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Ind. Eng. Chem. Res. 2005, 44, 1960-1966
CORRELATIONS Simultaneous Correlation of Saturated Viscosities of Pure Gases and Liquids Using the Significant Structure Theory† Gustavo Cruz-Reyes,‡ Gabriel Luna-Ba´ rcenas,§ J. Francisco Javier Alvarado,| Isaac C. Sa´ nchez,⊥ and Ricardo Macı´as-Salinas*,‡ ESIQIE-Chemical Engineering Department, Instituto Polite´ cnico Nacional, UPALM-Zacatenco, Me´ xico, D.F. 07738, Me´ xico, CINVESTAV-IPN Unidad Quere´ taro, Fracc. Real de Juriquilla, Quere´ taro, Qro. 76230, Me´ xico, Chemical Engineering Department, Instituto Tecnolo´ gico de Celaya, Av. Tecnolo´ gico S/N, Celaya, Gto. 38010, Me´ xico, and Chemical Engineering Department, University of Texas at Austin, Austin, Texas 78712-1062
The significant structure theory (SST) for liquid viscosities, originally proposed by Eyring, coupled with a cubic equation of state was used for the simultaneous correlation of gas and liquid viscosities of pure fluids (polar and nonpolar) at saturated conditions. The SST visualizes a liquid as having both “solidlike” and “gaslike” degrees of freedom with “fluidized vacancies” of molecular size randomly distributed throughout a quasi-lattice structure. In this context, the viscosity of a liquid is calculated from two main components: a gaslike ηg and a solidlike ηs contribution. The first viscosity contribution ηg represents the viscosity of a pure fluid at dilute gas conditions (low-pressure viscosity). The method of Chung et al. based on the ChapmanEnskog kinetic theory of gases was used to calculate ηg. The second contribution ηs captures the solidlike effects on viscosity. This quantity was calculated by means of Eyring’s absolute rate theory. All the thermodynamic properties required in the viscosity model were computed via the use of a well-known cubic equation of state (Soave-Redlich-Kwong or Peng-Robinson) thus allowing the simultaneous correlation of gas-liquid viscosities along their coexistence curve. The resulting model was satisfactorily validated in the representation of experimental saturated gas and liquid viscosities of a highly polar compound (water) and a nonpolar fluid (propane) over a wide range of temperatures (from near the triple point up to the critical region of the fluid of interest). 1. Introduction Viscosity represents one of the most important transport properties in process design and development. Viscosity of pure gases has been well described by the kinetic theory of gases, namely the Chapman-Enskog theory.1 In fact, current viscosity models for pure gases suited for practical engineering use are modified versions of the Chapman-Enskog theory. For dense gases and liquids, however, a theoretical description is much more difficult for such fluids due to their wide diversity of intermolecular forces as well as the structure and degree of disorder between the molecules. Despite numerous efforts that have been reported in the literature dealing with the development of reliable models for the correlation and/or prediction of viscosities for pure liquids, there is not yet a widely accepted theoreti† Paper presented at the 2003 AIChE Annual Meeting, November 16-21, 2003, San Francisco, CA. * To whom correspondence should be addressed. Tel.: +52(55)5729-6000 x55261. Fax: +52(55)5586-2728. E-mail:
[email protected]. ‡ Instituto Polite ´ cnico Nacional. § CINVESTAV-IPN Unidad Quere ´ taro. | Instituto Tecnolo ´ gico de Celaya. ⊥ University of Texas at Austin.
