Ind. Eng. Chem. Res. 2006, 45, 8199-8206
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Investigation of Critical Properties and Surface Tensions for n-Alkanes by Perturbed-Chain Statistical Associating Fluid Theory Combined with Density-Gradient Theory and Renormalization-Group Theory Dong Fu,*,† Xiao-Sen Li,‡ ShuMei Yan,† and Tao Liao† School of EnVironmental Science and Engineering, North China Electric Power UniVersity, Baoding 071003, People’s Republic of China, and Guangzhou Institute of Energy ConVersion, Chinese Academy of Sciences, Guangzhou 510640, 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 nonuniform fluids. In the bulk phases, the nonuniform EOS reduces to PC-SAFT and the phase equilibria inside the critical region are calculated by combining the renormalization-group theory (RGT). In the vapor-liquid surface, the density profile, surface thickness, and surface tension are calculated by using the nonuniform EOS with the equilibrium bulk properties as input. Investigation shows that PC-SAFT is able to provide accurate critical properties by combining the RGT and correlating the surface tensions for both the light and heavy n-alkanes satisfactorily by combining the DGT. 1. Introduction The vapor-liquid phase equilibrium and interfacial behavior of fluids are important for the development, design, and simulation of many industrial processes such as chemical engineering and environmental protection. In recent years, interests increased rapidly in establishing a self-consistent thermodynamic framework under which both the phase equilibria and interfacial properties can be investigated within a single set of molecular parameters For the calculation of phase equilibria, it is now well-known that the classical equation of state (EOS) that can be analytically expressed always overestimates the critical temperature and the critical pressure because, in such an EOS, the long-wavelength density fluctuation is ignored. To accurately determine the critical properties, one may combine the classical EOS with the renormalization-group theory (RGT)1-12 or crossover theory.13-22 Both theories yield good critical properties compared with the experimental data because the long-wavelength density fluctuation is correctly taken into account. For the calculation of surface tensions, the density-gradient theory (DGT)23-31 provides an efficient but convenient framework under which the classical EOS can be extended to vapor-liquid surface and can model the surface tension by adjusting an additional influence parameter. The perturbed-chain statistical associating fluid theory (PCSAFT) proposed by Gross and Sadowski32 has been widely used to investigate the phase equilibria for both pure fluids and mixtures.33-42 Compared with other EOSs like SAFT,43,44 PCSAFT is more precise for correlation of experimental data from low temperature up to the critical region and more predictive when applied to mixtures. However, similar to other analytically expressed EOSs, PC-SAFT is not able to accurately correlate or predict the critical properties. Moreover, its ability to fit and correlate the surface tension for light and heavy n-alkanes has not been well documented. The main purpose of this work is to develop an approach that can accurately describe both the phase equilibria and surface * Author to whom correspondence should be addressed. E-mail:
[email protected]. Tel.: 86-312-7523127. † North China Electric Power University. ‡ Chinese Academy of Sciences.
