Ind. Eng. Chem. Res. 2005, 44, 8409-8418
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Friction Theory and Free-Volume Theory Coupled with Statistical Associating Fluid Theory for Estimating the Viscosity of Pure n-Alkanes Sugata P. Tan, Hertanto Adidharma,* Brian F. Towler, and Maciej Radosz Department of Chemical and Petroleum Engineering, University of Wyoming, Laramie, Wyoming 82071-3295
Two viscosity models, the friction theory and the free-volume theory, are coupled with the statistical associating fluid theory equations of state, such as SAFT1 and PC-SAFT, to model the viscosity of pure n-alkanes. The parameters of both viscosity models are found to scale linearly with molecular weight so that the models can predict the viscosity of long n-alkanes. Introduction Viscosity is one of the most important transport properties in reservoir simulation. Together with the thermodynamic properties of fluids, viscosity is used in flow simulations that predict the movement of oil and gas underground. Consequently, there is a need for accurate models for viscosity calculations and predictions. Most Newtonian viscosity models are based on empirical approaches that are limited to the correlation range, for example, the widely used Lohrenz-BrayClark (LBC) correlation,1 which is a polynomial in reduced density. Similarity P-v-T and T-η-P relationships lead to viscosity models of fluids that use an equation of state (EOS) as the mathematical reference, for example, the work of Little and Kennedy,2 which used van der Waals EOS, and the more recent work by Guo et al.,3 which used Patel-Teja and Peng-Robinson EOSs. There also exist semitheoretical models, such as corresponding states (CS) models, in which one or two reference fluids are used. In these models, dimensionless properties of a substance are supposed to be equal to those of the reference substance if calculated at the same reduced conditions. An example of a CS method is the transport property prediction (TRAPP) model originally proposed by Ely and Hanley.4 Reviews on those viscosity models can be found elsewhere.5,6 However, there are fewer theoretically based viscosity models. Martins et al.7 proposed a model based on the absolute rate theory by Eyring8 that needs the residual Helmholtz energy, which in turn is calculated from a cubic EOS. Unfortunately, this model is applicable to liquids only. Perhaps the most successful model in the past few years is the friction theory9 (FT). This model has been coupled with various empirical EOSs, such as cubic EOSs, and applied to various types of fluids and fluid mixtures. Another promising model is the recent version of the free-volume theory10 (FVT), which has also been applied to calculate self-diffusion coefficients.11 Though free-volume expressions based on some EOSs exist,12 this model explicitly relates the freevolume fraction to the system density, so that it can directly utilize experimental density data to estimate the viscosity.10 * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: (307) 766-2500. Fax: (307) 7666777.
Instead of applying empirical EOSs with FT9 and using experimental density data with the recent version of FVT,10 we want to explore how these models work with a theoretically based EOS, such as the statistical associating fluid theory (SAFT) EOS, and how the parameters scale with molecular weight, for example, for n-alkanes. We choose two versions of SAFT, i.e., SAFT113 and PC-SAFT,14 as examples. Viscosity Models. Both FT and FVT divide the dynamic viscosity into two parts,
η ) η0 + ∆η
(1)
where η0 is the viscosity of dilute gas given by Chung et al.,15 which is common for both FT and FVT, and ∆η is the dense-state correction term, the form of which depends on the viscosity model we use. The correction term vanishes when the fluid system approaches the dilute gas limit. The dilute gas model by Chung et al.15 is based on the Chapman-Enskog kinetic theory of gases, which has been applied successfully in predicting the dilute gas limit of various fluids in wide ranges of temperature:
xMT Fc [µP] η0 ) 40.785 2/3 vC Ω*(T*)
(2)
where M is the molecular weight [g/mol], T is the absolute temperature [K], vC is the critical volume [cm3/ mol], and Ω* is the reduced collision integral as a function of dimensionless temperature T* (T* ) 1.2593Tr; Tr ) T/TC) with Tr and TC being the reduced temperature and critical temperature, respectively. The expression for Ω* was first given by Neufeld et al.16 as Ω(2,2)*. The empirical factor Fc for nonpolar substances is
Fc ) 1 - 0.2756ω
(3)
where ω is the acentric factor. (A) Friction Theory (FT).9 The dense-state correction term is the friction contribution derived from friction concepts in classical mechanics and the van der Waals theory of fluids:
∆η ) κr(T)Pr + κa(T)Pa + κrr(T)Pr2
(4)
where Pr is the repulsive pressure and Pa is the attractive pressure; both pressures come from the EOSs
10.1021/ie050723x CCC: $30.25 © 2005 American Chemical Society Published on Web 10/04/2005
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Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005
Figure 1. Typical proportion of attractive and repulsive pressures for SAFT-like EOSs.
