Ind. Eng. Chem. Res. 2008, 47, 5723–5733
5723
Cubic Two-State Equation of State for Associating Fluids Milton Medeiros* and Pablo Te´llez-Arredondo Facultad de Quı´mica, Departamento de Fisicoquı´mica, UniVersidad Nacional Auto´noma de Me´xico, Me´xico DF 04510, Me´xico
The Kontogeorgis’ approach (CPA; Ind. Eng. Chem. Res. 1996, 35, 4310) was employed for the extension of the Soave-Redlich-Kwong (SRK) equation to associating systems. The two-state association model (TSAM) of Cerdeirin˜a et al. (J. Chem. Phys. 2004, 120, 6648) was modified and employed for describing the association contribution. The modification of TSAM consisted of developing an expression for the association contribution for the Helmholtz free energy as an explicit function of temperature and density, on the basis of the two-state approach. Similar to the CPA equation, the resulting cubic two-state (CTS) equation of state (EoS) has five parameters, three of them related to the nonspecific (physical) contribution and the other two related to the association. The CTS EoS is polynomial in volume and preserves the shape of a cubic equation in the region of positive volumes greater than the covolume, having one or three roots in this zone. For pure substances the equation is quartic, and it has analytical solution. The equation was fit to experimental vapor pressure and liquid density data of water, alcohols, and phenols, with very low deviations. The model was also employed to predict second virial coefficients of pure compounds, and phase equilibria of alcohol-alkane mixtures, using conventional mixing rules and one binary parameter, with very good agreement with experimental data. Introduction One of the most important tools for chemical processes and engineering calculations is a reliable equation of state (EoS) able to describe the thermodynamic behavior of a specific system over a large range of temperature, pressure, and composition conditions. The most commonly employed EoS’s are the cubic ones, specially the Soave-Redlich-Kwong (SRK) and the Peng-Robinson (PR) EoS’s, derived from van der Waals’ theory. Such equations have been used extensively in simulation of chemical processes. The mathematical simplicity of those EoS turn them into a very attractive choice for massive calculations required in many applications such as real time and oil field simulations, for example. However, such equations’ main drawback lies on the fact that they normally require nontrivial and many parameter mixing rules when dealing with systems presenting highly directional specific interactions (association) such as hydrogen bonding. To overcome this constraint in associating systems, many explicit association term EoS’s have been proposed, based either on chemical theory of association, on lattice theory or on statistical perturbation models. Two excellent reviews of those approaches are presented in refs 1 and 2. A representative example of the coupled physical-association approach is the cubic plus association EoS (CPA EoS), proposed by Kontogeorgis et al.3 This EoS combines the simplicity of SRK EoS with an explicit association term coming from Wertheim firstorder perturbation theory, common to all SAFT1 family EoS. In the past decade, a series of papers were presented on phaseequilibria calculations with CPA,3–22 including binary and multicomponent vapor-liquid and liquid-liquid equilibria of single and cross-associating systems. The resulting mathematical expression for molar volume, however, is not polynomial and requires iterative procedures for root finding. Clearly, the description of the physical part with a cubic EoS improves the calculation speed of SAFT-like EoS, but the usage of CPA is * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: (+5255) 5622-3528. Fax: (+5255) 5622-3521.
still not competitive with pure cubic equations. Although this is not an impeditive drawback, it makes CPA less attractive for process simulation, especially when a large number of calculations is involved. Recently, a new association scheme, the so-called two-state association model (TSAM), was presented.23–26 This approach to association produces a very simple expression for the association NpT partition function. The TSAM states that a molecule can reside in two energy levels, associated or monomeric. Within the mean field approximation, the association contribution to the isothermal-isobaric partition function was isolated and, by means of thermodynamic relationships, the association contribution to some liquid properties was presented: constant-pressure heat capacity,24 vapor pressure,25 thermal expansion coefficient,26 and isothermal compressibility.26 The agreement with experimental data was good, and the two adjustable parameters (association enthalpy ∆has. and entropy ∆sas.) followed the expected trend as a function of the hydrogen bond type (OH-O, SH-S, NH-N) and molecular size and structure. Particularly, the calculated association enthalpy agreed with the ones previously reported for hydrogen bonding3 and with quantum-chemistry results.24,26 Thus, it seems that TSAM is able to describe the effect of association over liquid properties, making it