Article pubs.acs.org/IECR
Predicting Fluid Viscosity of Nonassociating Molecules Lawrence T. Novak* Department of Chemical and Biomedical Engineering, Cleveland State University, 2121 Euclid Avenue, SH455, Cleveland, Ohio 44115-2214, United States ABSTRACT: Previous work by the author demonstrated that a new predictive corresponding-states viscosity model, based on entity Chapman−Enskog scaling, correlated pure n-alkane component viscosity data with entity scaled residual entropy. With five fitting constants, a 5.2% average absolute relative deviation (AARD) was obtained over the entire fluid region for a group of 17 nalkanes, ranging from methane to 1280 Mw linear polyethylene. The work reported here demonstrates that this same model and fitting constants predicts fluid viscosity of other molecular classifications over a wide range of temperature and pressure. Molecular structure was incorporated to improve the predictability of the earlier corresponding-states viscosity model. With the inclusion of molecular structure, an AARD of 6.6% was obtained over the entire fluid region for ten classifications of nonassociating molecules and a wide range of temperature (60−800 K) and pressure (1−46 800 atm). Thus, a predictive corresponding-states fluid viscosity model for the more general class of nonassociating molecules was developed to provide a practical approach to determining viscosity in process and petroleum engineering applications.
■
original predictive corresponding-states viscosity model5 to the more general class of nonassociating molecules. A summary of the author’s correlative and predictive viscosity model equations is given below to assist readers in understanding results from this study. Variables are defined in the Nomenclature section.
INTRODUCTION
It was originally found1,2 that scaling pure component transport coefficients of n-alkanes with Chapman−Enskog3 values resulted in highly correlated relationships between the scaledtransport coefficients and residual entropy for viscosity, selfdiffusion coefficient, and thermal conductivity. Subsequent work4,5 replaced Lennard-Jones parameters with PC-SAFT segment parameters and then entity parameters. With entity Chapman−Enskog scaling and entity residual entropy scaling, all n-alkane viscosity data was found to correlate to a single semilog line over the entire fluid region, providing a predictive corresponding-states viscosity model. The entity mixture version of this model was also proven useful in predicting the viscosity of several Qatari natural gases over a wide range of temperature (250−450 K) and pressure (99−642 atm).6 Detailed background, model development, and evaluation for correlative segment Chapman−Enskog scaling4 and predictive entity Chapman−Enskog scaling5 was provided previously. Evaluating these models4,5 over a wider range of conditions (components, temperatures, and pressures) demonstrated performance comparable with other popular models.5 These Chapman−Enskog type scaled models are based on the use of PC-SAFT for calculating residual entropy. Although the use of PC-SAFT to determine residual entropy does not require critical properties, it does require molecular properties called segment parameters.7 Literature sources for these segment parameters are available,7,8 and missing segment parameters may be determined from density and vapor pressure data, or estimated by group contribution methods.9,10 In this work, n-alkanes are considered to move as simple chains over the entire fluid region. However, molecules such as neopentane, cyclohexane, and N2, for example, deviate from a simple chain structure. So, nonchain molecular movement and transport coefficients would be expected to deviate from molecules with chain-like movement. Structural models are proposed below (eqs 10 and 11) to extend the capability of the © XXXX American Chemical Society
⎛ η/N ⎞ seg # ⎟ = ln(A ) + B (S res/(N k )) ln(ηseg ) = ln⎜⎜ seg seg seg B ⎟ ⎝ (ηseg )CE ⎠ (1)
(ηseg )CE =
5 mseg kBT /π 16 σseg 2 Ω(2,2) *
(2)
Correlative Segment Chapman−Enskog Scaled Viscosity Model.4 With the above model, Aseg and Bseg values are determined by fitting component viscosity data over a range of conditions. Examples of typical raw component viscosity data− temperature plots have been illustrated previously for nalkanes.2,5 All of the raw component viscosity−temperature data are reduced to a single semilog line with component parameters Aseg and Bseg, in accordance with eq 1. Aseg and Bseg values vary with the component. However, Aseg and Bseg values for methane are close to 1.0 and −1.0, respectively. For methane, the segment and entity values are close in value and fit the predictive corresponding states viscosity model (eq 3). A 3.9% average absolute relative deviation (AARD) was obtained for correlating 17 n-alkanes.4 Predictive Entity Chapman−Enskog Scaled Viscosity Model.5 In the entity Chapman−Enskog scaled viscosity model, segments are replaced by entities.5 Received: April 22, 2015 Revised: May 11, 2015 Accepted: May 15, 2015
