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Ind. Eng. Chem. Res. 2008, 47, 4490–4495
Investigation of Vapor-Liquid Surface Tension for Carbon Dioxide and Hydrocarbon Mixtures by Perturbed-Chain Statistical Associating Fluid Theory Combined with Density-Gradient Theory Dong Fu* and Yuanzhen Wei School of EnVironmental Science and Engineering, North China Electric Power UniVersity, Baoding, 071003, People’s Republic of China
The perturbed-chain statistical associating fluid theory (PC-SAFT) and density-gradient theory (DGT) are used to construct an equation of state (EOS) applicable for the phase behaviors of supercritical carbon dioxide (CO2) and hydrocarbon binary mixtures. In the bulk phases, the nonuniform EOS reduces to PC-SAFT, which is able to accurately describe the vapor-liquid equilibria (VLE) below the critical region. In the vapor-liquid surface, this EOS is able to predict the surface tensions for binary mixtures with the bulk properties and the influence parameters of pure components as input. The surface tensions of CO2-butane, CO2-decane, CO2-benzene, CO2-cyclohexane, and CO2-tetradecane binary mixtures are predicted, and the results are satisfactory compared with the experimental data. 1. Introduction Supercritical CO2 is a particularly attractive oil displacement fluid due to its low-cost availability and its high degree of solubility in oil. In recent years, the phase behaviors of CO2-hydrocarbon mixtures have been a subject of considerable interest.1–6 In particular, because the carbon dioxide flooding technique is considered to be promising in the development of efficient and economical methods for secondary or tertiary recovery of petroleum reservoir fluids, the solubility of CO2 in hydrocarbons is important in the simulation and design for recovery processes. The surface tensions of CO2-hydrocarbon mixtures are also important. For example, in the displacements of oil by CO2, as the efficiency of the recovery process can be significantly affected by the surface tension between the fluid phases present in the reservoir, the qualitative description of the effect of surface tension on recoveries from CO2 flooding becomes an important issue. Developing an approach that can well describe both the bulk and interfacial properties of CO2-hydrocarbon mixtures is meaningful either in practice or in theory. In recent years, a lot of work has been focused on the description of vapor-liquid phase behavior for CO2-hydrocarbon mixtures by thermodynamic models.7–18 The perturbed-chain statistical associating fluid theory (PCSAFT) proposed and developed by Gross and Sadowski13 has been widely used to investigate the phase equilibria for pure fluids and mixtures.13,15,18–34 Compared with other equations of states such as SAFT, PC-SAFT is more correlative when applied to mixtures. For example, our previous studies showed that PC-SAFT is able to accurately describe the vapor-liquid equilibria (VLE) for both CO2-light hydrocarbon and CO2-heavy n-alkane (up to C44) binary mixtures by optimizing the binary interaction parameters (kij).18 Besides its applications in bulk phases, PC-SAFT is also applicable for the investigation of the structure and properties of the interface between coexisting phases. Recent studies33,34 show that by combining the density-gradient theory (DGT)16,17,33–47 PC-SAFT is able to correlate the surface tensions for both light and heavy n-alkanes and associating fluids with sufficient accuracy and provide * To whom correspondence should be addressed. Tel.: 86-3127523127. E-mail:
[email protected].