adequate for coupling with an EoS that could represent the physical (nonspecific) contribution to total pressure. However, as originally stated, the TSAM does not take into account the effect of pressure in the association entropy. Although the first-order expansion of r [r ) ln(∆sas./ R), where R is the gas constant] in terms of temperature and pressure is presented (see eq 9 in ref 26), no theoretical clue was given to determine the arising coefficients ∆Cp ) (∂∆has./ ∂T)p and (∂∆sas./∂p)T. Then, unless additional adjustable parameters are included in the TSAM equations, the model cannot express the effects of changing pressure or volume on the association entropy. In this work, the core ideas of TSAM were employed to develop an expression for the fundamental equation for Helmholtz free energy of association, where the dependence with
10.1021/ie071397j CCC: $40.75 2008 American Chemical Society Published on Web 07/02/2008
5724 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 Table 1. Set 5P of CTS Pure Component Parametersa
water methanol ethanol 1-propanol 2-propanol 1-butanol 2-methyl-1-propanol 2-methyl-2-propanol 1-pentanol 2-pentanol 3-pentanol 2-methyl-1-butanol 2-methyl-2-butanol 3-methyl-1-butanol 3-methyl-2-butanol 2,2-dimethyl-1-propanol cyclopentanol 1-hexanol cyclohexanol phenol 2-methylphenol 3-methylphenol 4-methylphenol 1-heptanol 1-octanol 1-nonanol 1-decanol 1-undecanol 1-dodecanol 1-hexadecanol
a0, Pa · m6 · mol-2
b × 106, m3/mol
c1
–E11/R, K
V11 × 106, m3/mol
Tr(min), K
AAD ps, %
AAD Fs, %
0.3027 0.5105 0.8409 1.257 1.218 1.745 1.703 1.495 2.280 2.222 2.254 2.232 2.170 2.111 2.099 2.189 1.992 2.852 2.546 1.905 2.312 2.573 2.272 3.493 4.142 4.836 5.414 6.291 7.047 10.02
14.70 31.78 47.37 63.19 64.30 79.77 79.62 80.40 96.64 95.34 94.63 94.03 94.17 94.32 94.09 95.24 81.35 112.8 95.37 78.24 93.04 95.96 92.61 132.9 151.4 170.9 182.3 209.0 227.8 295.5
0.5628 0.5137 0.6332 0.7775 0.9106 0.8815 0.8175 0.9766 0.8600 0.7557 0.7476 0.7072 0.6273 0.6894 0.7442 0.9426 0.6378 0.9381 0.6935 0.6597 0.6953 0.7782 0.5860 0.9720 1.031 1.113 1.173 1.250 1.299 1.451
2062 2405 2493 2396 2168 2333 2434 2239 2520 2645 2616 2764 2726 2853 2517 2257 2694 2514 2709 2445 2291 2188 2901 2608 2664 2714 2972 3117 3463 4463
1.422 0.6958 0.5030 0.5615 0.6672 0.5944 0.4272 0.3551 0.4454 0.2211 0.1596 0.2364 0.1197 0.3295 0.2964 0.2958 0.3647 0.3886 0.2915 2.249 3.354 2.988 1.637 0.3772 0.3388 0.2717 0.1042 0.07535 0.02989 0.002630
0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.60 0.55 0.55 0.55 0.55 0.56 0.55 0.55 0.55 0.57 0.55 0.55 0.56 0.57 0.61 0.58 0.55 0.55 0.55 0.55 0.55 0.55 0.55
0.24 0.29 0.22 0.08 0.05 0.08 0.10 0.02 0.18 0.08 0.09 0.21 0.07 0.30 0.27 0.18 0.14 0.09 0.14 0.45 0.38 0.09 0.52 0.15 0.07 0.06 0.11 0.13 0.17 0.23
0.52 0.11 0.11 0.13 0.18 0.25 0.27 0.23 0.46 0.37 0.34 0.41 0.37 0.46 0.57 0.48 0.49 0.35 0.48 0.41 0.32 0.18 0.31 0.49 0.47 0.47 0.44 0.53 0.67 1.10
a All five parameters were left adjustable. The maximum Tr ) 0.9 for all substances. AAD’s are the average absolute deviation of saturation pressure (ps) and of saturated liquid density (Fs).
Table 2. Set 4P of CTS Pure Component Parametersa
water methanol ethanol 1-propanol 2-propanol 1-butanol 2-methyl-1-propanol 2-methyl-2-propanol 1-pentanol 2-pentanol 3-pentanol 2-methyl-1-butanol 2-methyl-2-butanol 3-methyl-1-butanol 3-methyl-2-butanol 2,2-dimethyl-1-propanol cyclopentanol 1-hexanol cyclohexanol phenol 2-methylphenol 3-methylphenol 4-methylphenol 1-heptanol 1-octanol 1-nonanol 1-decanol 1-undecanol 1-dodecanol 1-hexadecanol
a0, Pa · m6 · mol-2
b × 106, m3/mol
c1
–E11/R, K
V11 × 106, m3/mol
Tr(min), K
AAD ps, %
AAD Fs, %
0.2839 0.5030 0.8465 1.267 1.241 1.748 1.706 1.513 2.279 2.221 2.253 2.229 2.165 2.101 2.099 2.178 1.996 2.848 2.546 1.895 2.276 2.481 2.296 3.493 4.130 4.800 5.348 6.218 6.967 9.968
14.24 31.39 47.25 62.95 63.82 79.41 79.40 79.77 96.52 95.40 94.65 94.26 94.34 94.72 93.95 94.76 81.49 112.7 95.47 77.92 92.25 94.89 93.36 132.9 151.4 170.9 182.4 209.2 228.1 295.8
0.3058 0.3817 0.5610 0.6619 0.6590 0.7471 0.7298 0.7298 0.8198 0.7810 0.7582 0.7940 0.7137 0.8190 0.6965 0.7301 0.6880 0.8902 0.7570 0.5940 0.5699 0.6263 0.7148 0.9770 1.054 1.106 1.130 1.198 1.240 1.410
2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600 2600
0.6682 0.4762 0.3956 0.3706 0.2626 0.3804 0.3195 0.2025 0.4002 0.2344 0.1618 0.2975 0.1359 0.5109 0.2644 0.2218 0.4021 0.3693 0.3071 1.829 2.269 2.031 2.443 0.3776 0.3730 0.4200 0.4341 0.4939 0.5197 0.4564
0.55 0.55 0.55 0.55 0.55 0.55 0.55 0.60 0.55 0.55 0.55 0.55 0.56 0.55 0.55 0.55 0.57 0.55 0.55 0.56 0.57 0.61 0.58 0.55 0.55 0.55 0.55 0.55 0.55 0.55
0.52 0.44 0.30 0.24 0.42 0.30 0.27 0.31 0.26 0.05 0.11 0.07 0.23 0.10 0.38 0.75 0.08 0.22 0.19 0.51 0.48 0.19 0.44 0.13 0.18 0.26 0.40 0.38 0.44 0.42
0.59 0.15 0.04 0.16 0.38 0.28 0.25 0.41 0.45 0.39 0.34 0.51 0.44 0.59 0.54 0.42 0.51 0.33 0.52 0.38 0.28 0.24 0.45 0.49 0.49 0.49 0.49 0.58 0.74 1.14
a Four parameters were left adjustable. The association energies (E11/R) were set to 2600 K. The maximum Tr ) 0.9 for all substances. AAD’s are the average absolute deviation of saturation pressure (ps) and of saturated liquid density (Fs).
