20546
J. Phys. Chem. B 2005, 109, 20546-20553
Prediction of Global Vapor-Liquid Equilibria for Mixtures Containing Polar and Associating Components with Improved Renormalization Group Theory Jianguo Mi,† Yiping Tang,‡ Chongli Zhong,*,† and Yi-Gui Li† Department of Chemical Engineering, The Key Lab of Bioprocess of Beijing, Beijing UniVersity of Chemical Technology, Beijing 100029, China, and Honeywell Process Solutions, 300-250 York St., London, Ontario N6A 6K2, Canada ReceiVed: July 14, 2005; In Final Form: September 9, 2005
Our recently improved renormalization group (RG) theory is further reformulated within the context of density functional theory. To improve the theory for polar and associating fluids, an explicit and complete expression of the theory is derived in which the density fluctuation is expanded up to the third-order term instead of the original second-order term. A new predictive equation of state based on the first-order mean spherical approximation statistical associating fluid theory (FMSA-SAFT) and the newly improved RG theory is proposed for systems containing polar and associating fluids. The calculated results for both pure fluids and mixtures are in good agreement with experimental data both inside and outside the critical region. This work demonstrates that the RG theory incorporated with the solution of FMSA is a promising route for accurately describing the global phase behavior of complex fluids and mixtures.
1. Introduction An accurate prediction of thermodynamic properties of supercritical carbon dioxide, ketones, water, alcohols, and their mixtures is of particular interest in many industrial chemical processes. As an example, extraction of ethanol using the fermentation method with supercritical carbon dioxide is a simple process, which is environmentally benign. Carbon dioxide has no dipole moment because of its molecular symmetry, while n-ketones have a strong dipole moment due to the existence of orientation angles of electrostatic forces, and water and nalcohols tend to form hydrogen-bonded networks for their highly directional attractive interactions. Therefore, it is difficult to deal with the global phase behavior of systems in which the components may differ appreciably in properties. Toward the goal of establishing an accurate equation of state (EOS) for the phase diagram both near and far from the critical region, many efforts have been made in recent years. While significant developments have been achieved in the description or correction of their vapor-liquid equilibria (VLE), a particular challenge in these endeavors is to develop a general model to predict mixture behavior based solely upon the knowledge of the molecular structure and intermolecular forces from pure components. The statistical associating fluid theory (SAFT)1,2 is proved to be one of the most sophisticated, versatile, and successful EOSs to describe the thermodynamic properties of fluid mixtures. Compared with simple cubic EOSs,3-5 SAFT is very advantageous, explicitly in yielding chain bonding and associating terms. Of those SAFT models, the version of SAFT proposed by Huang and Radosz (SAFT-HR)6,7 is the first success for a practical application. Since then, many modifications have been developed such as the Lennard-Jones SAFT (LJ-SAFT),8,9 softSAFT,10,11 perturbed-chain SAFT (PC-SAFT),12-14 SAFT of * To whom correspondence should be addressed. E-mail: zhongcl@ mail.buct.edu.cn;
[email protected]. † Beijing University of Chemical Technology. ‡ Honeywell Process Solutions.