cal method to estimate the viscosity of liquids, due perhaps to the poor understanding of the liquid state itself, and specifically to the mathematical complexity of reproducing the nonequilibrium distribution functions (although some progress has been recently made through simulations of nonequilibrium molecular dynamics). In consequence, the majority of the viscosity models published in the literature range from semi-theoretical to entirely empirical. For pure fluids, Reid et al.,2 Mehrotra,3 Monnery et al.,4 Mehrotra et al.,5 and Poling et al.6 have published excellent reviews and evaluations of the most important modeling approaches of the viscosity of gases and liquids. These works reveal that the Chapman-Enskog theory, the corresponding-states principle, and Eyring’s absolute rate theory stand as the most popular and utilized approaches in the viscosity modeling of pure substances. Further adaptations of these theories have yielded dedicated models for the calculation of viscosities of pure gases and liquids within a restricted range of temperatures, usually over two temperature regions: from Tr ≈ 0.7 down to the triple point and from Tr ≈ 0.7 up to near the critical point.2,6 Other viscosity models based on the geometric similarity between P-V-T and T-η-P relationships have been also proposed from well-known cubic equations of state (Little and Kennedy,7 Wang and Guo,8 Guo et
10.1021/ie049070v CCC: $30.25 © 2005 American Chemical Society Published on Web 02/18/2005
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al.9,10). These viscosity models, although being applicable to both liquid and gas phases over a wide pressure range, make use of a significant number of generalized constants which are only valid to hydrocarbon substances. In addition, they fail to give an adequate description of gas-liquid viscosities along the saturation line. Other viscosity modeling efforts based on the use of well-known equations of state include the so-called friction theory (f-theory) approach by Quin˜ones-Cisneros et al.11 On the basis of friction concepts of classical mechanics and the van der Waals theory of fluids, the authors developed the f-theory model to obtain highly accurate modeling of viscosity of pure fluids and mixtures from low to elevated pressures. The capabilities of the f-theory model were verified using two well-known cubic equations of state to reproduce the viscosity of pure alkanes from methane to decane with absolute average deviations (AADs) varying from 1.2 to 4.15% using the five-parameter f-theory model,11 and from 1.06 to 4.44% using the one-parameter f-theory model.12 For the latter, an additional set of compounds (from dodecane to octadecane) was considered by the authors. One of the most interesting viscosity model developments reported in the literature is that of Eyring and co-workers13 with their so-called significant structure theory (SST). This theory represents an intermediate model approach between formal theories of statistical mechanics and purely empirical correlations. It is based on a “quasi-lattice” arrangement aimed to represent the liquid structure thus allowing the development of equations not only for the transport properties but also for the thermodynamics properties of pure fluids and mixtures. Despite its attractive characteristics, to the best of our knowledge, very few applications have appeared dealing with the use of the SST approach in the calculation of transport properties of pure fluids. For example, Eyring and Ree14 developed a SST-based model to calculate the viscosity of argon at different temperatures. During their modeling effort, they introduced two nonadjustable model parameters obtaining AAD values ranging from 6.5 to 60%. Later, Ree et al.15 applied the SST approach to calculate the liquid viscosities of argon, nitrogen, chlorine, ethane, and benzene within a somewhat narrow range of temperatures. Although their resulting model contains no adjustable parameters, the agreement between calculated and experimental data was satisfactory, as reported by the authors. Recently, Hildwein and Stephan16 developed an equation of state and a viscosity model for pure fluids under the basis of the SST approach. Their viscosity model requires 5 adjustable parameters to be determined from experimental data. The authors reported individual values of these parameters for 14 nonpolar and 3 polar compounds. Their correlating results yielded an AAD value varying from 0.2 to 1.6% for dilute gas viscosities. Saturated liquid and vapor viscosities were correlated with AAD values of 2-4.7% and 1.6-6.9%, respectively. On the other hand, single-phase liquid and vapor viscosities were correlated with AAD values of 1.2-4.7% and 1.1-4.7%, respectively. The accuracy of a viscosity model often depends on how well the molar volume of the fluid is estimated or determined at given values of temperature and pressure. This quantity is usually not readily available, thus requiring the use of an equation of state to estimate it. Accordingly, equations for viscosity are often closely
