990
Ind. Eng. Chem. Res. 2007, 46, 990-996
Applications of Associating Martin-Hou Equation of State to Vapor-Liquid Equilibria in the Critical Region or the Near-Critical Region Li-Sheng Hao and Yan-Qing Nan* Department of Chemistry, Hunan Normal UniVersity, Changsha, Hunan 410081, People’s Republic of China
The associating Martin-Hou equation of state (AMH EOS) has been extended to high-pressure vaporliquid equilibria calculations of alcohol-containing systems in the critical or near-critical region. The old mixing rules used in previous studies have been revised. The high-pressure vapor-liquid equilibria of some alcohol + hydrocarbon systems can be represented satisfactorily by the AMH EOS with the old mixing rules. However, the high-pressure vapor-liquid equilibria of alcohol + carbon dioxide systems and some other alcohol + hydrocarbon systems should be correlated well by the AMH EOS with the revised mixing rules. High-pressure vapor-liquid equilibria of two ternary systems that contain two alcohols and two nonassociating ternary systems are predicted from binary data. Introduction
Table 1. Pure-Component Parameters for the Martin-Hou Equations of State (AMH EOS and MH EOS)
The correlation and prediction of thermodynamic properties and phase equilibria with an equation of state (EOS) is an important goal in the design and operation of separation processes in chemical and related industries. The use of supercritical fluid solvents and near-critical fluid solvents in chemical processes has been given more attention recently. However, the estimation of phase behavior for systems that contain supercritical components and near-critical fluid components has been one of the most difficult tasks of the thermodynamics for a long time, because of the diverse and complex behavior. During the past 20 years, efforts have been made in regard to developing alternative mixing rules. For example, the use of multiple interaction parameters in the quadratic mixing rules,1 the development of DDLC (densitydependent local composition) mixing rules based on the twofluid theory,2,3 the connection between excess Gibbs free-energy models and the EOS,4,5 and the use of nonquadratic mixing rules6 have been presented in the literature. In a review about the state of the cubic EOSs, Valderrama7 indicated that, for mixtures that contain supercritical components, the use of Gibbs free-energy models in the EOS parameters and nonquadratic mixing rules with interaction parameters in the volume constants of the EOS give the best results. Valderrama also indicated that more than one interaction parameter must be included, even in complex models.7 In our previous papers,8-10 using an approach similar to that proposed by Heidemann and Prausnitz,11 and that used by Ikonomou and Donohue12 in 1986, an associating Martin-Hou equation of state (AMH EOS) has been developed, based on the MH EOS by incorporating the chemical association. The AMH EOS is a four-parameter associating EOS; it has been satisfactorily applied to calculations of physical properties of pure associating compounds, simultaneous calculations of vapor-liquid equilibria, and excess enthalpies of binary alcohol + alcohol, alcohol + hydrocarbon, carboxylic acid + hydrocarbon and carboxylic acid + carboxylic acid mixtures, as well as the prediction of ternary vapor-liquid equilibria for systems that contain alcohol and hydrocarbon components or carboxylic acid and hydrocarbon components from binary data. The * To whom correspondence should be addressed. E-mail: nanyq@ 21cn.com.
