). It is observed that the thermodynamic behavior is very similar to that observed for the CO2 + ethanol system and that the EOS model gives a good qualitative and quantitative description of it. The high-pressure VLE data for the system propane + methanol were correlated using the NRTL model in the WS mixing rule. Figure 14 shows the results of the VLE correlation of three different isotherms with temperature-independent parameters. The results obtained for the prediction of Hex for this mixture are reported in Table 2. These results show that the WS mixing rule in combination with the NRTL model is able to give
Figure 9. Excess enthalpy for the mixture system cyclopentane + tetrachloroethylene. The symbols represent the experimental data, and the lines represent the prediction using the PRSV EOS with the WS mixing rule including the NRTL model.
Figure 10. VLE correlation of the system CO2 + ethanol. The symbols represent the experimental data, and the lines represent the results using the PRSV EOS with the WS mixing rule with the NRTL model.
accurate predictions of the excess enthalpy of this system at 348.15, 363.15, and 373.15 K at 50 and 150 bar. The first two temperatures are above the gasliquid equilibrium line for propane and between the liquid-liquid and gas-liquid critical lines of the mixtures, so both components are liquids. At 373.15 K, propane is a supercritical fluid. Figures 14 and 15 show that the model is able to a give reasonable description of the effects of temperature and pressure on the thermodynamic properties of this system. It is interesting to note that the 150-bar isobar data are wellrepresented, even though these data are almost 100 bar higher than the VLE data used to fit the parameters in the model. Along the isobar at 50 bar, the model predicts a higher maximum value of Hex than is seen in the experimental data. For this system, we compared the results obtained with the traditional van der Waals mixing rule with kij set equal to the value of 0.056 obtained from correlating the VLE data for the three isotherms of Figure 14. Figure 16 shows a comparison between the performance of the WS mixing rule and
Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001 1267
Figure 11. Excess enthalpy for the mixture CO2 + ethanol at 373.15 K. The symbols represent the experimental data, and the lines represent the prediction using the PRSV EOS with the WS mixing rule including the NRTL model.
Figure 12. VLE correlation of the system ethane + methanol. The symbols represent the experimental data, and the lines represent the results using the PRSV EOS with the WS mixing rule with the UNIQUAC model.
that of the van der Waals mixing rule at 348.15 K and 150 bar. These results show the poor performance of the van der Waals mixing rule for the prediction of the highpressure excess enthalpies of this system. For this particular system, the parameters obtained by fitting the VLE data using the UNIQUAC model in the WS mixing rule were not able to produce even a qualitatively correct description of the Hex of this mixture, as indicated by the entry in Table 2. For all of the temperatures considered in these calculations, it is observed that the isotherms at 50 bar are less symmetrical than those at 150 bar and that the maximum is shifted toward higher mole fraction of propane. It has been pointed out that this shape of the Hex line can be related to the effect of a negative contribution to Hex that results from the condensation of the supercritical propane into the liquid methanol (Sipowska et al.).35 This effect is more important at low concentrations of the supercritical component, as indicated, for instance, in the CO2 + ethanol system. In contrast, the high-pressure, high-CO2-concentration behavior is quite similar to that observed in liquid
Figure 13. Excess enthalpy for the mixture ethane + methanol at 348.15 K and 75 bar. The symbols represent the experimental data, and the lines represent the prediction using the PRSV EOS with the WS mixing rule including the UNIQUAC model.
Figure 14. VLE correlation of the system propane + methanol. The symbols represent the experimental data, and the lines represent the results using the PRSV EOS with the WS mixing rule with the NRTL model.
mixtures at temperatures considerably below the critical temperature, such as those considered earlier. This suggests that the condensation effect is less important at higher pressures. This is expected because the supercritical component should be more “liquidlike” at higher pressures. The high-pressure behavior of the system methanol + CO2 was also investigated. Figure 17 shows the results of the correlation of the experimental VLE data along five isotherms with temperature-independent parameters. For this mixture, excess enthalpies were predicted at conditions close to its critical loci and are shown in Figure 18. The EOS model was able to give a reasonably good estimate of Hex for this mixture. Again, we see that the model is able to represent the condensation effect. At higher temperatures and pressures, the effect of condensation of the supercritical component is less important, resulting in a parabolic Hex curve. In contrast, at lower temperatures, the model predicts negative Hex values that are in good agreement with
1268
Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001
Figure 15. Excess enthalpy for the mixture propane + methanol at 373.15 K. The symbols represent the experimental data, and the lines represent the prediction using the PRSV EOS with the WS mixing rule including the NRTL model.