A
DOI: 10.1021/acs.iecr.5b01526 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Industrial & Engineering Chemistry Research ⎛ η/N ⎞ entity # ⎟ = −(S res/NentitykB) ln(ηentity ) = ln⎜⎜ ⎟ ( η ) ⎝ entity CE ⎠ (ηentity )CE =
5 mentity kBT /π 16 σentity 2 Ω(2,2) *
■
RESULTS AND DISCUSSION Scope of Study. The experimental viscosity data scope and sources used in this study are listed in Table 1. The larger bold print in the left column defines the chosen molecular categories. And, the smaller print lists the components contained in the categories. The gray background identifies components considered to be nonchain-type 1, and the diagonally cross print background represents nonchain-type 2 that needed another set of 4-parameter constants for more accurate prediction in this category. Parameters used for prediction in this study are listed in Table 2. The five parameters (a1−a5) for chain-type molecules were determined previously for n-alkanes5 and initially used to predict viscosities of all compounds in this study. To improve predictions for some compounds, it was found necessary to consider model parameters for nonchain type molecules, as listed in Table 2 for eqs 10 and 11. Attempts at defining molecular structure in terms of some well-known structural models34,35 and Monte Carlo simulations of single alkane molecules, by the author, did not appear to be a better approach to incorporating molecular structure into the viscosity model. Although a strict mathematical definition for chain and nonchain categories was not used, it was observed that there is a qualitative difference in molecular structure been the chain and nonchain molecules listed in Tables 1 and 3. For example, with 2,2-dimethyl groups on the second carbon of shorter alkanes, the n-alkane chain model parameters (a1−a5) underpredict viscosity, and a new set of four parameters (a6−a9) are needed. This nonchain-type 1 set of four parameters was also found applicable to other nonchain molecules such as benzene, some cyclics, and N2, O2, CO, CO2. Using nonchain-type 1 parameters, neopentane and cyclohexane viscosities were under predicted by roughly 10%. With nonchain-type 2 parameters (a10−a13), AARD’s were reduced to 2.8−4.5%, respectively. Based on current data, the use of chain and nonchain parameters appears to be a reasonably practical approach for introducing molecular structure to broaden the capability of previous work.5 Segment-Based Viscosity Model Correlation for Nonassociating Molecules. Table 3 lists viscosity component correlations obtained by fitting component viscosity data to eq 1. The AARD for correlating viscosity data with this model for all 73 components and 6259 data points is 3.6%, with average R2 = 0.9990. The range of temperature and pressure given in Table 1 illustrates that this model handles the entire fluid region for nonassociating molecules, over a wide range of temperature (60−800 K) and pressure (1−46 800 atm). Corresponding-States Viscosity Model Predictions for Nonassociating Molecules. Using only the five parameters (a1−a5) listed in Table 2, the ARD’s and AARD’s were −4.0% and 9.9%, respectively for all of the viscosity data from 73 components. The worst predictions were for the C4−C10 2,2dimethylalkanes, C5−C10 cyclics, and N2, O2, CO, and CO2. The ARD’s and AARD’s for these three classes were (−16.4%, −21.1%, and −16.8%) and (20.7%, 23.2%, and 16.8%), respectively. These findings demonstrate that the chain model was generally under predicting viscosity in these molecular classifications. It was postulated that certain molecular structures and substructures could be influencing momentum transfer in a manner different than chain molecules (n-alkanes). Thus, the
(3)
(4)
It was proposed5 that the following relationships existed between entity and segment parameters and a molecular attribute (ξ), thereby providing predictability for the entitybased model. (σentity /σseg)2 = f (ξ)
(5)
(Nseg /Nentity ) = g (ξ)
(6)
Equations 3−6 combine to rewrite eq 3 in terms of segment variables and ξ. # ln(ηseg f (ξ) g (ξ) ) = −g (ξ)(S res/NsegkB)
(7)