variable range (SAFT-VR),15,16 etc. After taking care of structure information, primarily obtained from computer simulation, they yield much better results than the original SAFT both for pure compounds and for mixtures. However, since the important radial distribution function (RDF) has to be extracted numerically, these EOSs developed for pure component systems require the use of appropriate mixing rules for both the thermodynamic variables and the parameters of the EOS. The mixing parameters regressed from the experimental data have severely degraded the predictive capability of the SAFT model. A more promising SAFT was presented by Tang and Lu,17-21 in which the RDF for both pure LJ fluid and their mixtures is obtained from an analytical solution of the Ornstein-Zernike equation using the first-order mean spherical approximation (FMSA), and the troublesome mixing rules are no longer required in the FMSASAFT. In the critical region, however, the FMSA-SAFT fails to predict PVT properties of fluids just like almost all other SAFT or mean-field models. When approaching the critical point, the density fluctuation is regarded as one of the fundamental parameters to characterize the critical properties of fluids. The local density inhomogeneities have been observed unambiguously in many experiments.22,23 Those SAFT models, however, based on a homogeneous hypothesis, are developed for uniform fluids, where the density is taken to be spatially uniform. The fluctuation distance in the RDF is only several times the molecular diameter, which does not exhibit any nonanalytical macroscopic scaling characteristic. To reproduce the macroscopic behavior associated with the long-range fluctuation in density presented in the critical region of real fluids and their mixtures, the homogeneous theory is valid only after the inhomogeneity is taken fully into account. At present, density functional theory (DFT), in which the free energy of the system is expressed as a functional of the spatially varying density, is one of the most powerful tools to describe inhomogeneous systems, and the renormalization group (RG) theory has been proven to be the best way to convert a fluid from an “inhomogeneous” state to a “homogeneous” state and
10.1021/jp053871+ CCC: $30.25 © 2005 American Chemical Society Published on Web 10/11/2005
Global Vapor-Liquid Equilibria to interpret further the nonclassical exponents around the critical point. On the basis of the fundamental results of the RG theory, a crossover method is developed by Sengers and co-workers24-26 and by Kiselev and co-workers.27,28 This approach treats the critical point as the starting point, and upon moving away from the critical point to the low-density regions, the asymptotic quantities from the Ising model are modified to reflect the transition from scaling behavior to classical behavior. Very recently, a SAFT-VRX equation has been proposed by Kiselev and co-workers29-31 by combining the classing SAFT-VR with the crossover function. The equation was shown to provide an excellent description of the PVT and global phase behavior for both pure fluids and alkane-alkane binary mixtures. However, several crossover parameters are required to fit experimental data for each system. Moreover, the signification of density fluctuation has not been expressed very clearly. Another crossover approach was developed by Gospodinov and Escobedo32 with the histogram reweighting method. It was shown that the histograms provide a good approximation to simulation results. Application to real fluids and mixtures needs to be further investigated. In our previous work,33,34 an improved RG theory based on White’s theory35-38 was developed for chain bonding fluids. The long-range density fluctuation in the critical region was fully considered, and the contribution to Helmholtz free energy density is explicitly decomposed into the mean-field term, the local density fluctuation from the amplitude, and the nonlocal density fluctuation from the wave vector. The RG transform is carried out by incorporating the nonlocal term into the local term successively. The initial energy density for RG iteration was calculated from the FMSA-SAFT. Good results were obtained for the critical specific heat and phase coexistence curves. The resulting critical exponents were in good agreement with the reported nonclassical values. However, the nonlocal term remains to be the mean-field type. Further modification was made by reformulating the RG transform with the density functional method, and the analytical effective range of the potential was derived through Fourier transform. In addition, the direct correlation function (DCF),39,40 instead of the van der Waals’s (VDWs) original intermolecular potential, was used to account for the nonlocal behavior in the critical region. The method is fundamentally rigorous, and it provides both qualitative and quantitative insight into anomalous critical behavior. Furthermore, the results extend to chain fluids and mixtures that can be implemented directly into many engineering applications. The new method is highly predictive since no adjustable parameters and no mixing rules are needed for both real pure fluids and mixtures and only the original parameters for pure fluids obtained outside the critical region are needed. Our previous works33,34 were focused on nonpolar fluids, while in the industry, polar and associating systems are widely encountered and their global descriptions are of many practical interests. In this work, we reformulate our RG transform for these compounds and mixtures within the context of DFT. Similar to our previous work, DCF, instead of VDWs original intermolecular potential, is used to account for the nonlocal contribution and the FMSA is taken as the reference of the RG iteration. To modify the predictive accuracy of polar and associating fluids, we expand the density fluctuation up to the third-order term instead of the original second-order term. The new model needs no other adjustable parameters for mixtures except the original microscopic parameters of pure fluids obtained outside the critical region.