Figure 16. Comparison of WS and van der Waals predictions of the excess enthalpy for the mixture propane + methanol at 348.15 K and 150 bar. The symbols represent the experimental data, and the lines represent the predictions using the PRSV EOS with the van der Waals (vdW) and Wong-Sandler (WS) mixing rules.
the experimental observations. Although the predictions show some deviation from the experimental data, the model is able to provide a good qualitative description of the thermodynamic behavior of this system considering that no Hex data were used in the parameter estimation. Conclusions We have used an EOS mixing rule that includes a Gex model for the prediction of excess enthalpy using parameters determined solely from VLE data for highly nonideal mixtures. In each instance, the model satisfactorily reproduces the VLE behavior of the mixtures involved using temperature-independent parameters. Furthermore, the model is also able to provide reasonable predictions of the excess enthalpy extended over a wide range of temperatures and pressures, including extrapolations out of the range used for the parameter estimation. It is important to stress that, in this case, we used temperature-independent parameters in the
Figure 17. VLE correlation of the system methanol + carbon dioxide. The symbols represent the experimental data, and the lines represent the results using the PRSV EOS with the WS mixing rule with the NRTL model.
Figure 18. Excess enthalpy for the mixture methanol + carbon dioxide. The symbols represent the experimental data, and the lines represent the prediction using the PRSV EOS with the WS mixing rule including the NRTL model.
mixing rule; in contrast, most of the previous attempts to represent simultaneously VLE and Hex data reported in the literature have used other mixing rules and have had to resort to the use of temperature-dependent parameters. The results also show that, generally, the Wong-Sandler mixing rule gives comparable results when used in combination with either the NRTL or the UNIQUAC models. The model is able to give very accurate predictions of the Hex values of mixtures of hydrocarbons at atmospheric pressure. This is an interesting result because the temperature-independent parameters of the model were fitted only to low-pressure VLE data and were then used to successfully predict Hex for liquid mixtures at atmospheric pressure and over a range of temperatures. We also used the Gex/EOS model to obtain accurate high-pressure predictions of Hex for mixtures of methanol and ethanol with a supercritical component. The
Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001 1269
results confirm the usefulness of the model for the description of supercritical fluids, including when used for a homogeneous supercritical phase and in the highpressure VLE region. A comparison with the results obtained with the classical van der Waals mixing rules shows the inadequacy of the latter model for the description of these systems. It should also be noted that an activity coefficient model alone could not be used to accurately describe the behavior of these supercritical mixtures. It is only when an activity coefficient (or Gex) model is combined with an equation of state, as here, that a satisfactory description is obtained. Another important aspect of this work is that Hex predictions can provide a physically meaningful criterion for choosing the most appropriate parameter set for a mixture. In this work, it was observed that, because of the nonlinearity of the dependence of the predictions on the parameter values, several parameter sets might give equivalent correlations of the VLE data of a particular mixture; however, some of these parameter sets yielded unrealistic predictions of the excess enthalpies. Clearly, the best set of parameters is the one that also gives good predictions of other thermodynamic properties. Finally, we should point out that we have found the excess free energy/equation of state model proposed here is less successful for mixtures of components that both self- and cross-associate. The ethanol + water system is an example of such a mixture. For such systems, temperature-dependent parameters are needed to obtain a quantitative description of Hex, so that the model becomes correlative rather than predictive. This failure is a result of the underlying Gex models imbedded into the Gex/EOS model. We know this because, for lowpressure systems that can be described using activity coefficient or Gex models directly (that is, without an equation of state), temperature-dependent parameters are needed to correlate VLE and Hex data simultaneously. Thus, although a Gex/EOS model extends the range of application of equations of state, it does not overcome essential failures of Gex models. Acknowledgment The authors thank the National Science Foundation (CTS-9903536) for financial support of this research and Prof. L. A. Galicia-Luna of the Instituto Polite´cnico Nacional of Me´xico for useful discussions. Literature Cited (1) Voustas, E. C.; Boukouvalas, C.J.; Kalospiros, N. S.; Tassions, D. P. The Performance of EoS/GE Models in the Prediction of Vapor-Liquid Equilibria in Asymmetric Systems. Fluid Phase Equilib. 1996, 116, 480. (2) Orbey, H.; Sandler, S. I. Modeling Vapor-Liquid Equilibria. Cubic Equations of State and their Mixing Rules; Cambridge University Press: New York, 1998. (3) Escobedo-Alvarado, G. N.; Sandler, S. I. Study of EOS-Gex Mixing Rules for Liquid-Liquid Equilibria. AIChE J. 1988, 34, 1178. (4) Adachie, Y.; Sugie, H. A New Method to Predict Thermodynamic Properties of Mixtures by Means of a Cubic Equation of State. J. Chem. Eng. Jpn. 1988, 21, 57. (5) Peng, D.; Robinson, D. A New Two Constant Equation of State. Ind. Chem. Eng. Fundam. 1976, 15, 59. (6) Chen, G.; Wu, Z.; Chen, Z.; Hou, Y. Correlation of Excess Enthalpies and Prediction of Vapor-Liquid Equilibria from Excess Enthalpies by means of an Equation of State. Fluid Phase Equilib. 1991, 65, 145.
