Ind. Eng. Chem. Res. 2003, 42, 5383-5391
5383
Modeling Some Alcohol/Alkane Binary Systems Using the SAFT Equation of State with a Semipredictive Approach J.-P. Passarello* and P. Tobaly Laboratoire d’inge´ nierie des mate´ riaux et hautes pressions, 99 Avenue J.-B. Cle´ ment, F-93430 Villetaneuse, France
An approach proposed in an earlier work for modeling n-alkane + n-alkane systems (Benzaghou, S.; Passarello, J. P.; Tobaly, P. Fluid Phase Equilib. 2001, 180, 1) is here extended to the VLE (vapor-liquid equilibrium; vapor pressure and liquid-phase molar volume) representation of n-alkane + 1-alcohol binary systems using the SAFT equation. The original approach was based on molecular assumptions that consider all n-alkanes to be composed of identical segments. These assumptions are here extended to the 1-alkanol chemical series. The thermodynamic representation of the chemical series of pure 1-alkanols thus requires only four adjustable parameters. The values of the corresponding parameters were determined by fitting the data of the series of light 1-alkanols (1-propanol to 1-octanol). The extrapolation capability of this method was then tested. The VLE thermodynamic properties of pure heavy 1-alcohols (up to 1-octadecanol) and 1-alcohol + n-alkane mixtures were predicted without any additional binary parameters. The model predicts the experimental data within a few percent. Introduction A great number of compounds are used in chemical processes. In many cases, thermodynamic experimental data are scarce or even not available. Thus, predictive models are needed for the calculation of phase equilibria. A few attempts in this respect have been made over the past two decades, such as in the works of Fermeglia et al.,1,2 some of which deal with associating substances (see Comments and Discussion section below). A more comprehensive review of these approaches can be found in Tamouza et al.3 The goal of this paper is to present an extension of a previous work dedicated to the thermodynamic modeling of the vapor-liquid equilibria of some hydrocarbon systems.4 Our final objective is to develop a method for the reliable estimation of thermodynamic properties of chemical compounds on the basis of their molecular structure using an equation of state. For this purpose, it was decided to treat several series of compounds that belong to the same chemical family rather than each compound alone. It was shown then that the series of n-alkanes (from n-C2 to n-C38) could be treated using the SAFT equation of state (EOS) within a reasonable accuracy and with few adjustable parameters, namely, three. An extension of this methodology to other hydrocarbons (branched alkanes, alkenes, aromatic compounds) was also tested and provided encouraging results. The representation of some hydrocarbon mixtures was also investigated somewhat successfully. However, such a method is most useful if it can be extended to more complex compounds and, for instance, to associating compounds. This is the reason why we intended to test the methodology with 1-alkanols and their mixtures with n-alkanes. These compounds were chosen for two main reasons: First, they are important in petroleum and chemical industry. Second, alkanols * To whom correspondence should be addressed. Tel.: +33 1 49 40 34 06. Fax: +33 1 49 40 34 14. E-mail: passarel@limhp. univ-paris13.fr.
are typical representatives of associative compounds, and they have often been studied experimentally in the past. However, this work was restricted to 1-alkanols because only these alcohols have been systematically investigated several times. SAFT Model Used for the Calculations Since the first development of the SAFT equation initiated by Chapman et al.,5 there have been a large number of attempts to improve the original EOS. Such efforts have led thus to various versions of this equation.4 In our previous work, we chose the original equation (slightly modified) because it is one of the simplest versions and it provides good results for the representation of vapor-liquid equilibria. We have used this same version in this work. In the SAFT approach, a molecule is seen as a chain of spherical segments that can interact through an addition of repulsion, dispersion, and association terms. The general expression for the residual free energy is given by
a - a° ) aseg + achain + aassoc
(1)