information about the surface thickness and the density profiles across the vapor-liquid surface. However, the ability of PCSAFT to correlate the surface tensions of CO2-hydrocarbon mixtures has not been well documented by far. The main purpose of this work is to extend PC-SAFT to vapor-liquid surfaces of CO2-hydrocarbon mixtures and then predict the surface tensions. To this end, PC-SAFT and DGT are used to construct a nonuniform equation of state (EOS). The molecular parameters of carbon dioxide and hydrocarbon are directly taken from the work of Gross and Sadowski.13 The binary interaction parameters for CO2-butane, CO2-decane, CO2-benzene, and CO2-cyclohexane mixtures are taken from the work of Gross and Sadowski13 and Fu et al.,18 and the binary interaction parameter for CO2-tetradecane mixtures is regressed by fitting to the experimental data.48 By using the influence parameters of pure fluids (regressed from experimental data of surface tensions) and bulk properties as input, the surface tensions of five CO2-hydrocarbon mixtures are predicted. 2. Theory For a binary mixture with two equilibrium phases separated by an interface, the Helmholtz free energy density f[F1(r),F2(r)] could be described by expanding in a Taylor series.36–47 The expansion is expressed as f [F1(r), F2(r)] ) f0[F1(r), F2(r)] +
∑ 21 κ ∇F (r) ∇F (r) + · · · ij
i
j
i,j
(1) where Fi(r) is the local number density of molecule i at position r and ∇[Fi(r)] is the corresponding local density gradient. f0[F1(r),F2(r)] is the free energy density for the bulk phase. κ11 and κ22 are respectively the influence parameters for components 1 and 2; κ12 ) (κ11κ22)1/2 is the cross-influence parameter. Keeping the lowest order term in the expansion, the Helmholtz free energy can be expressed as A[F1(r), F2(r)] )
∫ [f [F (r), F (r)] + ∑ 21 κ ∇F (r) ∇F (r)] dr 0
1
10.1021/ie0716520 CCC: $40.75 2008 American Chemical Society Published on Web 06/10/2008
2
ij
i
j
i,j
(2)
Ind. Eng. Chem. Res., Vol. 47, No. 13, 2008 4491
In the equilibrium state, the density profiles across the interface must satisfy the following Euler-Lagrange equation:
∑ j
1 ∇[κij∇Fj(r)] 2
∑∑ k
j
∂κkj ∇Fk(r) ∇Fj(r) ) ∂Fi(r) ∂Ω[F1(r), F2(r)] ∂Fi(r)
∑ F(r)µ
0 i
dF1 √κ22[∂µ1/ ∂ F2]T,p,F1 - √κ11[∂µ1/ ∂ F2]T,p,F1 ) dF2 κ [∂µ / ∂ F ] - κ [∂µ / ∂ F ]
√
∂{f0[F1(r), F2(r)]} T,V,Fj(r) ∂Fi(r)
is the chemical potential of component i of the bulk phase. Because the density varies only in the direction perpendicular to the interface (z direction), F(r) can be replaced by F(z). Equation 3 is then rewritten as ij
j
]
∂Fj(z) 1 ∂z 2
∂κkj ∂Fk(z) ∂Fj(z) ) ∂z ∂z i
∑ ∑ ∂F (z) k
j
∂Ω[F1(z), F2(z)] ∂Fi(z)
(4)
Assuming that the density dependence of the influence parameters can be neglected, eq 4 becomes
∑κ
d2Fj(z)
ij
dz2
j
) µi[F1(z), F2(z)] - µ0i
(5)
where µi[F1(z),F2(z)] is the chemical potential of component i in the interface. Multiplying eq 5 by dFi(z)/dz, summing over i, and integrating, we obtain
∑∑ i
j
2 1 dFi(z) d Fj(z) κij ) Ω[F1(z), F2(z)] + pcoex ) 2 dz dz2 ∆Ω[F1(z), F2(z)]
∆Ω[F1(z), F2(z)] κ
(6)
(7)
κ is the influence parameter for the binary mixture, which can be formulated as κ ) κ11
[ ] dF1(z) dF2(z)
2
+ 2κ12
dF1(z) + κ22 dF2(z)
(8)
Dividing the vapor-liquid surface into N0 thin layers (no less than 2000), the number density in the kth layer can be expressed as F2(zk) ) F2(zk-1) + ∆F2
11
2
1 T,p,F2
(11)
(9)
where F2(z0) and F2(zN0+1) respectively equal the equilibrium vapor density F2V and the equilibrium liquid density F2L. ∆F2 is defined as
