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A Modeling Framework for Predicting and Correlating Viscosities of Liquids in Wide Range of Conditions Ilya Polishuk* Department of Chemical Engineering & Biotechnology, Ariel University, 40700 Ariel, Israel S Supporting Information *
ABSTRACT: This communication reports a modeling framework for estimating viscosities in a wide range of conditions that couples the critical point-based revised perturbed chain statistical association fluid theory equation of state (CP-PC-SAFT EoS) and the modified Yarranton−Satyro correlation (MYS), yet is generalized by SAFT molecular parameters. The proposed approach requires as an input just critical constants and the triple point liquid densities, while the same triplet of MYS parameters can predict viscosities of wide variety of compounds including n-alkanes (from methane and until n-octadecane), various aromatic and nonaromatic hydrocarbons, and oxygenated organic substances. In addition, it can yield particularly accurate predictions for saturated and single-phase viscosities of asymmetric mixtures. Consequently, this modeling framework is characterized by an enhanced predictive character.
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the MYS parameters. Later on, Abolala et al.23 have successfully coupled MYS with SAFT-VR-Mie24 EoS for modeling viscosities of additional imidazolium-based ionic liquids.
INTRODUCTION Quantitative correlation of liquid viscosities in wide range of conditions may present difficulties even for the multiparameter reference correlations.1−3 Consequently, predicting these data is an exceptionally challenging task. The phase diagrams of viscosities are much more sophisticated in comparison to equilibrium thermodynamic properties such as densities or sound velocities. Unlike these properties, viscosities can rapidly change in the relatively narrow ranges of temperatures. This phenomenon becomes particularly pronounced at high pressures and in vicinity of solidification. In addition, the values of viscosities are strongly dependent on molecular size and geometry. As a result, light and heavy compounds with similar densities may have viscosities varying by orders of magnitude. This feature indicates that a major progress in predicting viscosities can be achieved by considering the molecular parameters of theoretically based approaches such as molecular simulations or various versions of the statistical association fluid theory (SAFT) equations of state (EoS). Among the innovative studies implementing the molecular parameters for estimating viscosities, the molecular approach of de Wijn et al.4,5 should be noticed. In this respect, the contributions of Galliero,6 Zhao et al.,7 Quiñones-Cisneros et al.,8 Tan et al.,9 and Quiñones-Cisneros and Deiters10 should be acknowledged as well. It has also been demonstrated that qualitatively accurate results can be achieved by providing various viscosity models by densities or residual entropies generated by SAFT equations with further generalization of the viscosity coefficients not only by molecular weights,11,12 but also by SAFT molecular parameters.13−15 In the previous study,16 a modeling framework coupling SAFT+Cubic17,18 with the density-based modified Yarranton− Satyro19−22 correlation (MYS) comprising the molecular diameter σ of that EoS has been proposed. It has been shown that this approach allows predicting viscosities of sets of compounds, such as n-alkanes along with some heavy compounds, certain homologues series of ionic liquids, etc., while using the same triplets of © 2015 American Chemical Society
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METHODS This communication reports the preliminary results of an attempt of further enhancing the predictive character and transparency25 of the SAFT-MYS modeling framework, which have been achieved by two modifications. The first one is implementing the recently proposed26,27 critical point (CP)based version modifying the popular perturbed-chain (PC)SAFT EoS. The details of this model are provided in the Supporting Information. Unlike many SAFT approaches whose substance-dependent parameters are typically evaluated by fitting relatively large and sometimes vague experimental databases, CP-PC-SAFT applies their standardized numerical solution at the characteristic states, namely the critical points. In addition, it requires one liquid density point datum, normally at the triple point. In other words, this approach