The first two terms take into account segment interactions (repulsive and attractive) and chain formation, respectively. For their exact expressions, the reader is referred to the papers of Chapman et al.5 and Benzaghou.4,6 In these expressions, three adjustable parameters are involved per species: the diameter, σ, and energy, , of a segment, together with the chain length m. At this stage, one should notice that, in this equation of state (EOS), a chain molecule is composed of identical segments, which is a current limitation of the model. Thus, a priori and strictly speaking, heteroatomic molecules (or maybe, more reasonably, molecules made of significantly different chemical groups) cannot be represented using this EOS. In practice however, this limitation can sometimes be overcome.7 For instance, we were able to provide reasonable results for the series
10.1021/ie030306p CCC: $25.00 © 2003 American Chemical Society Published on Web 09/18/2003
5384 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
of alkylbenzenes.4 When modeling nonassociating compounds, the third term, aassoc, equals 0, but this is not the case with alcohols. In the latter case, the molecule is assumed to have so-called association sites that can interact with one another at short range through a square-well potential. The third term is then written as5
aassoc RT
)
[(
1
ln xA - xA ∑i xi ∑ 2 A i
)
i
i
1 + Mi 2
]
(2)
The first sum in the expression above is over species i. The second sum inside the brackets is over the association sites Ai on molecule i. As usual in the SAFT equation, xAi stands for the mole fraction of molecules i nonbonded at site Ai and obeys the following implicit system of equations
1
x Ai ) 1 + Nav
xjFx ∑j ∑ B
(3) Bj
∆
AiBj
j
It must be well understood that there are indeed as many equations as sites Ai for all molecules i involved in the mixture. Only in a few cases (see below) can this expression be solved analytically, i.e., to give an explicit expression for each xAi. Here, ∆AiBj is the association strength between the two sites Ai and Bj. Its value depends on the parameters of the corresponding interaction potential, namely, the depth AiBj and the bonding volume κAiBj, through the approximate relation AiBj [exp(AiBj/kT) - 1] ∆AiBj ) dij3 gHS ij (dij)κ
(4)
where dij ) (dii + djj)/2. dii and djj are the hard-sphere diameters of the segments in molecules i and j, respectively. Here, the hard-sphere diameters dii and djj are not adjustable parameters of the model because they are related to the corresponding diameters σii and σjj through a relation given by Cotterman et al.5 Parameters AiBj and κAiBj are treated as adjustable parameters. One should note that association is treated through a square-well potential. This allows for the very shortrange effect of the hydrogen bond to be taken into account, while neglecting the long-range effect. This truncation is not always completely justified, and in some cases, it can lead to nonnegligible errors. Hypotheses and Determination of the Parameters Pure 1-Alcohols. As a starting point, it is useful to recall some earlier results4 and the procedure followed to treat the n-alkanes series so that an extended procedure for the alkanols can be proposed. Modeling the n-alkane series is not a difficult task with the SAFT EOS, which treats molecules as being composed of identical segments. Indeed, to a first approximation, the chemical groups -CH3 and -CH2- can be regarded as similar. n-Alkane molecules can thus be considered as nearly uniform in terms of chemical groups, with this assumption being more justified for molecules with large numbers of carbon atoms. This means that only one diameter, σ, and one energy, , are necessary to handle the thermodynamic properties of n-alkanes. The differ-
ence between two n-alkanes was made only through the parameter m. This parameter was correlated with the carbon number nC using the linear correlation m ) AnC + 1 - A. These assumptions were validated with the following results. The VLE data of the series n-C2 to n-C8 were regressed within an average accuracy in the pressure data of 2.46%. The parameters thus-determined were used to predict the vapor pressure of the series from n-C9 to n-C38. The agreement between the experimental and predictive calculations was not as good, i.e., within 13.61%. This result can, however, be considered satisfactory with regard to our hypotheses and objectives: to provide a realistic prediction of the properties of an entire chemical family with a minimum number of parameters. Furthermore, the data become less accurate with increasing number of carbons, as already noticed and reported, for instance, by Lemmon and Goodwin.9 In fact, for high-molecular-weight alkanes, vapor pressures can be as low as fractions of a pascal and are therefore very difficult to measure accurately. In such cases, the best fit of the experimental data might not be the most adequate for our purposes. In the case of alcohols, the situation is somewhat different. For simplicity, the chemical structure of such compounds can be modeled as a hydrocarbon skeleton with an additional -OH group. The latter group is responsible for the