temperature and volume are explicit. Therefore, following Kontogeorgis’ approach,3 the resulting association EoS was coupled with the SRK EoS, giving rise to the CTS EoS. If association is absent, CTS will reduce to SRK EoS. As it will
be shown in following sections, the isotherms of CTS are polynomials in volume. For pure autoassociating compounds and their binary mixtures with nonassociating substances (inerts) this equation is fourth degree, with a meaningless negative root,
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5725
First, assuming that the Helmholtz free energy (A) of an associating system can be split as a sum of three contributions, namely, ideal (id), nonspecific or physical (ns), and association (as.): (1) A(N,V,T) ) Aid + Ans + Aas. To easily isolate the association contribution, we assume that Ans is negligible at low densities, and the only significant contributions are Aid and Aas.. The association contribution will be later coupled with an EOS that includes the nonspecific contribution Ans. For this “associating ideal gas”, the mean field approximation is made: Q(N,V,T) )
(qi)Ni Ni !
∏ i
(2)
A molecule labeled i in a fluid that obeys the ideal gas plus two-state condition can only be found at two energetic states, nonassociated with null intermolecular potential energy and associated with a j type molecule with energy Eij. If the numbers of microstates resulting in association and in monomeric states are ωij and ωi, respectively, then the one-particle configurational partition function qi is given by Figure 1. Nonspecific parameters (b, a0, and c1) of aliphatic noncyclic alcohols from four-parameter estimation (set 4P) as a function of the number of carbon atoms in the chain.
qi ) ωi +
∑ω e
-βEij
(3)
ij
j
where β ) 1/kBT and kB is the Boltzmann constant. Consequently, eq 3 can be rewritten as
[
qi ) Wi 1 +
ωij
∑ W (e
-βEij
]
- 1)
i
j
(4)
where Wi ) ωi + ∑j ωij is the total number of configurational microstates, associated or not, available for an i type particle, i.e., the volume of one particle configurational space. Assuming that Wi can be approximated by the configurational phase space volume of a particle i in an ideal gas state, then qi can be expressed as as. qi ) qid i qi
qas. i )1+
ωij
∑ W (e j
Figure 2. Characteristic volume of association (V11) of pentanol isomers. Lines are a guide for the eye.
and it can be analytically solved. This EoS has two new association parameters that were adjusted to experimental vapor pressures and liquid molar volumes for water, alcohols, and phenols. Finally, to check its predictive power, the equation was employed for the calculation of second virial coefficient and of vapor-liquid equilibria behavior of alcohol-alkane binary mixtures. Conclusions and future work are outlined in the last section. Canonical Two-State Association Model and CTS EoS The original development of TSAM was made in the framework of isothermal-isobaric (NpT) ensemble. The main ideas of NpT-TSAM and the corresponding equations for thermoproperties calculations are described elsewhere.24–26 In this work, the same concepts are applied in the context of canonical ensemble. The basic idea, as previously mentioned, is that association can be modeled as a result of the molecules transitions between two energetic levels, namely, associated and nonassociated levels.
