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Ind. Eng. Chem. Res. 2009, 48, 5472–5480
Thermodynamics of Triethylene Glycol and Tetraethylene Glycol Containing Systems Described by the Cubic-Plus-Association Equation of State Martin P. Breil and Georgios M. Kontogeorgis* Department of Chemical and Biochemical Engineering, Technical UniVersity of Denmark, DK-2800 Kgs. Lyngby, Denmark
A thorough investigation of triethylene glycol (TEG) containing systems has been performed. The introduction of a new six-site association scheme for the TEG molecule has shown to be advantageous. Glycols are often modeled using a four-site scheme (abbreviated as 4C) hence ignoring the internal lone pairs of oxygen. The new association scheme also takes these sites into account. The new parameters of TEG are based on the vapor pressure data, liquid density data, and liquid-liquid equilibria (LLE) data (n-heptane), and they are tested for binary systems (methane, n-octane, n-nonane, n-decane, benzene, toluene, ethylbenzene, and water) and different types of phase equilibria (vapor-liquid equilibria (VLE) and LLE) and thermodynamic properties (heat of mixing, activity coefficients). A less extensive investigation has also been performed on tetraethylene glycol (TeEG) containing systems. Similarly, a new seven-site association scheme for the TeEG molecule has been investigated. The new parameters of TeEG are based on vapor pressure data, liquid density data, and LLE data (n-heptane). The performance is similar to that the 4C scheme. Introduction
internal oxygen lone pairs are accounted for by introducing three extra sites to the four-site (4C) scheme.
Triethylene glycol (TEG), which has the structure HO(CH2)2O(CH2)2O(CH2)2OH, is used in approximately 95% of the glycol dehydration units for natural gas streams.1 This is due to the chemical stability of TEG as well as its low cost and high affinity to water. Furthermore, the solubility of aromatic hydrocarbons in TEG is also very important due to the hydrocarbon emissions from glycol regeneration units. Compared to diethylene glycolswhich has been the standard dehydration solvent for many yearssTEG has become more popular because of its lower loss and degradation. The modeling of glycols as hydrate inhibitors has previous been performed with GE models such as UNIQUAC and NRTL models2 and with equations of state such as PC-SAFT3 and SRK with Huron-Vidal.4-6 The cubic-plus-association equation of state (CPA EoS)7,8 has also been used for modeling glycols including TEG and TeEG (tetraethylene glycol)scontaining systems.1,4,9 The reason for the re-estimation of CPA parameters of TEG and TeEG is the availability of new experimental data10 and the development of new and more elaborate association schemes in general. The performance of the CPA EoS on different thermodynamic properties of TEG containing solutions is the target of the investigation. CPA has been applied extensively to the modeling of vapor-liquid equilibria (VLE) for alcohol-hydrocarbon systems and in correlating liquid-liquid equilibria (LLE) for alcohol-hydrocarbon mixtures, as well as for binary aqueous systems containing hydrocarbons.7,8 Finally, we present the results of an investigation of a new association scheme for tetraethylene glycol (TeEG), which has the structure HO(CH2)2O(CH2)2O(CH2)2O(CH2)2OH, where the * To whom correspondence should be addressed. E-mail: gk@ kt.dtu.dk.
