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Investigation of Surface Tensions for Pure Associating Fluids by Perturbed-Chain Statistical Associating Fluid Theory Combined with Density-Gradient Theory Dong Fu* 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 a nonuniform equation of state for associating fluids. In the bulk phases, the nonuniform equation of state reduces to PC-SAFT that yields accurate bulk properties. In the vapor-liquid surface, this nonuniform equation of state is able to satisfactorily correlate the surface tensions for pure associating fluids by adjusting the influence parameters. The surface tensions of 13 pure associating fluids are correlated, and the results agree well with the experimental data. 1. Introduction In recent years, the perturbed-chain statistical associating fluid theory (PC-SAFT) proposed by Gross and Sadowski1 has been widely used to investigate the phase equilibria for both pure fluids and mixtures.2-12 Compared with other equations of state (EOSs) like SAFT,13,14 PC-SAFT is more precise for correlation of experimental data from low temperature up to the critical region and more correlative when applied to mixtures. 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. A recent study12 shows that, by combining the density-gradient theory (DGT),15-22 PC-SAFT is able to correlate the surface tensions for both light and heavy n-alkanes with sufficient accuracy and provide information about the density profiles across the vapor-liquid surface and surface thickness. However, the ability of PC-SAFT to correlate the surface tension of associating fluids has not been welldocumented. The main purpose of this work is to extend the PC-SAFT to vapor-liquid surface for pure associating fluids and then accurately describe the surface tensions. To this end, the PCSAFT and DGT are used to construct a nonuniform equation of state (EOS) for associating fluids, including water, n-alcohols (C1-C6, C8, and C9), 2-propanol, acetic acid, hydrogen sulfide, and ammonia. The molecular parameters of the former 10 associating fluids are directly taken form the work of Gross and Sadowski,3 while those of acetic acid, hydrogen sulfide, and ammonia are regressed by fitting to the experimental data of liquid densities and vapor pressure. In the bulk phase, the nonuniform EOS reduces to PC-SAFT and yields accurate vapor-liquid equilibria. The obtained bulk properties then can be used to derive information about the structure and properties of the surface between coexisting phases. The influence parameters for each associating fluid are obtained by fitting to the experimental data of surface tension below the critical region. The surface tensions for 13 pure associating fluids are correlated and compared with the experimental data. 2. Theory 2.1. Density-Gradient Theory. For a system with two equilibrium phases separated by an interface, the Helmholtz free * To whom correspondence should be addressed. E-mail: fudong@ tsinghua.org.cn. Tel.: 86-312-7523127.
energy density can be described by expanding in a Taylor series,15
f [F(r)] ) f0[F(r)] + κ1[∇F(r)]2 + κ2∇2F(r) + ...
(1)
where F(r) is the local number density of molecules at position r and f0[F(r)] is the free energy density of the uniform state without the interface. Keeping the two lowest-order terms in the expansion, the Helmholtz free energy can be expressed as
A[F(r)] )
∫{f0[F(r)] + κ[F(r),T][∇F(r)]2} dr
(2)
where κ[F(r),T] is the influence parameter that can be expressed as a function of temperature or treated as a constant for the vapor-liquid surfaces with small density gradients.15 By abbreviating κ[F(r),T] as κ and replacing F(r) with F(z), the density gradient dF(z)/dz for a vapor-liquid surface is expressed as15
dF(z) ) dz
x
∆f[F(z)] κ
(3)
where ∆f[F(z)] ) κ[dF(z)/dz]2 is expressed as15
∆f[F(z)] ) f0[F(z)] - F(z)µ + p
(4)
where µ and p are the chemical potential and the pressure for bulk phases, respectively. The equilibrium density profile F(z) is as follows,
z ) z0 +
κ ∫FF x∆f[F(z)] l
v
dF(z)
(5)
where F(z0) ) (Fv + Fl)/2, and Fv and Fl are equilibrium densities for vapor and liquid phases, respectively. By evaluating the integral numerically, a distance z may be determined for any F(z) lying between the bulk densities. Once the equilibrium density profile is obtained, the surface tension can be calculated from15