-βEij
(5) - 1)
(6)
i
Before resuming the development, it should be remarked that the aforementioned assumption is not strictly true. When a molecule becomes associated, some of its internal degrees of freedom (conformational or rotational) are likely to be constrained. Then, the total number of ideal gas configurational microstates is not exactly the sum represented by Wi. The approximation within eq 5 is neglecting this rotational and conformational hindering. Also, it is worthwhile to noting that eq 6 is similar to the NpT-TSAM partition function of association, considering that e-βEij - 1 ≈ e-βEij for strong associations (Eij , 0) and that ωi ≈ Wi. However, NpT-TSAM considers the ratio r ) ωij/ωi as a constant, and the resulting expression is not a function of the volume. In the present formulation, however, the dependence Wi(V) can be estimated for an ideal gas in the classical limit: Wi )
∫
all space
dri dΩi ) ciV
(7)
where ci is a constant related to the integral over rotational and internal degrees of freedom. Similarly for ωij ωij )
∫
association around j
dri dΩi ) cijVij
(8)
5726 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
Figure 3. Second virial coefficients of (a) water, (b) methanol, (c) ethanol, and (d) 1-propanol: continuous lines, five-parameter estimation; dashed lines, four-parameter estimation; diamonds, DIPPR32 fit of experimental data. Table 3. Soave-Redlich-Kwong Parameters for Alkanes
propane butane 2-methylpropane 2,2-dimethylpropane pentane 2-methylbutane hexane 2-methylpentane 2,2-dimethylbutane 2,3-dimethylbutane 3-methylpentane heptane octane 2,2,4-trimethylpentane nonane decane undecane dodecane tridecane pentadecane
a0, Pa · m6 · mol-2
b × 106, m3/mol
c1
0.9088 1.347 1.301 1.669 1.827 1.770 2.350 2.282 2.197 2.245 2.281 2.912 3.517 3.224 4.157 4.826 5.483 6.260 7.128 7.983
57.97 74.62 75.49 91.41 91.16 90.48 108.0 107.4 106.0 105.5 106.3 125.3 143.0 139.3 161.1 179.3 195.8 216.7 241.0 247.5
0.6554 0.7236 0.7069 0.7389 0.7894 0.7610 0.8537 0.8302 0.7709 0.7884 0.8255 0.9183 0.9774 0.8706 1.035 1.089 1.136 1.189 1.249 1.274
where Vij is a characteristic volume of the pair interaction i-j. If Vij is considered to be proportional to the number of j sites, then ωij cij Vij NjVij ) ) Wi ci V V
provided i has the correct orientation. Hence, similar to the TSAM-NpT r parameter, Vij puts a figure on the highly specific nature of association: the greater Vij is, the more microstates (positions, orientations, and conformations) will be compatible with association. Equation 6, then, becomes qas. i )1+
1 V
∑ N V (e
-βEij
(10)
j
The one-particle association partition function is now NVTdependent. Hence Qas.(N, V, T) )
∏ (q
as. Ni i ) )
i
∏ i
[
1+
1 V
∑ N V (e
-βEij
j ij
]
- 1)
j
Ni
(11) and the Helmholtz energy of association will be given by Aas.(N, V, T) )kBT
[
]
∑ N ln 1 + V1 ∑ N V f (T) i
i
j ij ij
j
(12)
wherefij(T) ) e-βEij - 1. As eq 11 indicates, the model has two parameters for each association type, the association energy (Eij) and the association characteristic volume (Vij). The resulting expressions for the association contribution to the pressure and to the residual chemical potential, in molar base are
(9)
where Nj is the number of j-type particles. The parameter Vijsthe association characteristic volumeshas two implicit contributions: (i) cij/ci, which quantifies the fraction of the total orientations and internal conformations compatible with association (the greater this ratio is, the more orientations and conformations will result in association); (ii) the volume around the association site j in which the particle i will be bonded,
- 1)
j ij
pas.(x, V, T) ) -RT
∑
∑ x V f (T) j ij ij
xi
i
[
µas. k (x, V, T) ) ln RT V+
[
V V+
V
∑xV
j kj fkj
j
j
]
∑ x V f (T)]
(13)
j ij ij
j
-
∑ i
xiVik fik V+
∑xV f
j ij ij
j
(14)
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5727
where x ) (x1, x2,...) is the vector of molar fractions. Finally, assuming that the nonspecific contributions can be described by SRK EoS, the expression of CTS EoS is
p(x, V, T) )
a(x, T) RT + pas.(x, V, T) V - b(x) V[V + b(x)]
(15)
Table 4. Binary Interaction Parameters for CTS Vapor-Liquid Equilibria Calculations of Alcohol-Alkane Mixtures k12 a