Model Description: CPA The formulation of the CPA EoS, which has been applied in this investigation, is the one proposed by Kontogeorgis et al.11 Two association sites interact with an association strength of
[ ( ) ]
∆AiBj ) gref(V) exp
εAiBj - 1 bijβAiBj RT
(1)
where bij ) 1/2(bi + bj) and gref(V) is the radial distribution function of the reference fluid gref(V) )
1 , 1 - 1.9η
η)
b 4V
(2)
where b is the covolume parameter and V is the molar volume. All calculations in this work are based on the simplified CPA model. The cross-association energy is given by ε
AiBj
{
εAi + εBj for z ·z e 0 Ai Bj 2 ) for zAi·zBj > 0 0
(3)
where zk is the sign of site k. The association site can be either a negative or a positive site. It follows that positive sites associate with negative sites, only. However, there is an extension to this: the acid site, which is both a negative and a positive site (zAi ) ( or 0), associates with both negative and positive sites. The cross-association volume factor depends on the combining rule assigned to each cross-association Table 1. Parameters for the Correlations of DIPPR19 and Yaws20 TEG
A
B
C
D
E -18
vapor pressure 152.48 -16449 -17.67 6.4481 × 10 6 DIPPR (Pa) liquid density 0.59672 0.26217 769.5 0.24631 DIPPR (kmol/m3) vapor pressure 13.355 -3838.7 -1.3933 1.4247 × 10-10 9.8759 × 10-7 Yaws (mm Hg)
10.1021/ie801412y CCC: $40.75 2009 American Chemical Society Published on Web 04/23/2009
βAiBj )
{
Ind. Eng. Chem. Res., Vol. 48, No. 11, 2009
√βA βB i
βcross
j
for CR-1 for mCR-1
(4)
The CR-1 combining rule depends on the pure-component association volumes, whereas the mCR-1 is fitted to each particular pair. A solvating compound like benzene associates with water according to mCR-1. Any other combining rule would lead to an association strength of zero since the association volume of benzene is zero. Normally, association between molecules of the same type is called self-association, as opposed to cross-association which is association between molecules of different types. In this work, we consider that different types of association sites exist on the same molecule. Consequently, we need to apply a combining rule even for a pure component. Scheme 4C. The 4C association scheme was introduced by Huang and Radosz.12,13 The scheme mimics a molecule that has two alcohol groups, like monoethylene glycol.
The properties of site A are identical to those of site C, and the properties of site B are identical to those of site D. Hence, sites A and C are of the same type, identified as Ai in the association table, and sites B and D, also of the same, are identified as Bi. The association-site table of a 4C molecule looks like the following
where Rk is the repetition number of site k, zk is the sign of site k, εk is the association energy of site k, and βk is the association volume factor of site k. For simplicity, however, all four sites are assumed to have the same value of ε and β. Consequently, there are only five pure-component parameters to be estimated (the three cubic parameters bi, Γi, and ci, in addition to the two association parameters εi and βi)sinstead of eleven. Scheme 6D. In set 3 (one of the parameter sets in this work), TEG has been assigned a new association scheme, 6D. The association scheme has three different site types, and each type appears twice. In other words, the lone pairs of the ether oxygen atoms of TEG are now considered by the model to be negatively charged. Applied to triethylene glycol, TEG, the sites appear as follows:
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Scheme 7D. In set 6, TeEG has been assigned an association scheme where the lone pairs on the three internal oxygen atoms also are accounted for.
The TeEG molecule has three different types of sites since sites A and C are considered to be identical, and sites B and D are considered to be identical as well as sites E, F, and G which are considered to be identical. The corresponding associationsite table looks like the following
For simplicity, however, all seven sites have the same value of ε and β. Systems. The performance of the CPA EoS is evaluated by calculating certain thermodynamic properties of a number of TEG containing systems. Besides pure TEG (vapor pressure and liquid volume), the model is applied to nine binary systems: TEG-n-heptane (LLE: mutual solubility as a function temperature),2 TEG-methane (VLE: mutual solubility as a function of pressure and temperature),14,10 TEG-water (ln γ, γ∞, hE, and bubble pressure data),15,16 TEG-benzene: (VLLE: dew and bubble temperature data as well as mutual solubility),1,17 TEG-toluene (VLLE: dew and bubble temperature data as well as mutual solubility),1,17 TEG-ethylbenzene (γ∞),18 TEG-n-octane (γ∞),18 TEG-n-nonane (γ∞),18 and TEG-ndecane (γ∞).18 All the liquid-phase systems are at temperatures below 400 K, as opposed to the pure-component vapor pressure data which span from 286 to 619 K. The performance of the CPA EoS is also evaluated by calculating a few properties of TeEG containing systems. The investigation is based on the pure-component properties (vapor pressure and liquid density) of TeEG and the LLE measurements of the TeEG-n-heptane system reported by Derawi et al.2 Correlations for the Vapor Pressure. Derawi et al.9 generated the vapor pressure and liquid volume data from the DIPPR correlations19 for TEG, eqs 5 and 6. ln P ) A +
B + C ln T + DTE 265.95 < T < 769.50 K T (5)
(6) Fliq ) A/B1+(1 - T/C) 265.95 < T < 769.50 K Another correlation for the vapor pressure of TEG is that of Yaws20 D
The corresponding association-site table of a 6D molecule looks like the following
B + C log10 T + DT + ET2 T 265.79 < T < 700.00 K (7) This correlation is included in this investigation because it is of interest. The parameters for the correlations, eqs 5-7, are given in Table 1. Experimental Data for the Vapor Pressure. In this work, we have applied the experimental data for vapor pressure and log10 P ) A +
For simplicity, however, all six sites have the same value of ε and β.