linked with the development of a suitable equation of state. Rather than developing a new equation of state, the basic motivation of this work is to incorporate the use of an existing cubic equation of state into the SST framework to derive a viscosity model capable of accurately representing experimental liquid and gas viscosities along the saturation line for pure fluids (polar and nonpolar) over a wide temperature range (from very near the triple point up to the critical point of the fluid of interest). 2. The Significant Structure Theory The basic ideas behind the Significant Structure Theory (SST) were first introduced by Eyring et al.13 about 4 decades ago. Previous X-ray diffraction studies reveal that a liquid exhibits a microscopy structure characterized by a short-range order rather than a longrange order usually displayed by a solid. Taking this into account, Eyring et al.13 attempted to improve the reaction rate theory by developing the SST approach. In this theory, liquid structure is visualized as a “quasilattice” where sites are partially occupied by molecules. All molecules in the lattice are assumed to have both solidlike and gaslike degrees of freedom. A molecule displays gaslike behavior when it jumps from one vacant site to another. The vacant sites are assumed to have molecular size and to move freely through the lattice. On the other hand, when a molecule remains on its site, for the time it acquires solidlike properties. If V and Vs are the molar volumes of the liquid and solid phases, respectively, thus the difference V-Vs stands for the number of vacancies (or the number of gaslike molecules). Assuming that both occupied and vacant sites are randomly distributed, the fraction of gaslike molecules is
xg )
V - Vs V
(1)
while the fraction of solidlike molecules is given by
xs ) 1 - xg )
Vs V
(2)
Eyring et al.13 assumed that gaslike molecules and solidlike molecules do not interact. The viscosity of any fluid can then be calculated as follows:
η ) xg‚ηg + xs‚ηs
(3)
where ηg is the contribution due to the gaslike behavior of the molecules, and ηs is the contribution of solidlike molecules representing the effects of external forces on the “jumping” process. 3. Formulation of the Model Gaslike Contribution. For the contribution of gaslike molecules ηg, equations based on the ChapmanEnskog kinetic theory of gases typically have been used. For example, Eyring and Ree14 and Ree et al.15 used the elementary gas model of the kinetic theory, whereas Hildwein and Stephan16 applied Sutherland’s equation, a slightly modified nonattracting sphere model (Hirchfelder et al.17). Unlike these previous works, the equations of Chung et al.,18 also based on the kinetic theory of Chapman-Enskog, were used here to calculate the
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viscosity of the gaslike molecules. The method of Chung et al. applies in the limit of dilute gas for nonpolar and polar fluids within a wide range of temperatures. This approach is highly convenient since it allows reliable calculations of dilute-gas viscosities of highly polar substances such as acids and alcohols. The equations of Chung et al. are given below:
xM‚T ηg ) 40.785 2/3 ‚Fc [)] µP Vc Ω*
(4)
ing a constant value for the “jumping” frequency γ, this parameter should vary with the number of vacancies as follows:
γ ) 1 × 1012‚γ0‚xg-1
where γ0 is an adjustable parameter and xg is given in eq 1. Applying these two changes to eq 9, we obtain an expression for the viscosity of the solidlike molecules
ηs )
where an empirical expression is used to estimate the reduced collision integral
0.52487 1.16145 + + (T*)0.14874 exp(0.7732T*) 2.16178 - 6.435 × exp(2.43787T*) 10-4(T*)0.14874 sin[18.0323(T*)-0.7683 - 7.27371] (5)
T* )
1.2593‚T Tc
(6)
The factor Fc in eq 4 serves to account for molecular shapes and polarities of dilute gases and is expressed as follows:
Fc ) 1 - 0.2756ω + 0.059035µr4 + κ
(7)
where ω is the acentric factor, µr is a dimensionless dipole moment (greater than zero for polar substances), and κ is a correction factor for the hydrogen bonding effect of associating substances. Values of κ have been given elsewhere.6,18 Several techniques exist to nondimensionalize a dipole moment.2,6 Among them, the following equation was adopted in this work:
µ µr ) 131.3 (VcTc)1/2
(8)