J. Phys. Chem. B, Vol. 109, No. 43, 2005 20547 2. Theory 2.1. Improved RG Transform. According to DFT, the nonlocal free energy with a density distribution of F(r) can be expanded efficiently to second order as
∫ f[F(r1)] dr1 ) βF0(Fb) + βµ0 ∫ ∆F(r1) dr1
βF[F(r)] ) β
(1) 1 2
-
∫c(r1 - r2)∆F(r1)∆F(r2) dr1 dr2
where subscript 0 stands for the reference or the homogeneous bulk fluid, in this case, ∆F(r) ) F(r) - F0, f[F(r)] the free energy density and µ0 the chemical potential. c(r) is the DCF, the expression of which can be found from the references.34,39,40 The corresponding expansion for the local free energy is
βF0(F(r)) ) β
∫ f0(F(r1)) dr1 ) βF0(Fb) + βµ0
-
1 2
∫ ∆F(r1) dr1
(2)
∫c(r1 - r2)δ(r1 - r2)∆F(r1)∆F(r2) dr1 dr2
in which δ(r) is the Dirac function, which is responsible for removing any wavelength fluctuation correlations. Focusing on the difference between the nonlocal and local free energy, as indicated by subscript
βFD[F(r)] ) βF[F(r)] - βF0[F(r)]
(3)
we obtain
∫
1 [c(r1 - r2) 2 c(r1 - r2)δ(r1 - r2)]∆F(r1)∆F(r2) dr1 dr2 (4)
βFD[F(r)] ) -
)-
1 1 2 (2π)3
∫[c˜(k) - c˜(0)]∆F˜ 2(k) dk
where the tilde represents the three-dimensional (3D) Fourier transform. As a function of r for DCF, its Fourier transform is given more explicitly by
c˜(k) )
∫∫∫c(r)e-ikr dr ) 4πk ∫0∞ sin(kr)rc(r) dr
(5)
or an even function of k. Applying the following expansion
(kr)3 (kr)5 + + ‚‚‚ 3! 5!
(6)
c˜(k) - c˜(0) ) -4πk2C4 + 4πk4C6
(7)
sin(kr) ) kr we can obtain
in which the nth moment of DCF is defined
Cn )
1 (n - 1)!
∫0∞ c(r)rn dr
(8)
In eqs 5 and 6, the DCF in 3D k-space is transformed into 1D r-space through sinusoidal function and the function is expanded into three terms instead of the previously used two terms.34 The density fluctuation in eq 4 can be assumed as an input of the cosine function
20548 J. Phys. Chem. B, Vol. 109, No. 43, 2005
∆F(r) ) x2 x cos(kc‚r)
Mi et al.
(9)
or
δ(k - kc) + δ(k + kc) ∆F˜ (k) ) (2π) x2 x 2 3
∆F˜ 2(k) )
(10)
where x is the fluctuation amplitude, and kc is the wave vector. By the use of the following relations
[
ka * kb 0 δ(k - ka)δ(k - ka) dk ) δ(ka - kb) ) δ(0 B) ka ) kb
and within a macroscopic volume of V δ(0 B) ) V/(2π)3, we can find that eq 4 is reduced to
(13)
The third term on the right-hand side of eq 2 is given by
1 2
∫c0(r1 - r2)δ(r1 - r2)∆F(r1)∆F(r2) dr1 dr2 ) 1 ∫c(r)4πr2 dr(2π)3x2V ) 2πC2x2V 2(2π)3
∫ ∫
0
∫ ... ∫ ...
FN
dt1...dtN exp{-Vn[βfn-1,D(F b,B) t + fA]}
0
(19)
fB ) 2π
∑i ∑j C4,att,ijmimjtitjkij,n2 2π ∑ ∑ C6,att,ijmimjtitjkij,n4 i j
b,B) t ) fn-1,D(F
1 Ln βVn
[
∫
0
F
0
f ) βF/V
(15)
f0(F) ) fref(F)
(16)
dx exp{-Vn[β fn-1,D(F,x) + 2πC2x2]}
(21)
and mi is the segment number of component i. The initial input of the reference free energy density fref(F) is calculated by the FMSA-SAFT,19 which has been tested to be more efficient far away from the critical point. 2.2. FMSA-SAFT. The reduced Helmholtz free energy density for a fluid mixture of chain associating molecules from the FMSA-SAFT can be expressed as a summation of contributions from the ideal gas, spherical segments, chain bonding, and chain association
fref ) fideal + fseg + fchain + fassoc
(23)
For a nonassociating polar fluid mixture, the chain association term will be replaced by a dipole-dipole interaction
fref ) fideal + fseg + fchain + fdd
(24)
n
fideal ) F
]
(17)
fseg ) mfref0
On the basis of eqs 15-18, a new improved RG EOS can be established that is applicable to complex systems. For N-component chain mixtures, the fluctuation variables will be molecular densities of all the components or b F ) {F1, ..., FN}. The RG transform derived above is equally applicable after substituting F with b F and