compound
m
T′r
HO (K)
SO
methanol ethanol 1-propanol 2-propanol 1-butanol 2-butanol 2-methyl-1-propanol 2-methyl-2-propanol 1-pentanol 1-octanol 1-decanol carbon dioxide, CO2 methane, CH4 chlorodifluoromethane, R-22 ethane, C2H6 propylene, C3H6 propane, C3H8 n-pentane, C5H12 n-hexane, C6H14 n-heptane, C7H16
1.546 1.232 0.850 0.882 0.730 0.700 0.695 0.705 0.630 0.432 0.383 1.770 1.450 1.060 1.125 0.900 0.784 0.550 0.480 0.380
0.6240 0.6605 0.6180 0.6625 0.6603 0.6930 0.6575 0.6550 0.6855 0.7285 0.7382 0.7632 0.7560 0.7617 0.7601 0.7599 0.7642 0.7762 0.7687 0.7760
3000 3000 3000 3120 3000 3000 3000 3000 3000 3000 3000
-11.90 -12.25 -11.25 -12.24 -11.40 -12.00 -11.40 -11.90 -11.40 -11.40 -11.40
aforementioned applications are focused on low-pressure or moderate-pressure regions. In the present study, the AMH EOS is extended to highpressure vapor-liquid equilibria calculation of binary systems that contain an alcohol component in the critical or near-critical region, and to the prediction of ternary vapor-liquid equilibria for systems that contain a supercritical component or near-critical fluid component from binary data. For the studied systems, the mixing rules used in our previous work have been revised. Equation of State In our previous work,8-10 the AMH EOS has been developed. The AMH EOS for mixtures can be expressed as follows:
P)
() nT
RT
n0 V - b
+
5
FL(T)
L)2
(V - b)L
∑
(1)
where nT is the true total number of moles of the system and n0 is the total number of moles if association were absent (i.e., the superficial number of moles). For nonassociating systems, nT/n0 ) 1, the MH EOS is recovered. Because monomeric molecules of a pure component associate to form other species,
10.1021/ie060884g CCC: $37.00 © 2007 American Chemical Society Published on Web 12/29/2006
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 991 Table 2. Binary Parameters for QEN (QEN ) aEN + bENT) and QNE (QNE ) aNE + bNET) QEN system
aEN (×
QNE bEN (×
-53.35 -77.70a -97.33 -309.70 -2.00 10.00 -197.93 25.74a 30.58 -42.50 0 -192.44 -317.14 -226.20a -168.17 -201.41 75.84a 116.90 233.52a 223.26 229.30a 163.05 67.50 -150.36 0 -247.21 -8.00a -25.00 -209.34 -490.38 -241.22a -225.04 -206.31a -206.55 0a 122.29 175.60a 170.60 -235.67 -5.00a -25.0 -284.95 -258.97a -279.96 -257.90a -261.06 -8.00a -24.00 -228.24 -180.05a -11.21 -155.17 -253.28 -181.23 -69.79
CH4 + CO2 CH4 + C2H6 CH4 + C2H6 CH4 + C3H8 methanol + CO2 methanol + C3H6 methanol + n-C3H8 C2H6 + CO2 C2H6 + CO2 ethanol + CO2 ethanol + CH4 ethanol + R-22 ethanol + n-C2H6 ethanol + C3H6 ethanol + C3H6 ethanol + n-C3H8 ethanol + n-C5H12 ethanol + n-C5H12 ethanol + n-C6H14 ethanol + n-C6H14 ethanol + n-C7H16 ethanol + n-C7H16 C3H8 + CO2 1-propanol + CO2 1-propanol + CH4 1-propanol + C2H6 1-propanol + C5H12 1-propanol + C5H12 2-propanol + CO2 2-propanol + C2H6 2-propanol + C3H6 2-propanol + C3H6 2-propanol + n-C3H8 2-propanol + n-C3H8 2-propanol + n-C6H14 2-propanol + n-C6H14 2-propanol + n-C7H16 2-propanol + n-C7H16 1-butanol + CO2 1-butanol + C5H12 1-butanol + C5H12 2-butanol + CO2 2-butanol + C3H6 2-butanol + C3H6 2-butanol + n-C3H8 2-butanol + n-C3H8 2-butanol + n-C5H12 2-butanol + n-C5H12 2-methyl-1-propanol + CO2 2-methyl-1-propanol + C5H12 2-methyl-1-propanol + C5H12 2-methyl-2-propanol + CO2 1-pentanol + CO2 1-octanol + CO2 1-decanol + CO2 a
103)
103)
0.355 0.340a 0.4127 1.021 0 0 0.6743 0.1476a 0.1508 0 0 0.3567 0.8875 0.5731a 0.4298 0.5790 -0.1028a -0.1938 -0.4200a -0.4000 -0.4125a -0.2760 0 0.3250 0 0.6050 0a 0 0.5321 1.3600 0.5662a 0.5405 0.4855a 0.5000 0a -0.2500 -0.3500a -0.3500 0.5580 0a 0 0.7087 0.5677a 0.6286 0.6050a 0.6250 0a 0 0.5412 0.3333a -0.0444 0.3397 0.5921 0.3900 0.1143
aNE (×
103)
bNE (× 103)
62.00
0
0.60 -142.50 -52.00 -23.65 -151.40
0.05 0.6315 0 0 0.4484
17.15 -62.50 -1376.64 -208.52 -502.76
0.1587 0 3.5700 0.4178 1.3250
-239.63 -205.25
0.5960 0.4980
30
0
133.25
-0.2050
233.92 52.00 -623.01 -1623.80 -552.36
-0.4200 0 1.6950 4.1230 1.3700
10.00 -544.35 -1899.99
0 1.5144 5.6800
-260.56
0.6081
-213.70
0.4925
0
0
138.87 -477.72
-0.2750 1.2634
13.80 -571.85
0 1.4698
-429.96
1.0257
-315.07
0.7125
1.00 -348.64
0 0.8273
-78.04 -366.52 -358.96 -232.21 -290.48
0.1556 1.0371 0.8772 0.5270 0.8130
Coefficient of eq 13 in the old mixing rules.