where the critical molar volume Vc is in cm3/mol, Tc is in kelvins, and µ is in debyes. Solidlike Contribution. The viscosity model for saturated liquids recently presented by Macı´as-Salinas et al.19 served as the framework to calculate the viscosity of the solidlike molecules. The model proposed by the authors is based on Eyring’s Absolute Rate Theory and is given by
ηl )
[(
)]
∆Hvap RT 1 - ∆Zvap ‚ exp R Vl γ RT
β
exp(Zl)
(9)
where γ represents the frequency of a molecule that jumps from its initial position to a vacant site, while R and β are two regression constants used in the powerlaw expression between the activation energy and the internal energy of vaporization ∆Uvap ) ∆Hvap RT∆Zvap. The other variables, Vl, ∆Hvap, and Zl, are the molar volume, the latent heat of vaporization, and the compressibility factor, respectively, at the temperature T and pressure of saturation. To make eq 9 suitable for solidlike viscosity calculations, the liquid molar volume Vl in eq 8 has to be replaced by the difference V - Vs (number of vacancies). In addition, rather than assum-
[(
)]
[(
)]
∆Hvap RT 1 - ∆Zvap ‚ exp R V - Vs γ RT
β
exp(Z) (11)
or
Ω* )
with
(10)
ηs )
∆Hvap RT 1 ‚ 12 exp R - ∆Zvap V 10 γ RT 0
β
exp(Z) (12)
Note that eqs 9 and 12 are almost identical. On the basis of the aforementioned viscosity expressions for ηg and ηs, eq 3 can be readily applied to calculate the viscosity of saturated liquids and gases provided that a suitable equation of state is chosen to compute all the equilibrium properties (V, ∆Hvap, ∆Zvap, and Z) as required by eqs 1, 2, and 12. For this purpose, the cubic equations of state (CEoS) proposed by Soave20 (SRK) and Peng and Robinson21 (PR) were used, thus yielding two modeling versions: the SST-SRK approach and the SST-PR approach. Another important property to determine in the model is the molar volume of the solid at the melting point Vs. This quantity was also conveniently estimated from the CEoS as close-packed by the following approximation:
Vs ≈ b
(13)
where b is the van der Waals co-volume. Its value depends on the CEoS chosen. A quick inspection of all the equations presented here indicates that the proposed model contains only three adjustable parameters, namely γ0, R, and β. These parameters should be determined from experimental saturated viscosities of pure liquids and gases. 4. Application of the Model Eqs 3, 4, and 12 contain the complete description of the viscosity model proposed in this work. The application of the two modeling approaches was carried out in the representation of experimental saturated gas-liquid viscosities for two pure fluids: a nonpolar substance (propane) and a highly polar compound (water). Table 1 gives the pure fluids considered in this work along with their corresponding temperature and pressure ranges, and the source of the experimental viscosity data. A nonlinear regression analysis of the LevenbergMarquardt type was performed to obtain the best set of model parameters γ0, R, and β that closely reproduce the experimental saturated viscosities of both phases over a wide range of temperatures (from near the triple point up to the critical point of the compound of Table 1. Pure Fluids Considered compd
temp, K
pressure, bar
source
propane 110-369.83 4 ×10-6-42.49 Vogel et al.22 water 283.15-647.15 0.0123-220.6 Beaton and Hewitt23
Ind. Eng. Chem. Res., Vol. 44, No. 6, 2005 1963 Table 2. Regressed Parameters and AAD Values compd
N
R
β
propane water overall
53 48 101
0.10942 0.00246
1.132 2.381
propane water overall
53 48 101
0.08812 0.00161
1.212 2.551
T, K
AAD ηl, %
AAD ηg, %
SST-SRK 0.352 1.115
110-369.83 283.15-647.15
3.63 3.72 3.67
2.87 4.02 3.42
SST-PR 0.373 1.223
110-369.83 283.15-647.15
4.62 4.93 4.77
2.17 3.23 2.67
γ0
interest). The minimization of the following objective function served for this purpose: N
min f )
[ ] [ ]
1∑ j)1
cal ηl,j
2
N
+
exp ηl,j
1∑ j)1
cal ηg,j
exp ηg,j
2
(14)
where N is the number of experimental points, and ηcal and ηexp stand for the calculated and observed saturated viscosities of both phases (liquid and gas), respectively. The results obtained from the regression analysis are shown in Table 2 for the two compounds considered in this work. Table 2 includes, for each compound, the adjusted parameters using the two CEoSs, the correlating temperature range, and the values of the absolute average deviation AAD between calculated and experimental viscosities of the two phases in equilibrium. The latter variable was determined using the following equation:
AAD )
100 N
∑ |1 j)1 N
‚
ηcal j ηjexp
|