(25)
(26)
where the superscript ref0 represents the LJ reference mixtures, n ximi. The and m is the average segment number with m ) ∑i)1 Helmholtz free energy density for the reference term has been well developed21
and
fn-1(F + x) + fn-1(F - x) - fn-1(F) (18) fn-1,D(F,x) ) 2
xi ln(FxiΛi3) - 1 ∑ i)1
where Λi is the de Broglie wavelength, xi is the mole fraction, and F stands for the total number density. The segment contribution is
dx exp{-Vn[β fn-1,D(F,x) + 2πC4x2kn2 - 2πC6x2kn4]}
∫
(20)
fn-1(F b + B) t + fn-1(F b - B) t - fn-1(F b) (22) 2
with
F
∑i ∑j C2,att,ijmimjtitj
The ideal gas contribution is given by
fn(F) ) fn-1(F) + δfn(F)
δfn(F) )
]
dt1...dtN exp{-Vn[βfn-1,D(F b,B) t + fB]}
0 FN
fA ) 2π
(14)
Eqs 3 and 14 can constitute White’s RG formulation after taking care of free energy contributions from increasingly long fluctuation wavelengths. After n renormalization iteration, the free energy density is given by
F0
0 F0
where
]
(12)
β FD[F(r)] ) 2πC4x2k2V - 2πC6x2k4V
[
1 Ln βVn
(2π)6 2 2 x [δ (k - kc) + 2δ(k - kc)δ(k + kc) + 2 δ2(k + kc)] (11)
∫
δfn(F b) )
fref0 ) frep + f1 + f2
(27)
with
frep ) f0 - 2πF2
f0 )
[
∑i ∑j xixjg0,ij(Rij)Rij2(Rij - dij)
(π2 ξ ξ - ξ /ξ ) + 3
1 2
2
∆
2
3
ξ23 ξ32∆2
+
ξ23 ξ32
]
(28)
ln ∆ - F ln ∆ (29)
Global Vapor-Liquid Equilibria
f1 ) -2πF2β
∑i ∑j
[(
xixjij k1,ij G0,ij(z1,ij)ez1,ijRij -
(
-k2,ij G0,ij(z2,ij)ez2,ijRij -
)]
1 + z2,ijRij z2,ij2
8πF2β -8πF2β
f2 ) -πF2β
)
1 + z1,ijRij
A3
z1,ij2
NkBT -
+
) 32π2
F
2
135
∑i ∑j xixjijRij3Iij,∞ 1,ijRij
∑i ∑j ∑k xixjxk
)-
k2,ij(G2,ij(z2,ij)ez2,ijRij)] (31)
∑i ∑j xixjijg1,ij(Rij)Rij3Iij,1
() () () () ()
1 σij Iij,∞ ) 9 Rij Iij,1 )
1 σij 9 Rij
12
-
ξn )
1 σij 3 Rij
fassoc ) F
6
1 σij 2 σij + 3 Rij 9 Rij 6
i
(32)
XAi ) [1 + (33) (34)
ref ln g(i,j)(i,j+1) (σ(i,j),(i,j+1)) ∑ ∑ i)1 j)1
(35)
in which the RDF can be given by
g0(i,j),(i,j+1)(σ(i,j),(i,j+1)) exp[g1(i,j),(i,j+1)(σ(i,j),(i,j+1))] (36) The dipole-dipole interaction term given by Twu and Gubbins43,44 does not contain any binary parameters. The twoand three-body terms are calculated with the Pade approximation proposed by Stell and Rasaiah45
A2 NkBT
)-
2π 3
F
∑i ∑j xixj
(
1 + Mi 2 2
(40)
xiFXB ∆A B ]-1 ∑j ∑ B j
i j
(41)
∆AiBj ) σij3gij(dij)κAiBj[exp(AiBj/kT) - 1]
(42)
with
κ A iB i + κ A jB j 2
κ A iB j )
(43) (44)
where is the energy parameter of the hydrogen bonding of the i component, and κAiBi is the interaction volume. We here assume that the association occurs only at the BarkerHenderson dij, which is computationally much more facilitating than its integral alternatives.9,20 According to molecular simulation result,46 we select a 4-site model for water and 2-site model for n-alkanols as well as for binary mixtures. After we calculate the Helmholtz energy density of the system, the chemical potentials in phase equilibrium are calculated through some numerical scheme, which is computationally much more facilitating than any analytical attempt for the present FMSA-SAFT. The scheme is implemented by
[
(
µi ) /2/∆t,i ) 1, ...N
g(i,j),(i,j+1)(σ(i,j),(i,j+1)) )
∑ xiimiF
) ]
AiBi
xi
fdd )
/2 /2 µ/2 i µj µk Kijk
AiBj ) xAiBiAjBj
π ξ 6 3
mi-1
n
T/kk
j
to the LJ potential parameters, Rij is the Barker-Henderson diameter,41,42 g0,ij(Rij),g1,ij(Rij) are the RDFs at contact, G0,ij(zij),G1,ij(zij) are the Laplace transforms of hard-sphere and first-order RDFs, respectively, and the reader is directed to the original references for full details.20,21 The chain bonding term is given by
