Predicting Natural Gas Viscosity with a Mixture Viscosity Model for the

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Predicting Natural Gas Viscosity with a Mixture Viscosity Model for the Entire Fluid Region Lawrence T. Novak* Department of Chemical and Biomedical Engineering Cleveland State University 2121 Euclid Avenue, SH455 Cleveland, Ohio 44115-2214, United States ABSTRACT: A new predictive corresponding-states mixture viscosity model for the entire fluid region and large range of temperature and pressure is reported here. Using previously published parameters for pure n-alkanes, viscosity predictions with this model were made for the methane−ethane system and twelve component Qatari synthetic natural gas mixtures. For the synthetic natural gas mixtures studied, the new model presented here has comparable predictability to other literature models and superior predictability at higher (11 mol %) levels of N2 and CO2 content. The integral use of the PC-SAFT equation of state with the fluid mixture viscosity model introduced here provides an attractive unified approach for modeling flow in porous media.

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INTRODUCTION In previous work,1−3 novel pure component models were proposed and evaluated for self-diffusion coefficient, viscosity, and thermal conductivity over the entire fluid region (liquid, gas, and critical fluid). An extension4 of previous work led to a predictive corresponding-states pure component viscosity model for the entire fluid region. It was shown that a new entity-based scaled viscosity model correlates pure n-alkane components to a single semilog line with entity residual entropy. With five fitting parameters for the n-alkane class, a 5.2% group average of the absolute relative deviations (AAD) was obtained over the entire fluid region for a group of seventeen n-alkanes, ranging from methane to 1280 molecular weight linear polyethylene. These same five fitting parameters also predicted pure component viscosity of some olefins, ethers, and branched alkanes.4 When considering how individual molecules move in fluid, the entity model recognizes that all parts of polyatomic molecules do not typically move in unison. With the proper selection of entity size and number of entities in a molecule, the component Chapman-Enskog viscosity becomes identical to actual, or extrapolated, experimental values in the ideal gas state. And, all component scaled viscosity data follow the same semilog relationship with entity residual entropy over the entire fluid region. Experimental data demonstrate that a simple relationship exists between entity quantities and the PC-SAFT equation of state segment parameters for n-alkanes: Nentityσentity = Nsegσseg.4,20 In this work, it will be shown that the pure component entity-based scaled viscosity model4 can be extended and applied to mixtures of similar n-alkanes and some other nonassociating molecules. The entity-based scaled viscosity model for mixtures is based on the same general idea used in previous work on pure components. For mixtures, the magnitude of mixture fluid viscosity is related to an appropriate measure of mixture average entity size, mixture average entity momentum, and mixture average entity disorder. Results presented here will illustrate that the same five fitting parameters, determined previously from pure n-alkanes, reduce viscosity data for the methane−ethane system and synthetic © 2013 American Chemical Society

natural gases to the same semilog correlating line as found previously for pure components.4 When the number of components is one, this mixture model reduces to the pure component n-alkane model. In the following section, the entity-based scaled mixture viscosity model is defined. Variable definitions are listed in the Nomenclature section.

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ENTITY-BASED SCALED MIXTURE VISCOSITY MODEL The entity Chapman-Enskog scaled mixture viscosity is defined in a manner analogous to the entity Chapman-Enskog scaled pure component viscosity previously introduced.4 # excess ln(ηentity,mixture ) = −(Smixture /Nentity,mixturekB)

(1)

Mixing rules are defined to provide mixture average values for entity size, number of entities, and entity Chapman-Enskog viscosity. The entity mixture viscosity (ηmixture/Nentity,mixture) relative to the entity Chapman-Enskog mixture viscosity (ηentity,mixture)CE is defined as follows: # = ηentity,mixture

ηmixture /Nentity,mixture (ηentity,mixture)CE

(2)

The dynamic viscosity for a mixture is ηmixture. And, the mixture average number of entities (Nentity,mixture) for nc components in the mixture is written in terms of mole fraction (x̃i) weighted number of entities (Nentity,i). nc

Nentity,mixture =

∑ xĩ Nentity,i i

(3)