√
22
1
(12)
1 T,p,F2
µ1 and µ2 are the chemical potentials for components 1 and 2, respectively. Suppose FV 2 corresponds to z ) 0; the value of z in each layer can be calculated from z)
∫
F2
F2V
κ dF ∆Ω[F1, F2] 2
(13)
By evaluating the integral numerically, a distance z may be determined for any F2 lying between the bulk densities; hence F1(z) and F2(z) can be determined. Once the equilibrium density profiles are obtained, the surface tension can be calculated from γ)2
∫
F2L
F2V
√∆f [F1, F2]κ dF2
(14)
3. Results and Discussion In this work, the Helmholtz free energy for binary mixtures is expressed by PC-SAFT. The details of the formulations can be found in the work of Gross and Sadowski.13 At a given temperature T, the VLE of CO2-hydrocarbon mixtures can be determined according to the requirement pI ) pIIµI1 ) µII1 ,
where µi0 and pcoex are respectively the chemical potential of component i and the saturated pressure in the bulk phase. During the determination of density profiles F1(z) and F2(z), one should first select a reference fluid. For example, when component 2 is selected as the reference fluid, the density gradient of the reference fluid is expressed as dF2(z) ) dz
F1(zN0+1) ) FL1
where the differentiation dF1/dF2 in the (k - 1)th layer is expressed as
and
∑ ∂z∂ [κ
dF1(zk-1) ∆F2 ; F1(z0) ) FV1 , dF2(zk-1)
(3)
i
µ0i )
(10)
The number density of component 1 in the kth layer is F1(zk) ) F1(zk-1) +
where the grand potential density is Ω[F1(r), F2(r)] ) f [F1(r), F2(r)] -
∆F2 ) (FL2 - FV2 )/N0
µI2 ) µII2
(15)
where I and II stand for the upper and lower phases of the mixtures, respectively. p is the pressure; µ1 and µ2 are the chemical potentials for CO2 and the hydrocarbon, respectively. In the calculations of VLE, there are three molecular parameters for each component: segment number m, soft sphere diameter of each segment σ, and dispersion energy parameter of each segment ε/k. To fit the VLE well, one should introduce an additional binary interaction parameter (kij) and regress it by fitting to the experimental data. In this work, the molecular parameters for CO2, butane, decane, benzene, cyclohexane, and tetradecane are directly taken from the work of Gross and Sadowski,13 as shown in Table 1. The binary interaction parameter values kij ) 0.12, 0.14, 0.11, and 0.13 for the CO2-butane, CO2-decane, CO2-benzene, and CO2-cyclohexane mixtures are taken from the work of Gross and Sadowski13 and Fu et al.;18 kij ) 0.13 is regressed for the CO2-tetradecane mixture by fitting to the experimental data.48 It is worth noting that, in our previous work,18 we have regressed the binary interaction parameters for the mixtures of CO2 and n-alkanes (C1-C7, C10, C12, C16, and C20) and obtained the relationship kij ) 0.030128 Log(n) + 0.0726141 (n is the carbon number). However, compared with the experimental data,48 this relationship yields poor VLE for the CO2-tetradecane mixture. kij ) 0.13 for the CO2-tetradecane mixture is not in accord with the mentioned relationship, but only this value can qualitatively reproduce the experimental data.48
4492 Ind. Eng. Chem. Res., Vol. 47, No. 13, 2008 Table 1. Molecular Parameters and Influence Parameters
carbon dioxide butane cyclohexane benzene decane tetradecane
m
σ/10-10 m
εk-1/K
κii/10-19 J · m5 · mol-2
2.0729 2.3331 2.5303 2.4653 4.6627 5.9002
2.7852 3.7089 3.8499 3.6478 3.8384 3.9396
169.21 222.88 278.11 287.35 243.87 254.21
0.14 1.04 1.95 1.18 4.72 9.04