Based on evaluating several molecular properties for ξ, the following five empirical parameter (a1−a5) ξ-model was chosen, using the number of segments (Nseg) as the molecular attribute (ξ).5 Using eqs 7−9, component viscosity data can be calculated according to eq 3. In theory, all component viscosity data should fall along the corresponding-states semilog line (eq 3) with slope and intercept of −1 and 1, respectively. For the chain model ⎧ ⎫ ⎪ ⎪ a4 ⎬ g (Nseg) = a3exp⎨ ⎪ ⎪ ⎩ (Nseg + a5) ⎭
(8)
⎡ g (Nseg) ⎤(1/ a2) ⎥ f (Nseg) g (Nseg) = ⎢ ⎢⎣ a1 ⎥⎦
(9)
A 5.2% AARD was obtained for predicting the viscosity of 17 nalkanes, ranging from methane to linear polyethylene with a molecular weight of 1280 g/mol.5 The above segment and entity viscosity scaling models have been primarily evaluated with n-alkanes. In this study the following nonchain models are proposed to broaden the capability of the corresponding-states viscosity model. Nonchain molecules in this study are viewed as being more ordered on a molecular or submolecular level and lack the freedom of movement and disorder of n-alkane carbon atoms. For the nonchain type 1 model g (Nseg) = a6 ln(Nseg) + a 7
(10)
⎛ ⎛ g (N ) − a ⎞ ⎞ seg 8 ⎟⎟⎟⎟ f (Nseg) g (Nseg) = exp⎜⎜ −⎜⎜ a ⎠⎠ 9 ⎝ ⎝
(11)
Article
For the nonchain type 2 model, the above parameters in eqs 10 and 11 are replaced by the set (a10−a13). In the next section, the idea of classifying molecular movement for nonassociating molecules into chain and nonchain movement will be evaluated by using the chain and nonchain models and the ten different classes of nonassociating molecules containing different molecular structures. B
DOI: 10.1021/acs.iecr.5b01526 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research Table 1. Scope of Experimental Conditions and Viscosity Data Sources
Table 2. Entity Chapman−Enskog Scaled Fluid Viscosity Model Parameters for Chain and Nonchain Molecules
Table 3. Corresponding-States Viscosity Model Predictions for Nonassociating Molecules
1
Defined in the Nomenclature section. aExtrapolation, by Gross and Sadowski.7 bReference 8. cLTN data extrapolation (2014).
a,b,c
Sources for PC-SAFT segment parameters were not tabulated in ref 7. aExtrapolation, by Gross and Sadowski.7 bReference 8. cLTN data extrapolation
11. Fifteen out of 73 nonassociating molecules were modeled as nonchain molecules, as listed in Tables 1 and 3. ARD’s and AARD’s. Table 3 lists the model performance for all 73 components and ten molecular categories. Considering molecular structure, the use of the nonchain
idea of chain and nonchain molecular movement was considered and nonchain parameters were determined and listed in Table 2 for the respective nonchain models, eqs 10 and C
DOI: 10.1021/acs.iecr.5b01526 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
states semilog plots found for n-alkanes.5 Figures 3 and 4 provided a more detailed picture of predictability for some
categories reduced the AARD’s for the above three categories from 17 to 23% to 4−7%, comparable to the 5.2% predictability for n-alkanes.5 Overall, Table 3 illustrates that ARD’s and AARD’s of 0.8% and 6.6% were obtained for all 73 components and 6259 data points, covering ten molecular classifications. The range of temperature and pressure given in Table 1 illustrates that this predictive corresponding states model handles the entire fluid region for nonassociating molecules, over a wide range of temperature (60−800 K) and pressure (1−46 800 atm). It is anticipated that a 2,2-dimethyl group on longer chain nalkanes would tend toward having a minimal effect on overall molecular momentum transfer. In this case, viscosity should be closer to the chain model prediction. The same reasoning may apply to cyclics and aromatics. For example, benzene structure and viscosity are consistent with nonchain behavior, but, alkyl groups on the benzene ring results in chain viscosity behavior. Limited data on polycyclic and polynuclear compounds suggests that chain model parameters may be useful for viscosity prediction for these structures. Some of the parameters obtained for the nonchain models are expected. With a7 and a8 and a11 and a12 close to 1, eqs 10 and 11 reduce to the values for methane (methane Nseg, Aseg, and −Bseg = 1). So, the nonchain models are close to being essentially two parameter models with two parameters based on theory. However, attempts to use the form of eqs 10 and 11 on n-alkanes resulted in increased prediction error. Entity Chapman−Enskog Scaled Fluid Viscosity vs Entity Residual Entropy. Figures 1 and 2 provide plots
Figure 3. Predictive corresponding-states viscosity model for some nonchain molecules of varying structures.