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liquid density reported by DIPPR.19 The experimental data for the vapor pressure of TEG fall in three categories: accepted,21-25 not used,22,26,27 and rejected.23,28-31 All of them will be used, but we will clearly indicate in the corresponding figures which of these data points are characterized as rejected by DIPPR.19 None of the liquid density data of TEG are labeled as rejected by DIPPR. Errors. When dealing with experimental points, the error is understood as the difference between the experimental data points and the model. Thus, the mean absolute percentage error (MAPE) for a given set of experimental points is calculated as MAPE )
100% M
M
∑ i
|
1-
xCPA i xexp i
|
(8)
where M is the number of observations (experimental points). It is noted that MAPE is dimensionless. Objective Function. The objective function is a summation of the relative error of the vapor pressure data, the liquid density data, and the binary systems with n-heptane and methane (for the sets of this work). OF )
Mj
∑ ∑ j
1 Mj
i
(
1-
xCPA j,i xexp j,i
)
2
(9)
where Mj is the number of experimental data points of data type j. The other binary systems are fitted one at a time by estimating a binary interaction coefficient (and a βcross for solvating compounds). In Table 4, it is indicated which experimental data sets that have been used in the parameter estimation. Results and Discussion The pure-component parameters of TEG and water, methane, n-heptane, n-octane, n-nonane, n-decane, benzene,
toluene, and ethylbenzene are presented in Table 2, and the nonzero binary interaction and solvation coefficients are in Table 3. The CPA parameters denoted “TEG, set 1” in Table 2 are those available in the open literature,7,8 whereas the parameters “TEG, set 2” and ‘TEG, set 3” are of this work. The pure-component parameters for methane are based on its critical properties, Tc ) 190.564 K, pc ) 45.99 bar, and ω ) 0.0115. Ethylbenzene is traditionally treated as a nonsolvating compound (unlike the other two aromatic compounds: benzene and toluene). Vapor Pressure. In order to determine how well the CPA EoS performs on the vapor pressure data of TEG, a figure for the error is calculated and summarized in Table 4. First, MAPE is calculated at all the experimental points available; thus assessing the performance of the model over the whole temperature range. Second, MAPE is calculated at the experimental points below 375 K. This is done because all of the liquid-phase properties are at temperatures below this temperature. The pure-component parameters of Derawi et al.9 were estimated based on the data points generated by the DIPPR correlations, eqs 5 and 6. In order to compare the performance of the parameters of TEG to previous (literature) estimations, MAPE is calculated with generated data points which are evenly distributed in a given temperature range. The third set of MAPE is a comparison of CPA and the DIPPR correlation, eq 5, in the reduced temperature range 0.37-0.80. This range is identical to the range of the experimental points. The different results of MAPE values are due to the location of the generated data points; there are more generated data points in the low-temperature end than there are genuine data points. The forth set of MAPE is a comparison of CPA and the DIPPR correlation in the reduced temperature range 0.50-0.90.
Table 2. Pure-Component Parametersa compound
Tc (K)
b (L/mol)
a0/(Rb) (K)
c1
ε/R (K)
β × 103
scheme
TEG, set 1 TEG, set 2 TEG, set 3 water methane n-heptane n-octane n-nonane n-decane benzene toluene ethylbenzene
769.5 769.5 769.5 647.13 190.564 540.2 568.7 594.6 617.7 562.16 591.8 617.2
0.1321 0.128926 0.128926 0.014515
3562.481 3622.49 3622.49 1017.34
1.1692 0.9676 0.9100 0.67359
1724.441 1697.13 1420.00 2003.25
18.8 19.8 20.0 69.2
4C 4C 6D 4C
0.12535 0.14244 0.16035 0.17865 0.07499 0.09214 0.10872
2799.76 2944.91 3094.22 3190.54 2867.19 3051.36 3192.84
0.91370 0.99415 1.04628 1.13243 0.75760 0.80370 0.85390
SL SL
a The parameters of TEG (set 1) and of the other components are from the literature,7,8 and those of TEG (sets 2 and 3) are from this work. Scheme SL indicates a solvating compound.