It is important to note that, for both compounds, the model is incapable of calculating the viscosity of the phases in equilibrium very near or exactly at the melting point Tf. This limitation comes from the assumption previously made through eq 13 in that the model uses an approximate value of the molar volume at the triple point Vs which is set equal to the van der Waals co-volume b obtained from the CEoS used. For
(15)
Table 2 reveals interesting results: for the two compounds considered here and according to their AAD values for liquid viscosity, the SST-SRK model performed better than the SST-PR approach in representing the experimental data. Although it is well established that the PR CEoS yields better volumetric predictions in the liquid phase, it is very likely that the temperature dependency of the thermodynamic properties calculated from the SRK equation of state is more ad hoc within the SST framework for pure fluids. For the gas (or vapor) phase, however, the AAD values obtained using the SST-PR model were lower than those produced by the SST-SRK model. Under these circumstances, it is difficult to ascertain which model approach prevails. Therefore, it will be necessary to incorporate more compounds into this study to clearly establish the modeling “beauties” of each model version. The correlating capabilities of the present model (SST-SRK) are also graphically depicted in Figures 1 and 2 in the representation of experimental viscosities of the liquid and gas (or vapor) along the saturation line for the case of propane and water, respectively, over a wide temperature range. Both figures show the typical behavior exhibited by the viscosity of pure fluids with respect to temperature, that is, a linear dependency between ln η and the reciprocal of the reduced temperature Tr from the normal boiling temperature Tb down to the triple point Tf, and a nonlinear functionality between ln η and Tr-1 within the region Tb f Tc. As evidenced by Figures 1 and 2, the model captures this behavior remarkably well in both regions, particularly in the vicinity of the critical point Tc.
Figure 1. Representation of experimental saturated viscosities for propane.
Figure 2. Representation of experimental saturated viscosities for water.
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Figure 3. Model performance near the critical point for propane.
Figure 4. Model performance near the critical point for water.
the majority of the pure substances, this quantity may not necessarily correspond to the true value of the molar volume at the melting point. For comparison purposes, Figure 1 also includes saturated liquid-gas viscosities for propane obtained from the f-theory model proposed by Quin˜ones et al.11 at the same temperature conditions. A similar comparison for water was not possible since the applicability of the f-theory model has not been yet extended for this type of fluids. As depicted by Figure 1, although both approaches (the f-theory SRK model in its five-constant parametric version and the present model) give comparable results, the f-theory formulation yields a discontinuity for the liquid phase near the triple point, thus failing to represent the liquid viscosity data as this point is approached. A brief performance comparison of our viscosity model was also made against other modeling efforts based on the SST approach. Among the very few SST-based models so far reported in the literature, the viscosity model by Hildwein and Stephan16 was chosen for this purpose. The authors applied their model to 14 nonpolar and 3 polar compounds including water. Their correlating results for water yielded AAD values of 3.2% for liquid viscosities and 6.9% for vapor viscosities along their saturation lines. On the basis of these AAD values and those reported in Table 2 for water, our model in its SST-SRK version gives comparable results for saturated liquid viscosities with an AAD value of 3.7%. For the vapor, the present approach does a better job yielding an AAD value of 4% despite the fact that a much wider temperature range was adopted in this study (283-647 K) than that used by Hildwein and Stephan (450-625 K). Furthermore, the HildweinStephan formulation requires 5 adjustable parameters; 2 parameters more than the viscosity model developed here. To demonstrate the capabilities of the proposed model within the vicinity of the critical temperature, two graphs were prepared showing a zoom in of the region near this point for propane (Figure 3) and water (Figure 4). Both figures show that, as the critical locus is approached, the model tends to underestimate the liquid viscosity and overestimate the gas (or vapor) viscosity. Conversely, the model is capable of successfully representing the critical viscosity of both fluids. This result
Figure 5. Viscosity contributions for liquid water using the SSTSRK model.