a pure associating component becomes a mixture of association species in equilibrium. Moreover, the true total number of moles in an associating system is smaller than the moles present if association were absent (i.e., nT < n0). V is the molar volume (V ) Vt/n0), Vt is the total volume, and b is the co-volume constant of AMH EOS. For a pure associating compound, b can be evaluated from the following equation:
(
b ) VC 1 4
β 75ZC
)
02
3
- 0.85
() VC Mr
in which VC is the molar volume at critical point, Mr the relative molecular mass, and Z0C the superficial compressibility factor at the critical point (Z0C ) (PCVC)/(RTC)). FL(T) is the temperature function of AMH EOS; for pure associating compounds,
(
FL(T) ) AL + BLT + CL exp
)
-5.475T TC (for L ) 2, 3, 4, 5) (4)
(2) 2
β ) 106531(Z0C) - 96381.3(Z0C) + 32596.4(Z0C) 4858.22(Z0C) + 270.88 (3)
where TC is the critical temperature. AL, BL, and CL are constants of AMH EOS, in which C4, A5, and C5 are zero. The other constants can be evaluated by the method introduced in the literature.8 For pure nonassociating compounds, the evaluation method for the constants of MH EOS has been introduced in
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Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007
the literature13 or is referenced in the Supporting Information of this work. For multicomponent systems, in our previous work,8-10 the following mixing rules were used:
alcohol E. The assumption is made that association species form via an infinite number of equilibria as follows:
(1 - Qij)xixj|F2(T)i| ∑i xi F2(T)i - 2∑i ∑ j>i
E1 + E1 ) E2, E1 + E2 ) E3, ..., E1 + Ej ) Ej+1, ... (10)
F2(T) )
2
K2
|F2
1/2
(T)j|1/2 (5) FL(T) ) (-1)L+1[
∑i xi|FL(T)i|1/L]L b)
∑i
(for L ) 3, 4, 5)
xibi
F2(T) RT
The second virial coefficient B is a useful indicator of the overall interaction between two molecules. Therefore, the F2(T) term also is related to the interaction between two molecules. The large amount of calculations indicate that the F2(T) term in AMH EOS has an important role. The revision of the mixing rule of F2(T) may improve the calculation precision of AMH EOS. The mixing rule of F2(T) has been revised to a nonquadratic expression, similar to that of Panagiotopoulos and Reid:6
F2(T) )
ln K )
(7)
(8)
[1 - Qij - (Qij ∑i xi2F2(T)i - ∑i ∑ j*i
Results and Discussion 1. Binary Vapor-Liquid Equilibria. For binary systems that contain one alcohol, the self-association of the alcohol should be taken into consideration. For simplicity, the infinite equilibria model is approximated for the self-association of
HO + SO T
(11)
where HO and SO are association constants. The units of K are atm-1, and the units for HO are K; the constant SO is dimensionless. The AMH EOS contains four parameters: m, T′r, HO, and SO. MH EOS contains two parameters: m and T′r. The method of how to determine these parameters has been introduced in the literature,8,13 for MH EOS, or one can refer to the Supporting Information for this work. The corresponding parameters of AMH EOS for alcohols are listed in Table 1. This table also gives the pure-component parameters of MH EOS for the nonassociating components used in this study. For the binary systems that contain one alcohol E and one inert compound N, the analytical solution of nT/n0 is
nT ) n0 1 +
2xE
x1 + 4RTKExE/(V - b)
+ xN
(12)
For alcohol + CO2 systems, CO2 is a nonassociating component, and, for simplicity, the cross-association between alcohol and CO2 is neglected. Therefore, eq 12 is also suitable for binary alcohol + CO2 systems. The AMH EOS has been used for the correlations of highpressure vapor-liquid equilibria for binary systems that contain one associating component (alcohol E) and one supercritical or near-supercritical nonassociating component (compound N) (such as CO2, R-22, and hydrocarbons). First, the old mixing rules are used; for those systems for which the old mixing rules are not suitable (for example, for a binary methanol + CO2 system at 313.2 K and for propanol + CO2 systems at 313.4 and 333.4 K, where the average deviations of pressure and vapor-phase composition, using the old mixing rules, are 10.42% and 0.0071, 13.46% and 0.0033, and 6.05% and 0.0096, respectively), the revised mixing rules are then used. The calculation results indicate that the binary parameters QEN and QNE are dependent on temperature. Typically, they are expressed by linear relationships:
Qji)xi]xixj|F2(T)i|1/2|F2(T)j|1/2 (9) where Qij and Qji are the binary interaction parameters between components i and j (usually, Qij * Qji; if Qij ) Qji, eq 5 is recovered). To distinguish between the two mixing rules, the mixing rules represented by eqs 5-7 are called the old mixing rules, whereas the mixing rules represented by eqs 6, 7, and 9 are called the revised mixing rules.