tensions for n-alkanes from low temperature up to the critical region. To this end, the PC-SAFT and DGT are used to formulate the Helmholtz free energy functional for nonuniform fluids. In the bulk phase, the phase equilibria below and inside the critical region are calculated by using PC-SAFT and RGT with the molecular parameters presented by Gross and Sadowski32 as input. In the vapor-liquid surface, the influence parameter for each n-alkane is obtained by fitting to the experimental data of surface tension. The surface tensions for 10 light n-alkanes and 3 heavy n-alkanes are correlated from low temperature up to the critical point. 2. Theory 2.1. Density-Gradient Theory. Cahn and Hilliard23 proposed that, for a system with two equilibrium phases separated by an interface, the Helmholtz free energy density could be described by expanding in a Taylor series. The expansion is expressed as
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. In principle, κ[F(r), T] is dependent on the local density F(r) and temperature T. For spherical molecules, it can be expressed as
κ[F(r), T] )
∫
kT 2 r c[r, F(r)] dr 6
(3)
where c[r, F(r)] stands for the direct correlation function. For the vapor-liquid and liquid-liquid surfaces with small density gradients, Cahn and Hilliard23 proposed that κ[F(r), T] can be expressed as a function of temperature or treated as a constant. By abbreviating κ[F(r), T] as κ and replacing F(r) with
10.1021/ie0607393 CCC: $33.50 © 2006 American Chemical Society Published on Web 10/24/2006
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F(z), the density gradient dF(z)/dz for a vapor-liquid surface is expressed as23
x
∆f[F(z)] κ
dF(z) ) dz
z ) z0 +
∫F
V
x
κ dF(z) ∆f[F(z)]
∫F
FL V
x∆f[F(z)]κ dF(z)
(7)
(8)
where Ahc and Apert are the hard-sphere chain and perturbation contributions, respectively. As PC-SAFT adopts a hard-sphere chain fluid as a reference fluid, Ahc for the nonpolar fluids is expressed as hc
id
I1(η,m) )
(
Zhc + F
(1 - m)
(10)
(11)
where /k is the energy parameter. The Helmholtz free energy from perturbation contribution, Apert, can be expressed as
A1 A2 Apert ) + NkT NkT NkT
(14)
(12)
where A1 and A2 are the free energy from first-order and secondorder perturbation terms, respectively, which can be expressed as
I2(η,m) )
bi(m)ηi ∑ i)0
(15)
20η - 27η2 + 12η3 - 2 η4 (16) [(1 - η)(2 - η)]2
where ai(m) and bi(m) in eq 15 can be found in the work of Gross and Sadowski.32 Once the Helmholtz free energy is obtained, the pressure and chemical potential can be expressed by
p ) F2
[
]
∂[A/(NkT)] ∂F
N,T
kT and µ )
[
]
∂[A/(NkT)] ∂F
V,T
kT
2.3 Renormalization-Group Theory. In the critical region, the long-wavelength density fluctuations are important and the correlation between larger and larger numbers of molecules makes an increasingly significant contribution to the Helmholtz free energy. The classical EOS fails to describe the critical properties because the long-wavelength density fluctuation is ignored. To represent the long-ranged fluctuations in the critical region, the RGT is employed for the correction of dispersion contribution to the Helmholtz free energy. For a closed system at temperature T with particle number N and volume V, the canonical partition function is expressed as
(9)
where η ) π/6mFd3 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))
2 3 σ kT
)
Q)
where Aid and Ahs stand for the Helmholtz free energy from ideal gas and hard-sphere contributions, respectively. m is the number of segments, and ghs is the radial pair distribution function. Ahs can be expressed by the Carnahan-Starling equation,45
4η - 3η2 Ahs )m NkT (1 - η)2
()
∂Zhc 8η - 2η2 ) (1 + m) + ∂F (1 - η)4
hs
A A A ) - (m - 1) ln ghs NkT NkT NkT
I2(η,m)
6
ai(m)ηi, ∑ i)0
(6)
2.2. PC-SAFT EOS. In the bulk phase, according to the perturbation theory, the Helmholtz free energy can be divided into two parts,
A Ahc Apert ) + NkT NkT NkT
-1
6
(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 from23
γ)2
)
(13)
with
where µ and p are the chemical potential and pressure for bulk phases, respectively. The equilibrium density profile F(z) is as follows, FL