is attributed to the mathematical form of the SST itself and not to the nature of the two CEoSs used in this work. Finally, Figures 5 and 6 graphically show how each term in eq 3 distinctively contributes to the value of the viscosity for the case of liquid water and vapor water, respectively. It is clear from Figure 6 that below the normal boiling point of water (Tr ≈ 0.6), the major contribution to ηl comes from the solidlike term. Above Tr ≈ 0.6 the gaslike contribution becomes more important, particularly near the critical point. In fact, at the critical locus, the gaslike term represents almost 50% of the total value of liquid viscosity. Further, as evidenced by Figure 5, it can be readily deduced that liquid viscosity behaves more like a solid over the entire temperature range; the gaslike term serves as a “finetuning” to adequately represent the experimental data. For the case of vapor water, Figure 6 shows contribution trends that somewhat differ from those of liquid water. From a reduced temperature of about 0.7 down to near the triple point (Tr ≈ 0.4), the contribution of
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proposed here may include the use of more accurate equations of state within the framework of the significant structure theory. Accordingly, PVT expressions such as Patel-Teja and SAFT must be strongly considered. Acknowledgment G. C.-R. and R. M.-S. gratefully acknowledge the Instituto Polite´cnico Nacional and COSNET for providing financial support for this work. List of Symbols
Figure 6. Viscosity contributions for vapor water using the SSTPR model.
the solidlike is almost negligible to the total value of ηv. Within this temperature range, the gaslike term prevails as the dominant contribution, thus suggesting that the behavior of vapor viscosity exactly resembles that of the gas at low pressure. As temperature increases from Tr ≈ 0.7 up the critical point, the contribution of the solidlike term becomes increasingly important. Accordingly, as shown in Figure 6, the curve for the total value of ηv mimics the same trend exhibited by the solidlike term within the region near the critical point. In this region, vapor viscosity presumably behaves like a solid or more precisely like a dense gas. As in the case for liquid water, the contribution of the gaslike term serves to accurately reproduce the experimental viscosity data in the vicinity of the critical point. 5. Conclusions A versatile viscosity model based on the significant structure theory coupled with a cubic equation of state has been developed for the simultaneous correlation of viscosities of pure liquids and gases (polar and nonpolar) at saturated conditions. The following conclusions can be drawn from this work. (i) The proposed approach unifies the calculation of saturated viscosities of pure fluids within two important temperature regions: from the normal boiling point Tb down to near the triple point and from Tb up to the critical point (it is a common practice to use different models for each region). (ii) The model describes remarkably well the viscosity behavior of both phases along the entire saturation line making use of a single set of mathematical expressions within the framework of the significant structure theory in combination with a cubic equation of state. (iii) Although the validation of the model was merely exploratory, the results obtained for propane and water demonstrate the potential of the proposed approach in simultaneously modeling gas-liquid viscosities of nonpolar and polar fluids along their saturation curve. (iv) The fitting capabilities of both model versions (SST-SRK and SSTPR) should be further verified by incorporating more substances into the regression process. (v) Further refinements and improvements of the viscosity model
b ) van der Waals co-volume f ) Objective function Fc ) Correction factor as defined in eq 7 ∆Hvap ) Latent heat of vaporization M ) Molecular weight N ) Number of data points P ) Pressure R ) Ideal gas constant T ) Temperature T* ) Reduced temperature as defined in eq 6 ∆Uvap ) Internal energy of vaporization V ) Molar volume x ) Mole fraction Z ) Compressibility factor ∆Zvap ) Difference in compressibility factors (Zv - Zl) Greek Letters R ) Regression constant in eq 9 β ) Regression constant in eq 9 γ ) Jumping frequency of a molecule γ0 ) Regression constant in eq 9 η ) Dynamic viscosity κ ) Association parameter µ ) Dipole moment Ω* ) Reduced collision integral Subscripts b ) Normal boiling point c ) Critical property f ) Melting point g ) Gaslike molecules l ) Liquid phase r ) Reduced property s ) Solidlike molecules v ) Vapor phase Superscripts cal ) Calculated value exp ) Experimental value Literature Cited (1) Chapman, S.; Cowling, T. G. The Mathematical Theory of Nonuniform Gases; Cambridge, New York, 1939. (2) Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. (3) Mehrotra, A. K. A Generalized Viscosity Equation for Pure Heavy Hydrocarbons. Ind. Eng. Chem. Res. 1991, 30, 420, 1367. (4) Monnery, W. D.; Svrcek, W. Y.; Mehrotra, A. K. A Critical Review of Practical Predictive and Correlative Methods. Can. J. Chem. Eng. 1995, 73, 3. (5) Mehrotra, A. K.; Monnery, W. D.; Svrcek, W. Y. A Review of Practical Calculation Methods for the Viscosity of Liquid Hydrocarbons and their Mixtures. Fluid Phase Equilib. 1996, 117, 344.