fchain ) -F
[(
∑i ∑ A
In eq 28 and eqs 30-34, k1,ij, k2,ij, z1,ij, z2,ij are constants related
∆)1-
T/jj
X Ai
ln XAi -
xi
3
FmRnm ∑ m
σijσjkσikT/ii
Here, µ/i ) µi/(iimiidii3)1/2, and µ is the experimentally measured dipole moment at the condition of dilute gas. For acetone, 2-butanone, and 2-pentanone, we take µ ) 2.8 D. J(F*,T/ij) and Kijk are integrals over two-body and three-body correlation functions related to pure fluids with Kijk ) (KijKjkKik)1/3, Kij ) K(F*,T/ij), and the readers are referred to those earlier references43,44 for their details. The contribution due to association is calculated from Wertheim’s theory for associating fluids. The term involves a sum over the contributions of all associating sites Mi for each pure substance i weighted by the mole fraction
and 12
σii3σjj3σkk3
(39) (30)
∑i ∑j xixjijg0,ij(Rij)Rij3Iij,1
∑i ∑j xixjij[k1,ij(G1,ij(z1,ij)ez -4πF2β
J. Phys. Chem. B, Vol. 109, No. 43, 2005 20549
A2
)
1 - (A3/A2)
σii3σjj3 σij3T/ii T/jj
(37)
∑ j+ -
xja F
∑ j+
(
∑ j+
xja F∑ xj, ∑ jj-
)
xi + ∆t xn , ..., , ..., xj xj xj
x1 xj,
∑ j+
∑ j+
xi - ∆t xn , ...., , ..., xj xj xj
x1
∑ j-
∑ j-
∑ j-
)
]
(45)
Where Fa(F,x1,...,xn) ) f(F,x1,...,xn), ∑j+ xj ) ∑j xj + ∆t, ∑j- xj ) ∑j xj - ∆t and the step size ∆t can be taken as 0.001. 3. Results and Discussion
/2 / µ/2 i µj J(F*,Tij) (38)
To evaluate the predictive ability of the new model, a number of calculations are performed in this work. We first fit the pure
20550 J. Phys. Chem. B, Vol. 109, No. 43, 2005
Mi et al.
TABLE 1: Regressed Parameters for Pure Fluids Concerned in This Work segment parameters compound
temp range (K)
critical temp (K)
m
σ (Å)
/kB (K)
κAB (× 10-2)
AB (kcal/mol)
carbon dioxide acetone 2-butanone 2-pentanone water methanol ethanol 1-propanol 1-butanol 1-pentanol
225.0-275.0 268.0-448.0 272.0-467.0 282.0-481.0 298.0-563.0 272.0-459.0 311.0-463.0 313.0-475.0 326.0-497.0 336.0-516.0
304.2 508.1 535.5 561.1 647.3 512.6 513.9 536.8 563.1 588.2
1.528 2.503 2.992 3.428 1.601 1.856 2.602 2.891 3.014 3.259
3.150 3.387 3.420 3.469 2.576 3.052 3.049 3.258 3.466 3.586
191.7 226.3 235.5 240.8 257.1 202.5 191.0 228.6 247.7 268.5
6.341 3.425 1.984 0.847 0.488 0.165
4.087 5.091 5.289 4.918 5.045 5.202
fluid microscopic parameters for the FMSA-SAFT EOS to selected experimental data. For CO2 and n-ketones, three parameters (m, σ, and ) need to be regressed to account for segment number, segment diameter, and segment dispersion energy; while for water and n-alkanols, two additional parameters, interaction volume κAB and associating energy AB, are also needed. The results are listed in Table 1. From the table, we can see that the selected temperature ranges for regression are far from the critical points, where the nonlocal density fluctuations are neglectable. The parameters thus obtained are also employed by the new EOS without any further rescaling. 3.1. Applications to Pure Fluids. The FMSA-SAFT EOS has been tested to be an accurate EOS, which can give a fairly accurate description of phase behavior for real fluids outside the critical region. Figure 1 shows the calculated results for
The calculated vapor-liquid coexistence p-F and T-F curves for acetone, 2-butanone, and 2-pentanone are plotted in Figures 2 and 3, respectively. It is well-known that the T-F curves make
Figure 2. Calculated vapor-liquid coexistence p-F curves for acetone, 2-butanone, and 2-pentanone.