Figure 4. Predictive corresponding-states viscosity model for C6 isomers.
nonchain molecules and C6 isomers. It is evident that introduction of nonchain parameters broadens the predictability of the original entity-based Chapman−Enskog scaled viscosity model5 to include the broader class of nonassociating molecules. The use of chain and nonchain-type 2 parameters provide an approach to bracketing viscosity estimation.
■
CONCLUSIONS The capability of the original predictive corresponding-states viscosity model5 has been broadened to estimate and predict fluid viscosity of nonassociating components over the entire fluid region, and a wide range of temperature and pressure. By recognizing molecular structure as a factor in molecular transport processes, the concept of grouping as chain and nonchain molecules was used in this work to provide improved predictive capability over a broader range of nonassociating molecular classifications. An analysis using the original model5 with chain and nonchain parameters produced a predictive ARD and AARD of 0.8% and 6.6%, respectively, for 73 nonassociating components and 6259 viscosity data points. Using the segment-based viscosity model for component correlations,4 a 3.6% AARD, with average R2 = 0.9990, was obtained for the same components and 6259 viscosity data points. The predictive corresponding-states viscosity model5 with chain and nonchain parameters provides a practical tool for predicting viscosity over the entire fluid region for a broad range of nonassociating molecules. The small number of
Figure 1. Predictive corresponding-states viscosity model for chain molecules.
Figure 2. Predictive corresponding-states viscosity model for nonchain molecules.
resulting for chain-type and nonchain-type molecules. It is evident that all classifications follow the same correspondingD
DOI: 10.1021/acs.iecr.5b01526 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research
(3) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960. (4) Novak, L. T. Fluid Viscosity-Residual Entropy Correlation. Int. J. of Chem. Reactor Eng. 2011, 9, A107 http://www.bepress.com/ijcre/ vol9/A63. (5) Novak, L. T. Predictive Corresponding-States Viscosity Model for the Entire Fluid Region: n-Alkanes. Ind. Eng. Chem. Res. 2013, 52, 6841−6847. (6) Novak, L. T. Predicting Natural Gas Viscosity with a Mixture Viscosity Model for the Entire Fluid Region. Ind. Eng. Chem. Res. 2013, 52, 16014−16018. (7) 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. (8) Tihic, A.; Kontogeorgis, G. M.; von Solms, N.; Michelson, M. L. Applications of the simplified perturbed-chain SAFT equation of state using an extended parameter table. Fluid Phase Equilib. 2006, 248, 29. (9) Vijande, J.; Pineiro, M. M.; Legido, J. L.; Bessieres, D. GroupContribution Method for the Molecular Parameters of the PC-SAFT Equation of State Taking into Account the Proximity Effect. Application to Nonassociated Compounds. Ind. Eng. Chem. Res. 2010, 49, 9394. (10) Emami, F. S.; Valid, A.; Elliott, J. R., Jr.; Feyzi, F. Group Contribution Prediction of Vapor Pressure with Statistical Associating Fluid Theory, Perturbed-Chain Statistical Associating Fluid Theory, and Elliott-Suresh-Donohue Equations of State. Ind. Eng. Chem. Res. 2008, 47, 8401. (11) Diller, D. E.; Van Poolen, L. J. Measurements of the Viscosities of Saturated and Compressed Liquid Normal Butane and Isobutane. Int. J. of Thermophys. 