Kj+1
Assuming K2 ) K3 ) ... ) Kj+1 ) ... ) K (for j ) 1, 2, ..., ∞), where K is the association equilibrium constant, which is dependent on temperature, the relationship is as follows:
(6)
where x is the superficial mole fraction of the components. Qij is the binary interaction parameter. However, the calculation results of high-pressure vaporliquid equilibria for some systems that contain supercritical carbon dioxide, supercritical methane, or ethane are not satisfactory using the aforementioned mixing rules. To improve the calculation precision, more-suitable mixing rules should be developed. The experience of the cubic EOSs to the applications of systems that contain supercritical components may be helpful to multiparameter EOSs. The nonquadratic mixing rules proposed by Panagiotopoulos and Reid6 are usually applied to the phase equilibria of systems that contain a supercritical component.14,15 As the first step that extends the AMH EOS to the calculations of high-pressure vapor-liquid equilibria of systems that contain a supercritical component, the mixing rules of the AMH EOS are revised with a method similar to that of Panagiotopoulos and Reid.6 The second temperature-function term F2(T) in AMH EOS is related to the second virial coefficient B by the following equation:
B)b+
K3
QEN ) aEN + bENT
(13)
QNE ) aNE + bNET
(14)
where aEN and bEN are coefficients of eq 13, and aNE and bNE are coefficients of eq 14. The values of the coefficients for the systems studied in the present work are given in Table 2. For isothermal vapor-liquid equilibria, the following equation is chosen as an objective function: N
OF )
∑ i)1
(|
1-
pcal i pexp i
| |
exp + ycal i - yi
)
|
(15)
where N is the total number of data points used in the correlation, p is the phase equilibrium pressure, and y repre-
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 993 Table 3. Calculation Results of the AMH Equation of State (AMH EOS) to High-Pressure Vapor-Liquid Equilibria for Binary Systems with the Old Mixing Rules and the Revised Mixing Rules Old Mixing Rules
Revised Mixing Rules
T (K)
Dev(P) (%)
∆y
Dev(P) (%)
∆y
data points
reference
210 230 250 270 207 210 230 250 270 333.1 353.1 368.0 422.6 465.4 500.0 473.15 483.15 493.15 503.15 483.15 508.15 523.15 468.15 483.15 498.15 513.15 333.1 333.1 353.1 353.1 370.1 370.1 313.1 333.1 353.1 353.1 483.15 493.15 503.15 483.15 498.15 508.15 523.15 468.15 483.15 498.15 513.15 333.1 353.1 368.1 328.1 348.1 368.1 468.15 483.15 498.15 513.15 468.15 483.15 498.15 513.15
2.58 1.86 0.94 2.16 1.12 1.32 1.23 1.12 0.84 2.32 2.27 1.90 0.54 1.22 1.57 0.95 1.20 1.16 0.66 1.44 1.08 1.70 2.60 1.80 2.29 1.40 3.95 3.62 2.75 2.00 4.24 3.80 3.33 2.04 3.38 3.70 1.52 0.83 0.36 2.04 1.32 1.11 0.81 2.70 3.02 2.85 2.52 1.48 1.09 1.30 2.48 1.66 1.32 1.31 1.60 2.24 0.83 2.42 1.94 1.49 1.91
