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Friction Theory Coupled with Statistical Associating Fluid Theory for Estimating the Viscosity of n-Alkane Mixtures Sugata P. Tan, Hertanto Adidharma,* Brian F. Towler, and Maciej Radosz Department of Chemical and Petroleum Engineering, UniVersity of Wyoming, Laramie, Wyoming 82071-3295
Friction theory is coupled with statistical associating fluid theory, SAFT1 and PC-SAFT, to predict the viscosity of mixtures. The universal friction theory parameters working with SAFT1 and PC-SAFT for pure n-alkanes, which were previously found to scale linearly with molecular weight, are used without any readjustment to predict the viscosity of symmetric and asymmetric mixtures of n-alkanes with accuracy that is adequate for engineering purposes. Introduction Viscosity is one of the most important transport properties needed to estimate the movement of oil and gas in underground reservoirs. Consequently, there is a need for accurate models for viscosity calculations and predictions. In compositional reservoir simulation models, viscosities of the phases were most often calculated from correlations even when phase compositions and densities were calculated from equations of state.1 The most common correlation used was the one due to Lohrenz et al.,2 but other methods based on corresponding states3-6 have also been found to be favorable. These methods rely on experimental data for accuracy but lack a firm fundamental basis. Predictions can only be made if the model used has a sound theoretical background. Perhaps the most successful model of this kind in the past few years is that of friction theory7 (FT), which has been coupled with various empirical equations of state (EOS), such as cubic EOSs, and applied to various types of fluids and fluid mixtures. Recently, we coupled FT with a theoretically based SAFTlike EOS,8 such as SAFT19 and PC-SAFT,10 to correlate the viscosity of pure n-alkanes. The FT parameters were made to scale linearly with molecular weight for reliable extrapolation to heavy hydrocarbons. The purpose of this work is to extend FT+SAFT-like EOSs to n-alkane mixtures. Friction Theory.7 Pure Component. FT divides the dynamic viscosity into two parts
η ) η0 + ∆η
(1)
where η0 is the viscosity of dilute gas given by Chung et al.11 and ∆η is the friction contribution term, which vanishes when the fluid system approaches the dilute gas limit. The dilute gas model by Chung et al., based on the Chapman-Enskog kinetic theory of gases, predicts the dilute gas limit as follows:
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 * To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: (307) 766-2500. Fax: (307) 766-6777.
the reduced temperature and critical temperature, respectively. The expression for Ω* was first proposed by Neufeld et al.12 as Ω(2,2)*. The empirical factor Fc for nonpolar substances is
Fc ) 1 - 0.2756ω
(3)
where ω is the acentric factor. The friction contribution term is derived from friction concepts in classical mechanics and the van der Waals theory of fluids:7
∆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 EOS explained below. Within 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:7
κr(T) ) a1 exp(Tr-1 - 1) + a2(exp(2(Tr-1 - 1)) - 1)
(5a)
κa(T) ) b1 exp(Tr-1 - 1) + b2(exp(2(Tr-1 - 1)) - 1) (5b) κrr(T) ) c2(exp(2Tr-1) - 1)
(5c)
For each pure substance, therefore, the model has five parameters: a1, a2, b1, b2, and c2. Quin˜ones-Cisneros et al.13 implemented the concept of corresponding states (CS) to reduce the number of component-specific FT parameters so that there are 16 universal parameters (for the whole n-alkane series) plus one substance-dependent scaling parameter, namely, the critical viscosity. To eliminate this substance-dependent parameter, an empirical equation with two parameters can be used, so that it ends up with a total of 18 universal parameters. In our recent paper,8 those five parameters in eqs 5 can be made to be linear functions of molecular weight for the n-alkane series, so that
a1(M) ) a10 + a11M
(6a)
a2(M) ) a20 + a21M
(6b)
b1(M) ) b10 + b11M
(6c)
b2(M) ) b20 + b21M
(6d)
c2(M) ) c20 + c21M
(6e)
10.1021/ie051110n CCC: $33.50 © 2006 American Chemical Society Published on Web 02/10/2006
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The 10 parameters on the right-hand side were fitted to experimental data for a series of n-alkanes from ethane to n-hexadecane, except n-undecane, n-tridecane, and n-pentadecane. The values of the parameters can be obtained from our previous work.8 We used only a limited number of data sources, which proves sufficient to demonstrate the ability of FT+SAFT. The data used for fitting came from Stephan and Lucas14 for ethane to n-dodecane and from Ducoulombier et al.15 for the rest. Both SAFT1 and PC-SAFT have common expressions for the repulsive pressure and the attractive pressure used in eq 4.