can be characterized by an advanced predictive capacity thanks to the substantial reduction of the required experimental data. A second modification made in the current revision of the MYS is a further introduction of the CP-PC-SAFT’s molecular parameters in its expression in empirical manner. In particular, by following the regularities exhibited by the results, it has been found that an MYS parameter c1 responsible for estimating the viscosity data at the ambient conditions can typically be correlated by √m (the effective chain length), while c2, a parameter responsible for the temperature dependence of viscosity, can often be related with m3. In addition, responsible for the pressure dependence parameter, c3 is apparently proportional to ε/kb (segment energy parameter divided by Received: Revised: Accepted: Published: 6999
April 18, 2015 June 1, 2015 June 23, 2015 June 23, 2015 DOI: 10.1021/acs.iecr.5b01468 Ind. Eng. Chem. Res. 2015, 54, 6999−7003
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replace the MYS reference temperature of 330 K by Ttriple point + 120 K. After some additional minor changes, the resulting revised MYS has been obtained as
Boltzmann’s constant), and even better correlation is achieved by ε/kbm1/4. And finally, to cover light and heavy compounds by the same triplet of c1-c3, it has been found expedient to
⎛ ⎧ ⎜ ⎪ MW 4 ⎜ ⎪ T ⎜ ⎪ c1 m + ln 1 + c2v 4m3 ln TTP + 120 η (mPa s) = 0.1⎜exp⎨ cP ⎫ ⎜ ⎪ ⎧ ⎪ 1.04v exp ε / k b −(∂P3 / ∂v)T mv ⎪ ⎬− 1 − ⎜ ⎪ exp⎨ 3 NAvm(σ*) ⎜ ⎪ ⎪ ⎪ ⎭ ⎝ ⎩ ⎩
{
}{
{
}
⎞ ⎫ ⎟ ⎪ ⎟ ⎪ ⎪ ⎟ ⎬ − 1⎟ + η0 ⎪ ⎟ 1⎪ ⎟ ⎪ ⎟ ⎭ ⎠
}
(1)
Here, v is the molar volume (L/mol) yielded by CP-PCSAFT, P is pressure (bar), T is temperature (K), σ is the segment diameter (Å), MW is the molecular weight (g/mol), and NAv is Avogadro number. η0 is the zero density viscosity, typically having a negligible small contribution to the viscosities of liquids. It can been obtained using the Chung et al’s correlation28 and implemented to mixtures according to the mixing rule of Wilke.29 It should be pointed out that the regular SAFT’s mixing rule for σ can yield satisfactory results for viscosities of symmetric mixtures. However, an overall accuracy of the current approach is improved by implementing a slightly modified mixing rule specific for eq 1: σ* =
3
∑i ∑j xixj(miimjj)1.05 σij 3 ∑i (ximii)2.1
(2)
and m = ∑ximii, TTP = ∑xiTTP,ii, MW = ∑xiMw,ii. Remarkably, eq 2 is closer the regular SAFT’s mixing rule (see the Supporting Information) than the previously proposed method.30 An additional advantage of the proposed approach is the fact that the following triplet of adjustable parameters
Figure 1. Schematic representation of the proposed modeling framework.
c1 = 0.27 c 2 = 2.5 × 1011 c3 = 2.1
(3)
yields satisfactorily accurate results for particularly large variety of hydrocarbons and their mixtures. Consequently, in many cases, it can be considered as an entirely predictive method, whose schematic representation is depicted by Figure 1.
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RESULTS AND CONCLUSIONS An improvement achieved by generalizing the MYS parameters by ε/kb, σ, and m instead of MW can be demonstrated while considering two compounds with different chemical backgrounds and the almost identical molecular weights. By chance, the density and viscosity data are available31,32 for a pair of such compounds, namely, n-octane and dipropyl ketone at the same temperature (∼283 K, see Figure 2). In this case, the success of generalizing the MYS parameters with MW is defined only by the mutual density−viscosity interrelation of the compounds under consideration. As seen, dipropyl ketone has a higher density, whose value at the atmospheric pressure is reached by n-octane only around 2800 bar. Coincidentally, the viscosity of dipropyl ketone is higher as well. However, its value at the atmospheric pressure is reached by n-octane at the much lower pressure around 400 bar. Consequently, an attempt to generalize c1−c3 for n-octane and dipropyl ketone only by
Figure 2. Comparison of densities and viscosities of n-octane (283.15 K) and dipropyl ketine (283 K). Points, experimental data.31,32 Lines on the density plot, predictions of CP-PC-SAFT. 7000
DOI: 10.1021/acs.iecr.5b01468 Ind. Eng. Chem. Res. 2015, 54, 6999−7003
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Figure 3. Saturated high pressure viscosities of asymmetric n-alkane systems. Points, experimental data.33−36 Lines, predictions of the proposed approach.