γ)2
∫FF x∆f[F(z)]κ dF(z) l
v
(6)
2.2. PC-SAFT EOS. In the bulk phase, according to the perturbation theory, the Helmholtz free energy can
10.1021/ie070906e CCC: $37.00 © 2007 American Chemical Society Published on Web 09/21/2007
Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7379
[
be divided into three parts,
1 + Zhc + F
A Aid Ahc Apert ) + + NkT NkT NkT NkT
(7)
where Aid, Ahc, and Apert are the ideal gas, hard-sphere chain, and perturbation contributions, respectively. As PC-SAFT adopts a hard-sphere chain fluid as a reference fluid, Ahc for the association fluids is expressed as
Ahc Ahs Aass ) - (m - 1) ln ghs + NkT NkT NkT
(9)
where η ) (π/6)mFd3 is the packing factor and d is the hardsphere diameter for each segment. The relationship between d and σ, the soft-sphere diameter, can be expressed as
d ) σ[1 - 0.12 exp(-3/kT)]
(10)
where /k is the energy parameter. The Helmholtz free energy contribution due to association is expressed as13,14
Aass ) M ln χA - MχA/2 + M/2 NkT
(11)
where M is the number of association sites on each molecule. χA is the fraction of molecules not bonded at site A, which is given by
1
χA ) 1+
(12)
∑B FχB∆
where ∆ ) ghs[ exp(a/kT) - 1](d3κa) is related to the association strength, a is the association energy, and d3κa is a measure of the volume available for bonding of any two sites on different molecules. The Helmholtz free energy from perturbation contribution, Apert, can be expressed as1
A1 A2 Apert ) + NkT NkT NkT
(13)
where A1 and A2 are the free energies from first-order and second-order perturbation terms, which can be expressed as
A1 ) -2πFI1(η,m)m2 σ3 NkT kT
[
]
A2 ∂Zhc ) -πFm 1 + Zhc + F NkT ∂F
-1
I2(η,m)
(14)
[]
2 3 2 σ m (15) kT
with 6
I1(η,m) )
ai(m)ηi, ∑ i)0
6
I2(η,m) )
bi(m)ηi ∑ i)0
(1 - m)
20η - 27η2 + 12η3 - 2η4 (17) [(1 - η)(2 - η)]2
where ai(m) and bi(m) in eq 16 can be found in the work of Gross and Sadowski.1 3. Results and Discussions
(8)
where Ahs and Aass stand for the Helmholtz free energy from the hard-sphere contributions and the association, respectively. m is the number of segments, and ghs is the radial pair distribution function. Ahs can be expressed by the CarnahanStarling equation,23
Ahs 4η - 3η2 )m NkT (1 - η)2
]
∂Zhc 8η - 2η2 )1+m + ∂F (1 - η)4
(16)
The phase equilibrium requires pressure and chemical potential in both phases to be equal:
{
pI ) pII µI ) µII
(18)
By solving eq 18, the equilibrium density and pressure at temperature T can be obtained simultaneously. In the calculations of vapor-liquid equilibria, there are five parameters for each association fluid: segment number m, softsphere diameter of each segment σ, dispersion energy parameter of each segment /k, association volume κa, and association energy a/k. The molecular parameters for water, 2-propanol, and n-alcohols (C1-C6, C8, and C9) are directly taken form the work of Gross and Sadowski.3 For hydrogen sulfide and ammonia, the molecular parameters are regressed by fitting to the experimental data of vapor pressure and liquid densities24-27 (including the critical points) under two assumptions: M ) 2 (in accordance with the work of Gross and Sadowski3) and M ) 3 (suggested by Huang and Radosz13). In the first case, the n average relative deviations (p% ) ∑i)1 |1 - pcal/pexp|/n × 100 n l l l and F % ) ∑i)1|1 - Fcal/Fexp|/n × 100, where n is the number of data points) are, respectively, p% ) 1.12, Fl% ) 5.18 for hydrogen sulfide and p% ) 2.32, Fl% ) 6.20 for ammonia. In the second case, the average relative deviations are, respectively, p% ) 1.24, Fl% ) 4.93 for hydrogen sulfide and p% ) 2.83, Fl% ) 6.25 for ammonia. For acetic acid, the molecular parameters corresponding to M ) 2 have been given by Gross and Sadowski,3 and those corresponding to M ) 1 are regressed by fitting to the experimental data of vapor pressure and liquid densities24,25 (including the critical point), with the average relative deviations p% ) 2.31, Fl% ) 2.37. The molecular parameters of the investigated 13 associating fluids are listed in Table 1. The vapor-liquid coexistence T-F and P-F curves for acetic acid, hydrogen sulfide, and ammonia are presented in Figure 1, which shows that the assumptions of M do not significantly affect the results of the phase equilibria for these three pure fluids. Table 1. Molecular Parameters