Pr ) RTF2 Pa ) RTF2
(
( ) hs
∂a˜ ∂F
(7)
)
∂a˜disp ∂a˜chain + ∂F ∂F
(8)
where F is the molar density, T is the absolute temperature, R is the universal gas constant, and the a’s are the residual Helmholtz energy terms. The superscripts of the energy terms stand for hard sphere (hs), dispersion (disp), and chain terms, which represent the corresponding molecular interactions. The details of these energy terms along with the EOS parameters for SAFT1 and PC-SAFT can be found elsewhere.9,10 Mixture. FT for mixtures is an extension of eq 1 as follows:
ηmix ) η0,mix + ∆ηmix
(9)
Mixing rules are used to estimate both terms. For the dilute gas limit term, we use the mixing rule proposed by Wilke:16
η0,mix )
xiη0,i
∑i
( x
η0,i
xj 1 +
∑j
η0,j
)
2
(10)
xMj/Mi
x
The friction contribution term in eq 9 is
(11)
where the coefficients K for mixtures are7
Kmix )
∑i
xiKi
Mi
(12)
∑j xj/Mj
p
EOS
C7/C6 (53)b
C7/C9 (57)b
SAFT1 PC-SAFT PR
3.7%/5.3% 1.3%/3.6% 4.4%/-
2.9%/4.4% 2.9%/5.8% 3.3%/-
a ARD ) (1/N)Σ|(η calc - ηexp)/ηexp|; N ) number of data points. Numbers in parentheses are the numbers of data available from the source.17
b
of carbon atoms in the molecule, so that, for example, n-decane is written as C10. The first example is taken from Assael et al.17 for symmetric mixtures of n-alkanes, such as C7/C6 and C7/C9. The experimental data were measured at 303.15 and 323.15 K in the range of 1 e P < 718 bar and at two compositions for each mixture, i.e., 40 wt % and 70 wt % of C7. The second example is taken from Iglesias-Silva et al.18-21 for pure C5, pure C8, pure C10, their ternary mixtures, and the corresponding three binary mixtures. These binary and ternary systems have a moderate degree of asymmetry. The viscosity of each binary was measured at nine compositions in the range of 1 < P < 256 bar and 297.95 e T e 373.35 K. The ternary data were measured at 15 compositions. The third example is taken from Queimada et al.22,23 for pure C7, pure C10, pure C20, pure C24, their binary mixtures (C7/C20, C7/C24, C10/C20, C10/C24, and C20/C24), and their ternaries (C7/ C20/C24 and C10/C20/C24). These binary and ternary systems represent more asymmetric mixtures of n-alkanes. The experimental data were measured at atmospheric pressure in the range of 293.15 e T e 343.15 K. Each mixture viscosity was measured at five different compositions, except for the C20/C24 viscosity which was measured at equimolar composition only. Results and Discussion
2 2(1 + Mi/Mj)
∆ηmix ) Kr,mixPr + Ka,mixPa + Krr,mixPr2
Table 1. ARD and Maximum Deviation of Viscosity Prediction for C7/C6 and C7/C9a
p
Subscripts i and j refer to the individual component in the mixture. In eq 12, Mi is the molecular weight of component i, while the power p depends on the EOS coupled with FT. If p ) 0, the denominator in eq 12 becomes unity so that the equation simply reduces to Kmix ) ΣxiKi, where xi is the molar fraction of component i in the mixture. In mixtures, Pr and Pa are still given by eqs 7-8, but of course, all the energy terms and the density are those of mixtures. Systems Used as Examples. We prefer to use examples from sources that include data of more than one system measured at different compositions, temperatures, and pressures. Such examples are presented in this work in the order of their degree of asymmetry and the number of components. For simplicity, all components will be named with Cn, where n is the number
All results are predicted using the universal FT parameters for n-alkanes determined in our previous work for SAFT1 and PC-SAFT.8 For n-alkane binaries, we set the interaction parameter kij ) 0, because the EOS can represent the phase equilibria of the binaries well enough without any kij term. The properties of pure components needed in eq 2 are taken from the literature.24 The power p in eq 12 is found to be 0.3 for SAFT1 and 0 for PC-SAFT. It was found that for cubic EOSs, p ) 0.3 (Peng-Robinson), p ) 0.15 (Soave-Redlich-Kwong), and p ) 0 (Peng-Robinson-Stryjek-Vera).7 However, p ) 0.3 for those three cubic EOSs if the general model with 18 universal FT parameters