methanol methanol methanol ethanol ethanol ethanol ethanol ethanol ethanol ethanol ethanol ethanol ethanol 1-propanol 1-propanol 1-propanol 1-propanol 1-propanol 1-propanol 1-propanol 1-propanol 2-propanol 2-propanol 2-propanol 2-propanol 2-propanol 2-propanol 1-butanol 1-butanol 1-butanol 1-butanol 1-butanol 1-butanol 1-butanol 1-butanol 1-butanol 1-butanol 2-methyl-1-propanol 2-methyl-1-propanol 2-methyl-1-propanol 2-methyl-2-propanol 2-methyl-2-propanol 2-methyl-2-propanol 1-pentanol 1-pentanol 1-pentanol 1-pentanol 1-pentanol 1-pentanol 1-pentanol 1-pentanol 1-pentanol 1-pentanol 2-pentanol 3-methyl-1-butanol 3-methyl-1-butanol 1-hexanol 1-hexanol 1-octanol 1-octanol 1-octanol 1-octanol 1-octanol 1-decanol 1-dodecanol 1-dodecanol 1-dodecanol a
propane hexane 2-methylbutane propane butane 2-methylpropane pentane hexane 2-methyl pentane heptane octane 2,2,4-trimethylpentane nonane butane 2-methylpropane hexane heptane octane 2,2,4-trimethylpentane nonane undecane hexane heptane 2,2,4-trimethylpentane octane nonane decane pentane hexane 2-methylpentane 3-methylpentane 2,2-dimethylbutane 2,3-dimethyl butane heptane octane nonane decane hexane heptane octane 2-methylbutane octane 2,2,4-trimethylpentane pentane hexane 2,2-dimethylbutane 2,3-dimethylbutane 2-methylpentane 3-methylpentane heptane octane 2,2,4-trimethylpentane decane heptane hexane heptane hexane 2,2,4-trimethylpentane hexane heptane decane undecane dodecane hexane hexane tridecane pentadecane
AAD p, % b
FIT
CR
0.0004 -0.0252 -0.0418 0.0089 0.0048 -0.0104 0.0013 -0.0057 -0.0056 -0.0101 -0.0138 -0.0372 -0.0218 0.0129 0.0018 -0.0009 -0.0056 -0.0065 -0.0190 -0.0064 -0.0165 0.0105 0.0031 -0.0129 0.0045 -0.0027 -0.0144 0.0090 0.0052 0.0002 0.0016 -0.0066 -0.0011 0.0027 -0.0012 0.0018 -0.0012 0.0054 0.0055 0.0091 0.0050 0.0072 -0.0120 0.0132 0.0097 -0.0016 0.0024 0.0034 0.0041 0.0076 0.0030 -0.0089 -0.0010 0.0087 0.0095 0.0090 0.0095 -0.0076 0.0098 0.0060 0.0194 0.0252 0.0215 0.0075 0.0108 0.0192 0.0269
-0.0300 -0.0533 -0.0593 -0.0170 -0.0142 -0.0298 -0.0187 -0.0274 -0.0304 -0.0327 -0.0398 -0.0552 -0.0514 -0.0069 -0.0158 -0.0194 -0.0224 -0.0289 -0.0365 -0.0303 -0.0446 -0.0070 -0.0120 -0.0271 -0.0146 -0.0239 -0.0393 -0.0068 -0.0115 -0.0159 -0.0108 -0.0187 -0.0138 -0.0158 -0.0196 -0.0218 -0.0233 -0.0116 -0.0095 -0.0086 -0.0042 -0.0048 -0.0215 -0.0009 -0.0060 -0.0122 -0.0086 -0.0106 -0.0065 -0.0089 -0.0146 -0.0215 -0.0201 -0.0051 -0.0028 -0.0060 -0.0051 -0.0195 -0.0024 -0.0043 0.0042 0.0094 0.0048 -0.0014 0.0031 0.0052 0.0119
FIT
CR
2.45 1.90
2.56 2.39
1.53 1.23 1.37 0.99 1.22
1.99 1.36 0.94 0.56 1.54
1.03 2.17 1.14 0.98 1.14 1.56 0.52 1.23 1.68 1.35 1.43 3.77 0.54 1.09 1.11 1.20 1.22 0.62 1.01 1.22 1.37 1.24 1.21 1.40 1.09 1.37 1.11 1.90 0.73
1.11 3.06 1.32 1.24 1.35 1.70 1.01 1.43 2.07 1.55 2.50 4.48 0.71 1.29 1.24 2.08 2.00 1.22 0.63 1.20 0.91 1.07 1.01 0.96 1.68 1.27 1.96 1.34 1.24
2.51 0.97 1.46 1.53 1.29 1.35 1.52 1.36 1.16 1.83 1.36 1.08 0.49 1.97 2.28 2.72 1.58 0.68 1.43 1.98 2.00 1.39 1.47 1.18 0.86 2.45 1.35
2.27 1.41 1.72 1.13 0.94 1.07 1.12 1.03 1.21 1.85 1.72 1.52 0.86 1.91 1.75 2.16 1.35 1.90 1.93 2.52 2.76 1.72 1.23 0.88 0.68 2.87 2.38
FIT
0.05
FIT, alkane parameters adjusted to reproduce saturated pressures and saturated liquid densities (Table 3). corresponding states correlations (eqs 29–31).
∆y, %
AAD T, %
0.06 0.11 0.32
CR
CR
1.59
1.57
0.70 0.21
0.68 0.22
0.70 0.76 1.10 1.03 0.94 0.19 0.21 0.12 0.26
0.92 1.06 1.65 1.58 1.40 0.33 0.23 0.12 0.35
0.94 0.86 0.55 0.57 0.56
1.45 1.24 0.93 0.60 0.62
1.20 0.70 0.46 0.08 0.13
1.52 1.15 0.69 0.16 0.16
0.45 1.91 1.09 0.61 0.80 0.59 1.05
0.74 1.87 1.42 0.90 0.87 0.82 1.57
0.53 0.95
0.82 1.21
0.60
0.59
0.39
0.67
2.17
2.37
0.03
0.03
0.04
0.09 0.14 0.35
0.12
0.17
0.18
0.18
0.26 0.04
0.18 0.07
0.22
0.22
0.08
0.13
0.09
0.14
b
FIT
CR, alkane parameters predicted from
5728 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
Figure 4. Phase envelope of propane (1)-methanol (2) binary mixture at 313.55, 327.95, and 343.21 K: continuous lines, CTS EoS with adjusted alkane parameters; dotted lines, CTS EoS with alkane parameters predicted from corresponding states correlations; dashed lines, CPA EoS with adjusted alkane parameters and k12 from ref 4; diamonds, experimental data.33 See Table 4 for CTS binary parameters k12.