Figure 1. Calculated vapor-liquid coexistence p-F curves for CO2.
carbon dioxide both from the FMSA-SAFT and from the new EOS. The experimental data are from Smith and Srivastava.47 From the figure, we can see that, at low temperatures, the two EOSs yield indistinguishable results. In this area, the local density of the fluid is homogeneous; consequently, the effect of RG transform vanishes. As the temperature gets close to the critical point, the local environment around a single particle can differ from the bulk and the degree of change is dramatically increased up to the critical region; the nonlocal density enhancement causes the system to become inhomogeneous. The original FMSA-SAFT overestimates the saturated vapor pressure and underestimates the saturated liquid density, and the deviation between the calculated and the experimental data increases with closing to the critical point. The manifest derivation demonstrates that the mean-field FMSA-SAFT EOS cannot correctly predict the phase behavior. The effect of the RG transform, on the other hand, remedies the phase diagram successfully. The new EOS gives much better results in the critical region.
the deviation more clear at low temperatures, while the p-F curves tend to stress more on the deviation in the critical region since it is on an exponential scale. Comparing the two kinds of curves helps us to analyze the predictive capability of EOS in a vast temperature range. We can see that the two EOSs give an excellent description far away from the critical point, as shown in Figure 3, and their predictions in the vicinity of the critical region of Figure 2 are quite different. With the help of the RG transform, a significant improvement over the original
Figure 3. Calculated vapor-liquid coexistence T-F curves for acetone, 2-butanone, and 2-pentanone.
Global Vapor-Liquid Equilibria
J. Phys. Chem. B, Vol. 109, No. 43, 2005 20551
Figure 6. Calculated pressure-density isotherms for water. Figure 4. Calculated vapor-liquid coexistence p-F curves for water.
Figure 5. Calculated vapor-liquid coexistence T-F curves for water.
FMSA-SAFT EOS description is achieved, leading to good agreement between the new EOS and experimental data.47,48 We now consider the ability of the new EOS to describe the phase behavior of associating fluids, namely, water and alkanols. In Figures 4 and 5, we present the calculated vapor-liquid coexistence p-F and T-F curves from the FMSA-SAFT and the new EOS for water over a large range of temperatures (295647.09 K49). We note that the calculated saturated liquid density and vapor pressure are very satisfactory. To further illustrate the role of the new EOS, the pressure-density isotherms are also calculated and plotted in Figure 6. The experimental data are from Brunner et al.50 It can be seen from the figure that good agreement is achieved between calculated and experimental results. The calculated results for 1-alkanols from C1 to C5 are shown in Figures 7-9. In both cases of water and alkanols, the new EOS has the average relative deviation of 1-3% for saturated pressure and saturated liquid density up to the critical point, while the original FMSA-SAFT predicts the critical pressure too high compared to that of the experimental data.47-49 These results indicate that the present RG transform is equally satisfactory for these associating compounds, extending its range of applications previously known only for chain molecules.33,34 3.2. Phase Equilibria for Binary Mixtures. In our previous work,33,34 alkane-alkane mixtures had been investigated systematically, and it is very tempting for us to make a similar investigation for mixtures containing polar and associating components. As mentioned earlier, the FMSA-SAFT is totally
Figure 7. Calculated vapor-liquid coexistence p-F curves for methanol and ethanol.
Figure 8. Calculated vapor-liquid coexistence p-F curves for 1-propanol, 1-butanol, and 1-pentanol.
predictive and mixing-rule free, which is inherited here; the RG transform could improve VLE prediction, especially inside the critical region. To validate these capabilities of the new EOS, several representative experimental data for asymmetric binary systems, including CO2/acetone, CO2/methanol, and CO2/ ethanol, were collected from the literature.51,52 The calculation of the phase equilibria for these systems is very challenging
20552 J. Phys. Chem. B, Vol. 109, No. 43, 2005
Figure 9. Calculated vapor-liquid coexistence T-F curves for 1-propanol, 1-butanol, and 1-pentanol.