for n-alkanes is used.13 The average relative deviation (ARD, the first percent entry) and the maximum deviation (the second percent entry) of the prediction for the first example are listed in Table 1. The result using the Peng-Robinson25 (PR) EOS with universal FT parameters for n-alkanes13 is also included for comparison. It has to be noted that FT+PR applies some binary parameters (kij) for the mixtures.13 For the examples presented in this work, no maximum errors for the results obtained from the PR EOS are available. The ARD and the maximum deviation of the prediction for the second example are listed in Table 2. The result using the PR EOS with universal FT parameters for n-alkanes13 is also included for comparison. The ARD distribution for the binaries with respect to composition is shown in Figure 1, which suggests that, as expected, the ARD gets larger with the increasing degree of
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Table 2. ARD and Maximum Deviation of Viscosity Prediction for the Second Example (Involving C5, C8, and C10) EOS
C5 (31)a
C8 (36)a
C10 (36)a
C5/C8 (295)a
C5/C10 (312)a
C8/C10 (324)a
C5/C8/C10 (530)a
SAFT1 PC-SAFT PR
2.9%/9.8% 3.2%/6.8%
2.0%/7.4% 2.1%/5.1%
2.6%/6.6% 3.0%/5.1%
3.8%/11.6% 6.2%/14.5% 4.0%/-
4.4%/10.9% 6.5%/15.3% 4.6%/-
2.0%/9.4% 2.1%/6.7% 3.4%/-
2.5%/13.5% 2.4%/7.4% 3.3%/-
a
Numbers in parentheses are the number of data available from the sources.18-21
Figure 1. ARD distributions for the binaries in the second example with respect to composition.
Figure 2. ARD of FT+SAFT1 (circles) and FT+PC-SAFT (squares) for C5/C8/C10.
asymmetry. The ARD distribution for the ternary in the composition space is shown in Figure 2. The relatively larger deviations in the intermediate compositions suggest that the mixing rules (eqs 11 and 12) still have room for improvement. However, we decided to use eqs 11 and 12 in this work, as proposed in the original papers of FT.7,13 Figures 3 and 4 show typical prediction of the mixture viscosity using FT+SAFT1. The largest deviation takes place at lower temperatures. A similar phenomenon was observed with
cubic EOSs and attributed to the difficulties with viscosity measurements at low temperatures and the inaccurate performance of cubic EOSs for liquid phase at low temperatures.23 The ARD and the maximum deviation of the viscosity prediction for the third example along with the ARD of density prediction are listed in Table 3. The result using the PR EOS with universal FT parameters for n-alkanes22,23 is also included for comparison. It is worth noting that FT+PR uses modified critical properties of the pure components fitted to the liquid density, except for C7, because the PR EOS with the standard tabulated critical properties predicts inaccurate liquid densities, which affects the viscosity calculation.22,23 In our previous work,8 FT+SAFT1 was found to predict the viscosity of C20 and C24 inaccurately. This is not caused by the densities calculated from SAFT1, which is the usual problem with the cubic EOS; the densities of C20 and C24 are well predicted by SAFT1 as shown in Table 3. Therefore, the performance of FT+SAFT1 can be improved by fixing the constant parameter external to the EOS that is needed in determining the FT temperature-dependent coefficients K in eqs 5, i.e., the critical temperature TC. The critical temperatures of heavy alkanes are usually approximated, because they are never reached in reality for measurement. For our viscosity calculation purpose using FT+SAFT1, the critical temperatures are then adjusted to 777 K for C20 (from the tabulated 767 K) and 829 K for C24 (from the tabulated 804 K), which improve the viscosity calculation from 9.1% to 6.3% for C20 and from 14.6% to 2.9% for C24 for these data source. We use the new TC for predicting the viscosity of the mixtures in Table 3.
Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 2119
Figure 3. Viscosity of C5/C8, C8C10, and C5/C10 at two temperatures and three pressures: 49.95, 148.15, and 246.26 bar (from top to bottom); (dotted lines) eye guides; (solid lines) FT+SAFT1.