associative bond, i.e., hydrogen bond. Such bonding is very important in the thermodynamic properties, as has been recognized by several authors. However, the -OH group is not comparable in size to the -CH3, -CH2-, and -CH< groups constituting the hydrocarbon skeleton. The corresponding molecules appear asymmetrical in terms of both geometry and interactions. Thus, rigorously, such a molecule should not be represented by a model such as SAFT that assumes homogeneously segmented molecules. As in the case of the n-alkylbenzene series, however, we decided to go around this limitation and to treat the 1-alkanols similarly to the n-alkanes even if it is less justified in this case. In the spirit of our previous work, we then assumed that a given 1-alcohol is made of a uniform chain of segments. Segment parameters were taken to be independent of the number of carbons for the entire series of 1-alcohols. Because there is only one -OH group in each alcohol molecule, the diameter σ was roughly estimated to be of the same order as that for n-alkanes and was then set to the corresponding value of 3.254 Å. However, it is expected that this hypothesis will be less valid for short alcohols. On the other hand, the segment energy parameter was fitted to experimental data. It might be thought that, in addition to the dispersion interactions, the resulting parameter values could also compensate for the long-range part of the potential for the H-bond, which is neglected in the association term. As the presence of the -OH group in the molecule leads to nonnegligible effects on the thermodynamic properties even at large carbon numbers (nC > 10), the value of is expected to be rather different from that of a segment in the n-alkane series. Similarly, the parameter m was again assumed to follow a linear correlation with the number of carbon atoms, specifically, m ) A′nC + B′. Parameters A′ and B′ were also fitted, as no evident relation to parameter A (determined in a previous work for n-alkanes) could be drawn. The rigorous association site model for alkanols is 3B, as denoted by Huang and Radosz7 and as illustrated in
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5385
Figure 1. Definition of association sites A-C on an alkanol molecule (model 3B). Table 1. VLE Data Correlation Using the SAFT Equation for the Series of 1-Alcohols from 1-Propanol to 1-Octanola database
correlation accuracy
compound
pressure range (Pa)
temperature range (K)
AAD% (P)
AAD% (vL)
1-propanol 1-butanol 1-pentanol 1-hexanol 1-heptanol 1-octanol
(2.6 × 103)-(3.4 × 106) (2.3 × 103)-(3.6 × 106) (2.2 × 103)-(1.7 × 105) (1.8 × 103)-(9.4 × 104) (9.6 × 102)-(8.9 × 104) (3.8 × 103)-(4.4 × 105)
297-509 312-548 327-426 339-428 343-445 381-532
1.58 0.75 1.11 1.21 0.94 1.81
4.74 3.10 0.82 1.65 2.06 2.71
a
c
Data from ref 10. AAD%(vL) ) 100
b
∑ data
.
AAD%(P) ) 100
L |vLexp - vcalc |
ndatavLexp
∑ data
|Pexp - Pcalc| ndataP exp
.
.
Figure 1. This means that on each molecule are found three sites labeled A, B, and C, two of which are identical (site B ) site A). In this model, it is further assumed that ∆AB ) ∆AA ) ∆BB ) 0. Only the interactions between sites A and C (or B and C) are in fact considered. In the spirit of this work, the two association parameters ∆AC and κAC are considered to be constant for the entire series of 1-alkanols. Indeed, they characterize the hydrogen bond, and any effect of the hydrocarbon chain on the -OH group is neglected. From all of these assumptions, it follows that only two association parameters, OH and κOH, are necessary to model the entire series of 1-acohols. Moreover, in this particular case, eq 3 can be solved analytically. One finds for pure compounds
xA )
Figure 2. Correlated and experimental vapor pressures of light 1-alkanols.
NF∆ - 1 + x(1 + NF∆)2 + 4NF∆ 4NF∆ xC ) 2xA - 1
(5) (6)
where ∆ ) - 1] for the 1-alkanol series. In this approach, the entire series of 1-alcohols is represented using five adjustable parameters. The parameters were adjusted by regression of vapor-liquid equilibrium (VLE) experimental data of vapor pressure and molar liquid-phase volume of the series from 1-propanol to 1-octanol according to the procedure already discussed.4 We obtained the following parameters values: /k ) 203.04 K, m ) 0.5454nC + 1.3566, OH/k ) 2150.23 K, and κOH ) 0.018 17. The regressed database together with the corresponding results are described in Table 1. Methanol was not included in this database because of its specific behavior. Ethanol was also discarded for a reason explained above (short alcohol). The results thus obtained are very satisfactory (agreement of a few percent; see Table 1 and Figures 2 and d3gHS(d)κOH[exp(OH/kT)
Figure 3. Correlated and experimental liquid-phase molar volumes of light 1-alkanols.