0.0081 0.0130 0.0040 0.0154 0.0139 0.0126 0.0073 0.0084 0.0052 0.0041 0.0079 0.0141 0.0055 0.0046 0.0104 0.0096 0.0139 0.0086 0.0033 0.0223 0.0187 0.0197 0.0184 0.0152 0.0155 0.0053 0.0075
0.0065 0.0114 0.0049 0.0154 0.0065 0.0047 0.0038 0.0045 0.0048 0.0036 0.0079 0.0140 0.0047 0.0047 0.0105 0.0093 0.0141 0.0091 0.0033 0.0230 0.0183 0.0181 0.0179 0.0126 0.0114 0.0053 0.0070
0.0093 0.0089 0.0028 0.0246 0.0101 0.0158 0.0117 0.0339 0.0188 0.0339 0.0320 0.0006 0.0008 0.0007 0.0009 0.0015 0.0017 0.0211 0.0199 0.0206 0.0051 0.0490 0.0454 0.0259 0.0162
1.90 1.08 0.90 2.16 0.54 0.64 0.73 0.65 0.56 0.77 1.10 1.00 0.37 1.12 1.26 0.85 0.93 1.17 0.65 1.27 1.12 1.57 0.84 0.92 2.04 1.66 2.49 2.15 2.17 2.32 3.88 3.81 2.19 1.79 2.35 2.27 1.39 0.60 0.30 1.84 1.20 0.71 0.89 1.27 3.30 2.26 1.33 0.66 0.82 1.19 1.46 1.02 0.93 1.21 1.01 1.87 1.11 0.91 0.75 1.46 1.68
14 18 7 15 10 17 16 20 6 16 16 11 11 15 6 14 13 12 7 12 13 6 17 14 11 8 6 9 4 8 8 9 6 8 3 5 16 20 10 13 13 9 7 9 9 9 7 12 12 9 12 12 12 14 12 10 7 16 14 10 9
16 16 16 16 16 16 16 16 16 17 17 17 18 18 18 19 19 19 19 20 20 20 21 21 21 21 22 22 22 22 22 22 22 22 22 22 23 23 23 24 24 24 24 25 25 25 25 26 26 26 26 26 26 25 25 25 25 21 21 21 21
average (all systems)
1.72
0.0134
1.22
0.0125
average (systems containing alcohol)
1.76
0.0141
1.25
0.0138
system CH4 + C2H6 CH4 + C2H6 CH4 + C2H6 CH4 + C2H6 C2H6 + CO2 C2H6 + CO2 C2H6 + CO2 C2H6 + CO2 C2H6 + CO2 ethanol + C3H6 ethanol + C3H6 ethanol + C3H6 ethanol + n-C5H12 ethanol + n-C5H12 ethanol + n-C5H12 ethanol + n-C6H14 ethanol + n-C6H14 ethanol + n-C6H14 ethanol + n-C6H14 ethanol + n-C7H16 ethanol + n-C7H16 ethanol + n-C7H16 1-propanol + C5H12 1-propanol + C5H12 1-propanol + C5H12 1-propanol + C5H12 2-propanol + C3H6 2-propanol + C3H6 2-propanol + C3H6 2-propanol + C3H6 2-propanol + C3H6 2-propanol + C3H6 2-propanol + n-C3H8 2-propanol + n-C3H8 2-propanol + n-C3H8 2-propanol + n-C3H8 2-propanol + n-C6H14 2-propanol + n-C6H14 2-propanol + n-C6H14 2-propanol + n-C7H16 2-propanol + n-C7H16 2-propanol + n-C7H16 2-propanol + n-C7H16 1-butanol + C5H12 1-butanol + C5H12 1-butanol + C5H12 1-butanol + C5H12 2-butanol + C3H6 2-butanol + C3H6 2-butanol + C3H6 2-butanol + n-C3H8 2-butanol + n-C3H8 2-butanol + n-C3H8 2-butanol + n-C5H12 2-butanol + n-C5H12 2-butanol + n-C5H12 2-butanol + n-C5H12 2-methyl-1-propanol + C5H12 2-methyl-1-propanol + C5H12 2-methyl-1-propanol + C5H12 2-methyl-1-propanol + C5H12
sents the mole fraction of the vapor phase. The superscripts “cal” and “exp” denote calculated and experimental values, respectively. For some binary systems, the high-pressure vapor-liquid equilibria can be represented quite well by the AMH EOS with
0.0024 0.0051 0.0053 0.0164
0.0017 0.0053 0.0049 0.0184 0.0093 0.0093 0.0031 0.0250 0.0113 0.0158 0.0090 0.0338 0.0249 0.0338 0.0352 0.0004 0.0007 0.0008 0.0005 0.0007 0.0009 0.0184 0.0175 0.0196 0.0046 0.0498 0.0466 0.0278 0.0168
