Ind. Eng. Chem. Res. 2006, 45, 6803-6810
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Modeling Phase Equilibrium of H2 + n-Alkane and CO2 + n-Alkane Binary Mixtures Using a Group Contribution Statistical Association Fluid Theory Equation of State (GC-SAFT-EOS) with a kij Group Contribution Method Chi Le Thi,†,‡ Sofiane Tamouza,†,‡ J-Philippe Passarello,*,‡ Pascal Tobaly,‡ and J-Charles de Hemptinne† Institut Franc¸ ais du Pe´ trole, 1 & 4 aVenue de Bois-Pre´ au, 92852, Rueil-Malmaison Cedex, France, and Laboratoire d’Inge´ nierie des Mate´ riaux et des Hautes Pressions (LIMHP), CNRS UniVersite´ Paris 13, 99 aVenue J. B. Cle´ ment, 93430, Villetaneuse, France
A group contribution (GC) method combined with a SAFT equation of state (EOS) [Tamouza et al., Fluid Phase Equilib. 2004, 222-223, 67 and 2005, 228-229, 409] is extended here to model vapor-liquid phase equilibria of binary mixtures of H2 + n-alkanes and CO2 + n-alkanes. Modeling these systems requires binary interaction parameters kij that are estimated here in the same spirit as pure compound GC-SAFT parameters, i.e., through a specific group contribution method. Molecule-group interaction parameters (kH2,CH2, kH2,CH3, kCO2,CH2, and kCO2,CH3) are used rather than molecule-molecule interaction parameters. Two versions of SAFT are tested here: the Perturbed-Chain SAFT (PC-SAFT) [Gross and Sadowski, Ind. Eng. Chem. Res. 2000, 40, 1244] and Variable-Range SAFT (VR-SAFT) [Gil-Villegas et al., J. Chem. Phys. 1997, 106, 4168]. The results are very encouraging, particularly for predicting binary mixtures of CO2 and heavy n-alkanes. Mixtures that contain H2 are modeled here with deviations that compare well with those of the classically used Grayson-Streed model. Introduction A group contribution method that was developed earlier1,2 for several versions of the SAFT equation of state (called GCSAFT EOS) is extended here to some mixtures of industrial interest. This method was applied successfully to model the vapor-liquid phase equilibrium of pure aliphatic and aromatic hydrocarbons, alkanols, esters, and some of their mixtures without binary interaction parameters (kIJ ) lIJ ) 0). For petroleum and chemical industry applications, extension of these GC-EOS to more-complex chemical systems is needed. Especially, systems that contain CO2 (supercritical extraction) and H2 (hydrotreatment) are of great importance today and have been considered for this reason in this work. However, these systems are difficult to model for the following reasons. First, CO2 and H2 cannot be treated in the frame of the group contribution (i.e., decomposed in several groups), because they do not belong to a well-identified chemical family. Furthermore, above 50 K, pure H2 is, in fact, a mixture of ortho- and para-H2, in proportions that are a function of temperature.3 These two hydrogen configurations may also have different properties at the macroscopic level. Finally, much available vapor-liquid equilibrium (VLE) data for mixtures with CO2 correspond to almost-critical conditions, which are difficult to represent accurately with an usual EOS. For all these reasons, phase equilibria modeling of H2 + n-alkane and CO2 + n-alkane binary systems requires a different treatment from that used previously.2 Hence, pure CO2 and H2 SAFT parameters were determined following a specific procedure that has been described below. Also, to obtain the best modeling accuracy, it was decided here to use nonzero binary interaction parameters (kIJ). As shown below, these kIJ values * Towhomcorrespondenceshouldbeaddressed.Tel.: +33149403406. Fax: +33 149 40 34 14. E-mail:
[email protected]. † Institut Franc¸ ais du Pe´trole. ‡ CNRS Universite´ Paris XIII.