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(6) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (7) Little, J. E.; Kennedy, H. T. A Correlation of the Viscosity of Hydrocarbon Systems with Pressure, Temperature and Composition. Soc. Pet. Eng. J. 1968, 8, 157. (8) Wang, L.; Guo, T. A Unified Viscosity Model for Hydrocarbon Gases and Liquids Based on Transposed Patel-Teja Equation of State. J. Chem. Ind, Eng. (China) 1991, 6, 38. (9) Guo, X.-Q.; Wang, L.-S.; Rong, S.-X.; Guo T.-M. Viscosity Models Based on Equations of State for Hydrocarbon Liquids and Gases. Fluid Phase Equilib. 1997, 139, 405. (10) Guo, X.-Q.; Sun, C.-Y.; Rong, S.-X.; Chen, G.-J.; Guo, T.M. Equations of State Analogue Correlations for the Viscosity and Thermal Conductivity of Hydrocarbons and Reservoir Fluids. J. Pet. Sci. Eng. 2001, 30, 15. (11) 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. (12) Quin˜ones-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. One Parameter Friction Theory Models for Viscosity. Fluid Phase Equilib. 2001, 178, 1. (13) Eyring, H.; Ree, T.; Hirai, N. Significant Structures in the Liquid State. I. Proc. Natl. Acad. Sci. U.S.A. 1958, 44, 683. (14) Eyring, H.; Ree, T. Significant Structures in the Liquid State. II. Proc. Natl. Acad. Sci. U.S.A. 1961, 47, 526. (15) Ree, T. S.; Ree, T.; Eyring, H. Significant Structure Theory of Transport Phenomena. J. Phys. Chem. 1964, 68, 3262.
(16) Hildwein, H.; Stephan, K. Equation of State and Equations for Viscosity and Thermal Conductivity of Non-Polar and Polar Pure Fluids Based on the Significant-Liquid-Structure-Theory. Chem. Eng. Sci. 1993, 48, 2005. (17) Hirschfelder, J. O.; Curtis, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids, 4th ed.; Wiley: New York, 1967. (18) 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. (19) Macı´as-Salinas, R.; Garcı´a-Sa´nchez, F.; Herna´ndez Garduza, O. Viscosity Model for Pure Liquids Based on Eyring Theory and Cubic EoS. AIChE J. 2003, 49, 799. (20) Soave, G. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197. (21) Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59. (22) Vogel, E.; Kuchenmeister, C.; Bich, E.; Laesecke, A. Reference Correlation of the Viscosity of Propane. J. Phys. Chem. Ref. Data 1998, 27, 947. (23) Beaton, C. F.; Hewitt, G. F. Physical Property Data for the Chemical and Mechanical Engineer; Hemisphere: New York, 1989.
Received for review September 23, 2004 Revised manuscript received December 22, 2004 Accepted January 12, 2005 IE049070V