Figure 5. Viscosity of C10 from different sources. Figure 4. Viscosity of C5/C8/C10 at two compositions and six temperatures: 297.95, 313.05, 328.05, 343.15, 358.25, and 373.35 K (from top to bottom); (dashed and solid lines) FT+SAFT1
The slightly larger deviation of FT+PC-SAFT for C10 in Table 3 is caused by the large difference of the experimental
data from different sources, as shown in Figure 5. The data from Queimada et al.23 used in this example are lower than the rest, while the FT+PC-SAFT curve, which was fitted to n-alkane data from Stephan and Lucas,14 is the highest. Consequently, this also affects the prediction for the ternary (C10/C20/C24).
Table 3. ARD and Maximum Deviation of Viscosity Prediction and ARD of Density Prediction for the Third Example (Involving C7, C10, C20, and C24) FT+SAFT1
FT+PC-SAFT
FT+PR
system
no. data [ref]
viscosity
density
viscosity
density
C7 C10 C20 C24 C7/C20 C7/C24 C10/C20 C10/C24 C20/C24 C7/C20/C24 C10/C20/C24
4 [22] 6 [23] 4 [22] 2 [22] 24 [22] 15 [22] 24 [23] 16 [23] 3 [22] 20 [23] 20 [23]
1.8%/2.8% 3.7%/5.6% 6.3%/13.5% 2.8%/3.3% 5.8%/13.1% 3.4%/8.6% 3.8%/11.7% 3.7%/9.7% 5.3%/9.0% 3.8%/11.7% 3.8%/8.1%
0.21% 0.05% 0.06% 0.31% 0.16% 0.21% 0.10% 0.17% 0.23% 0.10% 0.12%
2.2%/3.3% 10.1%/11.1% 3.5%/8.1% 3.0%/4.2% 4.6%/13.5% 3.5%/7.9% 4.1%/10.0% 1.9%/5.6% 2.7%/4.5% 4.1%/10.0% 7.9%/15.1%
0.13% 0.74% 1.0% 1.5% 0.65% 0.48% 0.44% 0.65% 0.75% 0.44% 0.28%
viscosity
density
7.1%/8.0%/4.2%/3.8%/-
0.8% 0.7% 0.52% 0.37%
9.1%/9.2%/-
0.7% 0.55%
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Figure 6. Viscosity isopleths for C7/C20 and C7/C24: (lines) FT+PC-SAFT.
Figure 7. Viscosity isopleths for C10/C20 and C10/C24: (lines) FT+PC-SAFT.
Figures 6-8 show typical predictions of mixture viscosity using FT+PC-SAFT. The variation of ARD in the composition space is not as large as that for the previous example, possibly because the data were all measured at a single pressure (1 bar). Again, particularly for the ternaries, the largest deviations are at the lowest temperatures. With accurate density calculations, as shown in Table 3, it is unlikely that the deviations are due to the limitations of the SAFT EOS. A typical density calculation for C7/C20 and C7/C20/C24, shown in Figure 9, does not indicate greater inaccuracies at low temperatures. On the average, both FT+SAFT models improve the performance of the viscosity predictions over the FT+PR model for n-alkane mixtures. Recalling that the universal parameters
for pure n-alkanes in FT+SAFT1 and FT+PC-SAFT were fitted to data taken only in a narrow range of temperature and pressure and from only two data sources, i.e., the Stephan and Lucas compilation14 for C2-C12, excluding C11, and Ducoulombier et al.15 for C14 and C16, a more extensive parameter fitting with more reliable experimental data is expected to result in more accurate predictions, as inferred by the FT+cubic EOS in the original papers of FT.7,13 Conclusion The universal FT parameters working with SAFT1 and PCSAFT for pure n-alkanes, which were previously found to scale
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Figure 8. Viscosity of C7/C20/C24 and C10/C20/C24 at five temperatures: 303.15, 313.5, 323.15, 333.25, and 343.35 K (from top to bottom); (lines) FT+PCSAFT.
Figure 9. Typical density prediction from PC-SAFT.
linearly with molecular weight, without any readjustment, are found to be able to predict the viscosity of symmetric and asymmetric mixtures of n-alkanes with accuracy that is adequate for engineering purposes. Acknowledgment University of Wyoming’s Enhanced Oil Recovery Institute provided funding for this work.