Table 3. Binary Interaction Parameters and Solvation Parametersa set 1
a
system
kij
TEG-water TEG-methane TEG-n-heptane TEG-n-octane TEG-n-nonane TEG-n-decane TEG-benzene TEG-toluene TEG-ethylbenzene
-0.201 0.000 0.094 0.097 0.090 0.083 0.032 0.038 0.044
set 2 cross
β
CR-1
0.083 0.048
kij - 0.10 - 40/T 3.03 - 855/T 0.110 0.109 0.100 0.091 0.032 0.036 0.050
set 3 β
cross
CR-1
0.070 0.032
kij
βcross
-0.05 - 39/T 2.90 - 819/T 0.103 0.103 0.093 0.085 0.015 0.030 0.039
CR-1
0.120 0.100
The parameters of set 1 are from the literature7,8 except for ethylbenzene, n-octane, n-nonane, and n-decane which are estimated in this work.
Ind. Eng. Chem. Res., Vol. 48, No. 11, 2009 Table 4. Mean Absolute Percentage Error, MAPE, for All Three Sets of TEG Dataa MAPE range Tr
set 1
set 2
set 3
53.1* 59.0 200 39.9 61.1
39.3* 76.7 204 28.1 63.3
3.1 4.9 4.3 2.0*
2.6* 3.1 4.6 2.9
3.1* 3.1 6.3 4.5
4.6* 10.2* 35.2 29.5
3.5* 9.6 15.8* 32.1
11.3* 19.9 3.7* 46.5
39.7* 19.9* 31.2*
35.7* 4.8* 30.3*
39.1* 11.1* 29.1*
29.7* 8.1* 27.6*
8.4* 9.7* 4.4* 5.6* 7.4* 10.1*
14.9* 3.8* 1.1* 3.0* 3.9* 3.3*
vapor pressure of TEG CPA CPA CPA CPA CPA
vs vs vs vs vs
exp, all T exp DIPPR DIPPR Yaws
CPA CPA CPA CPA
vs vs vs vs
exp, all T exp DIPPR DIPPR
0.37-0.80 0.37-0.49 0.37-0.80 0.50-0.90 0.37-0.80
13.2 62.6 18.8 11.2* 36.6
liquid volume of TEG 0.35-0.87 0.35-0.49 0.35-0.87 0.50-0.90 TEG-water ln γ Px hE ∞ γwater
0.43 0.43 0.39 0.39-0.49
VLE: Pxy LLE: Pxw ∞ γbenzene
0.46-0.56 0.36-0.37 0.39-0.52
VLE: Pxy LLE: Pxw ∞ γtoluene
0.50-0.58 0.36-0.45 0.42-0.51
TEG-benzene 40.7* 9.0* 34.6*
TEG-toluene 19.4* 4.1* 31.2*
TEG-hydrocarbons methane, VLE: Pxy n-heptane, LLE: Txw ∞ γethylbenzene ∞ γoctane ∞ γnonane ∞ γdecane
0.39-0.41 0.40-0.46 0.42-0.51 0.42-0.51 0.42-0.51 0.42-0.51
80.9 4.6* 3.7* 6.8* 7.7* 8.2*
a The asterisks (*) indicate which data have been used in the estimations.