Figure 5. Phase envelope of 1-butanol (1)-decane (2) binary mixture at 358.15, 373.15, and 388.15 K: continuous lines, CTS EoS with adjusted alkane parameters; dotted lines, CTS EoS with alkane parameters predicted from corresponding states correlations; dashed lines, CPA EoS with alkane parameters adjusted and k12 from ref 4; diamonds, experimental data.34 See Table 4 for CTS binary parameters k12.
Equation 14 can be easily shown to be polynomial in molar volume. It can also be readily solved by numerical methods for polynomials root finding, such as with the Laguerre algorithm.27 For nonassociating systems it will obviously be cubic, and its degree will be increased by one for each association type. For
instance, for pure alcohols, eq 15 becomes quartic provided all possible hydrogen bonds are considered equivalent. It is important to note that, when just one type of association may occur, the association term pas. will not introduce spurious roots in the region of positive volumes, at given p, T, and x. In this
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5729
Figure 6. Phase envelope of hexane (1)-methanol (2) binary mixture at 293.15, 313.15, and 333.15 K: continuous lines, CTS EoS with adjusted alkane parameters; dotted lines, CTS EoS with alkane parameters predicted from corresponding states correlations; dashed lines, CPA EoS with adjusted alkane parameters and k12 from ref 4; diamonds: experimental data.34 See Table 4 for CTS binary parameters k12.
Figure 7. Phase envelope of propane (1)-ethanol (2) binary mixture at 313.58, 333.99, and 349.78 K: continuous lines, CTS EoS with adjusted alkane parameters; dotted lines, CTS EoS with alkane parameters predicted from corresponding states correlations; diamonds, experimental data.33 See Table 4 for CTS binary parameters k12.
case, it can be easily shown that eq 15 will always have a negative real root. Therefore, eq 15 will keep the approximate shape of the cubic part and will have only one or three real positive roots. Moreover, CTS has the correct behavior for stability requirements. For many association systems, however, the shape of an isotherm will depend on the association
parameters. Nevertheless, preliminary calculations of crossassociating systems have shown that the CTS isotherms preserve the crucial characteristic of having only one or three positive roots. The remaining real roots are found in the negative volumes region. Further research work should be done in order to confirm and generalize this fact.
5730 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
Figure 8. Phase envelope of hexane (1)-ethanol (2) binary mixture at 101.32 kPa: continuous line, CTS EoS with adjusted alkane parameters; dotted line, CTS EoS with alkane parameters predicted from corresponding states correlations; diamonds, experimental data.34 See Table 4 for CTS binary parameters k12.
Since CTS reduces to the SRK for nonassociating compounds, the dependency on temperature of covolume parameter b and of nonspecific energy parameter a is assumed to be equal to the original SRK EoS; i.e., b does not depend on temperature and, for pure substances a(T) ) a0[1 + c1(1 - √Tr)]2
(16)
where Tr ) T/Tc, Tc is the critical temperature and a0 and c1 are pure compound constants. Therefore, the model has five parameters for pure substances capable of only one association type: (i) a0, b, and c1 for the physical or nonspecific contribution and (ii) Eii and Vii for the association one. In the next section, these parameters will be determined by fitting the model to experimental saturation pressures and liquid volumes of pure alcohols. Parameter Estimation: Water, Alcohols, and Phenols The model was fit to experimental28 vapor pressures and saturation densities of water, alcohols, and phenols. Two approaches for parameter estimation were performed: global five- and four-parameter regression. The two sets of parameters will be referred as 5P and 4P. In the latter, the association energy was considered constant among the alcohols and its value was the typical one reported in the literature (2600 K). The objective function to be minimized was FOBJ(a0, b, c1, V11, E11) )
∑
(
)
pCTS,s - pexptl,s 2 + pexptl,s FCTS,s - Fexptl,s Fexptl,s
∑
(
)
2
(17)
where F is the molar liquid density and the superscript s refers to saturation properties. Before describing the results of the fitting, some details should be outlined about the adjusting
strategy. The objective function expressed in eq 16 is the most commonly used in many works on EoS fitting. It would be desirable that the parameters coming from this fitting could be used in the prediction of the other thermoproperties, such as residual heat capacities, for example. However, as pointed out by Gregorowicz et al.,29 the description of all properties with a single set of parameters is a challenge for any EoS. For instance, the original TSAM parameters were determined by fitting Cp data, and this additional property could have been included in eq 17. Other objective functions can be used, such as the one proposed of Lafitte et al.,30 where the speed of sound and enthalpy of vaporization were included. Here, however, we decided that further properties evaluation other than phase equilibria will be left for future work. Tables 1 and 2 show the parameters obtained with the aforesaid fitting approaches. Both of them show very good agreement with experimental data, as shown by the low average absolute deviations (AAD). The simplex-downhill method27 was employed for minimization. No convergence problems were observed, whenever the initial guess for the parameters was in the range of physical meaning. For a0, b, c1, and E11 the initial guesses were taken from Kontogeorgis’ recommended values3 or from the analogous alkanes. The association energies agree with those previously reported for experimental3 and theoretical24,26 hydrogen bonding energy. They also matched SAFT31 and CPA3 equivalent parameters. In Figure 1, for the global estimation, the physical part parameters a0, b, and c1 of 5P are plotted against the numbers of carbon atoms in the aliphatic alcohols. There is a clear increasing trend of them with molecular size. This is in agreement with the physical meaning of those parameters: b is related to molecular volume, and a0 and c1 point toward the magnitude of the effective intermolecular forces other than association. The dispersion of c1 values among the C3-C5 isomers is due to their different shapes, in accordance with the