explained in the next section. At the dilute gas limit, Pr ) Pa ) 0, so that the correction term ∆η ) 0 as desired. The temperature-dependent coefficients can be expressed as follows:
κr(T) ) a1 exp(Tr-1 - 1) + a2(exp(2(Tr-1 - 1)) - 1) (5a) -1
κa(T) ) b1 exp(Tr
-1
- 1) + b2(exp(2(Tr
κrr(T) ) c2(exp(2Tr-1) - 1)
- 1)) - 1) (5b) (5c)
For each pure substance, therefore, the model has five parameters: a1, a2, b1, b2, and c2. The critical properties needed in the calculation are available in the literature. The quadratic term on the right-hand side of eq 4 is to ensure the validity of the model at high-pressure conditions, where the distance between molecules becomes smaller. In fact, for some EOSs, the contribution of Pr and Pa is of the same order of magnitude,17 which is also the case for the SAFT-like EOS, as shown in Figure 1 for n-heptane. For other substances, the pressure behavior is similar to that in Figure 1. Therefore, a quadratic term in Pa can be applied in addition to the quadratic term in Pr in eq 4. However, according to our experience with FT + SAFT, at least in the temperature and pressure range of interest to us, the performance of the model does not improve much with the inclusion of this additional term. We, therefore, restrict our attention to the three terms in eq 4 as used in the first paper on FT.9 (B) Free-Volume Theory (FVT).10 The dense-state correction term in eq 1 is connected to the molecular structure through a representation of the free-volume fraction (fv) based on an empirical relation proposed by Doolittle,18
∆η ) A exp(B/fv) al.10
(6)
Allal et relate the free-volume fraction with the intermolecular energy controlling the potential field in which the molecular diffusion takes place. The final
Figure 2. Data comparison: n-pentane from NIST and Stephan and Lucas (S-L).
expression of this contribution is
∆η ) FLE
x
103M E 3/2 exp B 3RT RT
(( ) )
[µP]
(7)
where
E ) RFM × 103 +
P F
(8)
R is the universal constant [equal to 8.3145 J/(mol K)], M is the molecular weight [g/mol], and P is pressure [MPa]. The molar density F [mol/cm3] is the only property derived from the EOS. Equation 8 is an approximation of the intermolecular energy. Its first term is considered to be the energy barrier a molecule has to overcome to diffuse, and the second term is considered to be the energy needed to form a vacant passage for the diffusion. For the record, the exponential of reciprocal temperature resembles that of the absolute rate theory proposed by Eyring,8 in which E is interpreted as the flow energy barrier only. The model has three parameters: L is the length parameter [Å], absorbing the average quadratic length, which is related to the structure of the molecules and the characteristic relaxation time, R is the proportionality between the energy barrier and the density [J m3/ (mol kg)], and B is a positive number characteristic of the free-volume overlap. SAFT Equations of State. The SAFT residual Helmholtz energy can be expressed as the sum of a reference term and perturbation terms representing different types of molecular interactions:
a˜ res ) a˜ ref + a˜ 1 + a˜ 2 + ...
(9)
From this residual Helmholtz energy, the compressibility factor of the system can be calculated from the relation
Z)
( )
∂a˜ res P )1+F FRT ∂F
T,x
(10)
The unity on the right-hand side of eq 10 is the ideal gas contribution, while the second term accounts for the
Ind. Eng. Chem. Res., Vol. 44, No. 22, 2005 8411 Table 1. Data Used in the Parameter Fitting cubic EOS + FT9
this work CH4 C2H6 C3H8 n-C4H10 n-C5H12 n-C6H14 n-C7H16 n-C8H18 n-C9H20 n-C10H22 n-C12H26 n-C14H30 n-C16H34
FVT10
T [K]
P [bar]
points
sources
T [K]
P [bar]
points
sources
T [K]
P [bar]
points
sources
240-400 300-480 240-480 280-480 320-490 380-480 300-480 320-480 300-460 300-480 300-460 293-373 313-373
40-600 40-600 50-350 40-600 40-500 40-500 40-500 40-500 40-500 40-350 40-500 1-600 1-600
65 99 100 96 80 48 100 79 129 120 66 20 16
1 1 1 1 1 1 1 1 1 1 1 1 1
97-478 150-500 173-478 200-478 213-548 273-548 292-548 273-569 300-470 278-478
0.5-699 1-552 1.1-552 1-692 0.98-1027 0.98-1000 0.98-1023 1-1021 1-690 0.1-1019
743 951 370 421 567 456 360 346 308 155
15 7 8 8 8 12 7 7 2 5
91-600
0.1-2000
885
1
90-600
0.1-1000
1085
1
273-423
1-2500
292
3
273-448
1-5055
347
4
298-373
1-5016
53
2
nonideality of the system due to molecular interactions. A complete review of SAFT can be found elsewhere.19 In this work, SAFT supplies the density for the FVT model. Though the SAFT-estimated density may not be accurate at some conditions, the FVT parameters absorb the inaccuracies. However, close to the critical point, due to the inherent limitations of the mean-field theory, SAFT will produce errors that may be too large to be absorbed. This is also the case for coupling SAFT with the FT model. To couple SAFT with the FT model, we need to briefly know the information about the SAFT versions being used. It is suggested that the reader refer to the original papers for details. (A) SAFT1.13 SAFT1 uses square-well (SW) segments as reference. Thus, the reference term is calculated by summing the hard-sphere term and the SW dispersion term. The dispersion term has universal constants fitted from the thermodynamic properties of pure ethane. For n-alkanes, there is only one perturbation term, i.e., the SW chain term. Hence, SAFT1 has four parameters that are well-behaved in terms of molecular weight. The segment number, m, is a linear function of molecular weight, as is the product of m and the other parameters: the segment volume (v00), the segment energy (u0/ k), and the reduced range of the SW potential well (λ). To couple SAFT1 with FT for viscosity calculations, the pressures Pr and Pa are defined from the corresponding repulsive and attractive terms of the residual Helmholtz energy. The repulsive interaction is represented by the hard-sphere interaction only.