(7) Lichtenstein, C.; Rauzy, E.; Berro, Ch. Simultaneous Representation of Vapor-Liquid Equilibria and Excess Enthalpies of Methanol-Hydrocarbons Systems Using an Equation of State. Fluid Phase Equilib. 1993, 87, 37. (8) Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975, 21, 116. (9) Orbey, H., Sandler, S. I. A Comparison of Various Cubic Equation of State Mixing Rules for the Simultaneous Description of Excess Enthalpies and Vapor-Liquid Equilibira. Fluid Phase Equilib. 1996, 121, 67. (10) Wong, D. S. H.; Sandler, S. I. A Theoretically Correct Mixing Rule for Cubic Equations of State. AIChE. J. 1992, 38, 671. (11) Orbey, H.; Sandler, S. I. On the Combination of Equations of State and Excess Free Energy Models. Fluid Phase Equilib. 1995, 111, 53. (12) Boukouvalas, C.; Spiliotis, N.; Coustikis, P.; Tzouvaras, N. Prediction of Vapor-Liquid Equilibrium with the LCVM Model. A Linear Combination of the Huron-Vidal and Michelsen Mixing Rules Coupled with the Original UNIFAC and t-mPR Equation of State. Fluid Phase Equilib. 1994, 92, 75. (13) Michelsen, M. L. A Modified Huron-Vidal Mixing Rule for Cubic Equations of State. Fluid Phase Equilib. 1990, 60, 213. (14) Dahl, S.; Michelsen, M. L. High-Pressure Vapor-Liquid Equilibrium with a UNIFAC Based Equation of State. AIChE J. 1990, 36, 1829. (15) Ohta, T. Representation of Excess Enthalpies by the PRSV EOS with the Modified Huron-Vidal First Order and WongSandler Mixing Rules. Fluid Phase Equilib. 1997, 129, 89. (16) Stryjek R.; Vera J. H.. PRSV: An Improved PengRobinson Equation of State of Pure Compounds and Mixtures. Can. J. Chem. Eng. 1986, 64, 4, 323. (17) Djordjevic, B. D.; Kijevcanin, M. L.; Sarbanovic, S. P. Simultaneous Presentation of VLE, Hex and Cpex by the PRSV Equation of State with the Modified van der Waals One-fluid and Huron-Vidal-Orbey-Sandler Mixing Rules. Fluid Phase Equilib. 1999, 155, 205. (18) Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J. 1965, 14, 135. (19) Dai, W.; Ochi, K.; Kojima, K. Excess Enthalpy Data in Binary Systems Containing a Supercritical Fluid and Their Correlation. J. Chem. Eng. Data 1998, 43, 687. (20) Orbey, H.; Sandler, S. I. Reformulation of the WongSandler Mixing Rule for Cubic Equations of State. AIChE J. 1995, 41, 683. (21) Reid, R. C.; Prausnitz J. M.; Polig, B. E. The Properties of Gases & Liquids; McGraw-Hill: New York, 1988. (22) IMSL. IMSL Library; IMSL Inc.: Houston, TX, 1979. (23) Michelsen, M. L. The isothermal flash problem. Part I. Stability. Fluid Phase Equilib. 1982, 9, 1. (24) Michelsen, M. L. The isothermal flash problem. Part II. Phase Split Calculation. Fluid Phase Equilib. 1982, 9, 21. (25) Boublik, T.; Fried, V.; Hala, E. The Vapor Pressures of Pure Substances; Elsevier Scientific Publishing Company: New York, 1973. (26) Jennings, D. W.; Rong-Jwyn, L.; Teja, A. S. Vapor-Liquid Equilibria in the Carbon Dioxide + Ethanol and Carbon Dioxide + 1-Butanol Systems. J. Chem. Eng. Data 1991, 36, 303. (27) Hong, J. H.; Kobayashi, R. Vapor-Liquid Equilibrium Studies for the Carbon Dioxide-Methanol System. Fluid Phase Equilib. 1988, 41, 269. (28) Ma, Y. H.; Kohn, J. P. Multiphase and Volumetric Equilibria of the Ethane-Methanol System at Temperatures between -40 °C and 100 °C. J. Chem. Eng. Data 1964, 9, 3. (29) Galivel-Solastiouk, F. S.; Laugier, S.; Richon, D. VaporLiquid Equilibrium Data for the Propane-Methanol and PropaneMethanol-Carbon Dioxide Systems. Fluid Phase Equilib. 1986, 28, 73. (30) Gmehling; Onken, U.; Arlt, W.; Greenzhauser, U.; Weilich, B.; Kolbe, J.; Rarey, J. Vapor Liquid Equilibrium Data Collection; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt/Main, Germany, 1977; Vol. I, Parts 4a, 2c, and 6c.