may be computed using a group contribution method that is consistent with that proposed by Tamouza et al.4-6 for estimating SAFT EOS parameters. Models Used for Fugacity Calculation 1. SAFT Versions. Two versions of SAFT were used here. The first one is PC-SAFT, which is taken from the work of Gross and Sadowski.7,8 This model was chosen for this work because it applies particularly well to hydrocarbons and is especially designed for long-chain molecules. The second EOS used is VR-SAFT, which was proposed by Gil-Villegas et al.9 In this version, a square-well potential with a variable width is used. An additional parameter (λ) accounts for the well width. The expressions of the EOS are not recalled here, and the interested reader is referred to the original papers for more details. 2. Mixing Rules. Many authors have shown that the van der Waals one-fluid model is a reasonable concept to extend the SAFT EOS to mixtures. This approach is used here for the PCSAFT EOS. The corresponding expressions are given in several references.7,8 For the VR-SAFT EOS, several mixing rules were proposed by Galindo et al.15 These mixing rules span from the usual van der Waals one-fluid prescription to a more-complete description in terms of the pair distribution function of the pair interactions that goes beyond the one-fluid level. In this work, the MX3b mixing rules are used, as in one of our previous papers.2 Group Contribution Parameter Method Used for Parameter Estimation The pure compound parameters for alkanes are computed using a GC method that was developed and tested earlier.1 The main features of this method are briefly reported below, and are then extended for our purposes. The following convention is used here: an uppercase subscript I, J, ... always refers to a
10.1021/ie060424n CCC: $33.50 © 2006 American Chemical Society Published on Web 09/06/2006
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Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006
Table 1. Parameters Values for the CH3 and CH2 Chemical Groups CH3 VR-SAFT PC-SAFT
CH2
/k (K)
σ (Å)
λ
R
/k (K)
σ (Å)
λ
R
202.9 190.0
3.54 3.49
1.47
0.80 0.79
136.4 261.1
3.42 3.93
1.90
0.47 0.38
Table 2. Values of the Adjustable Parameters
VR-SAFT PC-SAFT VR-SAFT PC-SAFT
λJ
J/k (K)
σJ (Å)
385.01 178.74
CO2 4.045 0.911 2.896 1.886
1.291
29.72 23.42
H2 1.187 1.306
2.141
mJ
3.689 2.601
kCH2J
kCH3J
0.178 0.089
0.045 0.150
-0.073 0.131
0.061 -0.041
molecule, whereas a lowercase subscript i, j, p, ... refers to a chemical group. 1. Pure Compound Parameters. The GC scheme used for the computation of the pure compound parameters is as follows. The relations are inspired from the Lorentz-Berthelot combining rules. The energy parameter I of molecule I is obtained by a geometric average (eq 1).
x∏
Consequently, they were treated by a different procedure detailed in the section below. 2. Mixture Parameters. Most of current EOS are applied to mixtures using the one-fluid concept. Binary mixtures parameters are computed from pure ones through mixing rules and combining rules. Original combination rules, such as those of Lorentz-Berthelot, are
I )
(
npp,I)
(1)
Both the segment diameter σI and the VR-SAFT range parameter λI of the molecule I are calculated as an arithmetic average (see eqs 2 and 3). ngroups
∑ np,Iσp
p)1
nGr.I
(2)
λI )
np,Iλp
nGr.I
(3)
The chain-length parameter (mI) is calculated as a sum of chain contributions Rp, where the subscript p denotes group p (eq 4).