3), keeping in mind the fact that crude approximations were made for the parameters evaluation. They compares favorably with the results obtained for the nalkane series n-C2 to n-C8. However to validate our hypotheses, we need to test this approach when used for extrapolation. Using these same parameter values, it was possible to predict the thermodynamic behavior of heavier 1-alkanols within a reasonable accuracy. Although greater, the deviations remained comparable to those obtained in the series from 1-propanol to 1-octanol. The detailed results are given in Table 2. The results obtained here are still good, especially if one remembers that they are pure predictions (see Figure 4). Of course, one might notice that the accuracy seems to decrease with increasing number of carbons. However, one should also keep in mind that the precision of the measurement is decreasing too, especially as the temperature increases, because of the lack of thermal stability of long alcohols.12 Furthermore, the results are better than in the case of n-alkanes of comparable size (accuracy of vapor pressure predictions for pure n-C9 to n-C18 was on the order of 10%).
5386 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 Table 2. Prediction of VLE for Pure 1-Alcohols Using the SAFT Equationa prediction accuracy
data compound
pressure range (Pa)
temperature range (K)
AAD% (P)b
1-nonanol 1-decanol 1-dodecanol 1-tetradecanol 1-octadecanol
(7.5 × 102)-(9.8 × 104) (8.5 × 102)-(9.1 × 104) (7.1 × 102)-(9.5 × 104) (7.8 × 102)-(8.2 × 104) (2.7 × 103)-(3.0 × 104)
365-485 380-500 400-535 425-560 495-570
2.54 3.22 3.83 4.40 7.61
a
.
Data from ref 11.
b
In this case, we have
AAD%(P) ) 100
∑ data
|Pexp - Pcalc|
3
σX )
XσX3 )
ndataPexp
∑i ∑j xixjmimi
( ) σi + σj
3
2
∑i ximi∑j xjmj
∑i ∑j xixjmimiσij3[ij(1 - kij)]1/2 ∑i ximi∑j xjmj
(7)
(8)
The associative term requires no empirical binary parameters. However, the expressions for the fractions of nonassociated compounds in the mixtures are slightly different than in the case of pure compounds because they involve the mole fraction of alcohol x1. We then have
xA1 )
NF1∆ - 1 + x(1 + NF1∆)2 + 4NF1∆ 4NF1∆ xC1 ) 2xA1 - 1
Figure 4. Predicted and experimental vapor pressures of heavy 1-alkanols.
Extrapolation to methanol and ethanol using this approach leads, as expected, to greater deviations (8.62 and 56.15%, respectively, in vapor pressure). A better fit can be obtained if one specifically adjusts parameters m, /k, and σ while retaining the values for the association parameters OH and κOH. The latter approach leads to the following results for methanol: σ ) 3.3329, /k ) 241.33 K, and m ) 0.5517 with AAD%(P) ) 1.09% and AAD%(vL) ) 3.31%. For ethanol we get: σ ) 3.2064, /k ) 209.64 K and m ) 1.3816 with AAD%(P) ) 0.80% and AAD%(vL) ) 1.28%. 1-Alcohol/n-Alkane Binary Mixtures. For the purpose of modeling mixtures, we used modified LorentzBerthelot mixing rules that can involve two kinds of binary parameters, kij and lij. It is necessary to emphasize that, here, the mixing rules involve the segments and not directly the species. It should be well understood that the subscripts i and j in eqs 7 and 8 below are for the type of segment in molecule i and the type of segment in molecule j, respectively. Because only two different segments (one for n-alkanes and the other for n-alkanols) are involved in the considered mixtures, only up to two binary parameters can be determined. The second parameter, lij (mixing rule for σ), must be 0 because the diameters of the segments in n-alkanes and in 1-alcohols are the same. However, a priori, there is no physical reason for the other parameter, kij, to be 0.