1270
Ind. Eng. Chem. Res., Vol. 40, No. 4, 2001
(31) Polak, J.; Murakami, S.; Lam, V. T.; Benson, G., C. Excess Enthalpy, Volume, and Gibbs Free Energy of CyclopentaneTetrachloroethylene Mixtures at 25 °C. J. Chem. Eng. Data 1970, 15, 323. (32) Wu, H. S.; Sandler, S. I. Vapor-Liquid Equilibria of Tetrahydrofuran Systems. J. Chem. Eng. Data 1988, 33, 157. (33) Cordray, D. R.; Izatt, R. M.; Christensen, J. J.; Oscarson, J. L. The Excess Enthalpies of (Carbon Dioxide + Ethanol) at 308.15, 325.15, 373.15 and 473.15 K from 5.00 to 14.91 MPa. J. Chem. Thermodyn. 1988, 20, 655. (34) Christensen, J. J.; Cordray, D. R.; Oscarson, J. L.; Izatt, R. M. The excess enthalpies of four (carbon dioxide + an alkanol) mixtures from 308.15 to 573.15 K at 7.50 to 12.50 MPa. J. Chem. Thermodyn. 1988, 20, 867. (35) Sipowska, J. T.; Grahm, R. C.; Neely, B. J.; Ott, J. B.; Izatt, R. M. Excess Enthalpies for (Ethane + Methanol) at 298.15 and 348.15 K and 7.5 and 15 Mpa, and at 323.15 K at 7.5, 10, and 15 Mpa. J. Chem. Thermodyn. 1989, 21, 1085. (36) Sipowska, J. T.; Ott, J. B.; Neely, B. J.; Izatt, R. M. Excess Enthalpies for (Propane + Methanol) at the Temperatures (298.15, 323.15, 348.15 and 373.15) K and Pressures (5, 10, and 15) MPa, and at 363.15 and (5 and 15) MPa. J. Chem. Thermodyn. 1991, 23, 551.
(37) Conti, G.; Gianni, P.; Matteoli, E. Excess Enthalpies and Excess Heat Capacities of (N,N-Dimethylformamide + Tetrahydrofuran + Cyclohexane) at the Temperature 298.15 K. J. Chem. Thermodyn. 1994, 26, 1249. (38) Conti, G.; Gianni, P.; Lepori, L.; Matteoli, E. Excess Enthalpies and Excess Hear Capacities of the Ternary System Ethanol + N,N-Dimethylformamide + Cyclohexane at 298.15 K. Fluid Phase Equilib. 1995, 103, 93. (39) Christensen, C.; Gmehling, J.; Rasmussen, P.; Weidlich, U. Heats of Mixing Data Collection; DECHEMA Chemistry Data Series: DECHEMA: Frankfurt/Main, Germany, 1984; Vol. III, Parts 1 and 2. (40) Bolinaga, R.; Gracia, M.; Gutierrez, C. Excess Enthalpies of Mixtures Containing Tetrahydrofurduryl Alcohol or Tetrahydrofurfurylamine. J. Chem. Thermodyn. 1978, 10, 667.
Received for review July 24, 2000 Revised manuscript received November 9, 2000 Accepted November 11, 2000 IE000682Z