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(5) Ely, J.; Hanley, H. Prediction of Transport Properties. 1. Viscosity of Fluids and Mixtures. Ind. Eng. Chem. Fundam. 1981, 20, 323. (6) Huber, M. L. User’s Guide: NIST Thermophysical Properties of Hydrocarbon Mixtures Database (SUPERTRAPP); U.S. Department of Commerce: Gaithersburg, MD, 1998. (7) Quin˜ones-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. The friction theory (f-theory) for viscosity modeling. Fluid Phase Equilib. 2000, 169, 249. (8) Tan, S. P.; Adidharma, H.; Towler, B. F.; Radosz, M. Friction Theory and Free-Volume Theory Coupled with Statistical Associating Fluid Theory for Estimating the Viscosity of Pure n-Alkanes. Ind. Eng. Chem. Res. 2005, 44, 8409-8418. (9) Adidharma, H.; Radosz, M. Prototype of an Engineering Equation of State for Heterosegmented Polymers. Ind. Eng. Chem. Res. 1998, 37, 4453. (10) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244. (11) Chung, T. H.; Ajlan, M.; Lee, L. L.; Starling, K. E. Generalized Multiparameter Correlation for Nonpolar and Polar Fluid Transport Properties. Ind. Eng. Chem. Res. 1988, 27, 671. (12) Neufeld, P. D.; Janzen, A. R.; Aziz, R. A. Empirical Equations to Calculate 16 of the Transport Collision Integrals Ω(l,s)* for the LennardJones (12-6) Potential. J. Chem. Phys. 1972, 57, 1100. (13) Quin˜ones-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. One parameter friction theory models for viscosity. Phase Fluid Equilib. 2001, 178, 1. (14) Stephan, K.; Lucas, K. Viscosity of Dense Fluids; Plenum: New York, 1979. (15) Ducoulombier, D.; Zhou, H.; Boned, C.; Peyrelasse, J.; SaintGuirons, H.; Xans, P. Pressure (1-1000 bar) and Temperature (20-100 °C) Dependence of the Viscosity of Liquid Hydrocarbons. J. Phys. Chem. 1986, 90, 1692. (16) Wilke, C. R. A Viscosity Equation for Gas Mixtures. J. Chem. Phys. 1950, 18, 517.
(17) Assael, M. J.; Charitidou, E.; Dymond, J. H.; Papadaki, M. Viscosity and Thermal Conductivity of Binary n-Heptane + n-Alkane Mixtures. Int. J. Thermophys. 1992, 13, 237. (18) Estrada-Baltazar, A.; Alvarado, J. F. J.; Iglesias-Silva, G. A.; Barrufet, M. A. Experimental Liquid Viscosities for Decane and Octane + Decane from 298.15 K to 373.15 K and Up to 25 MPa. J. Chem. Eng. Data 1998, 43, 441. (19) Estrada-Baltazar, A.; Iglesias-Silva, G. A.; Barrufet, M. A. Liquid Viscosities of Pentane and Pentane + Decane from 298.15 K to 373.15 K and Up to 25 MPa. J. Chem. Eng. Data 1998, 43, 601. (20) Barrufet, M. A.; Hall, K. R.; Estrada-Baltazar, A.; Iglesias-Silva, G. A. Liquid Viscosity of Octane and Pentane + Octane from 298.15 K to 373.15 K up to 25 MPa. J. Chem. Eng. Data 1999, 44, 1310. (21) Iglesias-Silva, G. A.; Estrada-Baltazar, A.; Hall, K. R.; Barrufet, M. A. Experimental Liquid Viscosity of Pentane + Octane + Decane Mixtures from 298.15 K to 373.15 K up to 25 MPa. J. Chem. Eng. Data 1999, 44, 1304. (22) Queimada, A. J.; Quin˜ones-Cisneros, S. E.; Marrucho, I. M.; Coutinho, J. A. P.; Stenby, E. H. Viscosity and Liquid Density of Asymmetric Hydrocarbon Mixtures. Int. J. Thermophys. 2003, 24, 1221. (23) Queimada, A. J.; Marrucho, I. M.; Coutinho, J. A. P.; Stenby, E. H. Viscosity and Liquid Density of Asymmetric n-Alkane Mixtures: Measurement and Modeling. Int. J. Thermophys. 2005, 26, 47. (24) Daubert, T. E.; Danner, R. P.; Sibul, H. M.; Stebbins, C. C. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation; Taylor & Francis: Washington, D.C., 1997. (25) Peng, D. Y.; Robinson, D. B. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59.
ReceiVed for reView October 4, 2005 ReVised manuscript receiVed December 21, 2005 Accepted January 24, 2006 IE051110N