the old mixing rules in which only one binary interaction parameter is required; the correlation results are summarized in Table 3. However, for the other binary systems studied, revised mixing rules in which two binary interaction parameters are required should be used; the correlation results are sum-
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Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007
Table 4. Calculation Results of the AMH Equation of State (AMH EOS) to High-Pressure Vapor-Liquid Equilibria for Alcohol-Containing Binary Systems with the Revised Mixing Rules system
T (K)
CH4 + CO2 CH4 + CO2 CH4 + CO2 CH4 + CO2 CH4 + CO2 CH4 + C3H8 CH4 + C3H8 methanol + CO2 methanol + CO2 methanol + C3H6 methanol + n-C3H8 methanol + n-C3H8 methanol + n-C3H8 ethanol + CO2 ethanol + CH4 ethanol + CH4 ethanol + R-22 ethanol + R-22 ethanol + R-22 ethanol + n-C2H6 ethanol + n-C2H6 ethanol + n-C3H8 ethanol + n-C3H8 ethanol + n-C3H8 ethanol + n-C3H8 ethanol + n-C3H8 ethanol + n-C3H8 C3H8 + CO2 C3H8 + CO2 1-propanol + CO2 1-propanol + CO2 1-propanol + CH4 1-propanol + CH4 1-propanol + C2H6 1-propanol + C2H6 2-propanol + CO2 2-propanol + CO2 2-propanol + CO2 2-propanol + CO2 2-propanol + CO2 2-propanol + C2H6 2-propanol + C2H6 1-butanol + CO2 1-butanol + CO2 1-butanol + CO2 1-butanol + CO2 1-butanol + CO2 1-butanol + CO2 1-butanol + CO2 1-butanol + CO2 1-butanol + CO2 1-butanol + CO2 1-butanol + CO2 2-butanol + CO2 2-butanol + CO2 2-butanol + CO2 2-butanol + CO2 2-methyl-1propanol + CO2 2-methyl-1-propanol + CO2 2-methyl-1-propanol + CO2 2-methyl-2propanol + CO2 2-methyl-2-propanol + CO2 2-methyl-2propanol + CO2 1-pentanol + CO2 1-pentanol + CO2 1-pentanol + CO2 1-octanol + CO2 1-octanol + CO2 1-decanol + CO2 1-decanol + CO2 1-decanol + CO2
∆y
230 230 250 270 270 230 270 313.15 313.20 298.15 313.55 327.95 343.21 313.2 313.4 333.4 343.25 361.45 382.50 313.40 333.40 325.1 350.1 375.1 313.58 333.99 349.78 230 270 313.4 333.4 313.4 333.4 313.4 333.4 293.25 298.15 308.15 316.65 323.15 308.15 313.15 314.8 325.3 337.2 313.2 293.15 303.15 313.15 324.15 333.15 343.15 353.15 313.2 331.9 341.6 351.3 331.9
Dev(P) (%) 1.81 1.64 1.37 1.67 1.64 1.33 1.62 1.77 3.87 0.80 1.49 1.36 1.36 4.14 0.75 2.09 2.18 2.16 1.41 2.47 2.99 1.02 1.66 1.99 1.64 0.77 2.11 2.55 1.71 4.90 4.16 2.93 0.57 1.62 1.11 1.46 1.51 1.18 1.57 1.31 1.68 2.10 0.58 2.09 1.78 2.37 2.73 3.34 2.11 4.40 3.54 2.86 3.85 2.67 2.38 1.76 0.84 1.58
data points 20 23 19 13 16 32 35 11 13 12 11 9 11 11 5 5 9 7 10 5 9 6 6 5 6 7 9 30 27 10 9 5 5 5 6 13 10 5 11 6 9 8 8 9 11 8 7 14 23 15 11 8 8 10 8 8 8 8
reference
0.0058 0.0064 0.0088 0.0094 0.0109 0.0047 0.0109 0.0029 0.0029 0.0037 0.0177 0.0139 0.0085 0.0038 0.0004 0.0008 0.0078 0.0123 0.0239 0.0030 0.0222 0.0042 0.0069 0.0106 0.0011 0.0103 0.0038 0.0125 0.0117 0.0026 0.0084 0.0001 0.0006 0.0006 0.0033 0.0007 0.0010 0.0013 0.0027 0.0037 0.0014 0.0050 0.0016 0.0046 0.0073 0.0037 0.0078 0.0050 0.0010 0.0028 0.0117 0.0022 0.0093 0.0042 0.0023 0.0089 0.0045 0.0119