methanol ethanol 1-propanol 2-propanol 1-butanol 1-pentanol 1-hexanol 1-octanol 1-nonanol water ammonia2B ammonia3B hydrogen sulfide2B hydrogen sulfide3B acetic acid1B acetic acid2B
m
σ/10-10 m
‚k-1/K
100κa
a‚k-1/K
1.526 2.383 3.000 3.093 2.752 3.626 3.515 4.356 4.684 1.066 1.3921 1.3970 1.5635 1.5613 1.8085 1.3403
3.230 3.177 3.252 3.209 3.614 3.451 3.674 3.715 3.729 3.001 2.9376 2.9537 3.1027 3.1038 3.4780 3.8582
188.90 198.24 233.40 208.42 259.59 247.28 262.32 262.74 263.64 366.51 273.76 273.75 237.49 237.48 270.99 211.59
3.5176 3.2384 1.5268 2.4675 0.6692 1.0319 0.5747 0.2197 0.1427 3.4868 1.8701 1.1042 0.1102 0.0656 3.8201 7.5550
2899.5 2653.4 2276.8 2253.9 2544.6 2252.1 2538.9 2754.8 2941.9 2500.7 898.2 802.4 698.2 628.5 4193.6 3044.4
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Figure 1. (a) Vapor-liquid coexistence T-F and (b) P-F curves for acetic acid, hydrogen sulfide, and ammonia. Symbols: experimental data; acetic acid, (2);24,25 hydrogen sulfide, (b);24,25 ammonia, (O),24,25 (4),26 and (]).27 Lines: calculated from PC-SAFT; (---), M ) 2; (s), M ) 1 for acetic acid, M ) 3 for hydrogen sulfide and ammonia.
Using the bulk properties as input, one can calculate the equilibrium density profile F(z) across the vapor-liquid surface from eq 5 with a given influence parameter. Consequently, other interfacial properties like the fractions of not-bonded monomer and surface thickness can be determined from F(z). Figure 2a presents the equilibrium density profiles and the corresponding not-bonded fractions for water from 283 to 583 K. Besides the well-known phenomenon that vapor-liquid surface broadens with the increase of the temperature, this figure shows (in the insert plot) that the not bonded fraction monotonously decreases from the vapor phase to the liquid phase, indicating that higher densities tend to make more molecules bond with each other. Figure 2b presents the surface thickness (t) calculated from 10 to 90% rule.28 One finds from this figure
Figure 2. (a) Equilibrium density profile, not bonded molecule fractions (insert plot) and (b) the surface thickness for water ((4), calculated results, (---), for guiding the eyes).
that the calculated surface thickness is several times the softsphere diameter and increases with the increase of temperature. Moreover, when the temperature is higher than 600 K, the surface thickness increases steeply with the increase of temperature. Once the equilibrium density profiles are obtained, the surface tension can be determined by eq 6. To describe the surface tensions satisfactorily, the influence parameter is expressed as κ/(10-19 × J‚m5‚mol-2) ) a(T/K) - b, and the adjustable parameters a and b are obtained by minimizing the objective n |1 - γcal/γexp|/n × 100. The optimized a function γ% ) ∑i)1 and b and the corresponding average relative deviations for 13 associating fluids are listed in Table 2. Figure 3 presents the calculated surface tensions for hydrogen sulfide, ammonia, 1-propanol, 2-propanol, and 1-hexanol. One finds from this figure that, by adjusting the influence parameter, PC-SAFT and DGT give satisfactory correlations for surface
Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7381 Table 2. Regressed Influence Parameters and the Corresponding Average Relative Deviations (ARD%) for Surface Tensions
methanol ethanol 1-propanol 2-propanol 1-butanol 1-pentanol 1-hexanol 1-octanol 1-nonanol water ammonia3B ammonia2B hydrogen sulfide3B hydrogen sulfide2B acetic acid1B acetic acid2B
a
b
ARDγ %
T range for γ/K
0.00224 0.00372 0.00958 0.00610 0.00448 0.00815 0.01357 0.02129 0.02760 0.00025 0.00041 0.00041 0.00141 0.00140 0.00288 0.00486
0.3829 0.6718 2.3313 1.2098 0.6796 1.4261 2.8048 4.1027 5.4954 0.0401 0.0655 0.0643 0.2068 0.2065 0.5539 0.6272
4.4 2.3 4.6 2.9 4.0 0.9 2.7 0.4 0.7 2.1 3.2 3.7 3.7 3.7 1.6 1.4
273-483 283-473 373-493 283-348 330-440 283-373 298-373 283-373 283-373 283-563 240-310 240-310 220-320 220-320 293-523 293-523