Figure 10. Predicted vapor-liquid equilibria for CO2 + acetone mixtures.
Figure 11. Predicted pressure-composition envelopes for CO2 + methanol mixtures.
since the properties of the constituents differ largely, one is nonpolar and the other is polar or associated. Figure 10 shows the predicted results of vapor-liquid equilibria for a binary CO2/acetone mixture at 333.2 K. It should be kept in mind that in the mixture calculations, all the parameters are from pure components with the original FMSA-SAFT
Mi et al.
Figure 12. Predicted pressure-composition envelopes for CO2 + ethanol mixtures.
Figure 13. Predicted pressure-density envelopes for CO2 + methanol mixtures.
far away from the critical point, without introducing any additional binary parameters. We note that the FMSA-SAFT yields accurate results below the critical pressures. Since the FMSASAFT EOS is designed to describe all kinds of substances, it is capable of modeling the phase diagram of these unlike mixtures far from the critical region. However, the deviation between the calculated and the experimental data increases with closing to the critical point. This deviation is caused by the overestimation of the critical temperature and pressure for pure CO2 and acetone by the mean-field method. On the contrary, the RG transform can suppress the sharpness of the mixture phase envelope, which coincides with its performance for each pure component. This strongly demonstrates the global capability of describing VLE for the molecular-based EOS. Moreover, we adopted the CO2/methanol and CO2/ethanol systems to further illustrate the performance of the new EOS. Due to the unlike nature of the pure components, it is quite difficult to accurately predict their global phase diagram from a given EOS based solely upon pure component parameters. SAFT models7,53 have been applied to these systems, but these attempts end up with a binary parameter fitting and even an introduction of special mixing rules.7 Compared with the FMSASAFT EOS, our new EOS yields satisfactory predictions for CO2/methanol and CO2/ethanol concentrations in the liquid phase, and around the critical point, as shown in Figures 11 and 12. Only at low pressures, the vapor phase results slightly
Global Vapor-Liquid Equilibria derivate from the experimental data, probably due to the uncovered temperature range in our regression as shown in Table 1. The pressure-density envelopes for the CO2/methanol system both from our new EOS and from the FMSA-SAFT EOS are shown in Figure 13. It shows that the proposed EOS is capable of reproducing the VLE data simultaneously with the p-x phase diagram. Overall, the developed EOS is reliable to predict the VLE of these highly asymmetry mixtures and its description can be made with the global confidence. 4. Conclusions In this work, the improved RG theory previously tested against simulation data and alkane-alkane VLE was further improved for polar and associating pure fluids and mixtures. In the RG transform, the third-order expansion for density fluctuation within DFT was carried out to describe the nonlocal free energy in the critical region of both polar and associating fluids. The new RG theory was formulated consistently inside and outside the critical region. The new EOS can successfully predict the phase behavior of pure carbon dioxide, n-ketones, water, and n-alkanols, as well as their mixtures using the pure-component parameters from the FMSA-SAFT EOS, and no binary interaction parameters are needed for the mixtures. Compared with other SAFT models, the new EOS is unique not only by its new RG transform but also by its mixing-rule-free FMSA foundation. The calculated results demonstrate that the thus obtained EOS can give satisfactory predictions even for highly unlike mixtures. This illustrates clearly the power of the RG theory inside the critical region as well as the power of the FMSA-SAFT outside the critical region, and their combination is a good approach for developing an EOS to handle highly nonideal fluids, both outside and inside the critical region, providing a powerful tool for process design and development for chemical and related industries. Acknowledgment. The financial support of the Natural Science Foundation of BUCT (Contract QN0402), the Natural Science Foundation of China (20476003), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (20040010002) is greatly appreciated. References and Notes (1) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Fluid Phase Equilib. 1989, 52, 31. (2) Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Ind. Eng. Chem. Res. 1990, 29, 1709. (3) Redlich, O.; Kwong, J. N. S. Chem. ReV. 1949, 44, 233. (4) Soave, G. Chem. Eng. Sci. 1972, 27, 1197. (5) Peng, D.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59.
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