(9) (10)
In these expressions F1 denotes the product x1F. The database used to validate our approach is listed in Table 3. One might note that, despite the large number of systems investigated in the past, unfortunately, mainly the bubble curve has been experimentally determined. Table 4 summarizes the results of two different tests. First, because we wanted to test the predictive capabilities of the approach, we predicted the vapor pressures of different systems after imposing a value of kij ) 0. The deviations are reported in the table, and it can be seen that they are generally lower than 8%. Second, to evaluate the effect of a nonzero value of kij, the data for each binary system were regressed, yielding distinct values of kij. Both the values of kij and the resulting deviations are reported in the table. The values of kij are usually very low, but they still vary and can even be negative. Of course, this allows for a significant improvement in the accuracy in most cases, but such an approach is no longer predictive, and in some sense, it is inconsistent with our goals. A more consistent approach would have been to determine a unique value of kij (recall that kij is for intersegment i-j interactions), but this has not been done because it is unlikely, in view of the variations of kij in Table 4, that the predictions for other systems would be greatly improved. This is a limitation of the present approach. Therefore, our conclusion is that, for predictive purposes, a value of kij ) 0 should be used. If no data are available, an uncertainty on the order of 8% can be regarded as acceptable. More detailed results for this predictive approach (kij ) 0) are presented in Table 3. Comments and Discussion Our model agrees both qualitatively (good prediction of azeotropes) and quantitatively with the experimental data (see Figures 5-9). In most cases, the agreement is within 3-7% for the bubble pressure, and there is no
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5387 Table 3. Representation of VLE for Alcohol + Alkane Binary Mixtures data mixture 1-propanol + propane + pentane + hexane
+ heptane
+ nonane + decane + undecane 1-butanol + pentane + hexane
+ heptane + octane + nonane + decane + undecane 1-pentanol + pentane + hexane + heptane
+ octane + decane 1-hexanol + hexane
prediction accuracy
temp (K)
pressure range (Pa)
number of data points
ref
AAD% (P)a
354.75 378.15 393.15 313.10 317.10 318.15 323.15 338.15 348.15 303.15 313.15 323.15 333.15 333.15 348.15 348.15 298.15 363.15 368.15 333.15 353.15
(4.5 × 105)-(2.3 × 106) (5.0 × 105)-(3.6 × 106) (4.5 × 105)-(4.1 × 106) (6.6 × 103)-(1.2 × 105) (8.2 × 103)-(1.4 × 105) (4.0 × 104)-(4.9 × 104) (1.2 × 104)-(5.8 × 104) (2.7 × 104)-(9.8 × 104) (4.3 × 104)-(1.2 × 105) (3.9 × 103)-(1.1 × 104) (7.0 × 103)-(1.7 × 104) (1.2 × 104)-(2.6 × 104) (2.0 × 104)-(4.0 × 104) (2.0 × 104)-(4.0 × 104) (4.9 × 104)-(7.3 × 104) (3.9 × 104)-(7.4 × 104) (2.4 × 103)-(3.0 × 103) (5.1 × 104)-(7.4 × 104) (1.6 × 104)-(9.2 × 104) (5.5 × 102)-(2.1 × 104) (1.7 × 103)-(5.1 × 104)
14 17 18 10 10 5 22 12 10 16 16 16 16 33 15 23 17 11 12 17 17
13 13 13 14 14 15 16 17 17 18 18 18 18 16 19 20 21 22 23 24 24
0.70 2.95 4.61 16.51 16.57 7.10 8.01 11.90 9.89 9.30 7.85 6.85 6.22 6.20 4.95 8.76 6.85 4.46 4.38 7.75 10.99
303.15 323.15 323.15 332.53 338.15 348.15 323.15 333.15 363.15 373.15 383.15 323.15 373.15 373.15 383.15 353.15 373.15
(1.2 × 103)-(8.2 × 104) (3.2 × 104)-(5.5 × 104) (2.1 × 104)-(5.4 × 104) (8.0 × 103)-(7.7 × 104) (5.4 × 104)-(9.0 × 104) (5.8 × 104)-(1.3 × 105) (1.4 × 104)-(2.0 × 104) (8.0 × 103)-(3.1 × 104) (3.4 × 104)-(9.0 × 104) (4.6 × 104)-(7.4 × 104) (6.4 × 104)-(1.1 × 105) (4.7 × 103)-(5.7 × 103) (9.7 × 103)-(5.2 × 104) (9.5 × 103)-(4.6 × 104) (1.4 × 104)-(6.9 × 104) (1.7 × 103)-(2.3 × 104) (4.4 × 103)-(5.3 × 104)