MW cannot be effective. Indeed, consideration of c1−c3 independently of the SAFT molecular parameters (which intrinsically reflect the molecular geometry and chemical background) is equivalent to application of c1−c3 attached by ε/kb, σ, and m one compound to another. Figure 2 demonstrates that the results of such a practice are particularly inaccurate. Remarkably, similar effects are observed while the viscosity parameters of one compound are transferred to another even within the homologues series of compounds such as n-alkanes also in the cases of other viscosity approaches.16 Predictions yielded by the proposed modeling framework for viscosities of representative pure compounds in wide variety of conditions, including various aromatic and nonaromatic hydrocarbons and oxygenated organic substances, can be found in the Supporting Information. These results are compared with the previous modeling framework.16 It should be pointed out that unlike CP-PC-SAFT, implementation of SAFT+Cubic comprises adjustment of the pure compound parameters. In particular, for n-alkanes, the parameter m and in the cases of other nonassociating compounds two parameters, namely m and c, are fitted to the experimental data. Consequently, a comparison between these modeling frameworks can hardly be considered as entirely fair. Nevertheless, to make this comparison more equitable, a triplet of parameters evaluated for n-alkanes has been implemented for all the considered compounds. Although the overall AAD% are typically similar for both approaches, it can be seen that the proposed modeling
framework is usually advantageous when predicting the higher values of viscosity. At the same time, it should be emphasized that eq 3 is still not universal. Numerous compounds, such as benzene, carbon dioxide, nitrogen, acetone, 1-alkanols, haloorganic substances, etc., require different values of c1−c3. Moreover, the current predictive parametrization scheme26,27 of CP-PC-SAFT can hardly be applied for modeling heavy compounds whose reported critical constants can often be deceptive. In such cases, the model could be fitted to the data similarly to other SAFT equations. Development of the pertinent generalized parametrization scheme is currently under progress, and the preliminary results indicate that the triplets of c1−c3 parameters covering heavy aromatic substances and series of ionic liquids could be proposed as well. Figure 3 depicts the predictions of eqs 1−3 for saturated viscosities of asymmetric n-alkane systems. The binary parameter k12 = −0.022 has been previously26 evaluated for the available in wide PVT range VLE of the system methane+ndecane. Particular fitting of other systems included by Figure 3 has not been performed since their VLE data cover much narrower condition ranges. However, the same value of k12 yields good results for VLE of methane+n-octadecane and ethane+n-tetradecane as well. In the case of the available VLE in ethane+n-octadecane, the zero k12 seems to be appropriate. The figure demonstrates that the proposed approach yields particularly accurate predictions of the challenging viscosity data under consideration. 7001
DOI: 10.1021/acs.iecr.5b01468 Ind. Eng. Chem. Res. 2015, 54, 6999−7003
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Figure 4. Single phase high pressure viscosities of mixtures at the elevated pressures. Points, experimental data.37−40 Lines, predictions of the proposed approach.
compared. Comprehensive comparison of the results for methane, n-dodecane, methyl n-hexyl ketone, and dipropyl ether. Details of the CP-PC-SAFT EoS and the values of its substance specific parameters. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b01468.
Figure 4 considers the single-phase viscosities of additional mixtures at the elevated pressures. Two of them, namely methane+n-decane37 and methane+toluene,38 so far have been treated by the previous approach,30 which contrary to the current one has applied different values of c1−c3 for all three compounds under consideration. However, the overall predictions of the first system are more accurate (AAD% = 10.25% vs 13.09%) and slightly disadvantageous in the case of the second one (AAD% = 6.623% vs 5.151%). As seen, the proposed approach requiring an input of critical constants and the triple point liquid densities can predict viscosities of a wide variety of compounds including n-alkanes (from methane and until n-octadecane), various aromatic and nonaromatic hydrocarbons, and oxygenated organic substances. In addition, it can yield particularly accurate predictions for saturated and single-phase viscosities of asymmetric mixtures. Further efforts will be focused on accumulating the high pressure viscosity data of additional compounds and mixtures to delineate the validity range of the proposed approach and develop the MYS c1−c3 parameters for groups of compounds mismatching the triplet given by eq 3.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: +972-3-9066346. Fax: +972-3-9066323. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The author would like to thank Professor Ilmutdin Abdulagatov from the National Institute of Standards and Technology for the fruitful discussion and kind supply of unreachable data sets.
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ASSOCIATED CONTENT
REFERENCES
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S Supporting Information *
Predictions of viscosities of various pure compounds. Performances of CP-PC-SAFT coupled with the proposed version of MYS eqs 1−3, and the previous modeling framework16 attached by a triplet of parameters evaluated for n-alkanes is 7002
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