1985, 6 (1), 43−62. (12) Agayev, N. A.; Yusibova, A. D. The Visocosity of Isobutane at High Pressures; NTIS, 1972; FTD-MT- 24-1605-71. (13) Aspen-Plus, version 13.2; Aspen Technology Inc: Cambridge, MA, 2004. (14) Ma, R.; Shis, L.; Duan, Y.; Han, L.; Liu, N. Saturated Liquid Viscosity of Cyclopentane and Isopentane. J. Chem. Eng. Data 2003, 48 (6), 1418−1421. (15) Gonzalez, M. H.; Lee, A. L. Viscosity of 2,2-Dimethylpropane. J. Chem. & Eng. Data. 1968, 13 (1), 66−69. (16) McCoubrey, J. C.; Singh, N. M. The Vapor Phase Viscosities of the Pentanes. J. Chem. Phys. 1963, 67 (2), 517−518. (17) Padua, A. A. H.; Fareleira, J. M. N. A.; Calado, J. C. G. Density and Viscosity Measurements of 2,2,4-Trimethylpentane (Isooctane) from 198 to 348 K and up to 100 MPa. J. Chem. Eng. Data 1996, 41 (6), 1488−1494. (18) Knapstad, B.; Skjolsvik, P. A.; Oye, H. A. Viscosity of Pure Hydrocarbons. J. Chem. Eng. Data 1989, 34 (1), 37−43. (19) Jonas, J.; Hasha, D.; Huang, S. G. Density Effects on Transport Properties in Liquid Cyclohexane. J. Phys. Chem. 1980, 84, 109−112. (20) Caudwell, D. R.; Trusler, J. P.; Vesovic, M.; Wakeham, W. Viscosity and Density of Five Hydrocarbon Liquids at Pressures up to 200 MPa and Temperatures up to 473 K. J. Chem. Eng. Data 2009, 54, 359−366. (21) Dullien, F. A. L. Predictive Equations for Self-Diffusion in Liquids: a Different Approach. AIChE J. 1972, 18 (1), 62−70. (22) Falcone, D. R.; Douglass, D. C.; McCall, D. W. Self-Diffusion in Benzene. J. Phys. Chem. 1967, 71 (8), 2754−2755. (23) McCool, M. A.; Collings, A. F.; Woolf, L. A. Pressure and Temperature Dependence of the Self-Diffussion of Benzene. J. Chem. Soc. Faraday Trans. I 1972, 68, 1489−1497. (24) Parkhurst, H. J., Jr; Jonas, J. Dense liquids. I. The effect of density and temperature on self-diffusion of tetramethylsilane and benzene-d6. J. Chem. Phys. 1975, 63 (6), 2699−2704. (25) Assael, M. J.; Papadaki, M.; Wakeman, W. A. Measurements of the Viscosity of Benzene, Toluene, and m-Xylene at Pressure up to 80 MPa. Int. J. Thermophys. 1991, 12 (3), 449−457. (26) Dymond, J. H.; Robertson, J. Transport Properties of Nonelectrolyte Liquid Mixtures-VI. Viscosimetric Study of Binary
viscosity parameters in this corresponding-states viscosity model allows for simplification of viscosity parameter databanks for process engineering and computational fluid dynamics (CFD) software. These capabilities are useful in process engineering, petroleum engineering, pipeline engineering, and fracking applications. Considering previous work by the author,1,2 the approach described in this paper also shows promise for self-diffusion coefficient and thermal conductivity prediction for nonassociating molecules over the entire region, accept near the critical point for thermal conductivity.
■
AUTHOR INFORMATION
Corresponding Author
*Tel.: (216)687-2569. Fax: (216)687-9220. E-mail: lt_novak@ yahoo.com. Notes
The authors declare no competing financial interest.