15 12 14 28 11 12 8 24 24 7 7 15 21 7 7 19 18
25 26 21 27 26 26 28 29 29 30 30 31 30 30 30 24 24
9.18 9.30 4.48 7.00 6.42 5.30 1.82 5.51 3.71 3.82 3.58 6.19 4.59 6.13 4.35 4.84 3.67
303.15 303.15 323.15 348.15 358.15 363.27 368.15 373.32 363.27 373.32 363.27 373.32
(4.4 × 102)-(8.3 × 104) (4.4 × 102)-(2.5 × 104) (1.7 × 103)-(5.5 × 104) (7.3 × 103)-(4.9 × 104) (1.2 × 104)-(6.8 × 104) (1.5 × 104)-(7.9 × 104) (1.9 × 104)-(9.3 × 104) (2.4 × 104)-(1.1 × 105) (1.5 × 104)-(3.4 × 104) (2.4 × 104)-(5.4 × 104) (6.4 × 103)-(1.6 × 104) (9.6 × 103)-(2.5 × 104)
15 15 15 21 20 14 20 12 15 15 13 13
25 25 25 31 31 32 31 32 32 32 32 32
8.98 9.29 6.35 3.54 2.97 1.68 2.75 2.07 1.97 1.61 2.35 2.17
293.15 298.23 303.15 303.15 308.15 313.15 313.22 318.21 323.15 323.16 328.21 333.15 333.16 338.18 342.82 343.15 353.15 363.15 373.15
(6.0 × 101)-(1.7 × 104) (5.8 × 103)-(2.0 × 104) (1.4 × 102)-(2.5 × 104) (7.1 × 103)-(2.5 × 104) (8.6 × 103)-(3.1 × 104) (3.2 × 102)-(3.8 × 104) (1.0 × 104)-(3.7 × 104) (1.2 × 104)-(4.5 × 104) (6.5 × 102)-(5.5 × 104) (1.4 × 104)-(5.3 × 104) (1.7 × 104)-(6.4 × 104) (1.3 × 103)-(7.7 × 104) (2.0 × 104)-(7.5 × 104) (2.3 × 104)-(8.8 × 104) (2.6 × 104)-(1.1 × 105) (2.3 × 103)-(1.1 × 105) (4.1 × 103)-(1.5 × 105) (6.9 × 103)-(1.9 × 105) (1.1 × 104)-(2.5 × 105)
23 9 23 9 9 23 9 9 23 9 9 23 9 9 9 23 23 23 23
33 34 33 34 34 33 234 34 33 34 34 33 34 34 34 33 33 33 33
9.81 8.33 7.95 7.61 6.96 6.08 9.95 5.77 4.77 5.27 4.83 3.54 4.38 3.97 3.63 2.54 1.97 1.41 0.95
AAD (y1)b
1.69 × 10-2 2.35 × 10-2 4.85 × 10-3 2.08 × 10-2 4.30 × 10-2 4.51 × 10-2
2.46 × 10-2 2.25 × 10-2 2.07 × 10-2 7.23 × 10-3 6.87 × 10-3 6.20 × 10-3 3.91 × 10-3 1.05 × 10-2
8.50 × 10-2 1.34 × 10-2
5388 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 Table 3. (Continued) data mixture 1-octanol + hexane + octane + decane 1-decanol + hexane
1-dodecanol + hexane
+ decane + undecane + dodecane + tridecane + tetradecane + pentadecane 1-tetradecanol + undecane a
AAD%(P) ) 100
∑ data
prediction accuracy
temp (K)
pressure range (Pa)
number of data points
ref
AAD% (P)a
313.15 373.15 383.15 373.15 383.15
(8.8 × 103)-(3.6 × 104) (2.5 × 103)-(4.7 × 104) (4.3 × 103)-(6.5 × 104) (2.5 × 103)-(1.0 × 104) (4.3 × 103)-(1.5 × 104)
10 14 13 14 14
21 35 35 36 36
5.47 3.40 2.22 3.31 1.62
283.16 293.15 298.15 303.15 313.15 323.15 333.15
(2.6 × 103)-(1.0 × 104) (4.6 × 103)-(1.6 × 104) (5.1 × 102)-(2.0 × 104) (6.3 × 103)-(2.5 × 104) (9.2 × 103)-(3.7 × 104) (1.3 × 104)-(5.3 × 104) (1.7 × 104)-(7.4 × 104)
10 10 10 10 10 10 10
37 37 37 37 37 37 37
9.54 8.54 6.85 6.11 4.88 3.81 2.91
298.15 298.23 303.15 308.15 313.22 318.21 323.16 328.21 333.16 338.18 342.82 393.15 413.15 393.15 413.15 413.15 433.15 433.15 453.15 453.15 473.15 453.15 473.15 393.15 413.15
(7.1 × 103)-(2.0 × 104) (4.8 × 103)-(2.0 × 104) (5.9 × 103)-(2.5 × 104) (7.2 × 103)-(3.1 × 104) (8.6 × 103)-(3.7 × 104) (1.0 × 104)-(4.5 × 104) (1.2 × 104)-(5.3 × 104) (1.4 × 104)-(6.3 × 104) (1.6 × 104)-(7.5 × 104) (1.9 × 104)-(8.8 × 104) (2.2 × 104)-(1.1 × 104) (4.3 × 103)-(1.9 × 104) (4.1 × 103)-(3.4 × 104) (4.9 × 102)-(9.9 × 103) (1.4 × 103)-(2.1 × 104) (2.3 × 103)-(1.2 × 104) (4.7 × 103)-(2.2 × 104) (5.2 × 103)-(1.3 × 104) (8.6 × 103)-(2.4 × 104) (8.6 × 104)-(1.4 × 104) (1.7 × 104)-(2.6 × 104) (8.6 × 103)-(1.1 × 104) (1.6 × 104)-(2.1 × 104) (1.2 × 102)-(9.9 × 103) (4.3 × 102)-(2.1 × 104)
10 9 9 9 9 9 9 9 9 9 9 20 25 20 21 13 12 12 13 13 13 14 14 16 18
31 38 38 38 38 38 38 38 38 38 38 38 39 39 39 39 39 39 39 39 39 39 39 24 24
7.46 7.66 6.92 6.25 5.59 5.00 4.47 3.93 3.46 3.02 2.64 9.51 12.59 7.12 9.56 5.58 7.31 2.79 3.85 6.08 5.64 4.68 3.76 7.28 10.71
|Pexp - Pcalc| ndataP exp
.
b
AAD(y1) )
∑ data
|y1,exp - y1,calc| ndata