(
A2 ∂Zhc ) -πFm3 Zhc + F NkT ∂F
(4)
where ∆f[F(z)] ) κ[dF(z)/dz]2 is expressed as23
∆f[F(z)] ) f0[F(z)] - F(z)µ + p
A1 ) -2πFI1(η,m)m2 σ3 NkT kT
1 N!Λ3N
∫e-βΦ dr1dr2.....drN
(17)
where Φ represents total potential energy and β ) 1/(kT). The partition function itself is exact, near or far from the critical point, provided that the configurational integral is evaluated exactly. In a standard statistical mechanical approach, the microstates of a classical system are defined by the positions of the individual particles. This approach is convenient for discussing Table 1. Molecular Parameters and Influence Parameters for n-Alkanesa
methane ethane propane butane pentane hexane heptane octane nonane decane eicosane docosane tetracosane ARD%
σ, (10-10 m)
/k, K
m
T range for γ, K
3.70 3.52 3.62 3.70 3.77 3.80 3.80 3.84 3.84 3.84 3.97 3.98 3.98
150.03 191.42 208.11 222.38 231.20 236.77 238.40 242.78 244.51 243.87 258.05 259.28 260.32
1.000 1.607 2.002 2.342 2.690 3.058 3.483 3.818 4.207 4.663 8.11 8.83 9.55
90-110 123-183 163-233 230-300 253-313 283-373 283-363 283-383 423-535 293-540 313-343 323-343 323-343
ARDγ κ/m2, J‚m5‚mol-2 % 2.0 1.1 2.4 2.8 2.1 2.2 2.3 2.8 2.8 2.8 0.4 0.3 0.3 1.9
0.109 0.256 0.501 1.043 1.360 1.839 2.389 3.056 3.790 4.724 19.020 22.907 26.543
a The experimental data of surface tension are taken from literature refs 48-50. The optimized influence parameters can be fitted as a function κ/m2 ) 0.041 795 3 + 0.022 237 7n + 0.045 692 6n2.
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Figure 1. (a) Vapor-liquid coexistence curves and (b) the saturation pressures for methane, ethane, propane, butane, and pentane [symbols: experimental data,46,47 (b) methane, (O) ethane, (9) propane, (0) butane, (2) pentane; lines: (s) with RG correction, (- - -) without RG correction].
fluids far away from the critical point, where correlations among particles are dominated by short-ranged repulsion. However, it is not an efficient way to describe critical properties that are characterized by long-ranged correlations. To describe the role played by long-ranged correlations, a wavelength λs that is comparable to the size of a molecule is used to distinguish the short wavelength and long wavelength. Thus, the partition function is rewritten as
Q)
∑ e-βA[Fj (r)]-βU s
s
(18)
Fs(r)
where A[Fs(r)] represents the contribution of the Helmholtz energy that is insensitive to density fluctuations with wavelength longer than λs, Us accounts for the remaining contribution, and Fs(r) is the portion of F(r) that remains when the density fluctuations with wavelengths λ < λs are omitted.
Figure 2. (a) Vapor-liquid coexistence curves and (b) the saturation pressures for hexane, heptane, octane, nonane, and decane [symbols: experimental data,46,47 (b) hexane, (O) heptane, (9) octane, (0) nonane, (2) decane; lines: (s) with RG correction, (- - -) without RG correction].
The essential idea of RGT is that the contributions from the long-wavelength density wave packets can be removed successively from Us and incorporated into A[Fs(r)]. Subsequently, the set of function summation in the partition function becomes smaller. At the end of this sequence, Us disappears and an expression for the partition function that includes all levels of fluctuations is obtained. The contribution to the partition function from density fluctuations with wavelength longer than λs comes primarily from intermolecular attractions. On the mean-field approximation level, for a given density function F(r), the potential energy due to intermolecular attractions can be written as
φ)
∫∫ dr dr′ F(r)F(r′)u(|r - r′|)
1 2
where u is the potential.
(19)
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Figure 3. Pressure-density coexistence curves for n-alkanes [symbols: experimental data,46,47 (b) methane, (O) propane, (9) pentane, (0) heptane, (2) nonane, (4) ethane, (1) butane, (3) hexane, (right-facing solid triangle) octane, (right-facing open triangle) decane; lines: (s) with RG correction, (- - -) without RG correction].