tensions compared with experimental data. With the regressed influence parameter as input, this approach succeeds in predicting the surface tensions in high temperatures and correctly captures the temperature dependence of surface tension in the critical region. Moreover, the assumptions of M for hydrogen sulfide and ammonia do not significantly affect the final results of the surface tensions. As shown in Table 2, no matter in the case of M ) 2 or M ) 3, the average relative deviations for these two pure fluids are very close. Figure 4 presents the surface tensions for methanol, pentanol, acetic acid, and nonanol, and Figure 5 presents those for ethanol, butanol, octanol, and water. Besides the phenomenon shown in Figure 3, these two figures provide a comparison of the predicted and experimental surface tensions in the critical region. One finds that, for methanol and ethanol, even in the critical region, e.g., 503-513 K, the predicted surface tensions match the experimental data satisfactorily because the critical temperatures are slightly overestimated by PC-SAFT. However, for water, as the critical temperature is significantly overestimated, the
Figure 4. Surface tensions for methanol, 1-pentanol, acetic acid, and 1-nonanol. Symbols: experimental data;29,30 (O), methanol; (b), 1-pentanol; (0), acetic acid; (9), 1-nonanol. Lines: from DGT; (s), M ) 2; (---), M ) 1 for acetic acid.
Figure 5. Surface tensions for ethanol, 1-butanol, 1-octanol, and water. Symbols: experimental data;29,30 (O), ethanol; (b), 1-butanol; (0), 1-octanol; (9) water. Lines: from DGT.
surface tensions in the temperature range of 583-647 K are significantly overestimated.
Figure 3. Surface tensions for hydrogen sulfide, ammonia, 1-propanol, 2-propanol, and 1-hexanol. Symbols: experimental data;29,30 (O), hydrogen sulfide; (b), ammonia; (4), 2-propanol; (0), 1-propanol; (]) 1-hexanol. Lines: from DGT; (s), M ) 2; (---), M ) 3 for hydrogen sulfide and ammonia.
It is worth noting that one may also define κ/(10-19 × J‚m5‚mol-2) ) a(T/K) and then regress a by fitting to the experimental data of surface tensions, as presented in the work of Kahl and Endersr.20 In this case, the temperature dependence of surface tension can also be well-captured. However, because there is only one adjustable parameter, the relative deviations between the calculated and experimental data are greater than those presented in this work.
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4. Conclusions
Subscripts
The surface tensions for 13 pure associating fluids are investigated by using PC-SAFT and DGT. The results show the following: (1) By combining with DGT, PC-SAFT is able to describe the structure and properties of the surface between coexisting phases. Starting from the accurately determined bulk properties, PC-SAFT and DGT can correlate the surface tensions of associating fluids with sufficient accuracy by adjusting the influence parameters. (2) With the optimized influence parameters as input, PCSAFT and DGT are able to correctly capture the temperature dependence of surface tension in the critical region. When the critical temperatures are slightly overestimated, the agreement of the predicted and experimental surface tensions inside the critical region is satisfactory.
cal ) calculated properties exp ) experimental properties
Acknowledgment The author appreciates 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 Research and Development Program of China (No. 2006AA05Z319), and the key program from NCEPU. Nomenclature A ) Helmholtz free energy, J A1 ) Helmholtz free energy of first-order perturbation term, J A2 ) Helmholtz free energy of second-order perturbation term, J d ) temperature-dependent hard sphere diameter, 10-10m f ) Helmholtz free energy density, J‚m-3 g ) radial distribution function k ) Boltzmann constant, J‚K-1 M ) association sites m ) segment number for one molecule N ) number of molecules p ) pressure, Pa T ) absolute temperature, K Z ) compressibility factor Greek Letters β ) 1/kT χA ) fraction of molecules not bonded at site A ∆ ) association strength ) dispersion energy parameter, J a ) association energy parameter, J γ ) surface tension, mN‚m-1 κ ) influence parameter, J‚m5‚mol-2 κa ) measure of association volume µ ) chemical potential, J‚mol-1 F ) mass density, kg‚m-3 or number density of molecules σ ) soft sphere diameter, 10-10m Superscripts ass ) association hc ) residual contribution of hard chain system hs ) hard sphere id ) ideal l ) liquid pert ) perturbation v ) vapor
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ReceiVed for reView July 2, 2007 ReVised manuscript receiVed August 29, 2007 Accepted September 7, 2007 IE070906E