particular evidence for the degradation of model accuracy with increasing chain length (see Figures 5-7). The model is slightly less accurate in the case of mixtures than in the case of pure components, but the quality of the predictions still remains reasonable, especially when one keeps in mind that no binary interaction parameter was used. Furthermore, one should also note that, sometimes, measurements are difficult to obtain, so they might not be as accurate as desired. Note, for instance, the two different datasets in Figure 6 for the system C3OH + C7. The latter remark illustrates the fact that, when the data are subject to uncertainty, a deviation on the order of several percent should not be considered as a failure of the prediction. Again, each set of data could be correlated with much better accuracy by the adjustment of kij, but this is not the purpose of this work. Such 1-alkanol + n-alkane systems and, more generally, systems containing associating compounds have been the subject of numerous studies in the past two decades. Following Yakoumis et al.,40 one can classify all of these attempts on the basis of the way they take association into account. Apart from semiempirical
AAD (y1)b
.
modifications of the energy parameter a(T) in a cubic equation,41 there are three main categories of approaches: (1) One way (and historically the first one) consists of assuming that so-called “associative chemical reactions” occur. This procedure requires one or several chemical constants that depend on temperature and are not easily measured, i.e., that, in practice, have to be determined by regression. For instance, this is the approach followed by Go´ral42 with the EoSC equation, Nagata et al.43 with UNIQUAC, and Neau and Rogalski44 with a modified CRG model. Only the model used by Neau and Rogalski was free of binary interaction parameters and used a built-in group contribution method. However, the original CRG model had to be modified to improve the modeling of VLE data for mixtures. (2) An alternative approach to account for H-bonds consists of treating them using a lattice or quasichemical theory, as done by Park et al.45 and Mattedi et al.46 In both of these cases, however, the data for pure compounds and mixtures were correlated together, and the model was not tested in a predictive manner except
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5389 Table 4. Influence of kij on the Representation of Binary Mixtures prediction representation with prediction of with kij ) 0 fitted value of kij Mattedi et al.46 mixture
AAD%(P)a kij value AAD%(P)a
1-propanol + propane + pentane + hexane + heptane + nonane + decane + undecane 1-butanol + pentane + hexane + heptane + octane + nonane + decane + undecane 1-pentanol + pentane + hexane + heptane + octane + decane 1-hexanol + hexane 1-octanol + hexane + octane + decane 1-decanol + hexane 1-dodecanol + hexane + decane + undecane + dodecane + tridecane + tetradecane + pentadecane 1-tetradecanol + undecane
a
AAD%(P) ) 100
AAD%(P)a
2.91 16.50 9.26 7.92 6.85 4.42 9.37
-0.003 0.022 0.014 0.014 0.009 0.009 0.012
2.36 2.96 2.36 3.08 1.36 1.10 3.35
9.18 6.55 4.21 3.70 6.18 4.85 4.27
0.010 0.011 0.008 0.009 0.009 0.007 0.007
1.26 2.23 1.44 0.64 0.68 3.27 3.35
8.98 7.81 2.72 1.79 6.01
0.009 0.007 0.006 0.004 0.004
1.28 2.05 1.79 1.27 5.07
14.01 4.77 3.12
4.86
0.005
2.53
4.13
5.47 2.83 2.46
0.004 0.000 0.000
1.99 2.83 2.46
6.09
0.005
2.92
5.15 11.21 8.37 6.41 3.34 5.85 4.22
0.004 -0.010 -0.011 -0.006 -0.002 -0.006 0.002
2.13 5.29 2.64 3.41 2.92 4.75 4.10
9.09
-0.008
4.27
∑ data
|Pexp - Pcalc| ndataP exp
7.84
3.25
2.01
Figure 6. Calculated and experimental phase boundaries for the mixture 1-propanol + n-heptane at 333.15 and 348.15 K (kij ) lij ) 0). Data: [, from ref 13; 2, from ref 16; and 4, from ref 17.