∑ np,I Rp p)1
σI + σJ 2
(9)
)
[x nGr.I
(
∏ p)1
] x 1/2
ngroups
npp,I) × J
ngroups
)
nGr.I
[(pJ)1/2]n ∏ p)1
p,I
(10)
which introduces the cross-interaction energy parameter, pJ ) (pJ)1/2, between group p (of molecule I) and molecule J. The computation of this parameter may be empirically modified in the following manner, which has been inspired by eq 8, previously described:
pJ ) (1 - kpJ)(pJ)1/2
(11)
This defines kpJ, which is the binary interaction parameter between group p and molecule J. The value of this parameter is assumed here to be dependent only on the group p and the molecule J. Thus, eq 10 may be modified as
ngroups
∑ np,I p)1
(8)
The classical evaluation method of these binary parameters involves data regression of mixtures. However, as shown below, the total number of adjustable parameters also may be reduced, using a GC method for the binary parameters kIJ. Assume that molecule I is described by a GC method (which means that, here, molecule I refers to a hydrocarbon and J refers to H2 or CO2). Using the relation described in eq 1, which is the original combination rule for the energy parameter, eq 6 then becomes
(4)
In these equations, np,I is the number of groups p in the molecule I, and ngroups is the number of different chemical groups in the considered molecule. The total number of groups in molecule I (nGr.I) is given by
nGr.I )
IJ ) (1 - kIJ)(IJ)1/2
(
ngroups
mI )
(7)
which allows pure prediction of the mixtures using pure compound parameters. This approximation is reasonable for simple mixtures and even for some complex mixtures (alkanol + alkane, for instance2). However, sometimes, they appear inefficient. Binary parameters (kIJ, lIJ) are then introduced to improve their applicability. In the case of energy and diameter, one can write
IJ )
ngroups
∑ p)1
σI + σJ 2
σIJ ) (1 - lIJ)
p)1
σI )
(6)
σIJ )
ngroups
nGr.I
IJ ) (IJ)1/2
x∏
ngroups
(5)
The GC parameters that have been used here for VR-SAFT and PC-SAFT EOS (p, σp, λp, and Rp)were determined for the CH2 and CH3 groups in earlier works1,10 by VLE data regression of the n-alkane family. These values were reused here without further adjustment. All the parameter values for the different groups CH2 and CH3 that are needed in this work are given in Table 1. As noticed in the introduction, CO2 and H2 are small molecules that do not belong to a particular chemical family.
IJ )
nGr.I
[(pJ)1/2(1 - kpJ)]np,I )
p)1
(IJ)1/2 ×
x
nGr.I
[
ngroups
(1 - kpJ)n ∏ p)1
p,I
] (12)
Comparison between eqs 12 and 8 leads to
x∏
ngroups
1 - kIJ )
nGr.I
(
p)1
(1 - kpJ)np,I)
(13)
Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006 6805 Table 3. Database for Binary CO2 + n-Alkane Systems, and Results of Regression and Prediction 100 × AADy1
AADP% temperature, T (K)
a
pressure range, P (Pa)
PC-SAFT
VR-SAFT
PC-SAFT
VR-SAFT
reference
12.41 5.21 2.81 4.31
3.04 0.68 1.00 1.45
4.46 1.95 3.89 4.62
12 12 12 12
Propanea
230.00 270.00 311.05 361.15