The calculated VLE results and the comparison with experimental data48–51 are shown in Figure 1. From this figure, one finds that the VLE of the CO2-butane, CO2-benzene, and CO2-cyclohexane mixtures can be well predicted using the available parameters. However, for the CO2-decane and CO2-tetradecane mixtures, the calculated VLE in the region of high pressures (near the critical region) deviate significantly from experimental data. In order to improve the VLE calculations in the region of high pressures, we have expressed kij as a function of temperature. However, the results show that such a treatment very slightly affects the VLE calculations. Actually, near the critical region, as the fluids become inhomogeneous and the long wavelength density fluctuations are very important to the Helmholtz free energy, the classical equations of state that can be analytically expressed always overestimate the critical points. Results for the VLE can be used to derive information about the structure and properties of the interface between coexisting phases. To obtain the interfacial properties for binary mixtures, one should first determine the influence parameters for each pure component. In this work, the influence parameters for butane and decane are taken from our previous work33 and those for carbon dioxide, cyclohexane, benzene, and tetradecane are regressed by fitting to the experimental data52–54 of their surface tensions. The influence parameters are also shown in Table 1. The comparison of calculated and experimental surface tensions for each pure fluid is shown in Figure 2. Using the bulk properties and the influence parameters of a pure component as input, one can calculate the equilibrium
Figure 2. Vapor-liquid surface tensions for pure fluids. Symbols: experimental data. CO253 (O), butane52,54 (b), cyclohexane53 (9), benzene54 (0), decane52,54 (2), and tetradecane54 (∆). Lines: calculated by PC-SAFT + DGT.
Figure 3. Equilibrium density profiles across the vapor-liquid surface of CO2-butane binary mixtures under different conditions. Lines: calculated by PC-SAFT + DGT. s, for butane; ---, for CO2. In the main plot, p ) 4 MPa. Corresponding to the increase of the solubility of CO2, T ) 319.3, 344.3, and 377.6 K, respectively. In the inset plot, T ) 319.3 K. Corresponding to the increase of the solubility of CO2, p ) 0.93 and 3.18 MPa, respectively.
Figure 1. VLE for CO2-hydrocarbon binary mixtures. Symbols: experimental data. CO2-butane51 (b, 319.3 K; O, 344.3 K; 2, 377.6 K), CO2-decane50 (9, 344.3 K; 0, 377.6 K), CO2-benzene49 ([, 344.3 K), CO2-cyclohexane49 (], 344.3 K), and CO2-tetradecane48 (4, 344.3 K) binary mixtures. Lines: calculated by PC-SAFT.
density profiles across the vapor-liquid surface by numerically evaluating eq 13. Figure 3 shows the equilibrium density profiles for CO2 and butane in the vapor-liquid surface of the CO2-butane mixture in the case of p ) 4 MPa, indicating that, at a given pressure, the vapor-liquid surface broadens and the solubility of CO2 increases with the increase of temperature. The inset in Figure 3 shows the equilibrium density profiles for CO2 and butane in the case of T ) 319.3 K, indicating that, at a given temperature, with the increase of pressure, the vapor-liquid surface broadens and the solubility of CO2
Ind. Eng. Chem. Res., Vol. 47, No. 13, 2008 4493
Figure 4. Surface tension of CO2-butane binary mixture. Symbols: experimental data.51 b, 319.3 K; O, 344.3 K; 2, 377.6 K. Lines: predicted by PC-SAFT + DGT.