The entity Chapman-Enskog mixture viscosity is written as follows: Received: Revised: Accepted: Published: 16014

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Table 1. Experimental Viscosity Data: Scope and Sources for Methane−Ethane and Qatari Synthetic Natural Gases methane−ethane9−11

QNG-S17

QNG-S27

QNG-S38

QNG-S48

QNG-S58

100−300 15−344 599

250−450 99−642 248

250−450 99−642 248

250−450 99−642 248

0−100 0−100

84.99 5.53 2.01 0.40 0.59 0.17 0.15 0.15 0.09 0.10 3.50 2.33

250−450 250−450 99−642 99−642 248 248 mol% composition 90.26 80.34 5.83 5.19 2.11 1.88 0.41 0.38 0.64 0.57 0.21 0.19 0.16 0.14 0.16 0.15 0.11 0.09 0.11 0.09 0.00 6.60 0.00 4.38

84.70 5.58 1.96 0.42 0.55 0.21 0.16 0.15 0.10 0.00 3.71 2.46

85.09 5.53 2.01 0.40 0.61 0.17 0.14 0.15 0.00 0.10 3.50 2.30

temperature, °K pressure, atm data points methane ethane propane isobutane n-butane isopentane n-pentane n-octane toluene methylcyclopentane nitrogen carbon dioxide nc

(ηentity,mixture)CE =

entropy in predicting tracer diffusion for mixtures of model molecules.21 In the following section, the capability of this new mixture model (eqs 1−5) will be evaluated using mixture viscosity data available in the literature.

2 (ηentity,i)CE ) ( ∑ xĩ Nentity,iσentity,i i

nc

×

−1

2 ) ( ∑ xĩ Nentity,iσentity,i i

(ηentity,i)CE

5 mentity,i kBT /π = 16 σentity,i 2 Ω i(2,2) *

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(4)

RESULTS AND DISCUSSION

Scope of Study. The experimental viscosity data scope and sources used in this study are listed in Table 1. Viscosity data over the entire region for pure methane, pure ethane, and methane−ethane mixtures are used to illustrate the corresponding-states property of this model (eqs 1−5) and predictability for mixtures of similar nonassociating molecules. Then, viscosity data for twelve component synthetic natural gas samples are predicted by the new model and compared with predictions reported for other literature models, using the same viscosity data. The synthetic natural gas mixtures were originally chosen to represent the Qatari North Field reservoir and to study the effects of various components on some literature viscosity model predictions.7,8 Mixture Fluid Viscosity. Because methane and ethane are major components of natural gas, we begin our model evaluation with pure methane, pure ethane, and methane− ethane mixtures over the entire fluid region and a large range of temperature, pressure, and composition. A plot of viscosity data against reciprocal temperature for the methane−ethane system is shown in Figure 1. The effect of composition and pressure on fluid viscosity is evident in Figure 1. Figure 2 demonstrates the improved understanding and modeling provided by the entity-based fluid mixture viscosity model (eqs 1−5). By plotting the log (entity Chapman Enskog scaled mixture viscosity) against entity excess entropy, all of the data in Figure 1 correlate to the single semilog relationship in Figure 2. This includes pure component and mixture viscosity data. The model is represented in Figure 2 as the line with a slope of −1 and an intercept of +1. Viscosity data from the methane−ethane system are represented by symbols. These data cover the entire fluid region over a wide range of temperature and pressure, as indicated in Figure 2. By definition, excess entropy (Sex) is zero in the ideal gas state and η#entity,mixture = 1. For 599 methane−ethane viscosity data points, covering the indicated range of conditions, an average relative deviation

(5)