(9.71 × 104)-(8.94 × 105) (4.30 × 105)-(3.20 × 106) (1.30 × 106)-(6.68 × 106) (3.63 × 106)-(4.97 × 106)
7.29 1.66 1.43 3.97
250.00 311.09 357.77 418.15
(3.11 × 104)-(1.79 × 106) (3.57 × 105)-(7.52 × 106) (1.48 × 106)-(7.93 × 106) (3.32 × 106)-(4.70 × 106)
3.93 2.20 0.48 4.03
5.01 1.51 3.33 6.17
0.38 0.52 0.91 1.61
0.46 2.14 6.37 6.43
12 12 12 12
252.67 311.59 377.71 463.15
(1.59 × 105)-(1.79 × 106) (4.10 × 105)-(7.14 × 106) (7.86 × 105)-(9.62 × 106) (2.99 × 106)-(4.70 × 106)
n-Pentanea 3.85 4.10 2.30 2.87
4.59 2.22 7.63 7.11
0.22 0.35 0.97 1.60
0.33 0.95 4.10 8.33
12 12 12 12
298.15 353.15 393.15
(4.44 × 105)-(5.21 × 106) (8.62 × 105)-(1.07 × 107) (9.00 × 105)-(1.16 × 107)
n-Hexanea 4.95 3.35 2.77
4.99 3.83 6.30
0.52 0.37 0.79
0.78 2.53 5.01
12 12 12
310.65 394.26 468.40 502.00
(1.86 × 105)-(7.56 × 106) (1.13 × 106)-(1.33 × 107) (3.09 × 106)-(1.04 × 107) (3.04 × 106)-(7.68 × 106)
n-Heptanea 4.52 9.02 3.46 5.54
4.43 13.84 12.36 20.16
0.49 0.91 1.32 2.07
0.75 2.55 7.89 13.55
12 12 12 12
298.20 348.15 383.15
(6.10 × 105)-(5.85 × 106) (2.00 × 106)-(1.14 × 107) (2.90 × 106)-(1.31 × 107)
8.43 4.04 16.46
0.22 0.14 1.36
0.27 1.31 2.79
12 12 12
277.59 310.93 344.26 377.59 410.93 444.26 477.59 510.93
(2.76 × 101)-(3.91 × 106) (5.03 × 102)-(8.00 × 106) (2.76 × 103)-(1.28 × 107) (1.10 × 104)-(1.65 × 107) (3.50 × 104)-(1.86 × 107) (9.30 × 104)-(1.88 × 107) (2.15 × 105)-(1.78 × 107) (4.46 × 105)-(1.53 × 107)
n-Decanea 8.08 6.08 5.82 5.21 5.38 4.13 4.45 5.77
5.98 3.15 2.26 2.45 4.39 6.49 9.31 10.09
0.00 0.08 0.17 0.18 0.32 0.27 0.47 1.12
0.01 0.08 0.76 1.22 1.61 2.14 3.25 4.79
13 13 13 13 13 13 13 13
288.71 305.37
(1.08 × 106)-(5.02 × 106) (1.24 × 106)-(7.12 × 106)
n-Dodecane 8.03 11.39
4.56 22.57
14 14
288.71 305.37
(1.30 × 104)-(5.41 × 106) (8.27 × 105)-(7.32 × 106)
n-Tetradecane 9.56 16.62
5.53 9.41
14 14
310.15 323.15 323.20 348.15 373.15 373.20
(5.00 × 105)-(7.50 × 106) (5.00 × 105)-(7.50 × 106) (6.21 × 105)-(5.77 × 106) (5.00 × 105)-(7.50 × 106) (5.00 × 105)-(7.50 × 106) (1.07 × 106)-(6.76 × 106)
n-Eicosanea 4.10 5.09 2.90 2.02 8.92 2.34
1.86 2.02 4.31 2.57 12.19 5.84
15 15 16 15 15 16
323.15 348.15 373.15
(1.11 × 106)-(7.18 × 106) (1.11 × 106)-(7.18 × 106) (9.62 × 105)-(6.56 × 106)
n-Docosane 2.96 2.50 2.24
2.90 2.47 5.91
17 17 17
348.20 373.20 423.20
(8.45 × 105)-(9.60 × 106) (1.07 × 106)-(6.76 × 106) (8.41 × 105)-(9.25 × 106)
n-Octacosane* 2.13 7.77 15.43 21.23 3.26 9.49
16 16 16
348.15 373.15 398.15
(9.52 × 105)-(7.21 × 106) (1.16 × 106)-(7.22 × 106) (9.46 × 105)-(7.23 × 106)
n-Dotriacontane 3.92 2.76 3.40
6.81 7.22 5.56
17 17 17
373.20 423.20
(5.29 × 105)-(5.88 × 106) (1.02 × 106)-(8.63 × 106)
n-Hexatriacontane 1.99 10.40 2.17 11.41
16 16
373.20 423.20
(5.79 × 105)-(6.11 × 106) (8.14 × 105)-(7.08 × 106)
n-Tetratetracontane 1.71 11.81 0.85 11.58
16 16
n-Butane
n-Octanea 9.35 6.49 17.50
CO2 2 Regressed data. AADP% ) ∑data|Pexp - Pcalc|/(ndataPexp), and AADy1 ) ∑data|yCO exp - ycalc |/(ndata), where y1 represents the mole fraction of CO2.