341.6 351.3 331.9
1.24 0.40 2.40
0.0072 0.0090 0.0043
8 8 6
40 40 40
341.6 351.3
2.34 3.36
0.0135 0.0165
8 9
40 40
314.6 325.9 337.4 403.15 453.15 348.15 403.15 453.15
1.24 1.52 0.67 0.28 0.93 2.56 3.84 2.83
0.0007 0.0069 0.0034 0.0011 0.0030 0.0043 0.0007 0.0031
5 8 9 9 7 8 8 8
41 41 41 14 14 14 14 14
average (all systems)
2.03
0.0061
average (all alcohol + CO2 systems)
2.38
0.0047
16 27 16 16 27 27 27 28 29 30 31 31 31 29 32 32 28 28 28 32 32 33 33 33 31 31 31 27 27 32 32 32 32 32 32 34 34 34 34 34 35 35 36 36 36 37 38 38 38 38 39 39 39 37 40 40 40 40
Figure 1. Comparison of the correlation results with the experimental results for the high-pressure vapor-liquid equilibria of the methanol + CO2 system at 313.2 K. Experimental data are represented by symbols ((4) Elbaccouch et al.28 and (O) Yoon et al.29), whereas the solid curve has been calculated with the AMH EOS.
Figure 2. Comparison of the correlation results with the experimental results for the high-pressure vapor-liquid equilibria of the ethanol + CO2 system at 313.2 K. Experimental data are represented by symbols ((O) Yoon et al.29), whereas the solid curve has been calculated with the AMH EOS.
marized in Table 4. Deviations of the calculated vapor-liquid equilibria are defined as follows:
Dev(P) (%) )
∆y )
1
1
N
∑ |1 -
N i)1
pcal i pexp i
| × 100
(16)
N
∑ |ycali - yiexp|
N i)1
(17)
The results in Table 3 indicate that, for the old mixing rules with one binary interaction parameter, the average deviations of pressure and vapor-phase composition for the 14 binary alcohol + hydrocarbon systems are 1.76% and 0.0141, respec-
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 995
differ greatly, i.e., the nonsymmetry of this type of system is quite large. Therefore, use of the inequality Qij * Qji may be helpful. 2. Vapor-Liquid Equilibria for Ternary Systems. To verify the performance of the AMH EOS for high-pressure systems in the critical region or the near-critical region, ternary vapor-liquid equilibria have been predicted from parameters adjusted to the constituent binary system data. For systems that contain two alcohols E and N, and one inert component M, the self-association of the two alcohols and the cross-association between alcohol E and alcohol N should be considered; for detailed information, refer to our previous paper.9 For this type of system, the approximate analytical solution of nT/n0 is
nT ) n0 1 + Figure 3. Comparison of the correlation results with the experimental results for the high-pressure vapor-liquid equilibria of the 2-propanol + CO2 system at different temperatures. Experimental data from Secuianu et al.34 are represented by symbols ((0) 293.25 K, (O) 298.15 K, (4) 308.15 K, (3) 316.65 K, and (]) 323.15 K), whereas the solid curve has been calculated with the AMH EOS.