During the renormalization-group iteration, the partition function remains invariant when the contributions of longwavelength densities are subtracted from Us and incorporated into the A[Fs(r)] term. Supposing that the density wave packets with the wavelength λs < λ < λl have been subtracted from Us and incorporated into A[Fjs(r)], the partition function becomes
Q)
∑ e-β∫dr f [Fj (r)]-βU ) ∑e-β∫dr f [Fj (r)]-βU′ s
s
s
Fs(r)
l
l
s
(20)
Fl(r)
where fs is the free energy density before RG correction and fl is the free energy density after one RG iteration. The difference in free energy densities, δfl[Fl(r)] is defined as
δfl[Fl(r)] ) fl[Fl(r)] - fs[Fs(r)]
(21)
which can be numerically calculated from
e-β
∫dr f [Fj (r)] ) ∑ e-β∫dr f -βU l
l
D
D
(22)
FD(r)
where
fD ) fl[Fl(r)] - fs[F2(r)] and UD ) Us - Ul
(23)
FD(r) ) x cos(k‚r) (k1 < k < ks)
(24)
where x is the fluctuation amplitude and k is the fluctuation wave vector. Equations 21-24 provide the basic recursion equation for the renormalization-group theory. The RG recursion must be performed numerically. A sequence of corrections δfn, n ) 1, 2, ‚‚‚, is added to f(F) to take into account the contributions of longer-wavelength fluctuations. Equations 25-294 are used to evaluate δfn,
Figure 4. (a) Density profile for methane and (b) surface thickness for (O) methane, (0) propane, (4) pentane, (3) heptane, and (right-facing open triangle) nonane [symbols: calculated from DGT and PC-SAFT; lines: (- - -) for guiding the eyes].
Ωsn δfn(F) ) -Kn ln l , 0 e F < Fmax/2 Ωn
(25)
where Fmax is the maximum possible molecule density, Kn ) 1/(23nβL3), and Ωln and Ωsn are the density fluctuations for the long-ranged attraction and the short-ranged attraction, respectively,
Ωsn(F) )
∫0Fdx e(-G (F,x)/K ), R n
R ) s, l
n
(26)
2GRn (F, x) ) hf Rn (F + x)fhRn (F - x) - 2fhRn (F), R ) s, l (27) hf ln(F) ) fn-1(F) + fMF hf sn(F) ) fn-1(F) + fMF
ψω j
2
2
2n+1 2
L
,
(28)
(Lh ) σL)
(29)
Ind. Eng. Chem. Res., Vol. 45, No. 24, 2006 8203 Table 2. Comparison between the Calculated Critical Properties and the Experimental Dataa Tc , K methane ethane propane butane pentane hexane heptane octane nonane decane ARD% a
Fc, kg‚m-3
pc, MPa
exp
cal1
cal2
190.6 305.5 370 425.2 469.8 507.85 539.71 569.4 594.6 617.6
191.4 308.95 375.12 432.45 479.29 519.34 552.11 583.08 609.25 630.61 1.8
189.03 303.21 367.16 423.12 468.1 506.67 538.37 568.48 594.45 615.61 0.4
exp
cal1
cal2
exp
cal1
cal2
4.64 4.91 4.26 3.8 3.37 3.03 2.74 2.5 2.29 2.1
4.67 5.16 4.6 4.22 3.82 3.54 3.25 3.02 2.8 2.59 14.0
4.62 4.89 4.25 3.86 3.42 3.11 2.82 2.6 2.39 2.18 1.4
162.3 212.18 225 227.01 231.5 234.4 234.1 234.74 236.41 235.9
148.03 192.14 208.67 219.39 223.4 228.69 230.12 231.43 232.01 230.49 4.2
159.8 207.3 223.28 230.97 235.04 238.44 239.7 241.49 241.4 239.02 1.8
The experimental data are taken from literature refs 46 and 47. The superscripts 1 and 2 stand for the results without and with RG correction, respectively.