This comparison is performed in order to investigate how well the model performs in the temperature range which is typically used in applications. Because we have included high-temperature (predicted) data in this comparison, there is a tendency for poorer MAPE values. Finally, a similar comparison between CPA and the Yaws correlation is performed over a wide temperature range. Because of the relative poor fit of the Yaws correlation at high temperatures, the MAPE are rather high. Liquid Volume. The five different versions of MAPE for thevaporpressuredataarerepeatedfortheliquidvolumesalthough with the exception of the Yaws correlation for the liquid density. Set 1: Literature Parameters. The first set of parameters is composed by the model parameters available in the open literature.7,8 However, the kij values for TEG-ethylbenzene, TEG-n-octane, TEG-n-nonane, and TEG-n-decane are fitted in this work for the purpose of comparison among the parameter sets. In the work of Derawi et al.,9 the parameters of TEG were estimated based on the pure-component data (predicted from the DIPPR correlations, eqs 5 and 6) and the LLE data of the TEG-n-heptane system as well (with an additional binary interaction coefficient). The reason for including this binary system was the lack of success (in describing this particular system) when leaving it out of the estimation of the purecomponent parameters.
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In Figure 1a, the data points marked by circles are the experimental data from the accepted and not used categories of DIPPR. The data points marked by squares are the ones from the rejected category of DIPPR. Apparently, the correlation of Yaws,20 eq 7, emphasizes the low-temperature vapor-pressure data to a greater extent than the DIPPR correlation, eq 5. However, the DIPPR correlation does a better job representing the high-temperature data. When we look at Figure 1a and b, we note that CPA apparently describes the vapor pressure data above 400 K quite well, whereas the liquid volume data are described with a lower degree of precision. This impression is supported by the values of MAPE in Table 4 for the two figures. The mean absolute percentage error (MAPE), given in Table 4, of the vapor pressure data is somewhat high because of the deviations at temperatures < 400 K. The MAPE% values that Derawi et al.9 obtained were 3.04 for the vapor pressure and 1.61 for the liquid volume (0.49 < Tr < 0.82). These values compare well with the MAPE’s calculated in the range 0.50 < Tr < 0.90. Set 2. In set 2, the pure-component parameters of TEG (see Table 2) and the binary interaction parameters have been reestimated; see Table 3 and Figure 2. The estimation of the TEG parameters of set 2 are based on the pure-component data: vapor pressure, liquid density, and the binary mixtures TEG-methane and TEG-n-heptane. The parameters of set 1 were obtained without the TEG-methane data. The argument for including these two binary systems in the pure-component parameter estimation is that the description of the two binary systems is not acceptable when they are excluded. The binary interaction parameters of TEG-methane and TEG-n-heptane were estimated simultaneously with the pure-component parameters. Generally, sets 1 and 2 produce similar resultssexcept for the vapor pressure of pure TEG where set 1 is better. Set 3. In set 3, TEG has been assigned a new association scheme, 6D. The association scheme has three different site types and each type appears twice. In other words, the lone pairs of the oxygen atoms of TEG are now considered by the model. For simplicity all three sites have the same value of ε and β, i.e., we have the same number of parameters (see Figure 3). The estimation of the TEG parameters of set 3 are based on the pure-component data: vapor pressure, liquid density, and the binary mixtures TEG-methane and TEG-n-heptane. The objective function is identical to that of set 2. The covolume parameter b and the energy parameter Γ of TEG are fixed (to the values of set 2), while the remaining parameters are estimated. If all five pure-component parameters and the binary interaction parameters (of TEG-methane and TEG-n-heptane) were estimated freely, convergence becomes difficult. Binary Systems. The performance of CPA for the nine binary systems is presented in Figures 4-9. The results of set 1 are depicted with dotted lines, those of set 2, with dashed lines, and those of set 3, with full lines. Even though the liquid volume appears to be described poorly in Figures 1b, 2b, and 3b, the description of the liquid-liquid equilibrium of the TEG-n-heptane system is excellent, Figure 4a, for all three sets. Consequently, the values of MAPE of the TEG-n-heptane are small (see Table 4). The description of the TEG-methane system, Figure 4b, is only satisfactory for sets 2 and 3 possibly because this system was included in the estimation of these sets. The performance of set 1 cannot be improved by changing the
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Figure 1. Pure-component data with set 1. (a) The full curve is CPA, the dashed curve is the DIPPR correlation, eq 5, and the dotted curve is the Yaws correlation, eq 7. (b) The full curve is CPA, and the dashed curve is the DIPPR correlation, eq 6. The experimental data are those of DIPPR. The temperature range is 0.35 < Tr < 0.87.