increases. It is worth noting that, in the current DGT approach, care must be taken in the selection of a reference fluid. The density profile of the reference fluid must be a monotonic function of z over the whole vapor-liquid surface; e.g., as shown in Figure 3, only the density profiles of butane are monotonic functions of z, hence butane should be selected as the reference fluid. Otherwise, the entire density profiles cannot be correctly determined. In this work, the hydrocarbon is selected as the reference fluid for the investigated five CO2-hydrocarbon binary mixtures. Once the equilibrium density profiles are obtained, the surface tension can be calculated from eq 14. Figure 4 shows the predicted surface tensions of the CO2-butane mixture at three temperatures. One finds that the predicted results satisfactorily capture the tendency of surface tension as a function of pressure and mole fraction. In the region of low pressures, as the bulk properties are accurately determined, the predicted values agree well with the experimental data. However, in the region of high pressures (especially in the case of T ) 377.6 K), as the VLE are not so satisfactorily described, the deviations between the predicted and experimental vapor-liquid surface tensions become distinguishable. Figure 5 shows the predicted surface tensions for CO2decane and CO2-benzene mixtures, and Figure 6 shows those for CO2-cyclohexane and CO2-tetradecane mixtures. For CO2-benzene and CO2-cyclohexane mixtures, similar to the CO2-butane mixture, as the bulk properties are well determined, the predicted surface tensions satisfactorily match the experimental data. For CO2-decane and CO2-tetradecane mixtures, the predicted results agree well with the experiments in the region of low pressure because, in this region, the bulk properties are accurately determined. However, in the region of high pressures, because the critical points of the VLE are significantly overestimated, the predicted surface tensions deviate seriously from the experiments. Taking into account the facts that the fluids are in the critical region and the surface tensions themselves are of very small values, the deviations of the predictions are reasonable. Actually, in the prediction of surface tensions for binary mixtures under the framework of DGT, as
Figure 5. Surface tension of CO2-decane and CO2-benzene binary mixtures. Symbols: experimental data. Decane50 (b, 344.3 K; O, 377.6 K); benzene49 (0, 344.3 K). Lines: predicted by PC-SAFT + DGT.
Figure 6. Surface tension of CO2-cyclohexane and CO2-tetradecane binary mixtures. Symbols: experimental data. CO2-cyclohexane49 (b, 344.3 K); CO2-tetradecane48 (O, 344.3 K). Lines: predicted by PC-SAFT + DGT.
the surface tensions for pure fluids can be accurately correlated, the quality of the prediction is dependent on the accuracy of bulk properties calculation; hence a precise equation of state, e.g., PC-SAFT, is very important. 4. Conclusions The bulk and interfacial properties of CO2-hydrocarbon binary mixtures are investigated by using the PC-SAFT and DGT. The results show the following: (1) For CO2-butane, CO2-decane, CO2-benzene, and CO2-cyclohexane binary mixtures, PC-SAFT is able to satisfactorily predict the VLE below the region of high pressures,
4494 Ind. Eng. Chem. Res., Vol. 47, No. 13, 2008
with the available binary interaction parameters and molecular parameters as input. For the CO2-tetradecane mixture, the VLE in the region of low pressures can also be well described by regressing the binary interaction parameter. (2) With the available molecular parameters as input, PCSAFT is able to accurately describe the surface tensions for pure fluids by combining the DGT and regressing the influence parameters. (3) By combining DGT, PC-SAFT is able to predict the surface tensions for CO2-hydrocarbon binary mixtures with the bulk properties and the influence parameters of a pure component as input. The accuracy of bulk properties calculation plays an important role in the predictions of surface tensions. Acknowledgment The authors appreciate the financial support from the National Natural Science Foundation of China (Nos. 20606009 and 20576030), the Program for New Century Excellent Talents in University (No. 06-0206), the National High Technology ResearchandDevelopmentProgramofChina(No.2006AA05Z319), and the key program from NCEPU. List of Symbols A ) Helmholtz free energy, J f ) Helmholtz free energy density, J · m-3 kij ) binary interaction parameter m ) number of segment N0 ) total number of thin layers for the vapor-liquid surface p ) pressure, Pa T ) absolute temperature, K x ) mole fraction Greek Letters γ ) surface tension, mN · m-1 ε ) dispersion energy parameter for each segment, J κ ) influence parameter, J · m5 · mol-2 µ ) chemical potential, J · mol-1 F ) number density of molecules σ ) hard core diameter for each segment, 10-10 m Ω ) grand potential density, J · m-3 Superscripts I, II ) upper and lower phases Subscripts i, j ) components
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ReceiVed for reView December 4, 2007 ReVised manuscript receiVed April 27, 2008 Accepted April 29, 2008 IE0716520