tively, and the corresponding average deviations for all of the 16 studied binary systems are 1.72% and 0.0134, respectively. Although the deviations of some systems in Table 3 seem to be somewhat large (for example, those for some alcohol + C5H12 systems), the correlation results are comparable to the correlation results by other EOSs given in the literature,21,25 such as the SAFT EOS with one binary interaction parameter, or the PRSV EOS with the Wong-Sandler mixing rules, in which three binary interaction parameters are required. For those binary systems in Table 3, if the revised mixing rules are used, better results could be obtained. The correlation results using the revised mixing rules are also summarized in Table 3. The results in Table 4 illustrate that the average deviations of pressure and vapor-phase composition for 11 binary alcohol + CO2 systems are 2.38% and 0.0047, respectively, and the corresponding average deviations for all of the 20 studied binary systems are 2.03% and 0.0061, respectively. Some of the correlation results are compared with the experimental data in Figures 1-3. The results illustrate that the correlation model is in agreement with the experimental data. For those binary systems in Table 4, two binary interaction parameters are required. The reason may be that the diversity between the critical temperatures of the two constituent components is quite large, and it corresponds to a large diversity between the properties (for example, pressure) of the two components. In reference to the mixture with an equimolar ratio, the properties of the systems at different composition ranges
2xE
x1 + 4RTKE(xE + xN)/(V - b)
+
2xN
1 + x1 + 4RTKN(xE + xN)/(V - b)
+ xM (18)
where KE and KN are self-association equilibrium constants of alcohol E and alcohol N, respectively. Equation 18, coupled with eq 1, was used for the methanol + ethanol + CO2 and 2-propanol + 2-butanol + C3H8 systems. For nonassociating systems (e.g., CH4 + C2H6 + CO2 and CH4 + C3H8 + CO2), nT/n0 ) 1. The prediction results are listed in Table 5. For the binary methanol E + ethanol N system, the binary interaction parameters are QEN ) QNE ) 3.037 × 10-3, as cited from our previous paper;8 for the binary 2-propanol E + 2-butanol N system, for simplicity, let QEN ) QNE ) 0 in this work. The results in Table 5 indicate that the prediction for the highpressure vapor-liquid equilibria of the four ternary systems is in agreement with the experimental data. Conclusions In this work, the associating Martin-Hou equation of state (AMH EOS) has been extended to the high-pressure vaporliquid equilibria calculation of alcohol-containing binary systems in the critical region or the near-critical region. For this purpose, the old mixing rules of the AMH EOS used in our previous study have been revised. The calculation results indicate that binary systems such as CH4 + C2H6, C2H6 + CO2, and alcohol + pentane (or + n-hexane, or + n-heptane), or parts of the alcohol + propylene (or + propane) systems, can be represented satisfactorily by the AMH EOS with the old mixing rules. However, for binary systems such as CH4 + CO2, CH4 + C3H8, C3H8 + CO2, alcohol + CO2 (or + methane, or + ethane), or the other alcohol + propylene (or + propane) systems, the revised mixing rules should be used. High-pressure vapor-liquid equilibria of four ternary systems are predicted from the binary data satisfactorily.
Table 5. Prediction Results of Ternary Isothermal High-Pressure Vapor-Liquid Equilibria from Binary Data system
T (K)
P (MPa)
Dev(P) (%)
∆y1
∆y2
∆y3
data points
reference
methanol (1) + ethanol (2) + CO2 (3) 2-propanol (1) + 2-butanol (2) + C3H8 (3) 2-propanol (1) + 2-butanol (2) + C3H8 (3) 2-propanol (1) + 2-butanol (2) + C3H8 (3) CH4 (1) + C2H6 (2) + CO2 (3) CH4 (1) + C3H8 (2) + CO2 (3) CH4 (1) + C3H8 (2) + CO2 (3)
313.2 328.1 348.1 368.1 230 230 270
2.0-8.0 1.5-1.8 2.2-2.7 3.0-3.7 1.15-6.59 0.8-7.0 2.8-8.0
6.48 1.51 1.03 0.76 1.52 2.48 1.18
0.0010 0.0001 0.0002 0.0003 0.0079 0.0087 0.0070
0.0001 0.0010 0.0012 0.0015 0.0056 0.0028 0.0073
0.0011 0.0011 0.0014 0.0016 0.0056 0.0074 0.0039
16 3 3 3 98 33 38
29 26 26 26 16 27 27
996
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007
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ReceiVed for reView July 10, 2006 ReVised manuscript receiVed November 9, 2006 Accepted November 20, 2006 IE060884G