Equation 5 defines the entity Chapman-Enskog viscosity (ηentity,i)CE for component “i” as a function of the entity mass (mentity,i = mi/Nentity,i) for component “i” with molecular mass mi, number of entities in component “i” (Nentity,i), entity collision diameter (σentity,i) of component “i”, and collision integral Ωi(2,2)*, based on (kBT/εseg,i). In this work, an empirical equation5 is used to calculate collision integral values. Differences, if any, between εseg,i and εentity,i are neglected. Equations 4 and 5 illustrate that the entity Chapman-Enskog mixture viscosity (ηentity,mixture)CE is a characteristic collision area averaged value, determined by calculating an average mixture characteristic momentum of components in the ideal gas region and dividing by the average mixture characteristic collision area. The entity variables for pure components in the mixture (Nentity,i and σentity,i) are determined from PC-SAFT segment parameters and the five parameter relationship found previously by evaluation of pure component viscosity data (3122 data points) for n-alkanes ranging from methane up to 1280 molecular weight linear polyethylene.4 The calculation of excess mixture entropy (Sexcess mixture) in eq 1 using the PC-SAFT equation of state has been defined previously.6 In this work, excess mixture entropy was calculated by adding a FORTRAN front-end to the PC-SAFT FORTRAN code provided at the Web site of G. Sadowski (Lehrstuhl fur Thermodynamik, Universitat Dortmund, Emil-Figge-Strasse 70, 44227 Dortmund, Germany, www.th.bci.tu-dortmund.de/ forschung/pc-saft/download). Because of the corresponding-states property found previously,4 the entity Chapman-Enskog scaled viscosity-entity excess entropy relationship (eq 1) should be followed for both pure component n-alkanes and mixtures of similar nonassociating molecules. An earlier study, based entirely on molecular simulation, demonstrated the value of using excess 16015

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Figure 3. Entity Chapman-Enskog scaled mixture viscosity vs entity excess entropy: Qatari synthetic natural gases.

Figure 1. Methane−ethane system fluid mixture viscosity vs reciprocal temperature.

NBS (TRAPP model), CLS, LBC, and PFCT literature models can be found elsewhere.7 These literature viscosity models were primarily developed for hydrocarbon gas and liquid viscosity. They need component critical properties, and acentric factor in some cases, and have the following number of viscosity model parameters, excluding equation of state and reference fluid parameters: NBS(8), CLS(41), LBC(17), and PFCT(4). In contrast to these four literature viscosity models, the 5parameter entity-based viscosity model for pure components4 and mixtures over the entire fluid region does not require critical parameters or acentric factors. On the basis of Figure 1,4 it has been shown that the entity-based viscosity model can be further simplified to a 3-parameter viscosity model for nalkanes.20 However, a performance evaluation of the 3parameter viscosity model is not currently available. The average of AADs in Table 2 illustrate that for samples (QNG-S1 to QNG-S5), the corresponding-states mixture viscosity model introduced in this work is essentially comparable in predictability to the NBS, CLS, and LBC models. Although the PFCT model has an overall edge for all five samples, the PFCT was found to have a higher AAD than the mixture viscosity model (eqs 1−5) for the higher levels (11 mol %) of N2 and CO2 in sample QNG-S3.s

Figure 2. Entity Chapman-Enskog scaled mixture viscosity vs entity excess entropy: Methane−ethane system.

(ARD) of −2.2% and average absolute relative deviation (AAD) of 6.4% was obtained, using the previously determined4 5-parameters. This is comparable to the 5.2% AAD (with 2.7% standard deviation) found previously for a group of 17 pure nalkane components ranging from methane to 1280 molecular weight linear polyethylene.4 Therefore, pure methane, pure ethane, and methane−ethane mixture viscosity data supports an earlier suggestion that the corresponding states property found for pure components should apply to mixtures of similar molecules.4 As pointed out previously, other viscosity models (rough hard sphere theory 18 and frictionfree-volume theory19) for the pure n-alkane class of molecules omit methane because inclusion of methane negatively affects overall model performance.4 Predicting Qatari Synthetic Natural Gas Viscosity. Qatari synthetic natural gas viscosity data7,8 will be predicted by the mixture viscosity model introduced here (eqs 1−5), and compared to predictions from several literature viscosity models previously evaluated using the same data. Table 1 provides the scope of temperature, pressure, and composition for the five synthetic natural gas samples (QNG-S1 through S5). By plotting the log (entity Chapman Enskog scaled mixture viscosity) against entity excess entropy, all of the viscosity data (symbols) for the five synthetic natural gas samples correlate to the single semilog relationship in Figure 3, with slope of −1 and intercept of +1. Thus, the Qatari synthetic natural gas viscosity data also supports the corresponding-states mixture viscosity model introduced here. Table 2 lists the AADs obtained for the various Qatari synthetic natural gas samples as a function of various viscosity models. Abbreviations in Table 2 for four literature viscosity models are defined below the table. Brief explanations of the