■
NOMENCLATURE AARD = (100/ND)∑N1 D|(viscositycalc − viscosityexp)/viscosityexp|, ND = number of datapoints ARD = (100/ND)∑N1 D((viscositycalc − viscosityexp)/viscosityexp) Aseg and Bseg = component viscosity fitting parameters (eq 1) kB = Boltzmann constant m = molecular mass N = number S = entropy T = absolute temperature
Greek Symbols
ξ = a molecular attribute σ = characteristic size ε = characteristic energy of attraction between identical molecules Ω(2,2)* = Lennard-Jones collision integral for viscosity, a function of (kBT/ε) η = Newtonian, or zero shear, dynamic viscosity for a pure component η/Nentity = entity viscosity5 (ηentity)CE = (5/16)((mentitykBT/π)1/2/σentity2Ω(2,2)*) = entity Chapman−Enskog viscosity5 η#entity = = η/Nentity/(ηentity)CE = entity Chapman−Enskog scaled viscosity5 Superscript
res = residual, difference between a pure component real state value and ideal gas state # = Chapman−Enskog viscosity scaling for entire fluid region, Sres/kB ≤ 0 Subscripts
CE = Chapman−Enskog value entity = entity basis or units seg = segment basis or units
■
REFERENCES
(1) Novak, L. Self-Diffusion Coefficient and Viscosity in Fluids. Int. J. of Chem. Reactor Eng. 2011, 9, A63 http://www.bepress.com/ijcre/ vol9/A63. (2) Novak, L. T.; Galloway, F. M. Scaled Transport Coefficient Correlations for Fluids. Annual AIChE Meeting, Pittsburg, PA, Oct 29, 2012; Paper 64c. E
DOI: 10.1021/acs.iecr.5b01526 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
Article
Industrial & Engineering Chemistry Research Mixtures of Hexafluorobenzene and Aromatic Hydrocarbons. Int. J. Thermophys. 1985, 6 (1), 21−41. (27) Dymond, J. H.; Robertson, J.; Isadale, J. D. Transport Properties of Nonelectrolyte Liquid Mixtures-IV. Viscosity Coefficients for Benzene, Perdeuterobenzene, Hexafluorobenzene, and Equimolar Mixture of Benzene + Hexafluorobenzene from 25 to 100°C at Pressures up to the Freezing Pressure. Int. J. Thermophys. 1981, 2 (3), 223−236. (28) Dymond, J. H.; Young, K. J. Transport Properties of Nonelectrolyte Liquid Mixtures-V. Viscosity Coefficients for Binary Mixtures of Benzene plus Alkanes at Saturation Pressure from 283 to 393 K. Int. J. Thermophys. 1981, 2 (3), 237−247. (29) Oliveira, C. M. B. P.; Wakeham, W. A. The Viscosity of Five Liquid Hydrocarbons at Pressures Up to 250 MPa. Int. J. Thermophys. 1992, 13 (5), 773−790. (30) Dymond, J. H.; Awan, M. A.; Glen, N. F.; Isdale, J. D. Transport Properties of Nonelectrolyte Liquid Mixtures. VIII. Viscosity Coefficients for Toluene and for Three Mixtures of Toluene + Hexane from 25 to 100 °C at Pressures up to 500 MPa. Int. J. Thermophys. 1991, 12 (2), 275−287. (31) 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. (32) Abramson, E. H.; West-Foyle, H. Viscosity of nitrogen measured to pressures of 7 GPa and temperatures of 573 K. Phys. Rev. E 2008, 77 (041202), 1−5. (33) Abramson, E. H. Viscosity of Carbon Dioxide Measured to a Pressure of 8 GPa and Temperature of 673 K. Phys. Rev. E 2009, 80, 021201 EPAPS Document No. E-PLEEE8-80-125908. (34) Weiner, H. Structural Determination of Paraffin Boiling Points. J. Am. Chem. Soc. 1947, 1 (69), 17−20. (35) Hosoya, H. Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons. Bull. Chem. Soc. Jpn. 1971, 44, 2332−2339.
F
DOI: 10.1021/acs.iecr.5b01526 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX
