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Modeling the Viscosity of Liquid Mixtures: Polymer-Solvent Systems Lawrence T. Novak† The Lubrizol Corporation, 29400 Lakeland Boulevard, Wickliffe, Ohio 44092-2298
A theory-based model for liquid mixture viscosity is presented and evaluated using data from polymer-solvent systems. The model is based on the Eyring mixture viscosity model and the NRTL model for describing deviations from ideality. Comparisons of predicted and actual viscosity over the entire concentration range illustrate a significant improvement over the conventional liquid mixture viscosity model. The model is structurally capable of handling multicomponent mixtures and could be incorporated into process modeling and simulation software. Introduction Viscosity is a physical property that is often embedded in equipment and process models used by process engineers to design and optimize equipment and chemical processes. On numerous occasions the author has been challenged by process engineering applications requiring a viscosity model for complex liquid mixtures covering a wide range of molecular weights. The purpose of this paper is to present a theory-based liquid mixture viscosity model and to illustrate its capability to predict polymer-solvent mixture viscosity. Data from polymer-solvent systems covering the entire range of composition and covering a range of temperature were used to evaluate the model.
tion, a subset of the DIPPR viscosity model for pure liquids:
ln(η) ) A(1/T)B
(4)
Model Development Equations 1 and 2 can be combined to give eq 5 when the Eyring empirical constant is set to 1.0.
ln(ηmVm) ) x1 ln(η1V1) + x2 ln(η2V2) - ∆GE/RT (5)
(1)
The first two terms on the right-hand side of eq 5 can be thought of as the “ideal part”. ∆GE/RT is the “nonideal part”. In the model evaluation reported here, the mixture molar volume (Vm) was calculated from the mixture composition and pure-component densities. The NRTL model2 was used to predict the effect of composition and temperature on the excess Gibbs free energy (∆GE) in eq 5. Like the Wilson and UNIQUAC models, the NRTL model is based on a local composition model representing the nonrandom distribution of molecules in a mixture. The NRTL model for binary mixtures can be written as
For mixtures of simple liquids, Eyring and co-workers1 proposed the following equation for a binary mixture:
∆GE/RT ) x1x2[τ21G21/(x1 + x2G21) + τ12G12/(x1G12 + x2)] (6)
Theoretical Background Eyring and co-workers1 combined the theory of absolute reaction rates with a vacancy model for liquids to derive an equation for the viscosity of pure liquids. Their well-known equation is
ηV ) Nh exp(∆Gq/RT)
ηmVm ) Nh exp[(x1∆G1q + x2∆G2q - ∆GE/2.45)/RT] (2) One can view eq 2 as consisting of an “ideal part” and a “nonideal part” (∆GE). For “not too imperfect” solutions, Eyring and co-workers1 reduced eq 2 to
ln(ηm) ) x1 ln(η1) + x2 ln(η2)
(3)
Equation 3 is the conventional liquid mixture viscosity model based on empirical observations.1 In multicomponent form, eq 3 is the default liquid mixture viscosity model in some commercial process modeling and simulation software. The polymer, solvent, and polymer-solvent mixture viscosity data used in this work were all found to be Newtonian and to be described by the following equa† Tel.: (440)347-5558. Fax: (440)347-5948. E-mail: ltn@ Lubrizol.com.
τ12 ) (g12 - g22)/RT
(7)
τ21 ) (g21 - g11)/RT
(8)
G12 ) exp(-R12τ12)
(9)
G21 ) exp(-R12τ21)
(10)
Equations 5-10 will be referred to as the EyringNRTL model. Parameters g12 - g22, g21 - g11, and R12 are the temperature-independent parameters known as the NRTL binary interaction parameters. In theory, these parameters should be the same binary interaction parameters as those used in the NRTL model to predict vapor-liquid equilibria in multicomponent systems. Variants of the above Eyring-NRTL model have been reported in the literature and evaluated with data from nonpolymer mixtures. Wei and Rowley3 used the Eyring and NRTL models to develop a liquid mixture model that is different from the above Eyring-NRTL model.
10.1021/ie030051f CCC: $25.00 © 2003 American Chemical Society Published on Web 03/26/2003
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Like Eyring, they incorporated the idea of excess free energy into the mixture activation free energy. They went further by expressing the excess free energy in terms of excess enthalpy and excess entropy. Excess enthalpies were determined from the NRTL model, and excess entropies were obtained by applying the NRTL local composition model to sum up intermolecular viscous interactions. Their analysis resulted in a mixture viscosity equation containing an “ideal part” with volume fraction weighting and a “nonideal part” containing NRTL parameters. Evaluations with 47 binary and 7 ternary systems resulted in mean viscosity deviations of 4.4% for the binary and 2.7% for the ternary. Cao et al.4 combined the Eyring and UNIQUAC models. Their analysis resulted in an “ideal part” with volume fraction weighting and a “nonideal part” containing UNIQUAC parameters. Evaluations on 288 binary systems resulted in root-mean-square relative viscosity deviations of 0.9%. However, binary parameters determined from viscosity measurements produced only qualitative predictions when used to predict vaporliquid equilibria. For mixtures containing more than two components, eq 5 can be generalized to
ln(ηmVm) )
∑i xi ln(ηiVi) - ∆GE/RT
Figure 1. Polymer A and n-heptane mixture viscosity at 50 °C.
(11)
and the NRTL model can also be generalized to multicomponent mixtures.
Figure 2. Polymer B and distillate mixture viscosity at 50 °C.
