Comment/Reply pubs.acs.org/jced
Reply to “Comment on ‘New Experimental Data and Reference Models for the Viscosity and Density of Squalane’” Kurt A. G. Schmidt,*,† Doug Pagnutti,‡ and J. P. Martin Trusler§ †
Schlumberger, Abingdon Technology Center, Lambourn Court, Wyndyke Furlong, Abingdon, OX14 1UJ, United Kingdom Schlumberger, DBR Technology Center, 9450-17 Ave, Edmonton, AB T6N 1M9, Canada § Department of Chemical Engineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom ‡
ABSTRACT: The authors would like to thank Professor Bair for his insightful comments on ultrahigh-pressure viscosities. The viscosity model used by Schmidt et al.1 was a Tait-like2,3 model that has been shown to correlate accurately the viscosity of many fluids in the original investigation’s pressure range (0.1−275.8 MPa (40000 psi)) of interest. The upper pressure is indicative of the high pressures found in the petroleum industry4. However, after discussions with Professor Bair, it became clear a reference model that can accurately model the viscosity of squalane at ultrahigh-pressures is of interest to those working in the area of tribology.
O
and the reference pressure was found to vary linearly with temperature:
n the basis of a comprehensive review of the literature data, new experimental measurements, and a novel regression technique, the model developed by Schmidt et al.,1 was shown to describe accurately the viscosity of squalane to a pressure of 467 MPa. In the robust regression technique of Schmidt et al.,1 data at pressures greater than 467 MPa were automatically excluded from the data set due to the large differences between the model and the experimental data. Recently, Bair5,6 pointed out that the Tait-like model is equivalent to the McEwen7 model and is insufficient to model the viscosities of fluids at ultrahigh-pressures. This conclusion was reached by analysis with the Stickel function5,6 and an analysis of the goodness of the fit with the McEwen model.7 On the basis of this analysis and the robust regression results, the inconsistency between our model and the data at pressures greater than 467 MPa must be attributed to the limitation of the Tait-like model’s performance at ultrahigh-pressures rather than a problem with those particular data. Bair6 has shown that a hybrid model, consisting of the models of McEwen7 and Paluch et al.,8 was able to correlate squalane9 viscosity data at pressures ranging from atmospheric to 1298 MPa. On the basis of the success of this hybrid approach, the Schmidt et al.1 model was extended from 467 to 1298 MPa with the Bair9,10 data (at pressures greater than 467 MPa). The extended model has the following formulation: η = ηref (T , p)
p∞ = pa T + pb
The optimal parameters and a summary of the regression’s statistics for the extension of the viscosity model are presented Table 1. Regression Analysis Results parameter
value 2.2664·100 1.2242·101 −1.7838·103
CF pa/MPa·K−1 pb/ MPa Regression Statistics number of points Δ bias std. dev. max(Δi) Tmin/K Tmax/K Pmin/MPa Pmax/MPa number of outliers
38 5.5 % −1.57 % 6.48 % 14.0 % 293.15 373.15 467 1298 0
in Table 1. Overall, the average absolute percentage deviation (denoted Δ),
p ≤ p0
⎛ ⎞ C Fp0 C Fp ⎟ p>p η = ηref (T , p) exp⎜⎜ − 0 p∞ (T ) − p0 ⎟⎠ ⎝ p∞ (T ) − p
Δ=
(1)
100 N
N
∑ i=1
yi − f (xi , c) yi
(3)
was found to be 5.5 %. Figure 1 shows the relative deviation of the model with the measured viscosities as a function of
where ηref is the viscosity of squalane at the temperature and pressure of interest from Schmidt et al.,1 CF is a fragility parameter, and p∞ is the pressure for unbounded viscosity. The reference, or crossover pressure, p0, was chosen to be the maximum pressure (467 MPa) of the Schmidt et al. model.1 Using this approach, the fragility parameter was held constant © 2015 American Chemical Society
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Received: February 18, 2015 Accepted: March 16, 2015 Published: March 30, 2015 1213
DOI: 10.1021/acs.jced.5b00157 J. Chem. Eng. Data 2015, 60, 1213−1214
Journal of Chemical & Engineering Data
Comment/Reply
Bair’s ultrahigh-pressure data is attributed to the limitation of the original model rather than an inconsistency of those particular data. The extension of the original model allows for the inclusion of the ultrahigh-pressure viscosity data of Bair9,10 to the accurate model of Schmidt et al.,1 which now provides reference-like viscosities to 1298 MPa.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors would like to thank Professor Bair, Georgia Institute of Technology, Center for High-Pressure Rheology, George W. Woodruff School of Mechanical Engineering, for his discussions in the areas of modelling the viscosity of fluids in ultrahigh-pressures.