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Table 4. Database for Binary H2 + n-Alkane Systems, and Results of Regression and Predictiona AADP% temperature, T (K)
pressure range, P (Pa)
248.15 273.15 298.15
n-Propaneb (1.52 × 106)-(2.03 × 107) (1.52 × 106)-(2.03 × 107) (1.52 × 106)-(2.03 × 107)
3.18 2.65 6.70
277.59 344.26 410.93 477.59
n-Hexane (7.65 × 103)-(1.09 × 105) (3.45 × 106)-(6.90 × 107) (5.86 × 105)-(6.90 × 107) (1.91 × 106)-(4.08 × 107)
463.15 483.15 503.15 523.15 543.15 553.15
a
AADP%
PC-SAFT VR-SAFT
temperature, T (K)
pressure range, P (Pa)
11.49 2.79 8.51
273.15 323.15 373.15
n-Pentane (1.03 × 106)-(2.76 × 107) (6.93 × 105)-(2.76 × 107) (1.04 × 106)-(2.76 × 107)
PC-SAFT VR-SAFT 7.68 9.79 4.37
14.02 7.08 4.26
17.28 11.83 13.38 4.11
75.65 26.94 4.88 1.88
424.15 471.65 498.85
n-Heptaneb (3.79 × 105)-(7.85 × 106) (9.45 × 105)-(7.26 × 107) (1.48 × 106)-(3.92 × 107)
5.41 7.63 1.74
4.04 4.17 3.74
n-Octane (4.55 × 105)-(6.90 × 106) (6.62 × 105)-(1.03 × 107) (9.38 × 105)-(1.03 × 107) (1.29 × 106)-(1.03 × 107) (2.07 × 106)-(1.03 × 107) (2.76 × 106)-(6.90 × 106)
0.79 1.30 0.83 1.19 1.11 0.82
1.20 1.55 1.60 2.25 5.74 8.60
308.15 323.15 343.15 383.15 413.15 433.15 453.15 473.15 493.15 513.15 523.15 533.15 543.15 553.15 563.15 573.15
n-Decaneb (3.66 × 106)-(1.62 × 107) (3.37 × 106)-(1.48 × 107) (3.04 × 106)-(1.67 × 107) (2.51 × 106)-(1.63 × 107) (2.22 × 106)-(1.42 × 107) (4.12 × 106)-(1.29 × 107) (1.94 × 106)-(1.18 × 107) (1.85 × 106)-(1.08 × 107) (1.79 × 106)-(9.94 × 106) (1.78 × 106)-(9.12 × 106) (1.80 × 106)-(8.73 × 106) (1.82 × 106)-(8.36 × 106) (1.86 × 106)-(8.00 × 106) (1.91 × 106)-(7.65 × 106) (1.98 × 106)-(7.31 × 106) (2.06 × 106)-(6.96 × 106)
9.12 7.67 6.09 3.17 1.64 0.96 0.57 0.47 0.55 0.93 1.14 1.41 1.64 1.83 1.95 1.95
4.61 5.95 3.92 2.82 1.62 1.09 0.71 0.56 0.68 0.63 0.54 0.49 0.53 0.88 1.64 2.61
344.30 377.60 410.90
n-Dodecane (3.36 × 106)-(1.20 × 107) (1.42 × 106)-(1.32 × 107) (1.77 × 106)-(1.15 × 107)
3.24 5.33 1.56
3.34 2.62 1.81
328.15 403.15 473.15
n-Tetradecane (4.05 × 106)-(3.04 × 107) (4.05 × 106)-(3.04 × 107) (4.05 × 106)-(3.04 × 107)
4.33 9.28 4.40
11.63 6.34 5.08
461.65 542.25 622.85 664.05
n-Hexadecane (2.03 × 106)-(2.53 × 107) (2.01 × 106)-(2.52 × 107) (2.02 × 106)-(2.53 × 107) (2.00 × 106)-(2.54 × 107)
0.64 0.62 5.90 9.98
3.26 2.58 6.27 6.04
323.20 373.20 423.20 373.35 473.55 573.25
n-Eicosaneb (3.26 × 106)-(1.29 × 107) (2.23 × 106)-(1.18 × 107) (2.81 × 106)-(9.30 × 106) (1.01 × 106)-(5.08 × 106) (1.00 × 106)-(5.04 × 106) (9.94 × 105)-(5.07 × 106)