Data and Model Parameter Determination The Eyring-NRTL model was evaluated with data from the following pseudobinary mixtures: (1) polymer A and n-heptane; (2) polymer B and distillate. Polymers A and B are branched polyolefins. Mixtures of these polymers with solvents are referred to as pseudobinaries because the polymers and distillate are actually mixtures instead of pure components. The above pseudobinary mixtures were prepared to cover the entire mole fraction range. A Lubrizol research cone-and-plate rheometer was then used to determine polymer, solvent, and polymer-solvent mixture viscosities over a range of temperature. The experimental viscosity values used in this study were taken from the respective viscosity data-temperature fits using eq 4. Density as a function of temperature was obtained from the DIPPR databank for n-heptane and from Lubrizol measurements on distillate and polymers. Polymer Mn was determined by GPC measurement, and distillate Mn was determined from composition measurement. The NRTL binary interaction parameters (g12 - g22, g21 - g11, and R12) were determined by a constrained Gauss-Newton least-squares algorithm using a fixed value of R12 ) 0.2. Because viscosity data ranged from 1 × 104 Pa‚s, the algorithm determined the NRTL binary interaction parameters that minimized the square error of log(viscosity). Results and Discussion In polymer-solvent systems, there are several conceptual problems with the Eyring-NRTL model. First, when there is an order of magnitude difference in the pseudobinary molecular weights, the same-sized billiard ball type model used by Eyring and co-workers is obviously not physically correct. Second, evidence5 sug-
Figure 3. Polymer A and n-heptane mixture viscosity at 0, 50, and 100 °C. Table 1. NRTL Binary Interaction Parameters from Fitting Mixture Viscosity Data polymer
polymer Mn
solvent
A B
2917 2273
n-heptane distillate
solvent g12 - g22 g21 - g11 Mn (kJ/mol) (kJ/mol) R12 100 273
-33.46 -24.74
9.90 0.0
0.2 0.2
gests that large molecules flow by a sequence of segment movements rather than as a single unit. Despite these conceptual problems, it is surprising how well the Eyring-NRTL model can fit polymer-solvent viscosity data (see Figures 1-3). The least-squares fit NRTL parameter values used to generate the model predictions in Figures 1-3 are listed in Table 1. Figure 1 illustrates the Eyring-NRTL model capability to fit polymer A and n-heptane mixture viscosity data over the entire composition range at a fixed temperature. Figure 2 illustrates the Eyring-NRTL model capability to fit polymer B and distillate mixture viscos-
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ity data over a more limited range of composition at a fixed temperature. For these polymer-solvent mixtures, the Eyring-NRTL model provides a significant improvement over the conventional liquid mixture viscosity model (straight line). The average absolute mixture viscosity deviation for the very limited data in Figure 2 is 0.2%. A mixture viscosity model must predict the viscosity as a function of composition and temperature. Figure 3 illustrates the Eyring-NRTL model predictions over the entire composition range and the 0-100 °C range for polymer A and n-heptane. Use of eq 4 to fit individual viscosity data points at fixed composition produced a temperature dependency (B) ranging from 1.0 for solvents to about 2.3 for polymers. The viscosity deviations in Figure 3 suggest a temperature dependency that is different from the excess free energy model. At 0, 50, and 100 °C, the average absolute mixture viscosity deviations are 63%, 28%, and 35%, respectively. Improved data fits were obtained by fitting NRTL parameters to isothermal viscosity data. In view of previously cited literature, the NRTL parameters listed in Table 1 should be viewed as parameters in a liquid mixture viscosity model and not as thermodynamic parameters for predicting vaporliquid equilibria with the NRTL model.
Nomenclature A ) empirical constant in eq 4 B ) temperature dependency empirical constant in eq 4 g12 - g22 ) NRTL binary parameter for temperature dependency g21 - g21 ) NRTL binary parameter for temperature dependency ∆Gq ) Gibbs free energy change of activation for a molecule to jump to a hole ∆GE ) excess Gibbs free energy h ) Planck constant Mn ) number-average molecular weight N ) Avogadro’s number R ) gas constant T ) absolute temperature V ) molar volume x ) mole fraction Greek Symbols R12 ) NRTL binary parameter η ) viscosity Subscripts i ) component or pseudocomponent i m ) mixture 1 ) polymer 2 ) solvent
Literature Cited Conclusions For the polymer-solvent systems studied, the Eyring-NRTL model represents a significant improvement over the conventional liquid mixture viscosity model. With a mixture viscosity binary databank, the multicomponent version of the Eyring-NRTL model could be incorporated into process modeling and simulation software. Acknowledgment The author acknowledges Ross L. Beebe and Herman F. George, The Lubrizol Corp., for providing experimental support in this study. The author also thanks The Lubrizol Corp. for supporting this work and providing the approval for publication.
(1) Powell, R. E.; Roseveare, W. E.; Eyring, H. Diffusion, Thermal Conductivity, and Viscous Flow of Fluids. Ind. Eng. Chem. 1941, 33, 430. (2) Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J. 1968, 14, 135. (3) Wei, I. C.; Rowley, R. L. A Local Composition Model for Multicomponent Liquid Mixture Shear Viscosity. Chem. Eng. Sci. 1985, 40, 401. (4) Cao, W.; Knudsen, K.; Fredenslund, A.; Rasmussen, P. Simultaneous Correlation of Viscosity and Vapor-Liquid Equilibrium Data. Ind. Eng. Chem. Res. 1993, 32, 2077-2087. (5) Kauzmann, W.; Eyring, H. The Viscous Flow of Large Molecules. J. Chem. Phys. 1940, 62, 3113.
Received for review January 16, 2003 Revised manuscript received March 3, 2003 Accepted March 6, 2003 IE030051F