where fMF ) m2RF2 is the Helmholtz free energy density obtained by the mean-field approximation. L is the initial cutoff length, which can be adopted as 2.0σ according to the work of Jiang and Prausnitz.4 ψ is the average gradient of the wavelet function, which can be fitted by the experimental data. R and ω j 2 are dependent on the potential u(r). For square-well potential,
ω j2 )
∫
(30)
∫
(31)
1 sw u (|b r - b′|) r db r 2
R)-
1 r2usw(|b r - b′|) r db r 2 3!Rσ
3. Results and Discussions The phase equilibrium requires pressure and chemical potential in both phases to be equal:
[
pI ) pII µI ) µII
]
(32)
By solving eq 32, the equilibrium density and pressure at temperature T can be obtained simultaneously. During the RG recursion, because the Helmholtz free energy has no analytical expression, the pressure and chemical potential should be obtained by the numerical method,
[
p ) F2
]
∂(f(F)/FNkT) ∂F
N,T
kT, µ )
[
]
∂(f(F)/NkT) ∂F
V,T
kT (33)
where f(F) is the Helmholtz free energy density that fully includes the contributions of long-wavelength fluctuations. At a given temperature, to obtain f(F), the iteration procedures, eqs 25-29, are performed five times for each bulk phase. In the calculations of phase equilibria, there are four parameters for each fluid: segment number m, soft sphere diameter of each segment σ, dispersion energy parameter of each segment /k, and the average gradient of the wavelet function ψ. The molecular parameters (m, σ, and /k) for n-alkanes with n e 20 are directly taken from the work of Gross and Sadowski.32 m, σ, and /k for n-alkanes with n > 20 are extrapolated from the linear relationship between the molecular parameters and the molecular weight Mw.32 The parameter ψ is obtained by fitting to the critical temperature and critical pressure from experiments.46,47 In this work, we find a single value ψ ) 13.5 that is applicable for the investigated 10 light n-alkanes and 3 heavy n-alkanes. In the calculation of surface tension, there is an additional influence parameter that should be obtained by fitting to the experimental data of surface tensions48-50 below
Figure 5. Surface tensions for (b) methane, (O) ethane, (9) propane, (0) butane, (2) pentane [symbols: experimental data;48,49 lines: (s) calculated results].
the critical region. The molecular parameters, the regressed influence parameters, and the average deviations (ARD%) for surface tensions are shown in Table 1. Figures 1, 2, and 3 present the vapor-liquid coexistence curves, the saturation pressures, and the pressure-density coexistence curves for light n-alkanes in comparison with the experimental data from the literature.46,47 From the figures, one finds that the PC-SAFT overestimates the critical temperatures because it ignores the long-wavelength density fluctuation. However, by combining the RG correction, it is able to calculate the phase equilibria from low temperature up to the critical point accurately. Table 2 shows the comparison between the critical properties before and after RG correction. One can find that the calculated Tc, pc, and Fc are greatly improved by RG correction. The ARD% for Tc, pc, and Fc before the RG correction are 1.8%, 14.0%, and 4.2%, respectively, while after RG correction they become 0.4%, 1.4%, and 1.8%, correspondingly. Using the bulk properties as input, one can calculate the equilibrium density profile F(z) across the vapor-liquid interface from eq 6; hence, the interfacial properties including surface thickness and surface tension can be determined. Figure 4a presents the interfacial density profile for methane from 80 to 160K, demonstrating that, with the increase of the
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Figure 6. Surface tensions for (b) hexane, (O) heptane, (9) octane, (0) nonane, (2) decane [symbols: experimental data;48,49 lines: (s) calculated results].