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CONCLUSIONS New research, reported here, has demonstrated that the entitybased corresponding-states fluid viscosity model for pure components can be extended to mixtures of similar nonassociating molecules to predict fluid viscosity over the entire region. For the methane−ethane system, the AAD is comparable to the group AAD obtained in previous studies with pure component n-alkanes. In addition, the viscosity of Qatari synthetic natural gases containing nitrogen and carbon dioxide were predicted with an average AAD of 4.1%, also comparable to earlier pure component study results. It is encouraging that the five viscosity model parameters determined previously from pure n-alkane viscosity data continue to show some extrapolation capability to nonassociating molecules that are not n-alkanes. Therefore, the predictive corresponding-states viscosity model for pure components and mixtures provides a practical tool in process engineering, product engineering, oil and gas reservoir engineering, and fracking applications. The integral use of the PC-SAFT equation of state with the fluid mixture viscosity model introduced here provides an attractive unified approach for modeling flow in porous media. 16016

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Table 2. % AAD for Predicted Qatari Synthetic Natural Gas Fluid Viscosity by Various Models sample ID

NBSa (%)12

CLSb (%)13,14

this work (%)

LBCc (%)15

PFCTd (%)16,17

QNG-S1 QNG-S2 QNG-S3 QNG-S4 QNG-S5 average

5.9 3.9 6.7 5.6 6.0 5.6

4.7 3.5 5.4 4.2 4.7 4.5

4.6 4.5 2.2 4.6 4.6 4.1

3.6 3.6 4.1 3.4 3.7 3.7

2.5 1.6 2.9 2.3 2.5 2.4

a

NBS = Ely-Hanley, or TRAPP model. bCLS = Chung-Lee-Starling. cLBC = Lohrenz-Bray-Clark. dPFCT = Pedersen-Fredenslund-ChristensenThomassen.

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AUTHOR INFORMATION

Corresponding Author

(1) Novak, L. Self-Diffusion Coefficient and Viscosity in Fluids. Int. J. Chem. React. Eng. 2011, 9, A63. (2) Novak, L. T. Fluid Viscosity-Residual Entropy Correlation. Int. J. Chem. React. Eng. 2011, 9, A107. (3) Novak, L. T.; Galloway, F. M. Scaled Transport Coefficient Correlations for Fluids. Annual AIChE Meeting, Pittsburg, PA, 29 October 2012, Paper 64c. (4) Novak, L. T. Predictive Corresponding States Viscosity Model for the Entire Fluid Region: n-Alkanes. Ind. Eng. Chem. Res. 2013, 52 (20), 6841. (5) Neufeld, P. D.; Janzen, A. R.; Aziz, R. A. Empirical Equations to Calculate 16 of the Transport Collision Integrals Ω(l,s)*for the Lennard-Jones (12−6) Potential. J. Chem. Phys. 1972, 57 (3), 1100. (6) 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 (4), 1244. (7) Atilhan, M.; Aparicio, S.; Alcalde, R.; Iglesias-Silva, G. A.; ElHawagi, M.; Hall, K. R. Viscosity Measurements and Data Correlation for Two Synthetic Natural Gas Mixtures. J. Chem. Eng. Data 2010, 55 (7), 2498. (8) Atilhan, M.; Aparicio, S.; Iglesias-Silva, G. A.; El-Hawagi, M.; Hall, K. R. On the Viscosity of Natural Gas from Qatari North Field Reservoir. J. Chem. Eng. Data 2010, 55 (11), 5117. (9) Diller, D. E. Measurements of the Viscosity of Compressed Gaseous and Liquid Methane + Ethane Mixtures. J. Chem. Eng. Data 1984, 29 (2), 215. (10) Friend, D. G.; Ely, J. F.; Ingham, H. Thermophysical Properties of Methane. J. Phys. Chem. Ref. Data 1989, 18 (2), 583. (11) Friend, D. G.; Ingham, H.; Ely, J. F. Thermophysical Properties of Ethane. J. Phys. Chem. Ref. Data 1991, 20 (2), 275. (12) Ely, J. F.; Hanley, J. M. Prediction of Transport Properties. I. Viscosity of Fluids and Mixtures. Ind. Eng. Chem. Fundam. 1981, 20 (4), 323. (13) Chung, T. H.; Lee, L. L.; Starling, K. E. Applications of Kinetic Gas Theories and Multiparameter Correlation for Prediction of Dilute Gas Viscosity and Thermal Conductivity. Ind. Eng. Chem. Fundam. 1984, 23 (1), 8. (14) Chung, T. H.; Ajlan, M.; Lee, L. L.; Staring, K. E. Generalized Multiparameter Correlation for Nonpolar and Polar Fluid Transport Properties. Ind. Eng. Chem. Res. 1988, 27 (4), 671. (15) Lohrenz, J.; Bray, B. G.; Clark, C. R. Calculating Viscosity of Reservoir Fluids from Their Composition. J. Pet. Technol. 1964, 16 (10), 117. (16) Pedersen, K. S.; Fredenslund, A.; Christensen, P. L.; Thomassen, P. Viscosity of crude oils. Chem. Eng. Sci. 1984, 39 (6), 1011. (17) Pedersen, K. S.; Fredenslund, A. Improved Corresponding States Model for the Prediction of Oil and Gas Viscosities and Thermal Conductivities. Chem. Eng. Sci. 1987, 42 (1), 182. (18) Sun, T.; Teja, A. S. Correlation and Prediction of the Viscosity and Thermal Conductivity of Dense Fluid. J. Chem. Eng. Data 2009, 54 (9), 2527. (19) Tan, S. P.; Adidharma, H.; Towler, B. F.; Radosz, M. Friction Theory and Free-Volume Theory Coupled with Statistical Associating