3.91 1.42 0.82 6.66 2.88 1.47
4.26 2.42 1.45 10.98 1.97 3.38
348.20 373.20 373.25 423.20 473.25 573.15
n-Octacosane (3.53 × 106)-(1.31 × 107) (4.02 × 106)-(1.24 × 107) (9.86 × 105)-(5.05 × 106) (2.86 × 106)-(1.12 × 107) (1.01 × 106)-(5.07 × 106) (1.01 × 106)-(5.07 × 106)
4.94 3.81 5.30 4.81 1.43 2.48
3.14 2.55 3.50 2.33 5.12 3.75
373.150 373.20 423.20 473.05 573.15
n-Hexatriacontane (1.02 × 106)-(5.07 × 106) (4.11 × 106)-(1.68 × 107) (3.56 × 106)-(1.20 × 107) (1.03 × 106)-(5.07 × 106) (1.02 × 106)-(5.07 × 106)
3.24 6.10 5.48 3.28 2.52
12.18 3.22 2.06 7.05 7.83
All data are from ref 12. b Regressed data. AADP% ) ∑data|Pexp - Pcalc|/(ndataPexp).
which defines a GC method for calculating kIJ between two molecules I and J. A similar rule could be developed for the other binary parameters. In the same spirit, one may deduce that
1 - lIJ )
1 nGr.I
ngroups
∑ np,I(1 - lpJ)
(14)
p)1
However, in this work, as will be seen, only kIJ parameters had to be considered. All the other possible binary parameters were set to zero, which means that, except for energy, the original combination rules were used. Determination of Pure CO2 and H2 and Binary Parameter Values 1. CO2 + n-Alkanes. Using the procedure described previously, CO2 + n-alkanes mixtures were represented using only two adjustable binary parameters: kCH2,CO2 and kCH3,CO2. The values of the pure CO2 parameters σCO2, mCO2, CO2, and λCO2 and the binary parameters kCH2,CO2 and kCH3,CO2 were determined
by the simultaneous regression of pure and mixture VLE data. The relative weights given in the objective function were as follows: 25% for pure liquid densities, 25% for pure vapor pressures, and 50% for mixture bubble pressures. Pure vapor pressures and saturated liquid densities for CO2 were taken from ref 11. Only the data of the following binary systems were regressed: CO2 + C3, CO2 + C5, CO2 + C6, CO2 + C7, CO2 + C8, CO2 + C10, CO2 + C20, and CO2 + C28. The values of the adjusted parameters are reported in Table 2. Description of the complete database, together with the calculation results, are given in Table 3. VLE predictions of other binary systems using the adjusted parameters are also given in Table 3. 2. H2 + n-Alkanes. The values of the pure H2 parameters σH2, mH2, H2, and λH2 and the binary parameters kCH2,H2 and kCH3,H2 were determined via the simultaneous regression of only mixture VLE data of H2 + C3, H2 + C7, H2 + C10, and H2 + C20 systems. Pure VLE data for H2 were excluded from the regressed database, because they correspond to a very low temperature (