Figure 7. Comparison of the PC-SAFT with SAFT proposed by Huang and Radosz43 for correlating the surface tension of ethane [symbols: experimental data;48,49 lines: (s) calculated from PC-SAFT, (- - -) calculated from the SAFT proposed by Huang and Radosz,43 with parameters σ ) 3.48 × 10-10 m, /k ) 175.1 K, m ) 1.66, and κ/m2 ) 0.23 J‚m5‚mol-2 as input].
temperature, the vapor-liquid surface broadens. Figure 4b presents the surface thickness (t) for methane, propane, pentane, heptane, and nonane from 10 to 90% rule.51 One finds from Figure 4b that the calculated surface thickness is several times the soft-sphere diameter and increases with the increase of temperature. Moreover, when the reduced temperatures (T/Tc) are ∼0.85, the surface thickness increases steeply with the increase of temperature. Figures 5 and 6 present the surface tensions for 10 light n-alkanes correlated by DGT. From these two figures, one can
Figure 8. Vapor-liquid coexistence curves (insert plots) and saturation pressures for (b) eicosane, (O) docosane, (9) tetracosane [symbols: experimental data;46,47 lines: (s) with RG correction].
Figure 9. Surface tensions for (b) eicosane, (O) docosane, (9) tetracosane [symbols: experimental data;50 lines: (s) calculated results].
find that, by regressing the influence parameter with experimental data, PC-SAFT and DGT give satisfactory results for surface tensions from low temperature up to the critical region. Figure 7 shows the comparison between PC-SAFT and the SAFT proposed by Huang and Radosz43 for correlating the surface tensions of ethane, showing that the PC-SAFT yields better results. Besides the phase equilibria and surface tensions of light n-alkanes, we also investigate those for 3 heavy n-alkanes, as shown in Figures 8 and 9. One finds from Figure 8 that the critical properties for heavy n-alkanes can also be fitted well once the PC-SAFT is combined with RGT. Figure 9 shows that the PC-SAFT is able to satisfactorily correlate the surface tensions for heavy n-alkanes by combining the DGT.
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4. Conclusions The critical properties and surface tensions for n-alkanes are investigated by using PC-SAFT. The results show the following: (1) Below the critical region, PC-SAFT yields satisfactory phase equilibria. Inside the critical region, PC-SAFT is able to correctly describe the critical properties and accurately correlate the critical points by combining the RGT. A single value of ψ ) 13.5 is applicable for the investigated 10 light n-alkanes and 3 heavy n-alkanes. (2) By combining the DGT, PC-SAFT is able to correlate the surface tensions for both light and heavy n-alkanes from low temperature up to the critical region. The average deviation is 1.9%. (3) Compared with other SAFT versions, PC-SAFT yields better results for surface tension. Acknowledgment The authors appreciate the financial support from the National Natural Science Foundation of China (Grant No. 20576030 and 20606009) and the key program foundation from NCEPU. X.S.L. is also grateful for the financial support from the Research Program of the One-Hundred Talent Project of the Chinese Academy of Sciences and the National Natural Science Foundation of China (Grant No. 20676133). 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, 10-10 m f ) Helmholtz free energy density, J‚m-3 g ) radial distribution function k ) Boltzmann constant, J‚K-1 L ) initial cutoff length, 10-10 m m ) segment number for one molecule N ) number of molecules p ) pressure, Pa T ) absolute temperature, K u ) potential, J Z ) compressibility factor Greek Letters β ) 1/kT ) dispersion energy parameter, J γ ) surface tension, mN‚m-1 κ ) influence parameter, J‚m5‚mol-2 µ ) chemical potential, J‚mol-1 F ) mass density, kg‚m-3, and number density of molecules σ ) soft sphere diameter, 10-10 m ψ ) average gradient of the wavelet function Superscripts cal ) calculated property chain ) chain formation exp ) experimental property hc ) residual contribution of hard chain system hs ) hard sphere id ) ideal L ) liquid l ) long-ranged correlation pert ) perturbation
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ReceiVed for reView June 9, 2006 ReVised manuscript receiVed August 29, 2006 Accepted September 18, 2006 IE0607393