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5731
original Soave’s corresponding states correlation for this parameter, in which c1 appears as a function of the accentric factor only. Figure 2 illustrates the effect of sterical hindering on the 5P association characteristic volume of C5-OH isomers. For the hydroxyl group, position 1 of the chain is the most accessible for association and then, the characteristic association volume must be greater. As hindering increases, as it takes place in 2- and 3-pentanol, V11 is expected to decrease. This is the observed behavior of V11. Then, all the adjusted parameters reached physically sound values. Table 1 also shows that the 5P approach produces both Vij’s tending to zero and a fast increasing behavior of Eij’s, as the chain length increases for n-alcohols beyond n-octanol. This fact cannot be physically explained. It seems that Eii and Vij correlation occurs. Then, for these compounds, the set 4P is preferred: the deviations are similar to 5P and the association parameters retain the likely values, similar to the ones corresponding to lower n-alcohols (see Tables 1 and 2). To check the reliability of such pure compound parameters, the CTS-predicted second virial coefficients were contrasted with experimental data.32 For pure substances, the second virial coefficient for CTS EoS is given by -V ( ∂F∂z ) ) b - a(T) RT
B(T) ) lim Ff0
11 f11(T)
T
(18)
where B is the second virial coefficient and z ) pV/RT is the compressibility factor. As mentioned by Kontogeorgis et al.,3 the prediction of second virial coefficients is a very strict performance test for EoS’s. Figure 3 shows the CTS predicted second virial coefficient for water, methanol, ethanol, and propanol, with both sets 5P and 4P, and the corresponding experimental data.32 The agreement is very satisfactory. Unfortunately, this test failed to indicate, in general, which set is the best one: for water, the experimental data are between the predictions with 5P and 4P; for methanol 5P performs better; for ethanol and 1-propanol, 4P parameters produce the most accurate prediction. Alcohol-Alkane Vapor-Liquid Equilibria To check the reliability of CTS EoS for describing the properties of associating mixtures, this equation was employed in the prediction of vapor-liquid equilibria of alcohol-alkane mixtures. In relation to the physical part, the one fluid theory with conventional mixing rules was used to determine the mixture parameters: b(x) )
∑xb
i i
(19)
i
a(x, T) )
∑ x x a (T) i j ij
(20)
i,j
aij(T) ) (1 - kij)(ai(T) aj(T))1⁄2
(21)
where kij is the binary interaction parameter and the pure component ai(T)’s are determined by eq 16. In the case of alkane-alkane and alkane-alcohol interactions, there is no strong association. Then, for the association part, there is no need for an additional mixing rule. The effect of composition on association pressure and on association residual chemical potential can be determined directly by eqs 13 and 14 by making falcohol-alkane ) falkane-alkane ) 0. As previously mentioned, for this class of systems, with single association type, the CTS equation is quartic in compressibility factor:
4
F(z) )
∑dz )0 i
(22)
i
i)0
The coefficients are given by d0 ) -x1βγ(R + x1β)
(23)
d1 ) x1γ[R - β(β + 1)] - Rβ
(24)
d2 ) R + x1γ(x1 - 1) - β(β + 1)
(25)
d3 ) x1γ - 1
(26)
d4 ) 1
(27)
In the previous equations, x1 is the alcohol mole fraction, and R, β, and γ are defined as follows: pV11 f11(T) pa(x, T) pb(x) ; γ) (28) ; β) 2 RT RT (RT) Equation 22 has always one negative real root, since F(0) ) d0 < 0, and lim F(z) ≈ z4>0. zf-∞ For alcohols, set 4P presented in Table 2 was used in phase envelope calculations, whereas for alkanes the physical part parameters were determined in two ways: (i) by fitting the SRK EoS to experimental vapor pressures and saturation densities (set FIT), presented in Table 3 and (ii) by the following corresponding states correlations36 (set CR): R)
R2Tc2 a0 ) 0.42748 pc b ) 0.08664
(29)
RTc pc
(30)
c1 ) 0.480 + 1.574ω - 0.176ω2
(31)
where subscript c refers to a critical property and ω is the Pitzer’s accentric factor. Using the previous equations, the single temperature independent binary parameters kij were determined for 70 alcoholalkane binaries,33–35 by minimizing the following objective function: FOBJ )
∑ m,k
[
exptl pCTS (Tm) k (Tm) - pk
2 yexptl 1,k (Tm)] +
pexptl (Tm) k
∑ m,k
[
]
2
+
∑ [y
CTS 1,k (Tm) -
m,k
exptl TCTS (pm) k (pm) - Tk
Texptl (pm) k
]
2
+
∑ [y
CTS 1,k (pm) -
m,k
2 (32) yexptl 1,k (pm)]