. Figure 7. Calculated and experimental phase boundaries for the mixture 1-propanol + n-undecane at 333.15 and 353.15 K (kij ) lij ) 0). Data taken from ref 17.
calculations (kij )0). It appears that the deviations are of the same order. (3) Another approach is based on perturbation theory and was first applied with SAFT. More recently, the corresponding association term was used in CPA,40 an extension of a cubic equation. The results of the latter equation are good (within 1-2%) for correlating available data, using kij’s fitted for each system at each temperature.
Figure 5. Calculated and experimental phase boundaries for the mixture 1-propanol + n-pentane at 313.10 and 317.10 K (kij ) lij ) 0). Data taken from ref 10.
for a few systems.46 Corresponding predictive results are reported in Table 4 for comparison with our predictive
The results obtained above compare rather well with those of previous significant work. The quality of these earlier modelings is generally comparable to that of our approach (around a few percent for bubble pressure). In the case when better results were obtained (within 1-2% or less40,42,44), additional pure or mixture parameters were also involved, sometimes individually adjusted using the data for each pure compound44 and sometimes determined by correlation of VLE mixture data.42
5390 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 X ) mean Lennard-Jones segment interaction energy (mixture of spheres), see eq 8 AiBj ) depth of square-well interaction potential accounting for association between sites Ai and Bj F ) molar density κAiBj ) association volume between sites Ai and Bj σi ) diameter of a segment in the molecule i σX ) mean diameter (mixture of spheres), see eq 7 Abbreviations AAD ) absolute average deviation, see Tables 1-3 CPA ) cubic plus association CRG ) chemical reticular group EOS ) equation of state SAFT ) statistical association fluid theory SRK ) Soave-Redlich-Kwong equation (also RKS) VLE ) vapor-liquid equilibrium
Literature Cited Figure 8. Calculated and experimental bubble curves for the mixture 1-pentanol + n-decane at 363.27 and 373.32 K (kij ) lij ) 0). Data taken from ref 13.
Figure 9. Calculated and experimental phase boundaries for the mixture 1-dodecanol + n-decane at 393.15 and 413.15 K (kij ) lij ) 0). Data: ], from ref 20; and 2, from ref 25.
Nomenclature Ai, Bi, Ci ) sites for association on molecules i a ) molar free energy a° ) molar free energy in the ideal state dii ) hard-sphere diameter of segment i gHS ii ) hard-sphere radial distribution function k ) Boltzmann constant ≈ 1.3981 × 10-23 J/K kij ) binary interaction parameter lij ) binary interaction parameter, see eq 8 mi ) number of segments in molecule i mX ) Σiximi ) mean number of segments (mixture of spheres) Nav ) Avogadro’s number ≈ 6.02 × 1023 molecules/mol T ) temperature vL ) liquid-phase molar volume xi ) mole fraction of molecule i in the liquid phase xiAi ) mole fraction of molecule i not associated to site Ai yi ) mole fraction of molecule i in the vapor phase Greek Letters ∆AiBj ) association strength between sites Ai and Bj i ) Lennard-Jones interaction energy of segment i
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Received for review April 7, 2003 Revised manuscript received July 22, 2003 Accepted August 4, 2003 IE030306P