*Tel: (216)687-2569. Fax: (216)687-9220. E-mail: lt_novak@ yahoo.com. Notes

The authors declare no competing financial interest.

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NOMENCLATURE AAD= 100 ND

ND

∑

viscositycalc − viscosityexp viscosityexp

1

,

ND = number of data points

ARD= 100 ND

ND

∑

viscositycalc − viscosityexp

1

viscosityexp

REFERENCES

,

ND = number of data points

kB=Boltzmann constant m=molecular mass N=number S=entropy T=absolute temperature V=volume x̃=mole fraction Greek Symbols

σ=characteristic size of an entity ε/kB=characteristic energy of attraction between identical entities Ω(2,2)*=Lennard-Jones collision integral for viscosity, a function of (kBT/ε) η=Newtonian, or zero shear, dynamic viscosity ηmixture/Nentity,mixture=entity mixture viscosity (ηentity,i)CE = (5/16)((mentity,ikBT/π)1/2/σ2entity,iΩ(2,2) *)=entity i Chapman-Enskog viscosity for component “i” # ηentity,mixture = (ηmixture/Nentity,mixture)/(ηentity,mixture)CE=entity Chapman-Enskog scaled mixture viscosity

Superscript

excess=excess, difference between a mixture in the real state value and ideal gas state #=Chapman-Enskog scaled viscosity for entire fluid region, Sexcess/kB ≤ 0 Subscripts

CE=Chapman-Enskog type value entity=entity basis or units i=molecular component “i” mixture=mixture of component molecules 16017

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Fluid Theory for Estimating the Viscosity of Pure n-Alkanes. Ind. Eng. Chem. Res. 2005, 44, 8409. (20) Novak, L. T. Correction to “Predictive Corresponding States Viscosity Model for the Entire Fluid Region: n-Alkanes. Ind. Eng. Chem. Res. 2013, 52 (38), 13886. (21) Krekelberg, W. P.; Pond, M. J.; Goel, G.; Shen, V. K.; Errington, J. R.; Truskett, T. M. Generalized Rosenfeld Scalings for Tracer Diffusivities in Not-so-Simple Fluids: Mixtures and Soft Particles. Phys. Rev. E. 2009, 80, 061205.

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