Figure 2. Properties of pure TEG as functions of temperature for set 2.
Figure 3. Properties of pure TEG as functions of temperature for set 3.
binary interaction coefficient, kij, since that will only change the high-pressure performance of the model. The kij parameter is made temperature-dependent in sets 2 and 3 in order to handle the TEG-methane system at the two temperatures. The TEG-water results of set 1 are calculated using a fixed kij. But for sets 2 and 3, a temperature-dependent kij is introduced. The kij parameter is primarily fitted to hE data, Figure 6a. The association strength of cross-association is performed by CR-1. It is apparent that the hE data, Figure 6a, is poorly modeled by sets 1 and 2sthis indicates that the temperature dependence of these sets is inadequate. Furthermore, we notice that even though the ln γ data are described quite well, the γ∞ (infinite dilution) data are not. This could indicate an inconsistency of
the experimental data. It is the same experimental data that are presented both as ln γ data, Figure 5a, and as Px data, Figure 5b. The inflection points on the γTEG curve in Figure 5a for sets 1 and 2 do not indicate a possible phase split. A Gibbs energy plot of the two sets (not shown) reveals that there is no phase split. Both the TEG-benzene and TEG-toluene systems, Figures 7 and 8, are modeled by fitting a temperature-independent kij and βcross (solvation, mCR-1) to the experimental data. The systems are described equally well by the three setssnone of them can handle both the VLLE data and the γ∞ data. This is confirmed by the values of MAPE for the VLE, the LLE, and the γ∞ data.
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Figure 4. (a) Solubility of n-heptane in the glycol-rich phase and the solubility of TEG in the alkane-rich phase as functions of the temperature. (b) Solubility of TEG (both vapor and liquid phases) as a function of pressure at two temperatures.
Figure 5. (a) Activity coefficients of TEG and water at 60 °C as a function of the mole fraction of water. (b) Only the liquid branch of the VLE calculation. The temperature is 60 °C.
Figure 6. (a) Excess enthalpy of the TEG and water mixture at 25 °C. (b) Activity coefficient of water at infinite dilution in TEG.
The descriptions of the activity coefficient at infinite dilution, Figure 9, are similar for all three sets, but set 3 (the full line) is somewhat better. For the TEG-ethylbenzene system, only γ∞ data are available. As seen also for the TEG-benzene and TEG-toluene systems, there is little assurance that a good fit to the VLLE data would give a good fit to the γ∞ data, and vice versa. Therefore the estimated parameter kij is probably not suitable for a VLLE diagram. For the remaining three systems, TEG-n-octane, TEG-nnonane, and TEG-n-decane, the binary interaction coefficient kij has been optimized. It seems that the value of kij decreases as the molar mass increases.
Investigation of TeEG. A smaller investigation is carried out for TeEG since there are only pure-component data and LLE data for the TeEG-n-heptane system. The parameters of set 49 are based on the data predicted from the DIPPR correlations (of vapor pressure and liquid density) over the entire temperature range of 0.50-0.90. The parameters of TeEG of set 5 are re-estimated with the experimental vapor pressure and liquid density data of DIPPR as well as the binary system of TeEG-n-heptane.2 In set 6, the new 7D scheme for TeEG is applied instead of the 4C scheme. It is noted that the values of the parameters of association in set 4 (from the literature9) are orders of magnitude different from what is usually expected. An association energy is
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Figure 7. (a) TEG-benzene system at 1 atm. (b) Activity coefficient of benzene at infinite dilution in TEG.
Figure 8. (a) TEG-toluene system at 1 atm. (b) Activity coefficient of toluene at infinite dilution in TEG.
Figure 9. (a) Activity coefficient of ethylbenzene at infinite dilution in TEG. (b) Activity coefficient of octane, nonane, and decane at infinite dilution in TEG. Table 5. Pure-Component Parameters of TeEG (Set 4) from the Work of Derawi et al.9 and Those of Sets 5 and 6 from This Worka
a
compound
Tc (K)
b (L/mol)
a0/(Rb) (K)
c1
ε/R (K)
β × 103
scheme
TeEG, set 4 TeEG, set 5 TeEG, set 6
795.0 795.0 795.0
0.1777 0.16658 0.16658
3157.85 3789.49 3738.69
2.0242 1.5911 1.6261
57.6137 1140.83 856.72
3790 34.3 23.4
4C 4C 7D
The parameters for heptane are presented in Table 2.
normally of a magnitude of 2000-3000 K, and an association volume factor is normally of a magnitude of 10-20 × 10-3; therefore, the new parameters of sets 5 and 6 seem more reasonable (see Table 5). The performance of set 6 in regards to the vapor pressure data is shown in Figure 10a.