( )
Pr ) RTF2
∂a˜ hs ∂F
(11)
The attractive interaction originates from dispersion interactions among the segments and the covalent bonds in the molecule chain.
Pa ) RTF2
(
)
∂a˜ disp ∂a˜ chain + ∂F ∂F
(12)
(B) PC-SAFT.14 PC-SAFT uses hard chains as reference. Thus, the reference term is calculated by summing the hard-sphere term and the hard-sphere-chain term. For n-alkanes, there is only one perturbation term, i.e., the Lennard-Jones chain-dispersion term, the universal constants of which were fitted to the properties of the n-alkane series. Therefore, PC-SAFT can represent n-alkanes very well. For a nonassociating substance, PC-SAFT has three parameters that are well-behaved in terms of molecular weight: the number of segments
per unit molar mass, m/M, and the potential parameters, /k (depth) and σ (hard-core diameter), have the same formulation in terms of molecular weight. To couple PC-SAFT with FT for viscosity calculations, the pressures Pr and Pa are also estimated using eqs 11 and 12, respectively. Note that the dispersion and chain terms in eq 12 are now the chain-dispersion and the hard-sphere-chain terms, respectively. Parameter Fitting. There are two issues that impact the parameter fitting. First, the experimental data of viscosity are available in the literature within wide ranges of accuracy, typically about 3-5%. Moreover, data from one source could be different from those from other sources. For example, the viscosity data for n-pentane from two well-known sources, the compilation of Stephan and Lucas20 and the smoothed data from NIST,21 are plotted in Figure 2; the three isothermal curves show disagreement between the two sets of data at high pressures, about 5% at 320 K and about 10% at 440 K, relative to the lower pressure data. Second, the fitting involves numerous parameters, which leads to multivariable optimization and a nonuniqueness issue; multiple sets of parameters can fit the viscosity equally well for each substance. Therefore, to search parameter sets for n-alkane series that conform to one another, we need to include as many substances as possible, while the physical meaning of the parameters, if any, can help the fitting process. Since the main purpose of this work is to show how the viscosity models work with SAFT-like EOSs, it is sufficient to use one data source for each substance, for example, the compilation by Stephan and Lucas.20 We use this source for C2H6 to C12H26 but skip C11H24. Methane is not included in the fitting because it usually behaves differently within the SAFT framework. However, we add the data for C14H30 and C16H34 taken from Ducoulombier et al.22 to the data pool. As we expect to apply the viscosity models to enhanced oil recovery (EOR) in the future, we select the appropriate temperature and pressure ranges, as shown in Table 1, but some limits are according to the data source. The low-pressure range is also excluded, which means that only the viscosity of the liquid phase or the supercritical phase is fitted. For C14H30 and C16H34, one additional pressure, 1 bar, is included. The exclusion of low-pressure data, as shown later, does not affect the performance of the model much. The experimental data used in the FT model9 and the FVT model10 are also shown in Table 1, including the number of points and the number of data sources.
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Table 2. Results of Fitting Individual Compounds this work FT + SAFT1/PC-SAFT
uncertaintyc
ARDa
FVT + SAFT1/PC-SAFT
ARDmax b
ARDa
4.75% (260 K, 150 bar) 1.61% 1.22%
FT + SRK9 ARD
ARD
4.81% (340 K, 600 bar)
1.80%
2.59% 14.8% (300 K, 2000 bar)
CH4
3%
2.67% 2.04%
C2H6 C3H8
3% 3%
2.74% 1.43% 9.40% (320 K, 60 bar) 1.08% 0.64% 10.84% (400 K, 80 bar)
5.75% (300 K, 80 bar) 6.96% (400 K, 80 bar)
3.60% 2.51%
n-C4H10 n-C5H12 n-C6H14
2% 4% 5%
1.15% 1.04% 12.67% (450 K, 60 bar) 2.14% 1.57% 20.06% (450 K, 60 bar) 1.40% 1.34% 13.90% (490 K, 40 bar) 1.52% 1.15% 23.48% (490 K, 40 bar) 0.31% 0.30% 0.92% (420 K, 400 bar) 0.71% 0.33% 1.83% (420 K, 40 bar)
3.12% 3.76% 1.92%
n-C7H16 n-C8H18