where the p and T are the bubble point pressures and temperatures and y’s are the corresponding bubble composition. The first two terms are the bubble pressure and composition errors along isotherms while the other two are bubble temperature and composition errors along isobars. For some binaries there were no available experimental data for the evaluation of all four terms in eq 32. The results of kij evaluation for the two sets of pure alkane parameters are summarized in Table 4. As expected, the kij’s are very small and their absolute value never exceeds 0.027. The deviations from experimental data were quite satisfactory results. The average AAD’s for bubble pressure are 1.40 and 1.56% for the FIT and CR sets, respectively, with maximum values of 3.8 and 4.5%. Both sets of alkane pure parameters produce similar departures from experimental data. Since no adjustable parameters for the alkanes are required, set CR is preferable for engineering purposes, at least for vapor-liquid equilibrium calculations. However, both sets have
5732 Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008
to be proven for the prediction of liquid-liquid behavior. Voutsas and co-workers5 have shown that CR set performs better in LLE when using CPA EoS and the same might occur for CTS. Figures 4–8 illustrate the CTS performance. Very good agreement with experimental data is obtained. For the sake of comparison, CPA results for some isotherms are also plotted. There is no significant difference between both models. However, CTS curves were generated using a single temperature independent kij, whereas CPA has different kij’s for each isotherm.4 Concluding Remarks An extension of the two-state association model was presented for its use as a pVT equation of state. The resulting association EoS was combined with SRK equation giving rise to the CTS EoS. This equation was successfully employed in calculation of phase equilibria of pure self-association compounds and their mixtures with nonassociating fluids. For these types of systems, the proposed equation is fourth degree in volume with a meaningless real negative root and it has analytical solution. This feature makes CTS suitable for massive VLE and pVT calculations, avoiding iterative procedures and convergence problems. For cross-association systems, CTS becomes a polynomial equation that can be solved by specific numerical methods. In order for CTS to remain quartic, even for many association mixtures, investigations on one fluid theory and on mixing rules are in progress. Moreover, further research is being made in order to employ CTS on the following: (i) thermocalculation of other self-associating compounds (polyols, acids, esthers, amines, and thiols, etc.); (ii) multicomponent crossassociation mixtures; (iii) in liquid-liquid equilibria; (iv) second derivative-related properties (heat capacity and speed of sound, for example) and energetic properties. With these extensions CTS will become a simple and reliable equation for chemical engineering thermodynamic calculations. Nomenclature A ) Helmholtz free energy a ) SRK mixture energy parameter ai ) SRK energy parameter for pure i aij ) energy parameter for the pair i-j a0 ) parameter in the energy term B ) second virial coefficient b ) mixture covolume parameter bi ) covolume parameter for pure i c1 ) parameter in the energy term Cp ) molar heat capacity ci ) constant related to the integral over rotational and internal degrees of freedom for nonassociated molecule i cij ) constant related to the integral over rotational and internal degrees of freedom for associated molecule i on site j di ) coefficients of the quartic equation (eq 22) Eij ) association energy between sites i and j fij ) Meyer function ) e-βEij - 1 FOBJ ) objective function for parameter adjusting h ) enthalpy kB ) Boltzmann’s constant kij ) binary interaction parameter N ) vector of number of molecules Ni ) number of molecules of component i p ) pressure pc ) critical pressure
Q ) canonical configurational partition function qi ) one particle configurational partition function r ) TSAM degeneracy ratio R ) gas constant ri ) position vector of particle i s ) entropy T ) absolute temperature Tc ) critical temperature Tr ) reduced temperature V, V ) volume, molar volume Vij ) association characteristic volume of the pair interaction i-j Vij ) association characteristic volume of the pair interaction i-j per molecule Wi ) total number of configurational microstates available for an i type particle x ) vector of molar fractions xi ) molar fraction (usually in liquid phase) yi ) molar fraction in vapor phase z ) compressibility factor Greek Letters R, β, γ ) dimensionless parameters defined in eq 28 β ) 1/kBT F ) molar density Ωi ) vector of rotational and internal degrees of freedom of particle i ω ) Pitzer’s accentric factor ωi ) number of microstates resulting in a monomeric state of particle i ωij ) number of microstates resulting in association of particle i with site j Subscripts and Superscripts i, j, k,... ) for components i, j, k exptl ) experimental as. ) association id ) ideal ns ) nonspecific s ) saturation AbbreViations AAD ) average absolute deviation CPA ) cubic plus association equation of state CTS ) cubic two-state equation of state EoS ) equation of state LLE ) liquid-liquid equilibria VLE ) vapor-liquid equilibria SAFT ) statistical associating fluid theory SRK ) Soave-Redlich-Kwong equation of state TSAM ) two-state association model
Acknowledgment We thank Dr. Miguel Costas and Dr. Claudio Cerdeirin˜a for useful and stimulating discussions and for their comments about the manuscript. Literature Cited (1) Mu¨ller, E. A.; Gubbins, K. E. Molecular-Based Equations of State for Associating Fluids: A Review of SAFT and Related Approaches. Ind. Eng. Chem. Res. 2001, 40, 2193. (2) Economou, I. G.; Donohue, M. D. Equations of State for Hydrogen Bonding Systems. Fluid Phase Equilib. 1996, 116, 518.
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ReceiVed for reView October 16, 2007 ReVised manuscript receiVed January 2, 2008 Accepted April 24, 2008 IE071397J