The binary interaction coefficient kij for the TeEG-heptane system is 0.101 for set 4 and 0.075 for sets 5 and 6, as seen from Figure 10b. In Table 6 it is noted that there is a great MAPE for CPA vs the experimental points. The reason for this is the inclusion of some
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Figure 10. (a) Vapor pressure of TeEG for set 6, including the rejected data points. (b) Solubility of n-heptane in the glycol-rich phase and the solubility of TeEG in the alkane-rich phase as functions of the temperature.
but only with heptane. So, the full importance of the 7D scheme is yet to be evaluated.
Table 6. Mean Absolute Percentage Error for All Three Sets of TeEG MAPE range Tr
set 4
set 5
set 6
27.7 0.7
24.3 3.4
24.7 3.0
7.0 2.4 12.2
1.5 6.6 9.2
1.5 7.0 8.9
vapor pressure of TeEG CPA vs exp, all T CPA vs DIPPR
0.37-1.00 0.50-0.90 liquid volume of TeEG
CPA vs exp, all T CPA vs DIPPR TeEG-n-heptane, LLE: Txw
0.36-0.52 0.50-0.90 0.34-0.44
experimental outliers which are rejected by DIPPR. There is, however, a very good agreement between the vapor pressure calculation of CPA and the DIPPR correlation, as seen in Figure 10a. It is obvious from Table 6 that sets 5 and 6 do a better job in describing the data points than set 4. By comparing the MAPE’s of sets 4-6, we see that sets 5 and 6 describe the three systems slightly better than set 4. Another positive feature is that the values of the parameters of sets 5 and 6 are more reasonable than those of set 4swhen comparing ε and β. There are no significant differences between sets 5 and 6 in terms of errors. Conclusions This investigation shows that the introduction of the new association scheme 6D improves the modeling of a number of TEG containing systems. This could be interpreted that the addition of the internal lone pairs of oxygen on the TEG molecule is essential in order to capture the behavior of TEG in water, in aromatic compounds, and in n-alkanes. It is, however, fair to state that the “traditional” 4C scheme is not a poor substitute to the 6D scheme. The difference between the performances of the two schemes is rather smallsthis can be appreciated only when many properties are tested. In the TEG-water system, we have a 6D-scheme component (TEG) and a 4C-scheme component (water), in total five different sites (each repeated twice). This means that the strength with which a TEG and a water molecule associate is ascribed to more sites than usual (when TEG is assumed to be a 4Cscheme component). There is no apparent difference in applying the 4C or the 7D scheme for TeEG. However, the parameters of sets 5 and 6 have not been tested for systems with other associating compounds,
List of Symbols a0 ) parameter in the energy term of SRK, a(T) ) a0(1 + c1(1 Tr1/2))2 Ai ) site A in molecule i Bj ) site B in molecule j b ) covolume parameter, L/mol c1 ) parameter in the energy term of SRK, see a0 kij ) binary interaction parameter g ) radial distribution function p ) pressure T ) temperature Tc ) critical temperature Tr ) reduced temperature R ) gas constant xi ) liquid mole fraction of component i yi ) vapor mole fraction of component i Greek Letters β ) association volume factor ∆ ) association strength ε/R ) association energy parameter, K List of AbbreViations MAPE ) mean absolute percentage error, eq 8 CPA EoS ) cubic-plus-association equation of state LLE ) liquid-liquid equilibria SRK) Soave-Redlich-Kwong TEG ) triethylene glycol TeEG ) tetraethylene glycol VLE ) vapor-liquid equilibria VLLE ) vapor-liquid-liquid equilibria
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ReceiVed for reView September 19, 2008 ReVised manuscript receiVed April 2, 2009 Accepted April 6, 2009 IE801412Y