Figure 1. Relative deviations Δη/η = {η(exp) − η(calc)}/η(calc) of the experimental viscosity η(exp) of Bair9,10 from the values obtained using eq 1 η(calc): ▲, 298.15 K; ◆, 313.15 K; ◇, 338.15 K; ●, 373.15 K.
pressure for the four isotherms of data. As shown in Figure 2, the overall model correlates the data very well at 313.15 K. The
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REFERENCES
(1) Schmidt, K. A. G.; Pagnutti, D.; Curran, M. D.; Singh, A.; Trusler, J. P. M.; Maitland, G. C.; McBride-Wright, M. New Experimental Data and Reference Models for the Viscosity and Density of Squalane. J. Chem. Eng. Data 2015, 60, 137−150. (2) Kashiwagi, H.; Makita, T. Viscosity of Twelve Hydrocarbon Liquids in the Temperature Range 298−348K at Pressures up to 100 MPa. Int. J. Thermophys. 1982, 3, 289−305. (3) Comuñas, M. J. P.; Baylaucq, A.; Boned, C.; Fernández, J. High Pressure Measurements of the Viscosity and Density of Two Polyethers and Two Dialkyl Carbonates. Int. J. Thermophys. 2001, 22, 750−768. (4) Belani, A.; Orr, S. A Systematic Approach to Hostile Environments. J. Pet. Technol. 2008, 60 (July), 34−39. (5) Bair, S. Comment on “New Experimental Data and Reference Models for the Viscosity and Density of Squalane. J. Chem. Eng. Data 2015, DOI: 10.1021/je501147a. (6) Bair, S. Choosing Pressure−Viscosity Relations. High Temp.− High Pressures 2015, in press. (7) McEwen, E. The Effect of Variation of Viscosity with Pressure on the Load Carrying Capacity of the Oil Film Between Gear-Teeth. J. Inst. Petroleum 1952, 38 (344−345), 646−672. (8) Paluch, M.; Dendzik, Z.; Rzoska, S. J. Scaling of High-Pressure Viscosity Data in Low-Molecular-Weight Glass-Forming Liquids. Phys. Rev. B 1999, 60, 2979−2982. (9) Bair, S. The High Pressure Rheology of Some Simple Model Hydrocarbons. J. Eng. Tribol. 2002, 216, 139−149. (10) Bair, S. Reference Liquids for Quantitative Elastohydrodynamics: Selection and Rheological Characterization. Tribol. Lett. 2006, 22, 197−206.
Figure 2. Viscosity η of squalane along the 313.15 K isotherm as a function of pressure: ▲, experimental data of Bair;8, solid black line, correlation of Schmidt et al.;1 dashed black line, revised correlation.
fragility parameter and the unbounded viscosity pressure determined from this model are similar to those obtained by Bair.6 At pressures below 467 MPa, the uncertainty in the viscosity is 1.4 %; however, the uncertainty increases to 5.5 % at pressures greater than 467 MPa. As previously discussed by Schmidt et al.,1 extrapolation of empirical equations should be made with caution as shown in Figure 2 where the original model was extrapolated to 1200 MPa. The extended model now expands the pressure range while correlating the data over the full pressure range very well. However, it should be noted that at the reference pressure (467 MPa), ∂η(T, p)/∂p of eq 1 is not continuous. This discontinuity will be rectified when a new model is applied, via the robust regression analysis of Schmidt et al.,1 to the full squalane viscosity data set in a subsequent article. On the basis of the available data, the extended model is valid at temperatures from 293.15 K to 373.15 K. Compared with the large amount of data available at pressures below 400 MPa, additional data are needed at pressures greater than 400 MPa.
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CONCLUSIONS An extension to the reference model of Schmidt et al.1 was developed in a pragmatic way to extend the pressure limitation of the model from 456 to 1298 MPa. The initial exclusion of 1214
DOI: 10.1021/acs.jced.5b00157 J. Chem. Eng. Data 2015, 60, 1213−1214
